Rolf Kindmann, Matthias Kraus Steel Structures Design

Rolf Kindmann, Matthias KrausSteel StructuresDesign using FEM

Rolf Kindmann/Matthias Kraus

Steel StructuresDesign using FEM

Prof. Dr.-Ing. Rolf KindmannRuhr-Universit�t BochumLehrstuhl f�r Stahl-, Holz- und LeichtbauUniversit�tsstraße 15044801 Bochum

Dr.-Ing. Matthias KrausIngenieursoziet�t SKPPrinz-Friedrich-Karl-Str. 3644135 Dortmund

Language Polishing by Paul Beverley, London

Cover: SIGNAL IDUNA PARK, Dortmund, � Professor Rolf Kindmann

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� 2011 Wilhelm Ernst & Sohn, Verlag f�r Architektur und technische Wissenschaften GmbH & Co. KG,Rotherstraße 21, 10245 Berlin, Germany

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Printed in the Federal Republic of Germany.Printed on acid-free paper.

ISBN 978-3-433-02978-7Electronic version available. O-book ISBN 978-3-433-60077-1

4

Preface

Steel structures are usually beam or plate structures consisting of thin-walled cross sections. For their design, deformations, internal forces and moments as well as stresses are needed, and the stability of the structures is of great importance. Generally, the finite element method (FEM) is used for structural analysis and as a basis for the verification of sufficient load-bearing capacity.

This book presents the relevant procedures and methods needed for calculations, computations and verifications according to the current state of the art in Germany and the rest of Europe. In doing so the following topics are treated in detail:

determination of cross-section properties, stresses and plastic cross section bearing capacity finite element method for linear and nonlinear calculations of beam structures solution of eigenvalue problems (stability) for flexural, lateral torsional, torsional and plate buckling verification of sufficient load-bearing capacity finite element method for open and hollow cross sections

The basis of the calculations and verifications are the German standard DIN 18800 and the German version of Eurocode 3. They are widely comparable, however, the final version of Eurocode 3 has just been published and the corresponding national annexes have to be considered.

This book has evolved from the extensive experience of the authors in designing and teaching steel structures. It is used as lecture notes for the lecture “Computer-oriented Design of Steel Structures” on the Masters’ programme “Computational Engineering”

German books – see [25], [31] and [42] – and therefore, the references at the end of the book contain many publications in the German language. Further information can be found at www.kindmann.de, www.rub.de/stahlbau and www.skp-ing.de.

The authors would like to thank Mr Florian Gerhard for the translations, Mr Paul

Matthias Kraus Rolf Kindmann

at the Ruhr-University Bochum, Germany. Large parts of the contents were taken from

Beverley for language polishing and Mr Peter Steinbach for the drawing of figures.

© 2011 Ernst & Sohn GmbH & Co. KG. Published by Ernst & Sohn GmbH & Co. KG.Steel Structures. Design using FEM. First Edition. Rolf Kindmann, Matthias Kraus.

Content

1 Introduction 11.1 Verification Methods 11.2 Methods to Determine the Internal Forces and Moments 21.3 Element Types and Fields of Application 41.4 Linear and Nonlinear Calculations 61.5 Designations and Assumptions 71.6 Fundamental Relationships 131.7 Limit States and Load Combinations 161.8 Introductory Example 191.9 Content and Outline 231.10 Computer Programs 24

2 Cross Section Properties 252.1 Overview 252.2 Utilisation of Symmetry Properties 292.3 Standardisation Part I: Centre of Gravity, Principal Axes and

Moments of Inertia 31

2.4 Calculation of Standardised Cross Section Properties Part I 402.4.1 Separation of the Cross Section into Partial Areas 402.4.2 Partial Areas of Thin-Walled Rectangles 432.4.3 Basic Cross Sections and Elementary Compound Cross Section

Shapes 46

2.4.4 Tabular Calculation of Cross Section Properties 512.4.5 Numeric Integration / Fibre and Stripe Model 532.5 Standardisation Part II: Shear Centre, Warping Ordinate and

Warping Constant 58

2.6 Warping Ordinate 632.7 Shear Centre M 67

3 Principles of FEM 723.1 General Information 723.2 Basic Concepts and Methodology 723.3 Progress of the Calculations 783.4 Equilibrium 803.4.1 Preliminary Remarks 803.4.2 813.4.3 83

Virtual Work Principle

© 2011 Ernst & Sohn GmbH & Co. KG. Published by Ernst & Sohn GmbH & Co. KG.

Principle of Minimum of Potential Energy

Steel Structures. Design using FEM. First Edition. Rolf Kindmann, Matthias Kraus.

ContentVIII

3.4.4 Differential Equations 843.5 Basis Functions for the Deformations 873.5.1 General 873.5.2 Polynomial Functions for Beam Elements 873.5.3 Trigonometric and Hyperbolic Functions for Beam Elements 913.5.4 Basis Functions for Plate Buckling 953.5.5 One-Dimensional Functions for Cross Sections 993.5.6 Two-Dimensional Functions for Cross Sections 103

4 FEM for Linear Calculations of Beam Structures 1084.1 Introduction 1084.2 Beam Elements for Linear Calculations 1084.2.1 Linking Deformations to Internal Forces and Moments 1084.2.2 Axial Force 1104.2.3 Bending 1134.2.4 Torsion 116

1204.3 Nodal Equilibrium in the Global Coordinate System 1234.4 Reference Systems and Transformations 1264.4.1 Problem 1264.4.2 Beam Elements in the X-Z Plane 1314.4.3 Beam Elements in a Three-Dimensional X-Y-Z COS 1344.4.4 Loads 1384.4.5 Warping Moment and Derivative of the Angle of Twist 1394.4.6 Finite Elements for Arbitrary Reference Points 1464.5 Systems of Equations 1474.5.1 Aim 1474.5.2 Total Stiffness Matrix 1474.5.3 Total Load Vector 1494.5.4 Geometric Boundary Conditions 1514.6 Calculation of the Deformations 1534.7 Determination of the Internal Forces and Moments 1544.8 Determination of Support Reactions 1564.9 Loadings 1574.9.1 Concentrated Loads 157

1574.9.3 Settlements 1584.9.4 Influences of Temperature 1594.10 Springs and Shear Diaphragms 1594.11 Hinges 164

4.2.5 Arbitrary Stresses

4.9.2 Distributed Loads

Content IX

5 FEM for Nonlinear Calculations of Beam Structures 1685.1 General 1685.2 Equilibrium at the Deformed System 1685.3 Extension of the Virtual Work 1715.4 Nodal Equilibrium with Consideration of the Deformations 1785.5 Geometric Stiffness Matrix 1805.6 Special Case: Bending with Compression or Tension Force 1855.7 Initial Deformations and Equivalent Geometric Imperfections 1895.8 Second Order Theory Calculations and Verification Internal Forces 1935.9 Stability Analysis / Critical Loads 2015.10 Eigenmodes / Buckling Shapes 2035.11 Plastic Hinge Theory 2065.12 Plastic Zone Theory 2105.12.1 Application Areas 2105.12.2 Realistic Calculation Assumptions 2105.12.3 Influence of Imperfections 2135.12.4 Calculation Example 214

6 Solution of Equation Systems and Eigenvalue Problems 2176.1 Equation Systems 2176.1.1 Problem 2176.1.2 Solution Methods 2186.1.3 Gaussian Algorithm 2196.1.4 Cholesky Method 2206.1.5 Gaucho Method 2206.1.6 Calculation Example 2226.1.7 Additional Notes 2246.2 Eigenvalue Problems 2246.2.1 Problem 2246.2.2 Explanations for Understanding 2256.2.3 Matrix Decomposition Method 2306.2.4 Inverse Vector Iteration 2366.2.5 Combination of the Solution Methods 241

7 Stresses According to the Theory of Elasticity 2457.1 Preliminary Remarks 2457.2 Axial Stresses due to Biaxial Bending and Axial Force 2477.3 Shear Stresses due to Shear Forces 2507.3.1 Basics 2507.3.2 Calculation Formula for 254

ContentX

7.3.3 Open Cross Sections 2557.3.4 Closed Cross Sections 2607.4 Stresses due to Torsion 2617.4.1 General 2617.4.2 Arbitrary Open Cross Sections 2647.4.3 Closed Sections 2707.5 Interaction of All Internal Forces and Verifications 2707.6 Limit Internal Forces and Moments on the Basis of the Theory of

Elasticity272

8 Plastic Cross Section Bearing Capacity 2738.1 Effect of Single Internal Forces 2738.2 Limit Load-Bearing Capacity of Cross Sections 2758.2.1 Preliminary Remarks 2758.2.2 Plastic Cross Section Reserves 2778.2.3 Calculation Methods and Overview 2818.3 Limit Load-Bearing Capacity of Doubly-Symmetric I-Cross

Sections 288

8.3.1 Description of the Cross Section 2888.3.2 Perfectly Plastic Internal Forces Spl 2898.3.3 Equilibrium between Internal Forces and Partial Internal Forces 2918.3.4 Combined Internal Forces N, My, Mz, Vy and Vz 2938.3.5 Interaction Conditions of DIN 18800 and Comparison with the

PIF-Method296

8.4 Computer-Oriented Methods 3038.4.1 Problem Definition 3038.4.2 Strain Iteration for a Simple Example 3048.4.3 Strain Iteration for Internal Forces 3078.4.4 Consideration of the Internal Forces 3148.4.5 Examples / Benchmarks 317

9 Verifications for Stability and according to Second Order Theory

319

9.1 Introduction 3199.2 Definition of Stability Cases 3219.3 Verification according to Second Order Theory 3239.4 Verifications for Flexural Buckling with Reduction Factors 3299.4.1 Preliminary Remarks 3299.4.2 Axial Compression 3309.4.3 Uniaxial Bending with Compression Force 3389.4.4 Modified Reduction Factors 340

Content XI

9.5 Calculation of Critical Forces 3429.5.1 Details for the Determination 3429.5.2 Replacement of Structural Parts by Springs 3489.5.3 Compression Members with Springs 3529.6 Verifications for Lateral Torsional Buckling with Reduction

Factors360

9.6.1 Preliminary Remarks 3609.6.2 Beams Not Susceptible to Lateral Torsional Buckling 3609.6.3 Scheduled Centric Compression 3629.6.4 Uniaxial Bending without Compression Force 3649.6.5 Uniaxial Bending with Axial Compression Force 3689.6.6 Reduction Factors according to Eurocode 3 3699.6.7 Accuracy of Reduction Factors 3739.7 3759.8 Verifications with Equivalent Imperfections 3819.8.1 Verification Guidance 3819.8.2 Equivalent Geometric Imperfections 3819.9 Calculation Examples 3939.9.1 Single-Span Beam with Cantilever 3939.9.2 Beam with Scheduled Torsion 3969.9.3 Two Hinged Frame – Calculation in the Frame Plane 3999.9.4 Two Hinged Frame – Stability Perpendicular to the Frame Plane 4049.9.5 Frame Considering Joint Stiffness 413

10 FEM for Plate Buckling 42010.1 Plates with Lateral and In-Plane Loading 42010.2 Stresses and Internal Forces 42010.3 Displacements 42210.4 Constitutive Relationships 42310.5 Principle of Virtual Work 42510.6 Plates in Steel Structures 42810.7 Stiffness Matrix for a Plate Element 42910.8 Geometric Stiffness Matrix for Plate Buckling 43210.9 Plates with Longitudinal and Transverse Stiffeners 43410.10 Verifications for Plate Buckling 43810.11 Determination of Buckling Values and Eigenmodes with FEM 44810.12 45110.12.1 Single Panel with Constant x and 1.5 45110.12.2 Beam Web with Longitudinal Stiffeners 45410.12.3 Web Plate of a Composite Bridge with Shear Stresses 45710.12.4 Web Plate with High Bending Stresses 459

Calculation of Critical Moments

Calculation Examples

ContentXII

11 FEM for Cross Sections 46111.1 Tasks 46111.2 Principle of Virtual Work 46411.3 One-Dimensional Elements for Thin-Walled Cross Sections 46911.3.1 Virtual work 46911.3.2 Element Stiffness Relationships 47211.3.3 Equation Systems 47411.3.4 Calculation of Cross Section Properties and Stresses 47611.3.5 Compilation 47911.4 Two-Dimensional Elements for Thick-Walled Cross Sections 48011.4.1 Preliminary Remarks 48011.4.2 Virtual Work for Thick-Walled Cross Section Elements 48211.4.3 Element Geometry 48411.4.4 Transformation Relationships 48611.4.5 Stiffness Relationships 48811.4.6 Numerical Integration 49011.4.7 Cross Section Properties and Stresses 49311.4.8 Performance of the Approximate Solutions 49511.4.9 Special Case: Rectangular Elements 49711.5 Calculation Procedure 501

50311.6.1 Preliminary Remarks 50311.6.2 50311.6.3 50811.6.4 Rectangular Solid Cross Section 51111.6.5 Doubly Symmetric I-Profile 51811.6.6 Crane Rail 525

References 528

Index 534

Single-Celled Box Girder Cross Section Bridge Cross Section with Trapezium Stiffeners

11.6 Calculation Exa mples

1 Introduction

1.1 Verification Methods

For civil engineering structures the ultimate limit state (structural safety) and ser-viceability limit state have to be verified, see for example DIN 18800 Part 1. Since components for steel constructions are usually rather slender and thin-walled, struc-tural safety verifications for constructions susceptible to losing stability regarding flexural, lateral torsional and plate buckling are of major significance and therefore constitute a main focal subject in static calculations. In this context, the determination of internal forces and moments, deformations and critical loads is a central task. Its solution is treated in this book using the finite element method (FEM).

The calculations and verifications have to meet the legal requirements as well as the state of the art. For steel structures the basic standard DIN 18800 and corresponding engineering standards, or Eurocode 3, have to be taken into consideration. Table 1.1 contains a compilation of the verification methods according to DIN 18800 and the verifications as they are generally applied. Eurocode 3 contains similar regulations.

Table 1.1 Verification procedures according to DIN 18800 and common verifications

Verification procedure

Calculation of stresses Sd

Calculation of resistances Rd

Verifications

Elastic-Elastic

Elastic theory stresses and

Elastic theory design value of yield

stress fy,d

Verification of stresses: R,d = fy,d

R,d = fy,d / 3v R,d = fy,d

Elastic-Plastic

Elastic theory internal forces and moments N, My, etc.

Plastic theory utilisation of the plastic bearing capacity of the

cross sections

e.g. My Mpl,y,d or using interaction conditions or the

partial internal forces method

Plastic-Plastic

Plastic theory internal forces and moments according

to the plastic hinge or plastic zone theory

Plastic theory utilisation of the plastic bearing capacity of the cross sections and the

static system

According to the plastichinge theory or according to the plastic zone theory (with

computer programs)

The use of a verification method implies that the individual cross section parts (webs and flanges) can carry the compression stresses, so that no buckling occurs and a suf-ficient rotation capacity is provided. Assistance for the checking of the b/t relations can be found in profile tables; see for example [29]. If only longitudinal axial stresses and shear stresses occur, it is 2 2

v 3 . The verification of the equivalent/ R,d and / R,d >


stress (verification method Elastic-Elastic) is only required for


1 Introduction 2

0.5. Perfectly plastic internal forces and moments for rolled sections can be found in profile tables [29], interaction conditions and verifications using the partial internal forces method in [29] and [25].

The subscript "d" for Sd and Rd in Table 1.1 indicates that the stresses must be de-termined using the design parameters of the loads and that the design resistance has to be applied; see Section 1.7. Section 1.4 “Linear and Nonlinear Calculations” includes specifications concerning the calculation of stress and resistance.

1.2 Methods to Determine the Internal Forces and Moments

As it is generally known, internal forces and moments in statically determinate sys-tems may be calculated with the help of equilibrium conditions and intersection methods. This is not possible with statically indeterminate systems and thus another solution procedure is used, such as the force method, which is the classical method of structural analysis. It is appropriate for hand calculation and very straightforward since it is easy to understand in engineering terms. However, the disadvantage is that for differing structural systems many approaches must be developed and, moreover, it is completely inappropriate for many tasks.

Figure 1.1 Unknown values of the force, displacement and reduction method for a selected example

Figure 1.1 exemplifies a singlely indeterminate girder. Hence, when using the force method, one unknown force value must be defined. After this, the moment distribu-tion can be determined using the equilibrium conditions. The basis of the method is always the choice of a statically determinate structure (primary structure). Since there are several possibilities for doing so, the two systems in Figure 1.1 are selected exam-ples.

1.2 Methods to Determine the Internal Forces and Moments 3

Generally, there are three methods for determining the internal forces and moments:

Force method

Displacement method FEM

Reduction method FEM

Moreover, there are numerous variations within these three methods, which cannot be discussed in detail here. Whereas when using the force method, the forces are the unknown variables of the emerging equation system, when using the displacement method, the unknown variables are the displacements, i.e. the displacements and rotations. If the structural system is divided into finite elements (e.g. beam elements or segments), the displacement method is extremely appropriate for a generalised formulation and so is applicable in many different situations. The ideas involved are not simple in engineer terms but are very mathematical because large amounts of data must be handled with sizable equation systems solved. The actual amount of data and the size of the equation system will, of course, depend on the system under consider-ation, but it will certainly be more than would be needed for the force method.

Figure 1.1 shows the application of the displacement method. Using this method, the unknown values are the deformations at the nodes, i.e. for the examined beam the displacement w and the rotation . Thus, there are two unknowns per node, so depending on the geometric boundary conditions, there will be between one and 19 unknowns in each example. Using the FE model with 10 elements, a relatively large number of unknowns (19) occur, but there is no need of further hand calculation, which is an advantage. For procedural reasons, all state variables (bending moments, shear forces, displacements, rotations) at the nodes, i.e. virtually in the entire system, are determined.

Due to the numeric complexity, the widespread use of the FEM with the displacement method is closely connected to the rapid development of high-capacity computers. Until about 1985, it was important to model structures using finite elements in such a way that the limited memory capacity was sufficient and that computing times did not escalate. Nowadays, these considerations are only important for exceptional structures and calculations. Then again, it is often seen that in static calculations exaggeratedly fine FE-modelling or the use of inappropriate finite elements create reams of paper. As shown in Figure 1.1, it can be very reasonable to calculate a single-span beam using an FEM program, since all values for the necessary verifications are directly obtained by the program and the corresponding pages for the static calculation can be printed out with minimal effort.

The third method mentioned above is the reduction method, which is suitable for continuous beams including instance sharp bends. The unknowns of the resulting equation system are the unknown internal forces and displacements at the beginningof the beam structure (see Figure 1.1), so that for beams, a maximum of seven

1 Introduction 4

unknowns results. Accordingly, the requirements for disc space and calculating time are low, which was, as already mentioned above, of vital importance until about 1985. The reduction method was often used to design plate-girder bridges, since even for multi-span girders only two unknowns arise (main beam, transfer of vertical loads). Computer programs using the reduction method are rare these days. However, the procedure can definitely be found in current FEM programs for beams and frameworks, though here it is first calculated with a relatively rough division into finite elements according to the displacement method. Subsequently, the individual beams are generally divided into five to ten elements in order to be analysed more closely using the reduction method. Further details on the reduction method can be found in [31].

1.3 Element Types and Fields of Application

For FEM calculations structures are idealised using structural systems (beams, frameworks, plates, etc.) and are then appropriately divided into finite elements – see Figure 1.3. A distinction is drawn between:

line elements (one-dimensional, straight or curvilinear)

area elements (two-dimensional, plane or circumflex)

volume elements (three-dimensional, block-shaped or with curved surfaces)

Figure 1.2 Element types and possible nodal degrees of freedom

1.3 Element Types and Fields of Application 5

a) Beam elements for frames c) Curvilinear boundary elements for cross sections

b) Rectangular elements for plate buckling

Figure 1.3 Examples for the discretisation of different problems of steel structures using finite elements

In Figure 1.2, corresponding elements are exemplified. If beams and frameworks are to be analysed, it may in some cases be useful to examine the cross section with the help of the FEM. Depending on the task, the following elements are used:

line elements (one-dimensional, straight or curvilinear) or

area elements (rectangular or triangular, straight or curvilinear boundaries) For the calculation of steel structures almost exclusively beam elements are used (see Figure 1.3a). These are often part of the following structural systems:

single-span and continuous beams

columns and plane frames

plane and three-dimensional trusses

three-dimensional frameworks

girder grids The quoted static systems mainly appear in structural, industrial and plant engi-neering. Due to different stresses, beam elements with up to seven deformation variables in each node (degrees of freedom) are required. The number of required de-formations per node is discussed in more detail in the Chapters 3 and 5.

Finite beam elements are also commonly used for the calculation of bridges. Area elements (plates, shells) are rarely used, whether for plate, beam-framework, bow or cable-stayed bridges. An essential reason for this is that the current standards and codes are almost exclusively designed to suit the needs of calculating beam

1 Introduction 6

structures. Moreover, apart from a few exceptions, the accuracy of these calculations is quite sufficient.

An interesting field of application for finite area elements in steel structures is plate buckling. For example, Figure 1.3b shows the upper flange of a beam which has been divided into finite elements in order to perform an analysis of plate buckling. This topic is discussed in Chapter 10, where a rectangular plate element for the determina-tion of eigenvalues and modal shapes is derived. Apart from that, area elements are of course used for specific scientific research and development. Since, as has been mentioned, area elements are not often used, and volume elements even less so, for steel structures, the following can be stated:

Steel structures are almost exclusively calculated by using beam elements.

A range of beam elements are needed to appropriately calculate all occurring structures and loads.

Finite elements for the analysis of cross sections are covered in Chapter 11. As an example, Figure 1.3c shows the finite element modelling of a rolled I-section usingarea elements with curvilinear boundaries.

1.4 Linear and Nonlinear Calculations

Theoretically and numerically, linear calculations (first order theory) constitute the starting point. The following assumptions are the basis:

The material provides a linear elastic behaviour in the whole structure, which means that Hooke’s law is valid without restrictions of any kind.

The influence of the deformations of the structure is so small that it may be neglected and the equilibrium conditions may be formulated for the undeformed structure.

Structural and geometric imperfections, i.e. residual stresses and initial defor-mations, may be neglected.

Nonlinear calculations usually require a higher effort than linear calculations. Con-cerning the nonlinearity, we need to distinguish between physical and geometric nonlinearities. Regarding physical nonlinearity, the assumption of a linear elastic material behaviour is renounced and the plastifications in parts of the construction are considered in order to obtain more economic structures, i.e. structures of less weight. As far as the plastification is only considered regarding the bearing capacity of the cross sections, this approach is to be assigned to the verification method Elastic-Plastic in Table 1.1. Internal forces and moments are determined according to the elastic theory (elastic calculation of the system) and only load cases are permitted where a maximum of one plastic hinge occurs. In comparison to that, the plastic

1.5 Designations and Assumptions 7

bearing capacity of the cross sections and the system are utilised with the verificationmethod Plastic-Plastic, i.e. the spread of plastic zones or the development of several plastic hinges is permitted.

While the physical nonlinearity is mainly considered for economic reasons, the geometrical nonlinearities for structures susceptible to losing stability are indispensable for safety reasons. In comparison to linear calculations, relatively large deformations lead to higher internal forces and moments. For that reason, verifications against flexural, lateral torsional or plate buckling have to be executed.

In conjunction with geometric nonlinear calculations, it should be mentioned that the verifications according to the valid standards and codes, as for instance DIN 18800 Part 2, rely on a linearisation according to second order theory. This approximation is therefore the basis for the determination of deformations, internal forces and moments as well as critical loads (eigenvalues) in conformity with the codes. As a general rule, the accuracy of calculations according to second order theory is sufficient in terms of applications in engineering practice since deformations for steel structures are usually relatively small. In exceptional cases, it may be necessary to perform precise geometric nonlinear calculations. This is always the case when large or even very large deformations occur.

Summing up, the following can be stated:

The verification method Elastic-Elastic is still most frequently used; see Table 1.1. For the calculation of the static system a linear elastic material behaviour is assumed with which the internal forces and moments as well as the corre-sponding stresses are determined. Using these stresses, the verification can be executed.

Recently, the verification method Elastic-Plastic has been used more often. With this procedure, the bearing capacity can be increased until reaching the first plastic hinge.

For structures susceptible to losing stability the geometric nonlinear problem is linearised and internal forces and moments are determined according to second order theory. This linearisation is also used for the determination of critical loads (eigenvalues).

1.5 Designations and Assumptions

In this section, descriptions and assumptions are compiled which are needed for beam and frame structures. Some also apply for plates and the FE analysis of cross sections. In the Chapters 10 and 11, other terms and assumptions are added relating to these topics. The basis for the designations is found in DIN 1080 and DIN 18800.

1 Introduction 8

Abbreviations

ODE ordinary differential equation COS coordinate system LCC load case combination SMI self moment of inertia PIF-method partial internal forces method tot total ult ultimate cr critical

Variables in the global X-Y-Z coordinate system

Beam structures are divided into beam elements, which are connected to each other at the nodes. As shown in Figure 1.2, nodes can also be arranged on the inside of an element (internal nodes). Nodes are defined in the global X-Y-Z coordinate system (COS) by using the coordinates Xk, Yk and Zk as shown in Figure 1.4. Moreover, all global deformations and loads at the nodes relate to this COS. For reasons of clarity, the subscript k has been neglected for these values in Figure 1.4.

The deformations in the global COS are marked by an overbar (horizontal line above the variable). This designation will also be used for vectors and matrices if they apply to the global COS.

Figure 1.4 Definition of deformations and loads in the global X-Y-Z coordinate system


Variables in the local x-y-z coordinate system Coordinates, ordinates and reference points

x longitudinal direction of the local COS y, z principal axes in the cross section plane (local COS)

standardised warping ordinate S centre of gravity M shear centre

beam axis x, principal axes y and z, centre of gravity S, shear centre M

Figure 1.5 Beam in the local coordinate system with displacements, internal forces and moments

Beam elements apply to a local x-y-z COS and, as longitudinal beam axis, the x-axis is defined through the centre of gravity S. The axes y and z are the principal axes of the cross section. According to Figure 1.5, some of the displacements and internal forces and moments apply to the centre of gravity S, others to the shear centre M (y = yM, z = zM). For warping torsion a standardised warping ordinate is used.

Deformation variables

u, v, w displacements in x, y and z-direction (local COS)

x = rotation about the x-axis (twist)

y w rotation about the y-axis

z v rotation about the z-axisderivative of the angle of twist

Figure 1.6 Definition of positive deformations in the local COS

1 Introduction 10

Loads

qx, qy, qz distributed loads mx distributed torsional moment M L single load warping moment

Figure 1.7 Positive directions and application points of local loads

Internal forces and moments N longitudinal, axial force

Figure 1.8 Internal forces and moments at the positive intersection of a beam

Vy, Vz shear forces My, Mz bending moments Mx torsional moment Mxp, Mxs primary and secondary

torsional moment M warping bimoment Mrr see Table 5.1 (page 172) Subscript el: Limit internal forces and

moments according to the elastic theory

Subscript pl: Limit internal forces according to the plastictheory

Subscript d: design value

If the common definition of positive internal forces and moments (internal force defi-nition I) is used, the forces at the negative beam intersection act in directions opposite to the ones specified in Figure 1.8. With the sign definition II, the direc-tions of actions at both beam intersections are in compliance with the ones in Figure 1.8. In Figure 1.9, both definitions are shown for uniaxial bending with axial force.


According to custom, further subscripts are used to distinguish beam elements and nodes.

Figure 1.9 Internal forces and moments of the beam element “e” for uniaxial bending with axial force and sign definitions I and II

Stresses

x, y, z normal stresses xy, xz, yz shear stresses v equivalent stress

Figure 1.10 Stresses at the positive intersection of a beam

Cross section properties

A area Iy, Iz principal moments of inertia I warping constant IT torsion constant (St Venant)Wy, Wz section modulus Sy, Sz static moments iM, ry, rz, r values for second order theory and stability; see Table 5.1

ipy zI I

Apolar radius of gyration (inertia)

1 Introduction 12

Further symbols and assumptions Material properties (isotropic material) E modulus of elasticity, Young’s modulus G shear modulus

transverse contraction, Poisson’s ratio fy yield strength, yield stress fu ultimate tensile strength

u ultimate strain

Partial safety factors

M factor for resistances (material) F factor for loads (force)

Figure 1.11 Assumptions for material behaviour

Matrices and vectors

s vector of internal forces and moments K stiffness matrix G geometric stiffness matrix v vector of deformations p load vector subscript e: element

An overbar above the matrices and vectors indicates that they refer to the global co-ordinate system (X, Y, Z).

1.6 Fundamental Relationships 13

As long as nothing else is stated, the following assumptions and conditions apply:

A linear elastic-perfectly plastic material behaviour as shown in Figure 1.11 is assumed.

In terms of the beam theory, occurring deformations are small. For that reason, geometric correlations may be linearised.

The cross section shape of a beam is sustained when exposed to loads and deformations.

For biaxial bending with axial force, Bernoulli’s hypothesis is assumed, which states that the cross sections remain plane and that the influence of the shear stresses on the deformations due to shear forces is neglected (beams with infi-nite shear stiffness).

For warping torsion, Wagner’s hypothesis is assumed and the influence of the shear stresses on the rotation due to the secondary torsional moment is ne-glected.

1.6 Fundamental Relationships

Displacements (linear beam theory)

As is common for beams, y and z are the principal axes of the cross section and is the standardised warping ordinate – see Chapter 2. The longitudinal displacement uSrefers to the centre of gravity S and the displacements vM and wM describe the displacement of the shear centre M. For the longitudinal displacement u of an arbitrary point of the cross section the following formula applies:

S z yu u y z (1.1)

The first component is the displacement due to an axial force load. The second and the third components result from the bending moments and describe the displace-ments as a consequence of cross section rotations y and z. Here Formula (1.1) only covers displacements for which the cross section remains plane. The fourth compo-nent comprises the longitudinal displacement due to torsional loads depending on the derivative of the angle of twist .

The displacements v und w in the cross section plane result from the displacement of the shear centre M and from additional components deriving from the rotation

about the longitudinal axis (twist):

M Mv v z z (1.2)

M Mw w y y (1.3)

1 Introduction 14

Strains

The strains are linked to the displacements by geometric relationships. According to [25], the following relations are valid for the linear beam theory. For the displace-ments, Formulas (1.1) to (1.3) are considered and in addition, by neglecting secondary shear deformations, it is M zv , M yw and .

x S z yu u y zx

(1.4a)

yv 0y

, zw 0z

(1.4b, c)

xy Mu v (z z )y x y

(1.4d)

xz Mu w (y y )z x z

(1.4e)

yzv w 0z y

(1.4f)

Constitutive equations and stresses

The constitutive equations describe the correlation between stresses and strains. Neg-lecting the transverse strain, with the use of Hooke’s law, a material law describing isotropic, linear elastic material behaviour, and the strains defined in Formulas (1.4), the following stresses can be stated:

x x S z yE E u y z (1.5)

xy xy MG G (z z )y

(1.6)

xz xz MG G (y y )z

(1.7)

Internal forces and moments

Stresses can be summarised to resulting internal forces and moments. It must be pointed out that the axial force and the bending moments act at the centre of gravity, while shear forces, the torsional moments as well as the warping bimoment are re-lated to the shear centre – see Figure 1.8.

1.6 Fundamental Relationships 15

Table 1.2 Internal forces and moments as resultants of stresses

Condition Internal force/moment Definition

xF 0 : axial force xA

N dA

yV 0 : shear force y xyA

V dA

zV 0 : shear force z xzA

V dA

xM 0 : torsional moment x xz M xy MA

x xp xs

M y y z z dA

M M M

yM 0 : bending moment y xA

M z dA

zM 0 : bending moment z xA

M y dA

warping bimoment xA

M dA

Division of linear beam theory (infinite shear stiffness) into four subproblems

Table 1.3 shows four subproblems – biaxial bending with axial force and torsion – associated with the linear theory of beams with infinite shear stiffness. The table contains an allocation of loads, displacements and internal forces/moments as well as information concerning the equilibrium of a beam element and the stress x.

Table 1.3 Division of the linear beam theory according to [25]

“Axial force” “Bending about the z-axis”

“Bending about the y-axis” “Torsion”

Loads xx F;q zLyy M;F;q yLzz M;F;q LxLx M;M;m

Deformations Suu Mvv

MvyuMww

Mwzu u

Mzzv

MyywInternal forces and moments

N zM

yVyM

zVM

xsxpx MMM

Equilibrium xqN z y

y y

M V

V qy z

z z

M V

V qxsMM

xx mM

x =

SuEAN z

z

M

My

IE y v

y

y

M

Mz

I

E z w E

MI

1 Introduction 16

1.7 Limit States and Load Combinations

Limit states

The limit states of structures to be analysed and the corresponding load combinations are defined in “load standards” such as DIN 1055 [7] and EC 1 [9]. For the applica-tion additional information is given in the standards (e.g. DIN 18800 [8], EC 3 [10]). In this context, the bearing capacity of a structure characterises the ability of the car-rying members to resist all loadings which may occur during the erection work and the service life. The ultimate limit state describes a load situation of the structure where a violation of the limit would lead to a calculative collapse or a comparable failure, for example a rupture or a loss of stability and stable equilibrium, respec-tively. The demands on the ultimate limit state are related to the safety of people and the safety of the building including its equipment and facilities. In general, the states which may have to be observed cover the loss of the position stability (lifting, over-turning, buoying upwards), the failure of the structure or its members including the foundation (rupture, changeover in a kinematic chain, loss of stability) and the failure due to fatigue influences on the material and other time-related effects. With regard to steel structures, the ultimate limit state to be verified depends on the verification method (see Table 1.1):

beginning of a plastification

cross section being fully plasticised at one position

formation of a kinematic chain

rupture

Other limit states that may be relevant are: flexural buckling, lateral torsional buckling, plate and shell buckling as well as fatigue. In general, it has to be verified, for the entire structure and its members, that the design value of the internal forces and moments or stresses Sd due to the design loading Fd is smaller than the design resistance Rd:

d dS R (1.8)

The servicability limit state describes the conditions of a building beyond which it can no longer be used for its designated purpose. The demands on the serviceability are related to the function of the building, the safety of people and the structural appearance. It has to be verified that the design value of stress at the serviceability limit state does not exceed the design value of a serviceability criterion (e.g. tolerable deformations). Limit states for the serviceability are not specified in DIN 18800 and they are usually arranged and agreed on individually if they are not specified in other basic or engineering standards.

1.7 Limit States and Load Combinations 17

Since the ultimate limit state is the basis of a safe design, ensuring that the structure and its parts do not fail, is primary focus of this book.

Design loads and resistances

The safety concepts of the German and European standards are very similar. Both use so-called partial safety factors F and M for the determination of the design loads and resistances. These factors increase the “actual” loads to the design level and decrease the resistances accordingly. The factor F considers a possible unfavourable deviation of the load in terms of the statistical spatiotemporal spread and, in addition, possible insecurities in the mechanical and stochastic model. The factor M includes the spread of the particular resistance value and also covers inaccuracies in the mechanical model related to the calculation of the resistances.

The design value of a load Fd is determined by:

d F kF F (1.9)

Here, F is the partial safety factor which is associated with the particular load and Fkis the characteristic value of the load. If necessary, a combination factor as stated in Eq. (1.9) may be considered.

The design value of the resistance parameters Md is calculated by dividing the char-acteristic value of the resistance Mk (e.g. strength of the material fy,k and fu,k) with the partial safety factor M:

d k MM M (1.10)

Load combinations and resistance at the ultimate limit state

For the verification of the bearing capacity of a steel structure at the ultimate limit state different load combinations have to be examined which are mainly classified as follows:

basic combinations

exceptional combinations For the basic combinations two separate cases with corresponding loads F have to be considered. According to DIN 18800, this results in the following combinations:

permanent (dead) loads G and all variable loads Qi acting unfavourably:

F,G k F,Q i i,ki 1

G Q (1.11a)

permanent (dead) loads G and one unfavourable variable load Qi at a time: F,G k F,Q i,kG Q (1.11b)

1 Introduction 18

To clarify that the loads in the combination are rather combined and not necessarily directly added to each other, possibly due to acting in different directions or even at different positions of the structure, the symbol “ ” is used.

The design value of the permanent loads Gd is determined by:

d F kG G with F F,G 1.35 (1.12)

If the permanent load reduces the stress due to the variable loads, the partial safety for the permanent load has to be set to F = 1.0. It should be mentioned that additional rules are specified in the standards concerning the reduction of stress due to parts of the permanent loads.

The design value of the variable loads Qi,d of the combinations with one unfavourable variable load at a time is

d F i,kQ Q with F F,Q 1.5 (1.13a)

and for all variable loads acting unfavourably it is:

d F i i,kQ Q with F F,Q 1.5 and i 0.9 (1.13b)

For exceptional combinations, design values of the permanent loads Gd, all variable loads Qi,d and one exceptional load FA,d have to be considered. In contrast to Formulas (1.12) and (1.13b), the partial safety factor is used with F = 1.0 here. The design value for the exceptional load FA,d is determined with a partial factor of F = 1.0 as well.

At the ultimate limit state, the partial safety factor for the resistance is usually taken with:

M 1.1 (1.14)

The factor is not only used for the determination of the design material strength but has to be used for the design stiffness as well, which is determined with the nominal values of the cross section properties and the characteristic values of the elasticity modulus or the shear modulus, respectively. If the stability of members is not decisive, the factor M may be taken as 1.0.

Load combinations and resistance at the serviceability limit state

The safety factors F, combination factors and load combinations to be considered for the verification have to be arranged individually if they are not specified in different basic or engineering standards. At the serviceability limit state a partial safety factor of M = 1.0 is usually valid.

1.8 Introductory Example 19

1.8 Introductory Example

The following example is aimed to give a first overview of the verification methods according to DIN 18800 given in Table 1.1. In doing so, the main focus is set to the ultimate limit state. Due to the significance of this state as the basis of a safe design, as mentioned previously, it is the main focus of this book. Figure 1.12 illustrates a two-span girder with a uniformly distributed load to be verified. The distributed load is considered to consist of two components, one due to the dead load and one component including the snow loads, as shown in the figure.

Figure 1.12 Structural system of the introductory example

The calculation of the design load values follows with the load combination of Eq. (1.11b) regarding the partial safety factors F = 1.35 for the permanent load and F = 1.50 for one variable load according to Eq. (1.12) and (1.13a). This leads to the fol-lowing design load qd:

qd = gd + sd = 1.35 · 30 + 1.5 · 20 = 40.5 + 30 = 70.5 kN/m

With the partial safety factor of M = 1.1, the design yield strength of steel S 235 is:

fy,d = 24.0/1.1 = 21.82 kN/cm2

Verification method Elastic-Elastic

First of all, the stress in the system is determined by calculating the internal forces and moments. The mid support plays a key role for the verification of the bearing capacity since here the internal forces and moments are at maximum (see Figure 1.13). Using the internal forces and moments, maximum stresses can be calculated, leading to the following verifications:

2 2m

web

V 264.38 21.827.96 kN / cm 12.6 kN / cmA 33.2 3

2 2M 31 725max 27.35 kN / cm 21.82 kN / cmW 1160

verification is not successful!

1 Introduction 20

The (necessary) verification of the equivalent stress

d,y22

v f3

cannot be successful due to max > fy,d.

Figure 1.13 Bending moment and shear force according to the elastic theory

Verification method Elastic-Plastic

In order to verify the system in Figure 1.12, the plastic capacities of the cross section bearing capacity can be taken into consideration. Using the Elastic-Plastic procedure, the internal forces and moments are calculated according to the elastic theory – see Figure 1.13. For the verification of a sufficient load-bearing capacity the interaction conditions (e.g. DIN 18800) or the partial internal forces method can now be applied (see Chapter 8).

The use of the interaction conditions according to DIN 18800 requires knowledge of the internal forces and moments at the perfectly plastic state. By using the profile tables [29], Mpl,d = 285.2 kNm and Vpl,d = 419 kN can directly be obtained. This leads to the following verification:

33.0631.0419

38.264V

V

d,pl and < 0.9

pl,d pl,d

M V0.88 0.37M V

317.250.88 0.37 0.631285.2

= 0.979 + 0.234 = 1.21 > 1

verification is not successful!

1.8 Introductory Example 21

Verification method Plastic-Plastic

As shown with the previous verification, it is not possible to verify the bearing ca-pacity of the system in Figure 1.12 if the plastic reserves of the cross section are regarded at one position of the beam, which is, in this case, at the mid support. However, after the bearing capacity is reached at that position, a plastic hinge will develop and the system will still be able to carry additional loads since it will not be kinematic at that load stage. With the development of the plastic hinge (cross section in a perfectly plastic state) at the mid support due to MB and VB, the interaction condition used with the Elastic-Plastic procedure has to be fulfilled exactly (“= 1” instead of “ 1”). With V/Vpl,d > 0.33, it is:

B B

pl,d pl,d

M V0.88 0.37 1M V

pl,d BB B

pl,d

M VM 1 0.37 324 0.287 V0.88 V

This formula describes what maximum bending moment the cross section is able to carry at B with regard to the acting shear force.

Figure 1.14 illustrates the structural system regarding the symmetry after the forma-tion of the plastic hinge. For reasons of clarity, the subscript “d” to point out the design loads is neglected here. With regard to the equilibrium of the beam, the fol-lowing formulas can be stated for the internal forces depending on the position x:

xqM2

q)x(V B BB

MqV2

2xqxMx

2q)x(M

2B

Figure 1.14 Structural system after insertion of a plastic hinge at the mid support

With the equilibrium, the shear force at the support VB can be determined in terms of MB, as shown above. By regarding this relationship in the previous equation for MB,which was gained from the interaction condition, a formula for the calculation of the moment can be stated, which is now independent of VB:

1 Introduction 22

B BM 324 0.287 V and BB

MqV2

BB

MqM 324 0.2872

B324 60.7M 251.3 kNm

1.0478

The formation of the plastic hinge at the mid support, i.e. the full plastification, corresponds to MB = 251.3 kNm. This moment is smaller than Mpl,d = 285.2 kN due to the action of the shear force. It now has to be checked whether the arising internal forces and moments within the beam span can be carried by the cross section. The decisive stress in the span is caused by the internal bending moment. It reaches its maximum at the position V(x) = 0. Using the equilibrium formulation for V(x), this leads to:

BMqV(x) q x 02

BMx 2.404 m2 q

At that position, the bending moment can now be calculated with the equilibrium equation of M(x):

2

F70.5 6 251.3 70.5 2.404max M 2.404 2.404

2 6 2508.4 100.7 203.7 204 kNm

For the verification within the beam span the internal forces and moments are considered with V = 0 and max MF = 204 kNm, leading to the following condition:

pl,d

M 204 0.72 1M 285.2

If the condition is fulfilled, there is no development of a plastic hinge within the beam span. Therefore, the system will not form a plastic mechanism (chain) as failure mode and the load-bearing capacity can be verified using the Plastic-Plastic method. How-ever, it should be mentioned that additional verifications are necessary:

local buckling of cross section parts and sufficient cross section rotation capac-ity with existing b/t limit b/t (conditions are fulfilled for an IPE 400)

lateral torsional buckling if the deformations v and are not sufficiently re-stricted (by bracings for instance)

load transmission of support reactions into the beam; where required, stiffeners may have to be installed

if necessary, verifications at the serviceability limit state

1.9 Content and Outline 23

1.9 Content and Outline

Figure 1.15 contains an overview of the chapters of this book showing their interrelationship. The aim of the figure is to show which chapters are based on one another. At the same time, it gives information about which basic knowledge is of advantage for the understanding of a given chapter.

Figure 1.15 Chapter structure and dependencies

As shown in Figure 1.15, Chapters 2 and 3 are of foundational character. In Chapter2 the cross section properties arising in beam theory are discussed. Their knowledge is of fundamental importance for the application of beam theory (Chapters 4, 5, 6 and 9) and for a further treatment of cross sectional issues (Chapters 7, 8, 11). Chapter 3gives information about the principles of the finite element method (FEM). The basic idea of the method is needed for the understanding of Chapters 4, 5, 10 and 11 deal-ing with the numerical approach for beams and frameworks, for plates and for cross sections of beams.

Chapters 4, 5, 6 and 9 deal exclusively with the topic “beams, frames and members”. Here, the numerical backgrounds and procedures, the solution methods and the verification of bearing capacity are dealt with in detail. Since beams have a special importance in steel construction, these chapters are a central part of the book. With regard to the formulation of finite beam elements, the cross section properties (Chapters 2 and 11) are of significance and for the verification of members the resistance of the cross sections (Chapter 7 and 8).

Figure 1.15 shows that Chapters 2, 7, 8 and 11 can be described by the umbrella term “cross sections”. While in Chapter 2 the cross section properties arising from beam theory are discussed, Chapters 7 and 8 give information about the bearing capacity

1 Introduction 24

of cross sections. It is shown how stresses are determined and how the plastic resis-tance of cross sections can be analysed. The contents of chapters 2 and 7 are very helpful for the understanding of Chapter 11 where the cross sections are treated us-ing the finite element method (FEM).

Chapter 10 deals with the finite element method for plane load-bearing structures. The main focus is set on deriving a finite element for plate buckling and on the verifi-cation procedures against this failure mode. Besides Chapter 3 about the FEM-principles, this chapter is also closely related to Chapter 6 referring to the solution methods for equation systems and eigenvalue problems.

1.10 Computer Programs

Calculation examples in this book and additional examinations have mainly been per-formed using the following computer programs:

FE-Beams FE-Frames FE-Plate Buckling CSP-Three Plates CSP-Table CSP-FE CSP-FE ML

These are programs of the Institute of Steel and Composite Structures at the Ruhr-University Bochum, Germany. Information on FE-Beams (for beams, columns etc.),FE-Frames, FE-Plate Buckling, CSP-Three Plates, CSP-Table (CSP: Cross SectionProperties) and a multitude of further programs can be taken from [33]; see www.ruhr-uni-bochum.de/stahlbau as well. Details on the programs CSP-FE and CSP-FE ML are included in [54].

For the purpose of comparison and for further analysis, calculations with the following programs have also been carried out:

RSTAB Ing.-Software Dlubal GmbH, Tiefenbach, Germany RFEM Ing.-Software Dlubal GmbH, Tiefenbach, Germany BT II Friedrich + Lochner GmbH, Stuttgart, Germany DRILL FIDES DV-Partner GmbH, München, Germany ABAQUS ABAQUS, Inc., Providence, Rhode Island, USA ANSYS ANSYS, Inc., Canonsburg, Pennsylvania, USA

2 Cross Section Properties

2.1 Overview

Cross section properties, as e.g. moments of inertia Iy and Iz, are used for the determi-nation of:

deformations internal forces and moments in statically indeterminate structures with nonuni-form cross sections stresses according to the elastic theory

Since standardised cross section properties are usually used, the cross section ordi-nates y, z and have to be defined in such a way that the conditions

y z yz y zA A A A A A 0 (2.1)

are fulfilled. The ordinates y and z are then the principal axes and is the standar-dised warping ordinate. This is also referred to as the y-z principal coordinate sys-tem if, furthermore, is included as the y-z- principal system. The cross section properties of the abovementioned conditions are defined as follows (for example):

yA

A y dA (2.2)

The general aim of the standardisation is the decoupling of the system equations compiled in Table 1.3. The split-up into the four subproblems shown (axial force, bending about the strong and the weak axis, torsion) is only possible with reference to the y-z- principal system. If arbitrary reference systems are used, the four sub-problems are not independent of each other. The basics and the background of the standardisation are explained in detail in [25]. For the determination of the y-z-principal system and the cross section properties it is advisable to distinguish two subtasks:

a) biaxial bending with axial force position of the centre of gravity S direction of the principal axes y and z (angle ) area A and principal moments of inertia Iy and Iz ordinates y and z in the principal coordinate system

b) torsion position of the shear centre M integration starting point or const. transformation value for the warping ordinate warping constant I torsional rigidity IT standardised warping ordinate


2 Cross Section Properties 26

Table 2.1 Further cross section properties

section modulus y yW I max z or yI minz

section modulus z zW I max y or zI miny

gyration radius y yi I A

gyration radius z zi I A

polar moments of inertia p y zI I I

polar gyration radius p pi I A

static moment yA s

S s z dA

static moment zA s

S s y dA

first degree area moment with A s

A s dA

values for second order theory and stability iM, ry, rz, r (see Section 5.3, Table 5.1)

Which values are effectively required always depends on the task. For rolled profiles all necessary values can be taken from tables; see for example [29]. Table 2.1 con-tains a compilation of further properties being needed depending on the task.

Notes on the designations

In the relevant literature, y and z are not always the principal axes of the cross section. Iy and Iz are then also not the principal moments of inertia and a value Iyzoccurs. The designations in the present book are chosen in such a way that y and z are always principal axes. Based on the preconditions, it follows that Iyz = 0. In arbitrary reference systems, the ordinates are designated by y and z .

The area integrals are designated by “A” and by the respective subscripts in all reference systems, e.g. Ay . The principal moments of inertia are

Iy = Azz and Iz = Ayy

The designations chosen here allow for a consistent depiction of the interrelations during the standardisations. Further, the formulas for the calculation of the stresses remain unchanged, as for example

y zx

y z

M MMN z yA I I I (2.3)

2.1 Overview 27

and all the relations for the system or deformation calculations containing Iy, Iz or Iremain valid as well.

Example: Single-span beam

For the single-span girder shown in Figure 2.1, the deflection at midspan, the hori-zontal displacement at the right beam end and the maximum axial stress x are to be calculated.

Figure 2.1 Introductory example single-span beam

4yz x

m right xy y

Mq F5 Nw ; u ; z384 EI EA A I

2x y zN F 200 kN ; max M q 8 112.50 kNm

2E 21 000 kN cm ; A and Iy from tables in [29]:

2 4 3y yA 53.81 cm ; I 8 356 cm ; max z 15.0 cm ; W 557.1 cm

4

m5 0.25 600w 2.40 cm

384 21 000 8 356

right200 600u 0.106 cm

21 000 53.81

2x

200 11 250max 3.72 20.19 23.91kN cm53.81 557.1

The bending stress is here calculated with Wy = Iy / max z = 557.1 cm3 (section modulus). The very simple example selected shows, what the cross section properties A, Iy and Wy are required for. Calculations are not necessary for that since the values can be taken from tables (e.g. see [29]).

If, in contrast, the cross section sketched in Figure 2.2 is considered, which consists of three plates, it becomes immediately clear that respective calculations have to be performed. The positions of the centre of gravity S, the shear centre M and the principal axes are unknown and have to be determined arithmetically just as the cross section properties. The methods for doing so are conveyed in the following sections.


At this point, however, Table 2.2 provides a first overview of which values have to be determined.

Table 2.2 Example for the calculation of cross section properties (program CSP-3plate)

2.2 Utilisation of Symmetry Properties 29

Figure 2.2 Asymmetric three-plate cross section

In the first part of Table 2.2, the cross section is described and the dimensions and the position of the three plates are entered. After that, the results are compiled according to the standardisation and the performance of the transformations. As can be seen, in the general case quite a large number of values have to be determined.

For the stress analysis due to internal forces and moments, the ordinates y, z and are required in the principal system. The corresponding values at the ends of the three plates are therefore calculated and issued by the RUBSTAHL program CSP-3plates. At the end of the output, the values ry, rz and r , are found, which are required for the calculations according to second order theory and stability analysis – see Table 5.1 in Section 5.3.

2.2 Utilisation of Symmetry Properties

Many cross sections feature symmetry properties which can be used for the simplifi-cation and shortening of the calculations.

Centre of gravity S

The centre of gravity is the origin of the y-z principal coordinate system, i.e. in S it is y = z = 0. Provided that for a selected coordinate system the static moments are equal to zero (Ay = Az = 0), the position of the centre of gravity is known.

It results from the conditions that the centre of gravity must be on the axes of the cross section symmetry. Symmetry means that the partial areas have to be arranged symmetrically, as it can be seen from some of the basic examples and the marked axes of symmetry in Figure 2.3. If the cross section has two symmetry axes, the


centre of gravity is at their intersection point (Figure 2.3 top). For cross sections with one symmetry axis the calculations become easier as well: only the position of S on the symmetry axis must be determined. For point-symmetric cross sections (Figure 2.3) the centre of gravity S is at the point of symmetry.

Figure 2.3 Positions of the principal axes, the centre of gravity S and the shear centre M

Principal axes

The position of the principal axes is characterised by the rotation angle of the princi-pal axes . For

yzA

A y z dA 0 (2.4)

it is = 0 and y and z are then the principal axes. From the sketches in Figure 2.3 it becomes obvious that the position of the principal axes is known if the cross section has at least one symmetry axis (symmetry axis = principal axis, second principal axis is perpendicularly to that).

Shear centre M

The position of the shear centre M is described in the y-z principal coordinate system, i.e. the point M has the coordinates yM and zM. Thus, the standardised warping ordi-nate always refers to the point M.

2.3 Standardisation Part I: Centre of Gravity, Principal Axes and Moments of Inertia 31

If, for the determination of , a reference point is chosen for which the area integrals are Ay = Az = 0, the chosen point is the shear centre. However, in addition, the con-dition A = 0 must be observed.

The conditions lead to the fact that the shear centre is always on a symmetry axis of the cross section. For (minimum) doubly symmetric cross sections it is in the inter-section of the symmetry axes and for point-symmetric cross sections in the point of symmetry – see Figure 2.3.

Standardised warping ordinate

Besides Ay = Az = 0, the condition A = 0 must be observed for the standardised warping ordinate . It is met for thin-walled open cross sections if, for the determina-tion of the warping ordinate, the point of origin of the integration and the integration directions are defined as follows:

starting point of the integration intersection of symmetry axis and profile centre line special case: M on profile centre line put starting point of the integration into M

integration directions symmetric, starting from the integration point

Note: If, for a point-symmetric cross section, the starting point of the integration is chosen to be in M, then A 0. This is true for the Z cross section in Figure 2.3 for example.

2.3 Standardisation Part I: Centre of Gravity, Principal Axes and Moments of Inertia

As explained in Section 2.1, the determination of the y-z- principal coordinate sys-tem and the cross section properties is divided into two subtasks. This section covers the first subtask, i.e. the calculations which can be assigned to biaxial bending with axial force.

Figure 2.4 Different cross sections: solid cross section, thin-walled cross section with hollow box and open thin-walled cross section


For the following considerations it is assumed that the cross section shape may be completely arbitrary. The cross sections in Figure 2.4 are used as examples for expla-nation. According to Figure 2.4, two cross sections have a vertical symmetry axis. However, this will not be taken into account here since the general case is being considered. As starting point, an arbitrary reference point B as origin of a y-zcoordinate system is defined. A second, parallel displaced y-z coordinate system shall have its origin in the centre of gravity S. This is rotated by the angle into the y-z principal coordinate system. The circumstance, along with transformation rela-tionships, is shown in Figure 2.5. For reasons of clarity, no cross section has been charted.

Transformation relationships:

S

S

y y yz z zy y cos z sinz z cos y sin

Figure 2.5 Coordinate systems for the standardisation of cross section properties

After the selection of the y-z coordinate system, the cross section properties

y z yz yy zzA, A , A , A , A and A

can be calculated. Methods for the calculation of integrals (for example)

yzA

A y z dA (2.5)

are explained in Section 2.4. The main focus here is supposed to be on the standardi-sation. The appropriate conditional equations are obtained from the conditions

Ay = Az = Ayz = 0, (2.6)

if the transformation relations according to Figure 2.5 are used for y and z. Since the position of the centre of gravity is independent of the directions of the cross section coordinates, one can also use conditions y zA A 0 instead of Ay = Az = 0:

y S y SA A

A y dA y y dA A y A 0 (2.7)

S yy A A and analogous S zz A A (2.8)


For Ayz the transformations lead to

yzA A

A y z dA y cos z sin z cos y sin dA 0 (2.9)

2 2yz yz yy zzA A cos sin sin cos A A 0 (2.10)

With 2 2cos sin cos2 and sin cos sin 2 2 it follows that:

yz yz

yy zz yy zz

2 A 2 A1tan 2 arctanA A 2 A A (2.11)

The derived relations for S Sy , z and serve the calculation of standardised cross sec-tion properties and two methods are given here. Method A is summarised in Table 2.3. Using this method, the standardisation procedure becomes very clear, because the transitions between the three coordinate systems become clearly visible through the coordinate transformation in calculation steps and . However, in practice, method A is used only for single-symmetric cross sections which consist of two or three individual parts. The method is very clear because if = 0, only the transfor-mations

S Sy y y or z z z (2.12)

are to be performed.

Table 2.3 Calculation of standardised cross section properties part I – method A

Calculation of y zA,A and A in the y-z coordinate system:

y zA A A

A dA ; A y dA ; A z dA

Position of the centre of gravity: S y S zy A A; z A A

Transformation of coordinates: S Sy y y ; z z z

Calculation of yz yy zzA , A and A in the y-z coordinate system: 2 2

yz yy zzA A A

A y z dA ; A y dA ; A z dA

Rotation angle of principal axes : yz

yy zz

2 A1 arctan2 A A

Transformation of coordinates: y y cos z sin z z cos y sinCalculation of the principal moments of inertia Iy and Iz:

2 2y zz z yy

A A

I A z dA; I A y dA


Table 2.4 Calculation of standardised cross section properties part I – method B

Calculation of y z yz yy zzA, A , A , A , A and A in the y-z coordinate system:

yA A

A dA; A y dA; etc.

Position of the centre of gravity: S y S zy A A; z A A

Transformation of cross section properties: 2 2

yz yz S S yy yy S zz zz SA A y z A ; A A y A ; A A z A

Rotation angle of principal axes : yz

yy zz

2 A1 arctan2 A A

Calculation of the principal moments of inertia Iy and Iz with transformations: 2 2

z yy yy zz yzI A A cos A sin 2 A sin cos2 2

y zz zz yy yzI A A cos A sin 2 A sin cos

Transformation of coordinates:

S Sy y y cos z z sin

S Sz z z cos y y sin

For more complex cross sections method B (see Table 2.4) is usually applied. In con-trast to method A, all cross section properties are calculated in the y-z coordinate system and then transformed step by step. At the very end, the cross section coordi-nates are transformed into the principal axes y and z. Amongst others, method B also forms the basis of the example in Table 2.2, i.e. for the calculations with the RUBSTAHL program CSP-3plates. Method B is recommended for the use in com-puter programs, because the programming is easier.

Example: Single-symmetric cross section

For the single-symmetric cross section in Figure 2.6 the position of the centre of gravity in the z-direction and the principal moment of inertia Iy (methods A and B) is calculated.

The values of the individual parts of the cross section are listed in Figure 2.6. The calculation of the cross section properties is performed with the methods explained in Section 2.4. For a y-z reference system in the centre of gravity of the IPE-profile one obtains:

3zA 53.81 0 20.0 15.0 0.5 14.14 15.0 0.5 529.17 cm

Centre of gravity: S zz A A 529.17 87.95 6.02 cm


Figure 2.6 Cross section for the example on the application of methods A and B

Iy with method A:

The centres of gravity of the individual parts are obtained with Si Si Sz z z . This

relationship is directly used for 3 3

2 2y y,SMI,i i Si

i 1 i 1A

I z dA I A z .

2 2 2y

4

I 8 364 53.81 0 6.02 20.0 15.5 6.02 14.14 15.5 6.02

13 382cm

Iy with method B:

zzA in the y-z coordinate system:

2 2 2 4zzA 8 364 53.81 0 20.0 15.5 14.14 15.5 16 566 cm

For the use of Table 2.4 the following is valid here: 2

y zz zz SI A A z A

2 4yI 16 566 6.02 87.95 13 379 cm

Additions on the practical application and the clarification of connections

For carrying out the standardisation, the formulas in Tables 2.3 and 2.4 suffice. The following additions are to facilitate practical usage and to serve the further clarifica-tion of connections.

Utilisation of symmetry properties

As explained in Section 2.2, the symmetry properties of the cross sections can be used to simplify the calculations. If symmetry axes exist, the calculations given in Ta-ble 2.5 are sufficient. The numbering refers to Table 2.4, i.e. the approach of method B.


Table 2.5 Utilisation of symmetry properties for the calculation of standardised cross section properties part I

a) Symmetry to the y-axis, choose y-z COS

b) Symmetry to the z-axis, choose y-z COS

Az = = 0 Ay = = 0

y, yy zzA, A A , A z zz yyA, A A , A

s yy A A s zz A A2

z yy sI A y A

y zzI A

2y zz sI A z A

z yyI A

sy y y sz z z

yy zzA , A and yzA as function of Iy and Iz

With the transformation relationships

y y cos z sin and z z cos y sin (2.13)

the following relationships result if one proceeds analogously as for yzA (see above):

2 2yy z y z y z y

1 1A I cos I sin I I I I cos22 2

(2.14a)

2 2zz y z z y z y

1 1A I cos I sin I I I I cos22 2

(2.14b)

yz z y z y1A I I sin cos I I sin 22

(2.14c)

For the second formulation, the trigonometric functions were converted with

2 1sin 1 cos22

(2.15a)

2 1cos 1 cos22

(2.15b)

1sin cos sin 22

(2.15c)


Equations (2.14a-c) allow for a clear depiction of the basic connections – see Figure 2.7. In that way, among others, the extremal properties of Iy and Iz become clear.

Figure 2.7 On the transformation of the moments of inertia

Invariances of the area-inertia tensor

The matrices of the components of the area-inertia tensor

z

y

I 00 I

and yy yz

yz zz

A AA A

have tensor properties. Thus, there are relationships being independent of the angle .The 1st invariance is related to the sum of the principal diagonal elements and the 2nd

invariant to the determinant of the above matrices.

z y yy zzI I A A (2.16a)

2y z yy zz yzI I A A A (2.16b)

The first invariant can be directly taken from Figure 2.7 or it can be obtained by sub-traction of Eq. (2.14a) and (2.14b). After a short intermediate calculation, the second invariant can be confirmed with these equations. As it can easily be verified, also, that

2 2 2 2 2y z zz yy yzI I A A 2 A (2.17)

is an appropriate transformation relationship.


Principal moments of inertia Iy and Iz (alternative calculation)

The formulas for Iy and Iz given in Table 2.4 are, because of the trigonometric functions, relatively impractical for hand calculation. With Figure 2.7 and the application of the theorem of Pythagoras one obtains:

2 2z y yy zz 2

yzI I A A

A2 2

(2.18)

Using the first variant of the area-inertia tensor,

y yy zz zI A A I (2.19)

can be replaced. From this2 2

yy zz yy zz 2z yz

A A A AI A

2 2(2.20)

results and

2 2z yy zz yy zz yz

1 1I A A A A A2 4

(2.21)

Iy can be calculated in the same manner. The allocation of the signs in front of the root should be conducted in such a way that max I is allocated to the larger value of

yyA and zzA

Thus, for hand calculation, the usual calculation method for the determination of Iyand Iz can be summarised as follows:

2 2yy zz yy zz yz

max I 1 1A A A A A2 4min I

(2.22a)

yy zz z yA A : I max I und I min I (2.22b)

yy zz y zA A : I max I und I min I (2.22c)

Rotation angle of the principal axis

The formula for the calculation of standardised cross section properties according to Table 2.4 are the basis of the RUBSTAHL program CSP-3plates – see Table 2.2. Here, the rotation angle of the principal axis is calculated with:

z~z~y~y~

z~y~

AAA2

arctan21

(2.23)


Figure 2.8 Possible positions of the y z coordinate system

The function “ARCTAN” in MS Excel provides values between /2 and /2, i.e. for the following scope of application is valid:

4 4 or 45 45 (2.24)

In some of the relevant literature, angles can be found that are considerably larger and lead to a major rotation of the coordinate system. Since the y-z coordinate system may be turned further 90°, 180° or 270° (Figure 2.8), it is not wrong to calculate with large angles. For the calculations, however, it is more advisable to stay in the range between 45° and +45°; see Figure 2.9.

a) positive angle

yzA 0 and yy zzA A or yzA 0 and yy zzA A

b) negative angle

yzA 0 and yy zzA A or yzA 0 and yy zzA A

Figure 2.9 Directions of the principal axes y and z (angle )

A basic difficulty occurs if the denominator in the formula for is equal to zero, i.e. z~z~y~y~ AA . For 0A z~y~ the quotient is indeterminate. In the MS Excel program

CSP-3plates, it is calculated with the following instruction:

yz yy zzIF ABS A 0.00001; 0; IF ABS A A 0.00001; 4; according formula


The function “If (check; then_value; else_value)” has been changed a little for rea-sons of better readability. In principle, for 0A z~y~ it is always set as = 0. Also, if

0AA z~z~y~y~ , then 4 is chosen (one could also choose 4 ). In a few special cases it can happen that changes from +45° to 45° with minor changes of the cross section.

Polar moment of inertia

The polar moment of inertia is equal to the first invariant of the area-inertia tensor

Ip = Iy + Iz (2.25)

The polar gyration radius becomes:

p pi I A (2.26)

2.4 Calculation of Standardised Cross Section Properties Part I

2.4.1 Separation of the Cross Section into Partial Areas

Most cross sections used for steel construction consist of several individual parts, so that a direct integration for the determination of the cross section properties (see Ta-bles 2.3 and 2.4) is not possible or at least is impracticable. This is also true for cross sections of solid (massive) construction where normally breaks occur in the cross section contour.

Figure 2.10 Separation of cross sections into partial areas (three examples)

2.4 Calculation of Standardised Cross Section Properties Part I 41

Therefore, for the calculation of the cross section properties the cross sections should be separated into partial areas. It is advisable that the partial areas correspond to basic cross sections for which areas, positions of the centre of gravity and moments of in-ertia can be calculated with known formulas or taken from tables. Figure 2.10 shows three examples for the separation into partial areas. For the separation, the following partial areas emerge:

rectangle (flat steel, plate, concrete) circle (round steel)fillet area (for rolled-steel sections) triangle (concrete) rolled cross section, here I-profile

Table 2.6 Calculation of cross section properties by summation over partial areas

Position of the partial area “i” in the y-z reference system

Transformations:

Siiy y y

Siiz z z

Si: centre of gravity of the partial area

Cross section properties of n cross section parts (n partial areas)Area and static moments (first degree moments of area):

n n n

i y Si i z Si ii 1 i 1 i 1

A A ; A y A ; A z A

Moments of inertia (second degree moments of area): n n

yz yz, SMI , i Si Si ii 1 i 1

A A y z A

n n2

yy yy, SMI, i Si ii 1 i 1

A A y A

n n2

zz zz , SMI, i Si ii 1 i 1

Self moments of inertia (SMI) parts

A A z A

Steiner

In principle, a cross section can be separated into arbitrary partial areas or partial cross sections. In the sketch of Table 2.6 one arbitrary partial cross section “i” and the y-z reference system are outlined. At the centre of gravity of the partial cross section, a coordinate system with the coordinates iy and iz is set up. The horizon-


tal lines should indicate that the coordinate system of the partial cross section has axes parallel to the y-z reference system. The following coordinate transformations can be taken from the sketch in Table 2.6:

Siiy y y (2.27a)

Siiz z z (2.27b)

These relations are now used for the calculation of the cross section properties. For the portion of the partial area “i” in the area integral yA one obtains:

i i i i

y,i Si Si Si ii iA A A A

A ydA y y dA y dA y dA 0 y A (2.28)

The first integral in Eq. (2.28) is zero because the origin of the ordinate iy is located

in the centre of gravity of the partial area (static moment of the partial area = 0). For the second integral it is Siy const. and therefore it can be written in front of the integral. Now, yzA is examined as an example of the second degree area moments. The partial area “i” has the following portion:

i i

i i i i

yz ,i Si Sii iA A

Si Si Si Sii i i iA A A A

yz,SMI,i Si Si i

A y z dA y y z z dA

y z dA y z dA z y dA y z dA

A 0 0 y z A

(2.29)

The first summand in Eq. (2.29) is the self moment of inertia (subscript SMI) of the partial area “i”. The next two terms are equal to zero since these are the static mo-ments of the partial area. The last summand is the so-called Steiner part.

The relations derived here can be set up for the other area integrals in an analogous manner. By summation across all cross section parts, the properties of the total cross section are obtained. The respective formulas are listed in Table 2.6. It can be seen that the integrals are replaced by equivalent sums.

For calculations with the help of Table 2.6, the self moments of inertia of the partial areas are required. Normally, we try to arrange the y-z reference system that parallel axes are principal axes of the partial cross sections. In this case it is:

yz,SMI,iA 0 (2.30a)

yy,SMI,i z,iA I (2.30b)

zz,SMI,i y,iA I (2.30c)


The principal moments of inertia Iy and Iz can be determined with the formulas in Ta-ble 2.9 or for rolled profiles they can be taken from tables, e.g. [29]. For the three cross sections in Figure 2.10, the reference systems are arranged so that the axes yand z are located horizontally or vertically, respectively. In this position, the local axes of the fillets and the triangles are not principal axes. Therefore, Table 2.9 shows the moments of inertia for this position of axes also.

For some applications, partial cross sections occur where the principal coordinate systems show rotation angles with respect to the y-z reference system. Then it is advisable to transform Iy = Azz and Iz = Ayy in terms of the directions of the y-z sys-tem. The comparison of the sketch in Table 2.7 with Figure 2.5 shows that it is a transformation of the y-z into the y-z system. Using the same method as explained in Section 2.3, the relations listed in Table 2.7 are obtained, also see Eq. (2.13) and (2.14).

In the sketch of Table 2.7, a doubly symmetric I-section has been selected as an example for the partial cross section “i”. This could be, for example, a rolled profile.

Table 2.7 Transformation of the principal moments of inertia of a partial cross section into the self moments of inertia for the y-z reference system

Self moments of inertia:

yz,SMI,i z,i y,i i iA I I sin cos

i2 2

yy,SMI,i z,i y,i iA I cos I sin 2 2zz,SMI,i y,i i z,i iA I cos I sin

2.4.2 Partial Areas of Thin-Walled Rectangles

Cross sections for steel construction often consist of individual thin-walled rectangu-lar parts (t << ), i.e. of plates or flat steels. The self moment of inertia of the individual parts about the weak axis is usually so small that it can be neglected. Un-


der this condition, thin-walled cross sections can be concentrated to their profile centre lines.

The sketch in Figure 2.11 shows an individual rectangular part in an arbitrary posi-tion. It should be the partial cross section “i”. However, for reasons of clarity, the subscript “i” is omitted here. It is only used for the local axes y(i) and z(i) in order to avoid confusion and to establish a reference to Tables 2.6 and 2.7. As can be seen, the I-section has been replaced by the rectangular plate. Due to the thin walls

y,iI 0. (2.31)

With3

z,iI t 12 (2.32)

it follows from the self moments of inertia of Table 2.7 (here without subscript “i”): 3

yz,SMI e a e aA y y z z A 12 t 12 sin cos (2.33a)

2yy,SMI e aA y y A 12 3 2t 12 cos (2.33b)

2zz,SMI e aA z z A 12 3 2t 12 sin (2.33c)

plate thickness: t

ordinates in a: aa z,y

ordinates in e: ee z,y

plate length: 2ae

2ae zzyy

ae zzsin

ae yycos

area: tA

Figure 2.11 Thin-walled rectangular cross section part

The first and second degree area moments are obtained via the integration of the linear functions y and z, where Table 2.8 establishes the connection to familiar inte-gral tables which are often used in the field of statics:

y e a z e a1 1A y y A ; A z z A2 2

(2.34a)

yz a a e e a e e a1A 2 y z 2 y z y z y z A6

(2.34b)


2 2 2 2yy a a e e zz a a e e

1 1A y y y y A ; A z z z z A3 3

(2.34c)

These formulas are especially suited for programming. With the abbreviations

m a ey y y 2 ; e ay y y (2.35a)

m a ez z z 2 ; e az z z (2.35b)

the separation into self moments of inertia and Steiner parts remains apparent. Addi-tionally, the resulting formulas are easier to use for hand calculations:

y m z mA y A ; A z A (2.36a)

yzA y z A 12 + m my z A (2.36b)

yyA 2y A 12 + 2my A (2.36c)

zzA 2

Self moments of inertia

z A 12 + 2m

parts

z A Steiner

(2.36d)

Table 2.8 Integrals of g(s) and g(s) h(s)


2.4.3 Basic Cross Sections and Elementary Compound Cross Section Shapes

The derivations in the previous sections show that the knowledge of the properties of basic cross sections is required. Table 2.9 contains a compilation which is sufficient for most calculations.

Table 2.9 Areas and moments of inertia of basic cross sections

a) in the - cos

Thin-walled plate:A t

3 2yyA t cos 12

3 2zzA t sin 12

3yzA t sin cos 12

Right-angled triangle:A b h 2

3 3yy zzA h b 36 ; A b h 36

2 2yzA b h 72 Sign:

Quarter circle:2 4 rA r 4 ; e

34

yy zz 264 rA A 1

169

4

yz32 rA 19 8

Rolled fillet (square quarter circle): 2 2 rA 1 r ; e4 3 4

4yy zz

11 3A A r9 4 16

4yz

28 9A r72 4


Table 2.9 (continuation)

b) in the y-z principal axes system

Rectangle: Circle/round bar steel: A b h

3yI b h 12

3zI h b 12

2A d 4

4

y zdI I

64

Rolled profile: Annulus/thin-walled tube: IPE 300:

2A 53.81 cm4

yI 8 356cm4

zI 603.8 cm see [29]

mA t d

3y z mI I t d 8

Segment of a thin-walled tube:

m mA 2 r t ; e r sin

3z mI t r sin cos

3 2y mI t r sin cos 2 sin

The angle is measured in radian!

If necessary, additional formulas can be found in the relevant literature, as for in-stance in [90]. In addition, a large compilation of properties for common steel profile series is given in [29].

Doubly symmetric I-sections

Doubly symmetric I-sections are often used for steel construction. They are separated into partial cross sections according to Figure 2.12 or idealised for reasons of simplification.

In order to capture rolled profiles exactly, the cross section is separated into seven parts: upper flange, web, lower flange and four fillets. With the help of Tables 2.6 or 2.10 (summation formulas) and Table 2.9 (basic cross sections), the following relations are obtained:

2g g sA 2 b t h 2 t t 4 0.2146 r (2.37a)


Figure 2.12 Idealisation of doubly symmetric I-sections

2 3 3 4y g g g s g

22g

I b t h t 2 h 2 t t 12 b t 6 0.03018 r

0.8584 r h 2 t 0.2234 r(2.37b)

3 3 4z g g s

22s

I t b 6 h 2 t t 12 0.03018 r

+ 0.8584 r t 2 0.2234 r(2.37c)

Based on these formulas, cross section properties of rolled profiles can be found in profile tables; see [29]. The accuracy of these exact values is often not required for practical application. The overlapping model of Figure 2.12 (third sketch on the top row) significantly simplifies the formulas. The overlapping of the web up to the mid-dle of the flanges represents the fillets approximately. In addition, the self moments of inertia about the weak axis may be neglected because the plates are thin-walled. Then, the plate thicknesses can be assumed as being concentrated in the centre lines of the profile so that the line model sketched in Figure 2.12 (top right) emerges.

The ratio of the web area to the total area is often captured by the parameter:

AAweb (2.38)


Table 2.10 Integration by summation (here A, Iy and Iz) in the y z principal axes system

n

ii 1A

A dA A

n n2 2

y y,SMI,i Si ii 1 i 1A

Self moments of inertia parts

I z dA I z A

Steiner

n n2 2

z z,SMI,i Si ii 1 i 1A

Self moments of inertia parts

I y dA I y A

Steiner

Then one obtains

sswebwebg thA:withAtb2A (2.39a)

2y sI A h 3 2 12 (2.39b)

121bAI 2z (2.39c)

The parameter of rolled profiles ranges between 0.2 and 0.45.

The separation of a welded I-section is given in Figure 2.12 at the bottom right. As it can be seen, the separation corresponds to the separation of a rolled profile without fillets. The above-mentioned formulas for A, Iy and Iz are therefore also valid for the welded I-section if all terms which include the fillet radius r are neglected or if r = 0, respectively. Of course, by approximation, the line model with or without overlapping can be used, as was explained for the rolled profile.

Three- and two-plate cross sections (type HVH)

In [26], the determination of the limit load-bearing capacity for arbitrary cross section combinations has been covered for three- and two-plate cross sections. The sketch in Table 2.11 shows the discussed cross section with which, via the variation of plates, amongst others, the cross section shapes shown in Figure 2.13 can be captured.

Figure 2.13 Different three- and two-plate cross sections


Table 2.11 Cross section properties of three- and two-plate cross sections (type HVH) in the y-z coordinate system (middle of the web)

Partial areas: o o o s s s u u uA b t ; A h t ; A b t

Area:o s uA A A A

First degree moments of area:

y o o u u z o o u uA y A y A ; A A z A z

Second degree moments of area:

yz o o o u u uA A y z A y z2 2

2 2o uyy o o u u

b bA A y A y

12 122

2 2szz s o o u u

hA A A z A z12

For the cross section in Table 2.11 the flanges are arranged horizontally and the web vertically. The y-z reference system is located in the middle of the web. Through the increase of the web height hs into the flanges, the overlapping model explained above can also be realised. Here, it is not compulsory that hs is up to the middle of the flanges.

The calculation of the cross section properties is carried out with Table 2.11 and the following standardisation with Table 2.4. An example realisation is shown in the upper half of Table 2.2. Below that, the results for the warping torsion follow, which are expanded in Section 2.5.

Example: Z cross section

The cross section Z 160 is idealised with the line model (with overlapping) as shown in Figure 2.14. From profile tables [29] one can take the following dimensions:

o ut t 11 mm ; st 8.5 mm ; o ub b 65.75 mm ; sh 149 mm

The centre of gravity is located in the point of symmetry, so that the calculation can start with the y-z system.

0AA zy and zyz~y~ AA , yy yyA A , zzz~z~ AA

For the calculation, Tables 2.11 and 2.4 are used.


Figure 2.14 Z cross section

2oouo cm233.7tbAA ; 2

sss cm665.12thA2cm13.27A (27.5 cm2)

4oooz~y~ cm27.3542zyAA

42o

2ooy~y~ cm44.208212byAA

42oo

2ssz~z~ cm16.03712zA12hAA

27.20 (19.65°)

Iy = 1 168 cm4 (1 180 cm4) Iz = 77.64 cm4 (79.5 cm4)

The values of the profile tables are written in brackets. For this example, the line model and the neglecting of the fillets lead to deviations between 1.0 and 3.2 %.

2.4.4 Tabular Calculation of Cross Section Properties

In some areas of steel construction, such as bridge building, cross sections occur which consist of many individual parts. For such cross sections, a tabular determination of the cross section properties is recommended. In this subsection, the cross section of a pedestrian bridge is analysed, Figure 2.15. The single-symmetric cross section is here depicted without cross girders and plate buckling stiffeners. The areas, the position of the centre of gravity in the z-direction and the moment of inertia Iy are calculated. In order to avoid large values, the areas are given in cm2 and the “lever arms” z in m.


Figure 2.15 Cross section of a pedestrian bridge

The calculation of the cross section properties in the y-z reference system is per-formed on the basis of Table 2.5. Due to the symmetry, it is not necessary to take the

yzA values of the inclined plates into account, as they cancel each other. The trapezoidal profile is taken from Figure 2.16 at the bottom right:

bo = 392 mm; bu = 202 mm; t = 6 mm

The tabular calculation is compiled in Table 2.12. It complies with the calculation of the RUBSTAHL-program CSP-table.

Figure 2.16 Dimensions of trapezoidal profiles according to [28]


Table 2.12 Tabular determination of cross section properties of the cross section in Figure 2.15

Cross section parts Ai Siz i SiA z zz,SMI,iA 2

i SiA z

[cm2] [m] [cm2 m] [cm2 m2] [cm2 m2]Cover plate WebsLower flange Cornice plate Trapez. stiff. Trapez. stiff.

4 288 10 2 x 2 015 15

1 200 20 2 x 350 12

8 x 275 6 4 x 202 6

428.80604.50240.0084.00

132.0048.48

0.0000.9901.9850.1450.1290.258

0.0598.5476.412.217.012.5

0.00195.50

0.000.860.730.00

0.00592.47945.65

1.772.203.23

Sums: 1 537.78 1 116.6 197.09 1 545.31

Centre of gravity: S i Siz A z A 0.726 m

Moment of inertia: 2 2 2 2y zz,SMI,i i Si SI A A z A z 931.6703 cm m

Note: Self moment of inertia of inclined plates: 3 2 2zz,SMIA t sin 12 A h 12

2.4.5 Numeric Integration / Fibre and Stripe Model

The calculation methods for the determination of the cross section properties shown so far are all suitable for hand calculation. To avoid arithmetic errors and to reduce the effort, one uses computer programs as support for complex cross section shapes. A valuable help is the spreadsheet program MS Excel, with which complex problems can also be solved quickly. Especially for the performance of numeric integrations it is extremely well suited. Numeric integrations can be used for directly solving the integrations occurring for the calculation of the cross section properties shown in Tables 2.3 and 2.4.

Figure 2.17 Division of the cross section with parabolic contour into rectangles


In Figure 2.17 a cross section with a parabolic contour can be seen. It is used to demonstrate the execution of a numeric integration. As Figure 2.17 shows, the cross section is divided into rectangles which all have the same height:

ih h n (2.40)

The width ib is defined that the rectangles subtend the parabola exactly in the middle. For part “i” we have

i ib 2 y b 1 i 0.5 n (2.41)

As it can easily be seen, the centre of gravity of the rectangles may be described by

Siz h n i 0.5 (2.42)

Moreover, the familiar formulas are used and evaluated in an Excel spreadsheet. Table 2.13 shows the approach for n = 20 and h = b = 1 m. Which division is se-lected, e.g. 10, 20 or 50 rectangles, is of no significance in terms of the effort. Marking of the row i = 1 just has to be “pulled down” far enough.

Table 2.13 Calculation of the cross section properties for the parabolic cross section with MS Excel

h = 1 Formulas for the last two columns:

b = 1 3zz,SMI,i i iA b h 12 in

n = 20 12bhI 3iiiz, in

i bi hi Siz Ai Sii zA 2Sii zA zz,SMI,iA Iz,i

1 Enter formulas 2···

20

Mark row i = 1 and pull it down (“filling in”)

Sum up

AzAz SiiS ; 2 2y i Si zz,SMI,i SI A z A z A

Table 2.14 contains a compilation for the use of 10 and 20 rectangles and a compari-son with the exact solutions for h = b = 1 m. As can be seen, an outstanding accuracy can be achieved with the numeric integration. The largest deviations occur for Iy:

n = 10: 1.78% and n = 20: 0.61%


Table 2.14 Comparison of the numeric integrations for n = 10 and n = 20 with the exact solutions

Exact solution 10 rectangles 20 rectangles A 0.666667 0.668384 100.26 % 0.667295 100.09 % Iy 0.045714 0.046530 101.78 % 0.045995 100.61 % Iy without SMI 0.045973 100.57 % 0.045856 100.31 % Iz 0.033333 0.033286 99.86 % 0.033321 99.96 %

In the row “Iy without SMI”, the self moments of inertia of the rectangular partial areas have not been taken into consideration. The error emerging through this is minor. Interestingly, neglecting the self moments of inertia leads to more accurate results for Iy. The reason for this is of course the given bending of the curve, and it results from the fact that Iy is approximated from above.

The comparisons when neglecting the self moments of inertia should show that for this approximation and a sufficiently fine division of the cross sections the desired accuracy can always be achieved. In computer programs, often so-called fibre or stripe models are used, for which the regular partial areas (rectangles) are only repre-sented by their area and the position of their centre of gravity. For the moments of inertia, therefore, only the Steiner parts are taken into consideration. This, of course, requires a correspondingly fine division.

Figure 2.18 Division of cross sections into fibres and stripes

In Figure 2.18, examples of the fibre and the stripe models are shown. If the division is sufficiently fine, the fibre model can be used for any desired application cases. The field of application of the stripe model depends on the cross section shape and on what is to be calculated. An example for the application of the stripe model is the parabolic cross section in Figure 2.17. If, for this example, the self moments of inertia of the partial areas are to be neglected consistently, another modelling is required for the determination of Iz. In this case, vertical stripes (rectangles) or the fibre model are chosen. Fibre or stripe models are frequently used in large computer programs for


solving complex problems. An example of this are programs with which limit load calculations based on the plastic zone theory are carried out; see Section 5.12. In doing so, effective cross section properties have to be determined in many steps (load increments) depending on the corresponding current state of plastification – also see Section 8.4. Furthermore, internal forces and moments and partial internal forcescan be calculated via numeric integration with the methodology mentioned here.

Figure 2.19 Cross sections with parabolic contour

In Figure 2.19, two variants are shown for the parabolic cross section in Figure 2.17. The variant on the left contains a trapeziform cut-out, so that a single-celled box cross section results. Best for the calculations is to assume the solid cross section and to consider the trapezium as a “negative” area.

On the right-hand side of Figure 2.19, a comparable steel cross section was constructed. Cross sections with parabolic curvilinear contour are indeed extremely rare. Nevertheless, they can be interesting if special emphasis is put on architectural design. For the calculation of the cross section properties it is sufficient to consider the profile centre line and to concentrate the plate thicknesses there.

Figure 2.20 On the numeric integration for thin-walled curvilinear plates with t = const.


Figure 2.20 shows the profile centre line and the division into straight sections from node to node. With the length and the area of the sections as well as the formulas in Section 2.4.2, all area integrals can be determined by numeric integration. As an ex-ample, the moment of inertia Iz is calculated for the parabolic profile centre line in Figure 2.19 on the right. In doing so, the self moment of inertia of the individual parts is neglected, so that a relatively fine division is necessary.

k

2k,mk

A

2z yAdAyI (2.43)

choose: ky b n const. y (2.44a)

1ky2byk for k = 1 to n +1 (3.62b) (2.44b)22

kk by41hz (“function of the parabola”) (2.44c)

2yyy k1kk,m (2.44d)

The evaluation with MS Excel provides for n = 50, t = 1.5 cm, b = 60 cm and h = 90 cm:

Iz = 124 672 cm4 (2.45)

The exact solution is Iz = 124 770 cm4. In many cases the numeric integration is eas-ier to perform and it is less error-prone than application of the direct integration methods.

In Chapter 11, the calculation of the cross section properties is performed on the basis of the finite element method. The fibre model of Figure 2.18 and the elementing shown in Figure 2.19 on the right can also be considered as a finite element mesh and therefore can be the basis of these procedures, which have further advantages in terms of computer-oriented calculations – see Chapter 11. In the associated context of isoparametric finite elements, the issue “numeric integrations” is also considered in Section 11.5.6. Due to the large numeric effort needed for the integration of finite elements, more efficient procedures would be useful. That is why the Gauss quadrature is dealt with in Section 11.5.6, which allows exact or at least very accurate solutions for the numeric integrations with minimum effort. However, with regard to the implementation and programming of the procedure, it is more extensive than the approaches shown in this chapter, which can directly be solved with fairly simple Excel sheets.


2.5 Standardisation Part II: Shear Centre, Warping Ordinate and Warping Constant

As explained in Section 2.1, the determination of the y-z- principal system and the cross section properties is divided into two subtasks. In part I (Section 2.3) the

centre of gravity, principal axes and Iy and Iz

were determined. The calculations are now continued with part II:

shear centre, standardised warping ordinate and I .

The results of part I are taken as an starting point for part II. For the example in Fig-ure 2.21 this means that the principal axes are assumed with the origin at the centre of gravity S. Further, the principal moments of inertia Iy and Iz are assumed to be known already.

Figure 2.21 Starting point for the standardisation part II (example)

For part II of the standardisation the following conditions must be met:

A

A dA 0 (2.46a)

yA

A y dA 0 (2.46b)

zA

A z dA 0 (2.46c)

As already mentioned, y and z are the principal axes of the cross section with refer-ence to the centre of gravity S, and is the standardised warping ordinate. Condition (2.46a) leads to a corresponding choice of the integration starting point on the profile centre line. From Eqs (2.46b) and (2.46c) the position of the shear centre M results.

If the positions of the reference points are not known, other reference points are se-lected for their determination and then respective transformations are performed. The easiest way of deriving the transformation relationships is by considering thin-walled

2.5 Standardisation Part II: Shear Centre, Warping Ordinate and Warping Constant 59

cross sections with a reduction to the profile centre line, as shown in Figure 2.21. On the centre line, a local coordinate s can be introduced. According to Figure 2.22, a “centre of rotation” D is chosen in the first step, which the “lever arm” tr refers to. The point A serves as integration starting point on the profile centre line for which

0 is assumed. From that, a warping ordinate can be determined, as described in Section 2.6:

s

t0

r ds (2.47a)

t D Dr y y sin z z cos (2.47b)

Figure 2.22 On the transformation of the warping ordinate into the standardised warping ordinate

For the example in Figure 2.21 reference points D and A were arranged in the centre of the upper flange and, starting from there, the profile ordinate s was chosen. The sketch in Figure 2.21 on the right shows the warping ordinate in qualitative terms.

In a completely analogous manner, one obtains for the reference points A and M in Figure 2.22:

s

t0

r ds (2.48a)

t M Mr y y sin z z cos (2.48b)

With Eqs (2.47b) and (2.48b), rt can also be formulated as follows:


t t M D M Dr r y y sin z z cos (2.49)

Used in Eq. (2.48a), this gives s s s

t M D M D0 0 0

r ds y y sin ds z z cos ds (2.50)

and with sin ·ds = dz as well as cos ·ds = dy (Figure 2.22)

A A

ys z

t M D M D0 z y

r ds y y dz z z dy (2.51)

In general, the position of the point A, and thus also the coordinates yA and zA at po-sition s = 0, are not known. They are not absolutely necessary since it is only a matter of a suitable transformation relationship for the warping ordinate. The desired pur-pose, namely the fulfilment of conditions (2.46a) to (2.64c), can be achieved with the transformation relationship

k M D M Dz y y y z z (2.52)

In Eq. (2.52), k is a transformation constant for the warping ordinate. With reference to Eq. (2.51), it contains three parts

first integral: Transformation of s into s

second integral: Consideration of z (s = 0)

third integral: Consideration of y (s = 0)

The constant k can be explained as follows: the warping ordinate determined with the arbitrary reference points D and A usually contains a mean axial displace-ment, which is not allowed due to Eq. (2.46a). This is corrected by k . In other words, k is an integration constant since it is 0 at the beginning of the integra-tion. By looking at k in Eq. (2.52), i.e. by calculating k( ) , the negative and positive parts of the warping ordinate are balanced (area replacement), representing the actual warping with reference to the rotation origin D. For its determination, Eq. (2.52) is inserted in Eq. (2.46a):

A

k M D M DA

k z M D y M D0 0

A dA

z y y y z z dA

A A A y y A z z 0

(2.53)

With regard to the principal system, Ay = Az = 0 and Eq. (2.53) leads to:

2.5 Standardisation Part II: Shear Centre, Warping Ordinate and Warping Constant 61

kAA

(2.54)

If Eq. (2.52) is inserted in Eq. (2.46b), one obtains

yA

2k M D M D

A

y k y yz M D z M D

0 0

A y dA

y y y z y y y z z dA

A A A y y I z z 0

yM D

z

Az z

I

(2.55a)

With the analogous procedure, the third condition leads to

zM D

y

Ay yI (2.55b)

As shown with Eq. (2.53), the relationships of Ay = Az = Ayz = 0 of the principal sys-tem have been regarded for Eq. (2.55) as well. The transformations performed here are compiled in Table 2.15. It is to be understood as a continuation of Tables 2.3 and 2.4 containing the respective relations for biaxial bending with axial force (centre of gravity, rotation angle of the principal axis, principal moments of inertia and principal axes).

Table 2.15 Calculation of standardised cross section properties part II

Conditions: A, Iy and Iz are known; y and z are principal axes Calculate A , Ay , Az and A for an arbitrary integration starting point and

centre of rotation: 2

A A A A

A A A Ay zdA; y dA; z dA; dA

Transformation constant for the warping ordinate: k A A

Position of the shear centre: yz

M D M Dy z

AAy y ; z zI I

Warping constant I :2 22 2

k M D y M D zA

I dA A A y y I z z I

Standardised warping ordinate:

k M D M Dz y y y z z


The distinction between method A and method B is not made here (in principle, Ta-ble 2.15 complies with method B). If the points and are interchanged and I is calculated with the standardised warping ordinate, so not with the transformations, method A is obtained.

For the cross section properties yA and zA , y and z are the principal axes of the cross section. Sometimes, for example for computer-oriented calculations, it can be more beneficial to use the arbitrary y-z reference system as well as yA and zA .Then yA and zA may be calculated with the following transformations:

y y S k z S kA A y A cos A z A sin (2.56a)

z z S k y S kA A z A cos A y A sin (2.56b)

Utilisation of symmetry properties

As explained in Section 2.2, the shear centre is on the axes of symmetry of the cross sections. Also = 0 is valid on the axes of symmetry. If one takes advantage of these circumstances and puts the reference points D and A on the axes of symmetry as well as choosing the s direction symmetrically, the standardisation part II is simplified considerably. The calculation procedure is compiled in Table 2.16 with reference to Table 2.15.

Table 2.16 Utilisation of symmetry properties for the calculation of standardised cross section properties part II

a) Symmetry to the y-axis b) Symmetry to the z-axis

Put reference points D and A on the axes of symmetry; choose s direction as symmetric

0AA yk 0AA zk

A,Az A,Ay

yzDM IAyy zyDM IAzz

y2

DM IyyAI z2

DM IzzAI

DM yyz DM zzy

2.6 Warping Ordinate 63

The example in Figure 2.21 (symmetry to the z-axis) shows the course of the warping ordinate . From the antisymmetric course it can be directly seen that the integrals A and zA must be equal to zero. Furthermore, it is not sufficient for the calcu-lation of yA and A or I , respectively, only to consider one half of the cross section and to take into account the other one with the factor 2. The methods for the calculation of

y z M M, , A , A , A , A , I , y and z

are explained in Sections 2.6 and 2.7 and in Chapter 11. Generally, the resulting integrations may be solved with the help of Table 2.8. Since the general proceeding is shown in Section 2.4.2 in conjunction with the standardisation part I, it will not be dealt with again here.

2.6 Warping Ordinate

The warping ordinate is needed for the calculation of different values. As the over-view in Table 2.17 shows, these are values that are connected to torsional stresses. Whether torsion occurs in a structural system can only be decided if the position of the shear centre is known. Thus, the shear centre is of overriding importance, see Section 2.7. The easiest way of determining its position is generally with the help of the warping ordinate, so here the warping ordinate is given greater significance than usual in the relevant literature.

Table 2.17 Warping ordinate as input parameter for various calculation formulas

Displacement u of a beam due to torsion u

Warping bimoment as stress resultant

xA

M dA

x due to M or xM EI

xs due to Mxsxs

xsM A s

I t s

Position of the shear centre

yzM D M D

y z

AAy y ; z zI I

Warping constant 2

A

I dA

St Venant‘s torsion constant T M M M M

A

I y y y y z z z z dAz y


The starting point for the warping ordinate are the displacements u in the longitudinal direction of the beam. According to Eq. (1.1),

u x,y,z y,z x with (2.57)

describes the warping of the cross sections, i.e. the deviation from a plane surface.In Figure 2.23, an example is given for the warping of the cross section. For the de-scription the warping ordinate is used. Since, according to Eq. (2.57), it complies with the warping for the derivative of the angle of twist 1, it is also frequently called the warping function. With the displacement u in cm and in 1/cm, results in cm2.

Figure 2.23 Warping u of an open cross section

The function used to describe the warping of a cross section complying with the torsional deformations u neglecting distortions due to secondary shear stresses caused by Mxs. The determination of the warping ordinate is often difficult. Analytic solutions for thick-walled general cross sections only exist for basic shapes (ellipse, circle, rectangle, equilateral triangle), which are gained by so-called stress functions. Details on this can be found in [25]. For that reason, numeric approaches for the de-termination of are very significant for thick-walled cross sections, which are dealt with in detail in Chapter 11.

However, cross sections applied in steel construction may often be treated as thin- walled with a reduction of the plate thicknesses to their centre line, as shown in Figure 2.12 for I-sections. With these models, formulas can be developed for the determination of . The warping ordinate is then approximated with the centre line only and assumed to be constant over the plate thickness (see Figure 2.23).

2.6 Warping Ordinate 65

Figure 2.24 Displacement v along the profile centre line due to

For thin-walled open cross sections the warping ordinate can be determined by considering distortions caused by primary (St Venant’s) shear stresses only. Due to the fact that, for open cross sections, these stresses have a linear course over the plate thickness, the stress at the centre line is zero. That is why an associated strain does not occur. On this basis, the warping ordinate can be determined as shown below.

With the reduction of the profile to its centre line, a local coordinate s is introduced describing the course of the centre line. It is assumed that shear stresses can only act in the direction of s described by xs. With regard to Eq. (1.4) it is:

xs xsvuG G

s x(2.58)

The displacement v (linearised) of an arbitrary cross section point P caused by a tor-sional rotation is illustrated in Figure 2.24. For the change of the displacement (derivative) in longitudinal beam direction of an element dx it is:

trdxdv

(2.59)

By introducing Eq. (2.59) into Eq. (2.58) and assuming that the strain is zero on the centre line for open thin-walled cross sections as mentioned previously, we have:

sttxs dsru0r

su

(2.60)

Taking Eq. (2.57) into account, the warping ordinate can now be described by

ts

r ds (2.61)

The value rt is the distance between the tangent of the profile centre line of an arbi-trary cross section point P and the shear centre M, as shown in Figure 2.25. The direction of rt corresponds to the normal at P.


Figure 2.25 Lever arm rt for the determination of

The definition of signs for the warping ordinate is shown in Figure 2.26. If the orien-tation of the rotation of a notional moment ds · rt is going along with a positive rotation about the longitudinal beam axis, the warping ordinate will increase. With an engineering diction, the sign convention might be even easier to memorise:

“Positive torsional moment ds · rt about M” larger warping ordinate (+ d )

Figure 2.26 Starting point of integration A and definition of signs for the warping ordinate

The warping ordinate according to Eq. (2.61) also depends on the starting point of integration A; see Figure 2.26. From a mathematic point of view, the starting point has to be at a position where = 0 since otherwise an integration constant would have to be introduced into Eqs (2.60) and (2.61).

2.7 Shear Centre M 67

It becomes clear that the knowledge of the position of the shear centre M and the starting point A is necessary in order to determine . Without that knowledge, Eq. (2.47) is applied and reference points D and A have to be considered. The local coordinate is described with s in that case and the corresponding lever arm tr refers to D, cf. Eq. (2.47b). The ordinate determined on that basis, for which the previous explanations of this chapter regarding apply as well, has to be transformed (stan-dardised) according to Section 2.5.

The background xs = 0 for the determination of shows that the warping of a thin-walled open cross section is not a result of strain due to stresses, but just caused by the derivative of the angle of twist of the beam; see Eq. (2.60).

For closed (hollow) cross sections this is not the case. Here, additional primary stresses due to the torsional stiffness of the hollow cells occur (Bredt’s shear flow), which have a constant distribution over the plate thickness. The shear stress at the profile centre line is thus not zero and the warping of closed cross sections consists of two components, which have a totally different mechanical origin. The first compo-nent corresponds to the warping of the open cross sections caused by , the second relates to strains due to the shear stresses of the hollow part. For the determination of

statically indeterminate calculations may be performed. Details are given in [25]; see Eq. (3.62) as well. Due to the fact that these calculations are relatively extensive, numeric procedures (FEM) are used for thin-walled cross sections as well. Besides the procedures for thick-walled (arbitrary) cross sections, Chapter 11 also deals with the finite element method for thin-walled cross sections.

2.7 Shear Centre M

The shear centre is an important reference point of cross sections. Its position gives information on whether the loading of beams causes torsional stress. If external loads Fy and Fz (concentrated loads) or qy and qz (distributed loads) act at the shear centre, torsion does not occur. This means that the beam does not rotate about its longitudinal axis. In addition, the internal forces Vy and Vz, the torsional moment Mxand the warping moment M act at the shear centre as internal forces and moments. This underlines the fact that the position of the shear centre is of overriding importance for beam structures.

The utilisation of symmetry has already been discussed in Section 2.2. There, it is illustrated that the shear centre is located on the axis of symmetry. If the cross section shows two axes of symmetry (doubly symmetric), the position of M is at the point of intersection. For point-symmetric cross sections it is located at the point of symmetry. If the position cannot be clearly indicated, it has to be calculated. There are different conditions serving this aim, leading to corresponding calculation procedures:


Figure 2.27 Conditions for the determination of the shear centre

a) Condition: Rotation = 0 Not a generally applicable procedure

b) Condition: Torsional moment Mx = 0 Procedure: “Use of shear stresses”

c) Condition: Warping moment or warping constant I minimal Procedure: “Use of warping ordinate”

The conditions and corresponding procedures are more or less practicable depending on the cross-sectional shape. Additionally, they are partially of engineer-like descrip-tive or rather of mathematic character. Figure 2.27 gives an overview. The sketch of 2.27 a) shows a cantilever beam. It shows that the deflection of the beam leads to a displacement of the cross section in the vertical direction; however, if the load acts at the shear centre, there will be no rotation. Calculations with the condition 0 are of very clear nature, but only suitable for a few cross sections. In [25], the application for a symmetric cross section is presented. Due to the fact that the procedure is not generally applicable and therefore not easily applicable to computer-oriented calculations, it is not focused on here.

Using condition b) Mx = 0, the position of the shear centre is determined from the shear stresses and the resulting forces arising because of internal shear forces Vz and Vy. This is the classic approach, which is very clear in terms of engineering understanding.


Condition c) leads to the calculation procedures shown in Section 2.5, which are used for the determination of the standardised cross section properties shown in Table 2.15. The warping ordinate is used here for the determination of the shear centre, which is perfectly suitable for computer-oriented applications. In many cases, calculations according to method c) are less time-consuming in comparison to method b). However, the disadvantage is the lack of clearity in terms of engineer-like thinking. Procedures b) and c) are now examined in more detail.

Use of shear stresses

The classic procedure for determining the position of the shear centre is to calculate the shear stress distributions due to internal forces Vz and Vy, which are then summarised as single forces of the cross section parts. With the condition that the moment equilibrium relative to the shear centre may not lead to a torsional moment (Mx = 0), the postion of M can be specified. Since it is not known a priori, an arbitrary point (yD, zD) is chosen, which serves as reference point for the formulation of Mx about D (Mx,D). Figure 2.28 leads to the following expression for a thin-walled partial plate rotated by :

x xs ts

xs t M D xs M D xss s s

x,D M D z M D y

M T r ds

T r ds y y T sin ds z z T cos ds

M y y V z z V

(2.62)

For the determination of yM and zM two equations are required. For this purpose, the action of shear forces Vy and Vz are individually taken into consideration leading to shear flows Txs(Vy) and Txs(Vz). With Eq. (2.62), this results in

stzxs

zDM dsrVT

V1yy (2.63a)

styxs

yDM dsrVT

V1zz (2.63b)

Thin-walled cross sections in steel construction mostly show straight profile centre lines with constant plate thickness (step by step). The lever arm tr is then constant for each segment. It can therefore be taken out of the integration and, with a split-up into n cross section parts, the integrations lead to the partial shear forces in the individual components j:

j

j z xs zs

V V T V ds and j

j y xs ys

V V T V ds (2.64a, b)

With this, Eq. (2.63) can be written as follows:


n

M D j z t, jj 1z

1y y V V rV (2.65a)

n

M D j y t, jj 1y

1z z V V rV (2.65b)

For the individual straight cross section parts j with the starting point a (ya, za) and the ending point b (yb, zb), the lever arm t, jr can be determined by

j,b j,a j,b j,at, j j,a D j,a D

j j

z z y yr y y z z (2.66)

This formula can be simplified by using the centre of gravity as rotation point D (yD = 0, zD = 0):

t, j a b b a jj

1r y z y z (2.67)

For the determination of the resulting shear forces Vj, the knowledge of the shear stress distributions is necessary. This issue is reconsidered in Chapter 7 and on the basis of the finite element method in Chapter 11.

Figure 2.28 On the calculation of the shear centre with shear stresses


Use of the warping ordinate

In general, the calculation of the position of the shear centre by using the warping ordinate has been shown in Section 2.5 in connection with the standardisation. The corresponding equations are derived with Formula 2.55. In this section, the use of the warping ordinate has been mentioned in conjunction with the condition that I is supposed to be the minimum second degree warping moment of inertia (“I : Min.”). If according to Eq. (2.52) is inserted into the corresponding formula of I in Table 2.15, it gives

2 22k k M D y M D z

y M D z M D

I A 2 A A y y I z z I

2 A z z 2 A y y(2.68)

This equation assumes that y and z are the principal axes going along with Ay = Az = Ayz = 0. The minimum can be found by setting the first derivates to zero:

M D y zM D

I2 y y I 2 A 0

y y

zM D

y

Ay yI

(2.69a)

M D z yM D

I2 z z I 2 A 0

z z

yM D

z

Az z

I

(2.69b)

A comparison with Equations (2.55) shows that with this derivation equal conditions for the shear centre are provided. In addition, it is clear that the reference of the warping ordinate to the shear centre leads to a minimum of I . These calculations require the warping ordinate, which is dealt with in Section 2.6 and in terms of the finite element method in Chapter 11.

3 Principles of FEM

3.1 General Information

As already outlined in Figure 1.1, a structural system is discretised into finite elements using FEM. This means that the system is divided into a suitable number of elements, which are connected at the nodes and may also show intermediate nodes. Subsequently, internal forces and moments as well as displacements can be deter-mined either using the displacement method or the field transfer matrix method. Because of the extreme significance of the displacement method, in the following, we will concentrate on this method exclusively. Occasionally, it is also referred to as deformation method. For the application of the field transfer matrix method refer to [31], where it is discussed with regard to a particularly appropriate field of appli-cation.

3.2 Basic Concepts and Methodology

When examining static systems using the displacement method, the process of cal-culations always follows a constant and highly schematic approach. In conjunction with its universal suitability for a wide variety of tasks, this results in the impressive success of the methodology. However, besides the relatively strongly mathematically oriented solution method, another disadvantage is that the basic ideas of the procedure are not immediately apparent. As these ideas are of major importance for the understanding, they are explained here in connection with the methodology.

The structural system in Figure 3.1a is used as an example. It shows a plane frame for which the internal forces and moments and deformations are to be determined ac-cording to linear beam theory. The frame has different cross sections along the horizontal and inclined parts and an in-plane loading of F and g. As is generally known, a basic idea of the FEM is the division of a structure into finite elements be-ing connected to each other at the nodes. As shown in Figure 3.1b, the plane frame can be separated into four beam elements with five nodes. For practical calculations a finer FE modelling would be used since internal forces and moments as well as de-formations in shorter intervals are required. With the rough division in Figure 3.1b, these would have to be determined by additional calculations.


3.2 Basic Concepts and Methodology 73

Figure 3.1 Basic example for the understanding of the FEM

3 Principles of FEM 74

It is necessary or advantageous to arrange for nodes to be

at bearings, concentrated loads and point springs at the beginning and end of distributed loads, distributed springs and shear dia-phragms at cross section jumps at direction changes of beams (sharp bends).

For the sake of completeness it should be noted that the arrangement of nodes for the loads can be altered if transformations are made as explained in Section 4.9. There-fore, node 2 in Figure 3.1b is not obligatory.

In terms of FEM, bearing conditions are geometric boundary conditions. Just as with concentrated loads and point springs, they always apply to the global coordinate sys-tem and hence they correspond to the X- and Z-direction. For that reason, the concentrated load F in Figure 3.1a has been divided into its components FX2 and FZ2 (Figure 3.1c) with positive signs if the directions of action comply with X and Z. Support reactions (indicated by the subscript “R”) are defined as positive contrary to X and Z.

In conjunction with the beam elements, local x-z coordinate systems are used and distributed loads as well as distributed springs are assigned to the beam elements. Hence, the “element load” qz4 in Figure 3.1c applies to the local z-ordinate of ele-ment 4. Later on, it will be transformed into the equivalent nodal loads, which correspond to the global X-Z coordinate system (see Figures 3.3 to 3.5). The support reactions in Figure 3.1c are unknown variables which have to be in equilibrium with the loads.

The next important step of the displacement method is the definition of the displace-ments, see Figure 3.1d. In each node the three variables u , w and y occur, whichdirectly correspond to the loads FX, FZ and MYL. Since local deformations are also needed later on, the labelling of the global variables is clarified by using an overbar. As can be seen, 5 3 = 15 unknown displacements (nodal degrees of freedom) occur. Due to the geometric boundary conditions (bearing conditions: restraint left and hinged bearing right) five nodal degrees of freedom are equal to zero and therefore 10 unknown variables remain to be calculated. It should be mentioned that the support forces at the hinged bearing have been replaced by horizontal and vertical components.

A further basic idea of FEM is to cut free the nodes of a structure and to formulate the equilibrium at the nodes with the help of virtual displacement principles or the virtual work, respectively:

W = Wext + Wint = 0 (3.1)


The principle is explained in Section 3.4.2. As an example, node 4 of the frame is considered in Figure 3.2. The concentrated loads lead to the external virtual work Wext, and the nodal internal forces and moments of the adjacent beam elements to

the internal virtual work Wint. Furthermore, it must be noted that the nodal internal forces and moments with an overbar as well as the loads refer to the global X-Z coordinate system and, accordingly, the virtual work will result from the global node displacements. The directions or signs of the nodal internal forces and moments result from the equilibrium with the internal forces and moments at the beam ends (same direction as the load!), which are incidentally defined as positive at both beam ends with given directions. The resulting sign definition II for the internal forces and moments at the beam ends is an essential component of the FEM and the systematic formulation of the equilibrium at the nodes.

Figure 3.2 Equilibrium at node 4

If the virtual works emerging from the virtual displacements 4u , 4w and Y4 are considered separately, three equilibrium conditions result; these are indicated in Figure 3.2. The terms in the round brackets must be equal to zero. This is in compliance with the conditions of the “classic” nodal equilibrium for FX = 0, FZ= 0 and MY = 0.

Since the equilibrium may be formulated in a similar manner for each node of a structure, the frame with five nodes in Figure 3.1 provides overall 15 conditions cap-turing the equilibrium of the frame. The requirement that the virtual work must be zero at each node of a structure is a major element of FEM.


Figure 3.3 Internal forces and moments at the beam ends of element 4

The equilibrium conditions at the nodes are not sufficient to determine the unknown internal forces and moments at the beam ends. For example, the 15 equations of the plane frame contain more than 15 unknown internal node forces and moments. For the solution of this problem another basic idea is required. For that reason, Figure 3.3 shows beam element 4, for which six relationships between local internal forces and moments of the beam ends and the corresponding displacements are formulated. Their derivation on the basis of the virtual work is a central part of FEM as well as of the displacement method. Beam elements are considered in more detail in Section 4.2. In matrix notation, the beam stiffness relation regarding “beam loads” is:

T T Te e e e e e ev s v K v v p (3.2)

In Eq. (3.2) all variables relate to the local x-z coordinate system of the beam element. The transformation into the global X-Z coordinate system is covered in Section 4.4. As a result, the following is obtained:

T T Te e e e e e ev s v K v v p (3.3)

Figure 3.4 Transformation of local nodal displacements of beam element 4 into the global X-Z coordinate system


A clear interpretation of Eq. (3.3) can be carried out with the help of Figure 3.4. The local nodal displacements of beam 4 are transformed into the global X-Z coordinate system, so that they can be replaced by the variables of the global system. Since this can be conducted for the internal forces and moments in a comparable manner (see Section 4.4.1), it is now possible to replace the internal forces and moments of the conditions for the nodal equilibrium. This approach is very transparent when, for example, the internal forces and moments at node 4 of element 4 in Figure 3.2 are examined. By comparing them with the internal forces and moments at the beam ends in Figure 3.3 and performing transformations as shown in Figure 3.4 this should become apparent. As a result, the equilibrium conditions at all nodes can be combined into one system of equations – see Figure 3.5:

T Tv K v v p (3.4)

Figure 3.5 System of equations for the plane frame of Figure 3.1

For the plane frame of Figure 3.1a we obtain the system of equations shown in Figure 3.5, which consists of 15 individual equations allocated to the virtual work displacements on the left-hand side and summarised in matrix notation. The totalstiffness matrix K contains the four beams, which are designated by 6 6 element matrices. As one can see, an overlapping of element matrices at the common nodes (connecting nodes) results, so that in these areas, their values are added up.


The point spring Cw3 at node 3 is also to be considered in the total stiffness matrix.The spring force FZC3 acts contrary to support reaction FZR3 in Figure 3.1c and can be replaced by

ZC3 w3 3F C w (3.5)

It corresponds to 3w , so that the value Cw3 has to be added to the main diagonal ele-ment in the eighth row.

The vector v contains 15 displacements at the five nodes, which are shown in Figure 3.1d. In the total load vector p , the loads and the reaction forces are included. As already mentioned, the signs of the forces and moments refer to the global X-Z coordinate system.

To solve the equation system in Figure 3.5, the geometric boundary conditions, i.e. the support conditions, must be taken into account. Due to

1 1 Y1 5 5u w u w 0 , columns 1 to 3 as well as 13 and 14 are dropped. This also applies for the corresponding rows since they belong to 1u , 1w , Y1 , 5u and

5w . Also, the load vector contains the unknown bearing forces in these rows. In Figure 3.5, the dropping of columns and rows is illustrated through the horizontal and vertical arrow pairs on the main diagonal. A 10 10 equation system remains. It can be solved according to Section 4.6. As a result, we obtain the 10 previously unknown deformations of the vector v .

With Figure 3.5, the calculation of the support reactions becomes obvious, because the rows 1 to 3 as well as 13 and 14 contain these values and now all the deforma-tions of the vector v are known. The determination of the internal forces and moments is slightly more complex since the single beams with their local x-z co-ordinate system have to be analysed. The calculation is carried out with the help of Eq. (3.2). However, in the vector ve, the local displacements for each beam have to be determined. The vector v of the system is used for this purpose. The required values have to be allocated to the element nodes and to be transformed into the local coordinate system. Details can be seen in Section 4.7.

3.3 Progress of the Calculations

In Table 3.1, the FEM procedure using the displacement method is compiled for the linear theory. Here, it does not make any difference whether the structural system is discretised using beam, plate or shell elements. Independent of the element types used, the same progress always applies for the calculations, so that with one and the same methodology numerous tasks may be solved. A further advantage is the strongly schematic procedure, which does not require any individual decisions, be-

3.3 Progress of the Calculations 79

cause only suitable finite elements and a reasonable separation into finite elements have to be chosen.

Table 3.1 Procedure for the displacement method (linear theory)

Nr. Task Details 1 Divide structural system into elements. Figure 3.1b

2 For each element: calculate element stiffness matrix. Section 4.2

3 For each element: transform the loads acting within an element into equivalent nodal loads (element load vector).

Section 4.2

4 Transform element stiffness matrix and integrate it into total stiffness matrix.

Section 4.4, Figure 3.4 Section 4.5.2, Figure 3.5

5 Integrate loads acting at the nodes into the total load vector of the system according to point 3.

Section 4.5.3, Figure 3.5

6 Where required, consider springs, shear fields and hinges.

Section 4.10, 4.11, Figure 3.5

7 Consider the geometric boundary conditions (support, restraints, etc.) in the total stiffness matrix and in the total load vector.

Section 4.5.4

8 The result of points 4 to 7 is the following equation system:

Section 4.6 Chapter 6 K · v = p

The deformations of the system at the nodes can be obtained by solving this equation system.

9 For each element: calculate the internal forces at the nodes using the element stiffness relations (stiffness matrices according to point 2 and load vectors according to point 3) and the now known nodal deformations (see point 8).

Section 4.7

10 For each element: where required, calculate the internal forces and moments within each element with the help of shape functions.

Section 3.5

In Table 3.1, it is assumed that in point 2 all the element stiffness matrices are calcu-lated and saved, because they are needed again for the determination of internal forces and moments under point 9 and that the allocation of point 4 takes place after the total completion of point 2. This method of presentation is beneficial for the un-derstanding. However, it does not correspond to general practice. Usually, the element stiffness matrices are determined and directly integrated into the total stiff-ness matrix without saving. For the determination of the internal forces and moments they are calculated again according to point 9. The procedure for the element stiff-ness matrices described here is similarly used for the element-related loads; see points 3, 5 and 9.


Table 3.1 can also be applied for calculations according to second order theory,and the system is initially analysed according to first order theory (first run). Sub-sequently, the procedure is restarted at point 2 and additional geometric element stiffness matrices are now determined, for which the internal forces and moments according to point 9 of the first run are needed, which means according to first order theory. In the second run, an additional total geometric stiffness matrix is generated at point 4. For that reason, the matrix of the equation system at point 8 consists of two matrices:

K G (3.6)

The matrix K representing the stiffness of the system belongs to the linear theory. Gcontains the additional components for second order theory. The consideration of initial deformations or geometric imperfections, respectively, is discussed in Sections 5.7 and 9.8 and the calculation of the internal forces and moments in detail in section 5.8.

For eigenvalue problems the approach described for calculations according to second order theory is applied as well. However, in the second run, the load vector is omitted and with p = 0 the eigenvalue problem is formulated:

KiK G v 0 (3.7)

Eq. 3.7 represents the starting point for the determination of the “critical load factor” (eigenvalue) and the modal shape. In general, the lowest positive eigenvalue and the corresponding modal shape are of interest. The solving of eigenvalue prob-lems is covered in Chapter 6; see also Sections 5.9 and 5.10.

3.4 Equilibrium

3.4.1 Preliminary Remarks

When structural systems are stressed, the loads lead to deformations. As a reaction, stresses and strains occur in the structural system, which correspond to internal forces and moments and deformations – the structural system is in a balanced state. The formulation of the equilibrium conditions is a major task for FEM. For that reason, appropriate principles and methods are needed. Common approaches are:

Principle of virtual work

Principle of minimum of potential energy

Equilibrium of differential elements/differential equations

3.4 Equilibrium 81

Ordinary differential equations (ODEs) are used in Section 3.5 to identify appropriate basis functions for the deformations. They are very helpful for this purpose since analytical solutions of the ODEs are known for some important special cases. For other cases, as for instance biaxial bending with torsion and axial force according to second order theory, solutions are not known. That is why it is convenient to use gen-eral principles for equilibrium formulations. Throughout this book, the principle of virtual work is used to formulate equilibrium conditions.

3.4.2 Virtual Work Principle

A structure is in the state of equilibrium when the sum of the virtual work is equal to zero. Therefore, the condition

W = Wext + Wint = 0 (3.8)

is the general requirement for equilibrium. In Eq. (3.8), Wext describes the virtual work of the external introduced forces (ext = external), and Wint the virtual work due to the resulting stresses (int = internal). As a reaction to the acting forces, the internal virtual work is negative.

Figure 3.6 Virtual work of a force

The familiar relation “work = force times displacement” is shown in Figure 3.6 with regard to a force F. It is displaced in the direction of its line of action and the distance of displacement is designated uF. This theoretically made displacement uF (“virtual displacement”) yields a virtual work W = F uF.

Figure 3.7 Virtual work as a result of axial force N and stress x


The virtual work, associated with an axial force in conjunction with a virtual displacement of a cross section, can be obtained in a comparable way. According to the agreement that N acts at the centre of gravity S, the virtual displacement in Figure 3.7 is designated as uS. On the right side, the axial stresses as a result of N are shown for the determination of the internal virtual work. As it is a reaction to the acting force it is negative, and the product x x has to be integrated for the entire beam. Where displacements correspond to forces, strains correspond to stresses concerning the virtual work.

The virtual work for beam can be obtained from [25], where it is derived in detail. Table 3.2 contains a compilation for the linear beam theory (first order theory). Note that all loads act at the centre of gravity or the shear centre, respectively, and the directions correspond to the principal axes. In cases where this does not apply, corresponding transformations have to be performed beforehand, as shown in Section 4.4. The virtual work for springs and shear diaphragms are compiled in Section 4.10 and additional work components for second order theory and stability in Chapter 5. The virtual work for cross sections of beams is illustrated in Chapter 11.

Table 3.2 Virtual work for beams according to first order theory (linear theory)

3.4 Equilibrium 83

3.4.3 Principle of Minimum of Potential Energy

The term energy describes the work stored by a system. If a body is displaced due to the acting of a force, the force performs work on the body, which is stored as potential energy. The total potential energy (total potential) of a system consists of the strain energy i (internal energy or internal potential) and the energy of the external forces (external potential) a :

i a (3.9)

Usually, structural systems in civil engineering are conservative systems, which means that they contain conservative forces. A force is described as conservative if the work it performs at a mass (material) point is independent of the path taken. In other words, the work which is performed through displacing the mass point from one position to another is always the same, no matter which path is being covered. If the path returns to its origin, which means a deformation back to the original state, the total work of conservative forces is equal to zero.

The principle of minimum of potential energy, which is also referred to as Dirichlet’svariational principle, states that for all geometrically possible states of deformation the true state of a conservative system, at which it will therefore be in equilibrium, leads to the minimum of potential energy:

Minimum (3.10)

With the first variation of the total potential, the essential condition

0 (3.11)

and with the second variation, the sufficient condition2 0 (3.12)

can be stated for the existence of a minimum.

Figure 3.8 shows the basic correlations according to [70]. A detailed description of the potentials and their variation is not done here, but can for example be taken from [2], [48], [70] or [64]. As a result, it can be seen that the first variation of the total potential of conservative systems is equal to the virtual work. Both principles are equivalent. The principle of minimum of potential energy provides an additional conclusion though, namely that due to its extremal character, every approximate solution for the state of deformation will estimate the potential energy “from above”. For nonconservative systems only the virtual work can be used to develop approximate solutions.


For that reason, the virtual work and the minimum of potential energy are equal as principles for the equilibrium formulation within the fields of application covered here. Since the virtual work principle is more universal and easier to understand, it is used throughout this book.

Figure 3.8 Essential correlations concerning the principle of minimum potential energy and types of equilibrium according to [70]

3.4.4 Differential Equations

As already mentioned, virtual work is used to formulate the equilibrium as well as the element stiffness matrices. Therefore, differential equations are not so important here. On the other hand, they promote the understanding of the mechanical and static coherencies and they can especially be used to rate the basis functions for the deformations, as shown in Section 3.5. In [25], the differential equations for linear beam theory are derived in detail. The virtual work, the definition of the internal forces and moments and the equilibrium formulated for a differential segment of the beam are used there. Also, the stresses x and , calculated according to the theory of elasticity, are considered. Table 3.3 contains a compilation of the differential equa-tions of [25] for linear beam theory; also see Table 1.3.

3.4 Equilibrium 85

Table 3.3 Differential equations of linear beam theory (biaxial bending with axial force and torsion)

Axial force Bending about the z-axis

Bending about the y-axis Torsion

SuEAN Mzz vEM I

Mzy vEV IMyy wEM I

Iz y MV E w

IEM

IEMxs

Txp GM I

xS quEA yMz qvEI zMy qwEI IE xT mGI

with: modulus of elasticity E, shear modulus G, cross section area A, principal moments of inertia Iy and Iz, warping constant I , torsion constant IT

For calculations according to second order theory and for stability (eigenvalues) further differential equations result. According to [42] and [44], the following rela-tions between displacements and loads are obtained:

S xEA u q (3.13a)

z M M M y yEI v N v z M q (3.13b)

y M M M z zEI w N w y M q (3.13c)

T y M z M M M M MEI GI M v M w N y w z v

rr y q M z q M xM q y y q z z m (3.13d)

Mrr: see Table 5.1

The ODEs (3.13) show how the four partial problems “axial force, bending about the y- and z-axis and torsion”, which can be handled separately for the linear beam theory, are connected when taking into account second order theory. The linking is a result of the internal forces and moments and the equation system (3.13) cannot be solved analytically. For certain cases for which solutions are known, the problem of “flexural buckling” is of general relevance. In the ODEs of the Eqs (3.13) the axial force N is defined as positive as tension force and the distributed load qx acts at the centre of gravity – also see Figure 1.9.

With Eq. (3.13c) the known ODE for flexural buckling about the strong axis can be gained when (x) = 0 and a constant stiffness EIy and axial force compression are assumed:


y D M zEI w N w q (3.14)

It is appropriate to consider beam segments with the length for the solving of the ODEs and to introduce a member characteristic . With that, the ODEs can be for-mulated as follows:

2D z

M My

qw wEI

(axial force compression) (3.15)

2Z z

M My

qw wEI

(axial tension force) (3.16)

with: DD

y

NEI

and ZZ

y

NEI (3.17)

The solutions of the ODEs (3.15) and (3.16) are given in section 3.5 since they are needed there in conjunction with the basis functions for the deformations.

Finally, the stability problem of plate buckling is addressed here. According to [42], the homogeneous ODE is

3

x xy y2E t w 2w w t w 2 t w t w 0

12 1(3.18)

In Formula (3.18), the raised dashes (primes) represent a derivation with respect to x, the raised dots with respect to y. The stresses x and y are defined as positive for compression stresses, which is in accordance with the compression forces for the flexural buckling of beams. If y = = 0 is assumed in Formula (3.18), the homogeneous ODE arising corresponds to the ODE of flexural buckling:

3

M x M2E t w t w 0

12 1 (3.19)

Due to this compliance, the homogeneous ODEs (3.15) and (3.19) have the same so-lution functions.

3.5 Basis Functions for the Deformations 87

3.5 Basis Functions for the Deformations

3.5.1 General

Basis functions for the description of the deformations must be capable of accurately capturing the possible structural deformation. Almost exclusively, polynomial func-tions are used for the description. In many cases, these are capable of exactly meeting this central requirement. For some practical challenges, however, they are only ap-proximations and therefore require an adequately fine FE modelling, so that the calculation results are sufficiently accurate. Examples are the flexural buckling of members and other cases described in Section 3.5.3.

Another important point is capturing the boundary conditions, and here geometric and physical conditions have to be distinguished. The basis functions should be chosen in such a way that the deformations in the nodes directly allow the geometric boundary conditions to be taken into account. In the case of bending about the y-axis, this means for example, that the displacement wM and the rotation y Mw should be included in the basis functions as degrees of freedom.

Physical boundary conditions are conditions for the internal forces and moments at the ends or boundaries of structural systems. As an example of bending, the boundary condition My = 0 at the end of a beam will be considered here. From Table 3.3 we have y y MM EI w , so that from My = 0 immediately Mw 0 follows. Naturally, the curvatures Mw or comparable variables, which correspond to the physical boundary conditions or the internal forces and moments, can be taken into account as degrees of freedom in the basis functions. This has sporadically been implemented in FE programs, but has not become widely accepted since certain disadvantages are connected to it. For the displacement method, only degrees of freedom correspond-ing to the geometric boundary conditions should normally be used.

3.5.2 Polynomial Functions for Beam Elements

For finite beam elements polynomial functions are of vital importance. As an exam-ple, the function

2 30 1 2 3f x a a x a x a x (3.20)

is considered. This polynomial function with four terms is of third degree with the coefficients a0, a1, a2 and a3. We usually just refer to them, in brief, as “polynomials”.


Loading with axial forces

The ODE for this load case can be taken from Table 3.3. Since exclusively beam elements with constant cross sections are going to be considered, EA can be assumed as being constant with the following ODE resulting:

S xEA u q (3.21)

Integrating the ODE twice gives:

2xS 0 1

q1u x a a x x2 EA

(3.22)

For the case of qx = 0, the exact solution is a first degree polynomial with a linearly varying distribution of uS(x). With the help of Figure 3.9 the integration constants a0und a1 can be replaced by the longitudinal displacements at the ends of the beam element. The boundary conditions uS(x = 0) = uSa and uS(x = ) = uSb lead to the inte-gration constants:

a0 = uSa (3.23)

Sb Sa x1

u u q1a2 EA

(3.24)

With the introduction of the nondimensional coordinate = x/ , the following func-tion for the longitudinal displacement results:

22x

S Sa Sbq1u 1 u u

2 EA(3.25)

Function for the longitudinal displacement under axial forces:

22x

S Sa Sbq

u ( ) 1 u u2 EA

Function for the rotation (twist) under St Venant’s torsion:

22x

a bT

m( ) 1

2 GI

Figure 3.9 Beam element and functions for the longitudinal displacement under axial force and the rotation (twist) under St Venant’s torsion

Since only deformations at the nodes are involved here (no derivations!), the shapefunctions in Figure 3.9 comply with the coefficient functions of the Lagrange inter-


polation polynomial for two nodes. Shape functions are commonly used in conjunction with FEM. Corresponding to the deformation at the given node, they adopt the value one, while the deformations at the other nodes describing the dis-placement function are zero.

St Venant’s torsion

For cross sections which are free of warping, the warping constant is I = 0, so that pure St Venant’s torsion occurs. For this particular case the ODE according to Table 3.3 with the assumption of GIT = const. leads to:

T xGI m (3.26)

In Eq. (3.21), i.e. the ODE for the longitudinal displacement, the formal conformance of both equations can be recognised. For that reason, the rotation of St Venantstorsion (twist) may be described using the following function for the development in the beam element (also see Figure 3.9):

22x

a bT

m112 GI

(3.27)

Bending about the z-axis

If a uniform cross section is assumed within a beam element, it is EIz = const. and, with the help of Table 3.3, the following differential equation can be stated:

z M yEI v q (3.28)

Four times integration leads to

y2 3 4M 0 1 2 3

z

q1v x c c x c x c x x24 EI (3.29)

For qy = 0, the function is a third degree polynomial with the integration constants c0to c3. Just as for loading with axial forces the constants are replaced by deformations, being more easily accessible for the engineering understanding. Mechanically, it is useful to choose the dieplacements and rotations at the element ends as degrees of freedom since the deflections have to be continuous going beyond the element boundaries, which means without sharp bends. As shown in Figure 3.10, the degrees of freedom at the nodes are: vMa, Ma zav , vMb and Mb zbv . By using the boundary conditions vM(x = 0) = vMa, Mv (x = 0) = za, vM(x = ) = vMb and Mv (x = )= zb, the integration constants in Eq. (3.29) can be replaced. With the nondimen-sional coordinate = x/ , the following function for the bending of the beam element in y-direction can be formulated:


2 3 2 3M Ma za

4y2 3 2 3 2 3 4

Mb zbz

v 1 3 2 v 2

q13 2 v 224 EI

(3.30)

This equation is a Hermitian interpolation polynomial of order 2 = 4 since besides the displacements at the points a and b also the first derivation is used to describe the bending.

Figure 3.10 Beam element and shape functions f( ) for the deflection vM( )

Bending about the y-axis

This load case is directly comparable to the bending about the z-axis, as shown in Table 3.3. With an analogical procedure, the following function for the bending wM( ) of the beam element in the z-direction can be stated:

42 3 4z

M 1 Ma 2 ya 3 Mb 4 yby

qw f w f f w f 224 EI

(3.31)

The shape functions f1 to f4 are given in Figure 3.10 since they are identical for the displacements wM( ) and vM( ). The negative signs in Eq. (3.31) are a result of the relations ya Maw and yb Mbw for the angles of rotation.


3.5.3 Trigonometric and Hyperbolic Functions for Beam Elements

Here we will show, for three exceptional cases, that the functions for the dis-placements may include

trigonometric functions: sin x and cos x hyperbolic functions: sinh x and cosh x.

Bending with axial compression force according to second order theory and the stability problem flexural buckling

According to Section 3.4.4, Eq. (3.15), the ODE for bending in the z-direction is: 2

D zM M

y

qw wEI

with DD

y

NEI (3.32)

As is generally known, the solution of this ODE consists of two parts:

M Mh Mpw (x) w (x) w (x) (3.33)

The first term describes the solution of the homogeneous ODE, i.e. for qz = 0, and the second one the particular solution. According to [42] and with x = , we have:

24z

M 0 1 2 D 3 Dy D

qw ( ) c c c sin c cos2 EI

(3.34)

As shown in Section 3.5.2, the integration constants c0 to c3 may be replaced by deformations in terms of “engineer-like thinking”. With

M Ma

M ya

M Mb

M yb

w 0 w

w 0

w 1 w

w 1

(3.35a-d)

the following function for the deflection of a beam element subjected to an axial com-pression force and a uniformly distributed load arises:

M D 1 Ma 2 ya 3 Mb 4 ybw , f w f f w f4

2D zD D D 3

D y D

cos 1 qcos 1 sinsin 2 EI

(3.36)


with:

1 D D D D D D nf 1 cos 1 cos sin sin k

2 D D D D D D D

D D D D n

f 1 cos sin sin 1 sin cos

cos cos 1 k

3 D D D D D nf 1 cos 1 cos sin sin k

4 D D D D D

D D D n

f 1 cos sin sin cos 1

cos 1 k

n D D Dk 2 2 cos sin

Equation (3.36) for flexural buckling is much more extensive in comparison to the element subjected to bending according to Eq. (3.31) and therefore much more complex for further application. Fortunately, it is rarely needed in conjunction with FEM. Here, we will only show what the exact solution of the displacement function for flexural buckling looks like. In Section 5.6, the exact solution of the ODE is again used to derive a stiffness matrix according to second order theory.

With N = 0 or D = 0, Eq. (3.36) can be transferred into Eq. (3.31). However, this is not directly possible through insertion since indeterminate terms of the form “0/0” occur. The solution using the limit value rule of Bernoulli and de l´Hospital,

D D

D D0 0D D

f ( ) f ( )lim limg( ) g ( ) (3.37)

is extensive and has to usually be applied several times. Using the series expansions

3 5D D D D

1 1sin ...3! 5!

(3.38)

2 4D D D

1 1cos 1 ...2! 4!

(3.39)

for the trigonometric functions, the coherence between Eq. (3.36) and (3.31) can be recognised. Since for D 0 Eq. (3.36) merges into Eq. (3.31), Eq. (3.36) can, by ap-proximation, be replaced using the polynomial functions for small member characteristics. A correspondingly fine FE modelling will always lead to D being small, because the element length affects this parameter. As will be shown in Section 5.6, the approximation with the polynomial function (3.31) is sufficiently exact if the condition


DD

N 1,0EI

(3.40)

is observed. Figure 3.11 shows the error of the polynomial function with regard to the exact function (3.36) for the deflection where D = 1. Since the discrepancies for the functions f3 and f4 match those of functions f1 and f2, they are not illustrated – also see Figure 3.10. As can be seen, the discrepancies are low with maximum of 2.5 %.

Figure 3.11 Comparison of the polynomial function Eq. (3.31) with Eq. (3.36) for D = 1

Bending with axial tension force according to second order theory

In comparison to the previous load case with axial compression force, the influence of an axial tension force is now analysed. The ODE (3.16)

2Z z

M My

qw wEI

mit ZZ

y

NEI (3.41)

has the solution 24

zM 0 1 2 Z 3 Z

y Z

qw ( ) c c c sinh( ) c cosh( )2 EI

(3.42)

All solutions for second order theory with compression force (ND) may be converted to solutions for second order theory with tension force (NZ). With

ND = NZ (3.43)

we obtain

D D z ZN EI N EI 1 i (3.44)

In Eq. (3.44), “i” is the imaginary unit where i2 = 1. The transformation can usually be achieved using the following relations:


2 2D Z (3.45)

cos D = cos( Z i) = cosh Z (3.46)

sin D = sin( Z i) = i sinh Z (3.47)

With these formulas, Eq. (3.36) can be converted without any problems. That is why it is not shown in detail here. For completion, the series expansions for the hyperbolic functions may be indicated as well:

53Z Z Z Z

1 1sinh( ) ( ) ...3! 5!

(3.48)

42Z Z Z

1 1cosh( ) 1 ( ) ...2! 4!

(3.49)

For second order theory with axial tension force the approximation using the polynomial function (3.31) can be used as well. For the choice of the element length the condition

ZZ

N 1,0EI

(3.50)

has to be considered. Further details for the use of the approximation are given in Section 5.6.

Warping torsion

With Table 3.3 and the assumption of a constant stiffness in the beam element, the following ODE can be stated for warping torsion:

T xEI GI m (3.51)

With the definition of a member characteristic

TT

GIEI

(3.52)

for torsion, the ODE is 2

T xmEI

, (3.53)

and formally corresponding to the ODE (3.41). Therefore, its solution leads to a func-tion ( ) including hyperbolic functions sinh( T ) and cosh( T ), just as in the case of “bending with axial tension force”. It formally corresponds to Eq. (3.42). The derivation of the element stiffness matrix for warping torsion is carried out in Section 4.2.4.


Lateral torsional buckling and further combined loadings

Apart from the previously shown special cases, there are no further solutions for other problems. However, it is most likely that for lateral torsional buckling and for com-bined loadings as well, which require a second order theory approach or a stability analysis, trigonometric and hyperbolic functions are needed for the description of the deformations. Since in these cases only polynomial approaches can be used, a suffi-ciently fine FE modelling has to be used. As an indication, the three special cases can be used. In case of doubt, a finer modelling has to be applied or a stepwise refinement has to be performed.

3.5.4 Basis Functions for Plate Buckling

For the structural analyses of plates triangular or rectangular plate elements with straight boundaries are usually used since problems with curvilinear boundaries are rare in civil engineering. Figure 3.12 shows triangular and rectangular elements, which differ in the number of nodes and the number of degrees of freedom.

Figure 3.12 Triangular and rectangular elements for plates

The plates are here considered to be in the x-y plane, and for the plate elements we use the nondimensional coordinates:


= x/ x (3.54)

= y/ y (3.55)

With these designations, the combinations of n m are obtained for plate elements as shown in Figure 3.13 if polynomial functions are introduced for the description of the deformations. Additionally, Figure 3.14 shows the terms of the polynomial approach for w( , ) using two, three and four terms for both directions. If these terms are to be completely covered, corresponding elements with nodes and degrees of freedom have to be chosen. This is of great importance for the appropriate description of the deflections. Therefore, area elements with 4, 9 or 16 degrees of freedom are reasonable – also see Figure 3.14.

Figure 3.13 Polynomial terms for polynomial functions of plate elements (Pascal’s triangle of polynomials)

The bending of plates corresponds to the load-bearing behaviour of beams if the ad-ditional load transfer in the y-direction is considered. As an example, girder grids may be mentioned, which are applied for plates stiffened longitudinally and transversely by using beams in the x- and y-directions. Therefore, the derivations of Section 3.5.2 for beams in bending can be applied.

Equation 3.29 shows that the solution of the problem is covered exactly with a four-term polynomial and therefore the approach for w( , ) at the bottom of Figure 3.14 for the bending of plates corresponds to the bending of beams. Due to the four terms of the approach, the multiplication leads to 16 polynomial terms. That is why a plate element should if possible include 16 degrees of freedom. The rectangular element in Figure 3.12 on the right with four degrees of freedom at each of the four nodes fulfils this demand best. An additional advantage is that this plate element can directly be combined with beam elements for uniaxial bending with warping torsion. This is of major importance for stiffened plates.


Figure 3.14 Approaches for w( , ) and arising polynomial terms

The buckling of plates can be compared with problems of beams, as was done pre-viously for plate bending. For this purpose, the homogeneous ODE (3.18) for plate buckling and the ODEs (3.15) and (3.16) of the member buckling or the bending with compression or tension forces according to second order theory are considered. The comparison shows that the plate buckling for xy = 0 for compression stresses x and

y, can be understood as a “two-dimensional buckling of members”. According to Section 3.5.3, the displacement function w( , ) for plate buckling includes the trigonometric functions sin and cos, as it did for the buckling of members. In addi-tion, hyperbolic functions sinh and cosh may be necessary to describe the shape of the buckled plate (eigenvalue) since in some areas tension stresses can occur. In this context, Petersen [67] has examined an interesting example. He solves the plate buckling problem for a so-called “single plate” in Figure 3.15 with a fixed support (restraint) of the longitudinal plate boundaries and proves that the approach

1 y 2 y 3 y 4 yx

m xw(x, y) sin C cosh C sinh C cos C sin (3.56)

is an appropriate solution. As can be seen, the functions sin, cos, sinh and cosh are included.


Figure 3.15 Buckling panel with restrained longitudinal edges

Because a finite element for plate buckling has to be universally applicable, an approach using polynomial functions for - and -directions as shown in Figure 3.14 at the bottom is advisable. Using a fine elementing, the previously mentioned sin, cos, sinh and cosh functions can be approximated with sufficient accuracy. As a matter of principle, it can be stated that the element mesh has to be fine enough in order to be able to describe the buckled shape properly. For an orientation involving the element dimensions, i.e. the element length, the conditions D 1.0 and Z 1.0 may be used, as shown in Section 3.5.3 for beam elements. In a similar way, the following condition for the element length x may be formulated:

2

x 2xcr

E t 107.4 t12 1

(3.57)

For x, the critical buckling stress xcr has to be inserted in kN/cm2. Since this value is supposed to be calculated and because it is not yet known when meshing the plate, it has to be estimated in advance. After the determination of xcr, condition (3.57) has to be rechecked.

Using the deformation function for a beam element according to Eq. (3.31) and with the help of the multiplication for a plate element in Figure 3.14, the displacement function w( , ) may be formulated as follows:

1 a 2 a x 3 b 4 b x 1

1 a y 2 a x y 3 b y 4 b x y 2

1 c 2 c x 3 d 4 d x 3

1 c y 2 c x y

w( , ) f w f ( ) w f w f ( ) w f ( )

f w f ( ) w f w f ( ) w f ( )

f w f ( ) w f w f ( ) w f ( )

f w f ( ) w 3 d y 4 d x y 4f w f ( ) w f ( )

(3.58)


The shape functions f1( ), f2( ), f3( ) and f4( ) were given in Figure 3.10. The func-tions f1( ), f2( ), f3( ) and f4( ) are gained by replacing the nondimensional coordinate with .

Figure 3.16 Rectangular plate element with four nodes and 16 degrees of freedom

The displacement function according to Formula (3.58) for the rectangular element shown in Figure 3.16 contains 16 degrees of freedom (four nodes, in each node w, w , w and w ). This element has proven its worth for the problem of plate buckling for many years. We would advise against the use of elements which only include the three degrees of freedom w, w and w in the nodes. For some problems, totally wrong solutions have been obtained with this element type since the degree of freedom (w ) , the derivative of the angle of twist, can not be neglected. The applicability of different elements for plate bending is discussed in detail in [48]. There, it is concluded that the consequent expansion of the one-dimensional to two-dimensional shape functions is the rectangular element including 16 degrees of freedom. For that reason, it is most suitable for the analysis of rectangular plates.

3.5.5 One-Dimensional Functions for Cross Sections

For the analysis of cross sections the displacements u(x,s) in the longitudinal direc-tion of beams must be determined. Figure 3.17 shows the deformation u of a C-section due to (x) . Only the centre line of the cross section is considered since the cross section is thin-walled and instead of using the cross section coordinates y and z, a profile ordinate s is used.

Usually, for different parts of the profile, the centre line is straight. For that reason, cross sections are divided into straight partial plates. In terms of finite element meth-ods, these are straight cross section elements with a constant plate thickness t. For the description of the displacements u in the cross section plane, one-dimensional Lagrangian functions are used since, due to the theoretical background, it is advisable to only deal with the displacements in single points (cross section nodes) and not their derivations.


Figure 3.17 Displacements due to the derivative of the angle of twist

Table 3.4 Lagrangian functions for line elements

Linearly varying course of function

Quadratic course of function

Cubic courseof function

121f 1

122f 1

2121f

2122f

23f 1

9 1 116 3 31f 1

9 1 116 3 32f 1

27 116 33f 1 1

27 116 34f 1 1


In Table 3.4 a cross section element is regarded in an arbitrary position. The profile ordinate s is replaced by the nondimensional ordinate = (2 s/ 1). Here is used as ordinate in the cross section plane. Principally, it is comparable to = x/concerning beam elements, whereas for beam elements the longitudinal direction is described. The origin of the nondimensional ordinate is assumed to be in the mid-dle of the cross section element, and in Table 3.4, line elements with two, three and four nodes are shown. Unlike with beam elements, the origin of is set to the middle of the element since it is a convenient position for two-dimensional elements requir-ing numeric integrations. For reasons of a consistent depiction, it is also used here for the one-dimensional cross section elements. The function for the displacement u re-sults in a line element with n nodes from the shape functions fi and the node displacements ui as follows:

n

i ii 1

u( ) f u (3.59)

Figure 3.18 shows the course of some selected shape functions. With the following thoughts about principles, appropriate functions for the different problems of Chapter 11 are shown.

Figure 3.18 Course of the shape functions using Lagrangian polynomials

Warping ordinate

One aim of the analysis of cross sections is to determine the standardised warping ordinate , which combines the displacements u of a cross section with the derivative of the angle of twist by

u ; (3.60)

see Sections 11.3 and 11.4.1. According to Section 2.6, the warping ordinate of thin-walled cross sections is determined with

ts

r ds (3.61)

and for closed thin-walled cross sections Eq. (3.61) has to be expanded as follows:


n

t ii 1s s

dsr dst(s) (3.62)

In Eq. (3.62), the parameters i are the torsional functions of the hollow cross section cells, [73]. If straight plates are assumed, the distance rt between the shear centre and the tangent at the profile centre line is constant for the plate and for the cross section element. This also applies for the factor of hollow cells. For that reason, integration over the ordinate s leads to a linearly varying course of for each cross section element. With regard to , which is constant for the cross section, Eq. (3.60) shows that u has a linearly varying course as well, see also Figure 3.17. With the use of Table 3.4 the displacement can therefore be described exactly by using the following function:

1 11 22 2u 1 u 1 u (3.63)

St Venant’s torsion

According to Section 11.4, the shear stresses of thin-walled cross sections of the StVenant’s (primary) torsion consist of two components. In the rectangular partial plates (open sections), shear stresses occur, which are equal to zero at large ranges of the centre line. Due to the reduction of the cross section to its centre line, they can not be covered with the degrees of freedom u of corresponding finite elements. They therefore have to be determined using different methods. In the hollow cells of thin-walled cross sections, the shear stresses have a constant distribution within the plate thickness and length. With xs = G xs and xs = u/ s + v/ x, the integration will lead to linearly varying deformations u. For that reason, the cross section element applied for the warping ordinate according to Table 3.4 on the left-hand side or Eq. (3.63), respectively, is sufficient for this problem. This subject is dealt more detailed in Section 11.3.

Shear forces and secondary torsion

As shown in Section 11.3, to calculate shear stresses arising from shear forces and secondary torsion, corresponding shear deformations are used. For thin-walled cross sections these stresses are usually determined by considering equilibrium statements, which lead to the following calculation formula as shown in Chapter 7:

y xszxs

y zs s s

V MV z(s) t ds y(s) t ds (s) t dsI t I t I t (3.64)

For straight cross section parts with a constant plate thickness the factors in front of the integrals are constant values as well. Therefore, the course of the shear stress distribution only depends on the integration. Since the ordinates z(s), y(s) and (s)


have constant or linear distributions, the shear stresses will have a linearly varying or quadratic course. With the definition of the shear stresses depending on the shear displacements according to Eq. (11.21) in Section 11.4.1, we have:

xs xss

u 1G u dss G (3.65)

For this reason, the deformations u(s) will have a quadratic or cubic course. An exact description of the deformation is gained with the functions on the right in Table 3.4:

9 91 1 1 11 216 3 3 16 3 3

27 271 13 416 3 16 3

u 1 u 1 u

1 1 u 1 1 u(3.66)

The course of the shape functions included in Eq. (3.66) is shown for the functions f1and f3 in Figure 3.18b.

3.5.6 Two-Dimensional Functions for Cross Sections

In Section 3.5.5, the functions for the deformations u(x,s) have been formulated. With them, thin-walled cross sections can be analysed. In comparison to thin-walled sections, it is not satisfactory to only consider the centre line of a profile when dealing thick-walled profiles. Therefore, two-dimensional interpolation polynomials are needed for these cross sections in order to describe the deformations u, with which also the course of the function can be acquired via the plate thickness.

At first, a two-dimensional rectangular element with edges coinciding with the y-z principal axes of a cross section is regarded. The ordinates of the principal axes can be replaced by nondimensional element ordinates:

= 2 (y – y0)/ y and = 2 (z – z0)/ z

Figure 3.19 shows three rectangular elements with 4, 9 and 16 degrees of freedom for the deformation u.

For thin-walled cross sections it was possible to directly state which polynomials ex-actly describe deformations u(x,s) for the different problems, i.e. the warping ordinate as well as the shear displacements u as a consequence of primary torsion, shear force and secondary torsion. This is not possible for thick-walled cross sections since the progress of the displacements u(x,y,z) depends on the geometry of the cross section.


Figure 3.19 Two-dimensional elements for cross sections and degrees of freedom

The progress can only be stated analytically for basic forms (e.g. for rectangles or equilateral triangles). That is why the most suitable degree of the polynomial func-tions as approach for the displacements can not be stated and generalised a priori. Generally, the chosen displacement functions can therefore only lead to an approxi-mate solution. However, with a refinement of the element mesh or an FE model of higher value, it will converge to the exact solution.

In order to guarantee this, the displacement approach must fulfil consistency require-ments. When looking at the constitutive equations of arbitrary thick-walled cross sections (see Section 11.5.2), it becomes obvious that the deformations u and the first derivatives of the deformations emerge in the virtual work. For the displacement ap-proach this means that it must be continuous concerning the functional values in order to ensure finite values for the first derivatives reflecting the strains. They may not take on infinite values, because this would correspond to a gap within the ele-ment. We speak of a C0-continuity here. At the same time, the approach for the displacements must also be differentiable at least once, without being zero. Both re-quirements are met through the use of the Lagrangian interpolation polynomials, which were dealt with in Section 3.5.5 in connection with the thin-walled cross sec-tions for one-dimensional problems.


Table 3.5 Lagrangian interpolation polynomials for area elements

Bilinearly varying function

Quartic courseof function

141f 1 1

142f 1 1

143f 1 1

144f 1 1

2 2141f

2 2142f

2 2143f

2 2144f

2 2125f 1

2 2126f 1

2 2127f 1

2 2128f 1

2 29f 1 1

The shape functions fi of the bilinearly varying and the quartic approach are compiled in Table 3.5 for the two-dimensional elements required here. Also, Figures 3.20 and 3.21 graphically show the course of the functions. The formulation of the functions is carried out using the nondimensional coordinates and . Their origin is located in the middle of the corresponding intervals 1 1 and 1 1. This leads to the course of the function to describe the deformations u for an element with n nodes:

1n

2e1 2 n i i

i 1

n

uu

u( , ) f f f f u f u

u

(3.67)

Figure 3.20 Shape function f3 of the bilinearly varying function


Figure 3.21 Shape functions f3, f6 and f9 of the quartic course

So far, elements of rectangular shape have been assumed in connection with Figure 3.19. However, Section 11.4 shows that Eq. (3.67) may also be applied to the dis-placement description of elements being not right-angled or with curvilinearboundary elements.

In principle, arbitrary Lagrangian polynomials, i.e. the bilinearly varying, quartic, bicubic or polynomials of higher order can be used in Eq. (3.67). Here, it is important to note that in order to achieve good solutions at the cross section calculations using the bilinear functions, lots of elements are required compared to the quartic ones; see Section 11.7.4. For this reason, the use of the bilinearly varying approach is not advisable.

For thin-walled cross sections it can be shown that the displacements due to shear force and secondary torsion are described exactly with a cubic polynomial and hence it is sufficient to describe a plate through a single element. Therefore, the choice of a bicubic approach for the two-dimensional elements seems obvious. However, it depends on the shape of the cross section how well this approach is able to describe the real deformations. Since more than one element is arranged with regard to the plate thickness, one can alternatively choose elements with a quartic course of the function. Moreover, the elements should numerically be as “stable” as possible. This is achieved by using rectangular, preferably quadratic, elements. Therefore, the steel cross sections are usually discretised by elements for which the quartic and the bicubic approaches lead to comparably good results. Even though a lower quantity of elements is needed while using the bicubic funtions, computing time is disproportion-


ately high due to the numeric integrations; see Section 11.5.6. For this reason, elements with a quartic course of the function are recommended.

Note: For a better understanding it may be added that besides C0-steady often C1-steady polynomials are required. This is the case if the constitutive equations, i.e. the virtual work, also include the second derivation of a deformation. In that case, the first derivativation of the polynomial function for the description of the deformation must also have a continuous course. One example of this would be beams with infinite shear stiffness, for which Hermite’s interpolation polynomials are used.

4 FEM for Linear Calculations of Beam Structures

4.1 Introduction

This chapter deals with the finite element method regarding the linear beam theory (first order theory). An important assumption is that the resulting deformations are small and that the equilibrium may be formulated for the undeformed structure as an approximation. Beam elements will then be considered and relationships which are required for the displacement method will be derived. Using these, equation systems can be formulated in order to calculate the displacements and internal forces and moments for beam structures (beams, columns, frames).

Calculations according to the linear theory are the basis for second order theory analyses; see Chapter 5. However, internal forces according to the linear theory are also needed for stability verifications where they are carried out with the -procedure (flexural buckling) or the M-procedure (lateral torsional buckling) according to DIN 18800 Part 2 or when the procedures stated in Eurocode 3 are applied.

4.2 Beam Elements for Linear Calculations

4.2.1 Linking Deformations to Internal Forces and Moments

The starting point for the following derivations is an element in the local x-y-z coordinate system. Figure 4.1 shows the beam with both nodes a and b as well as the nondimensional coordinate = x/ in the axial direction.

Figure 4.1 Beam element

assumed, y and z are the principal axes of the cross section and beam axes referring to the centre of gravity S and the shear centre M have to be distinguished. In this respect, Figure 4.1, which contains only one axis, is a simplified representation.

Figure 4.2 shows the definition of the positive deformations at both beam ends (in-dices a and b). Here, w and v are designated with a subscript “M” describing the deformation of the shear centre. Accordingly, uS is the longitudinal displacement at


Since here, as usually done for beams, standardised cross section properties are

4.2 Beam Elements for Linear Calculations 109

the centre of the gravity. In addition to the displacements, the rotations are designated by y, z and . They are explained in more detail in Figure 1.6. is the derivative of the angle of twist (rotation per unit length), a deformation variable which is required for the warping torsion. Overall, 14 displacements occur, 7 at each beam end.

Figure 4.2 Beam element with the definition of deformations at the elements ends

Figure 4.3 Beam element with the definition of internal forces/moments and loads qx, qy, qz and mx

The nodal internal forces and moments in Figure 4.3 directly correspond to the direc-tion and to the point of application of the 14 deformations. As can be seen, their directions are not consistent with the common sign definition. That is why the directions in Figure 4.3 are referred to as sign definition II. This kind of sign definition results from the formulation of the nodal equilibrium (see Figure 3.2 for example) and hence it is an essential element of FEM.

4 FEM for Linear Calculations of Beam Structures 110

Figure 4.3 also defines the positive direction of action of the loads qx, qy, qz and mxon a beam and the assumed points of application. If the loads act at different locations, they have to be displaced to the points S and M, and corresponding transformations have to be performed (see Section 4.4.4). In Figure 4.3, M is the warping moment corresponding to the derivative of the angle of twist (also see Section 4.2.4). The internal forces and moments at the beam ends in Figure 4.3 can be linked to the corresponding deformations using “stiffness relationships”. In matrix notation, the relation for the linear beam theory (first order theory) is:

e e e es K v pwith: es Vector of the nodal internal forces and moments

eK Element stiffness matrix

ev Vector of the nodal deformations

ep Vector of the loads due to qx, qy, qz and mx

(4.1)

In Eq. (4.1), the subscript “e” labels variables of beam elements. In the following Sections, the element stiffness matrix and the load vector are derived. For reasons of clarity and with regard to varying tasks in civil engineering, the following states are distinguished, as can be seen in Table 1.3:

axial force bending about the y-axis bending about the z-axis torsion

A first overview concerning the procedure for deriving the element stiffness rela-tionships is given in Table 4.1, where the bending about the y-axis is considered.

4.2.2 Axial Force

We are dealing with beam elements as shown in Figure 4.1 which are only loaded by axial forces (at the centre of gravity). In this case, displacements u in the axial direction are constant for the cross section and the nodal displacements uSa and uSb(see Figure 4.2) as well as the internal forces Na and Nb (see Figure 4.3) act at both beam ends. The distributed load qx also acts at the centre of gravity and it is assumed to be uniform within the element, i.e. qx = const.

As explained in Section 3.4, the equilibrium is formulated with the help of the virtual work principle. For a beam element we have

ext intW W W 0 (4.2)


with: ext Sa a Sb b S x0

W u N u N u q dx

int x x0 A

W dA dx

The axial stress and the virtual strain can be replaced by using Eqs (1.5) and (1.4a). Since here only the case “centric axial force stress” is considered, x SE u and

x Su are obtained. In Section 3.5.2, we determined that the polynomial function

S Sa Sbu 1 u u (4.3)

is the exact solution in order to describe the longitudinal deformations depending on the nodal displacements. The third term in Eq. (3.25), containing qx, is not considered in Eq. (4.3), since for that the internal virtual work results to zero. With the application of the chain rule, Eq. (4.3) gives

S Sa Sbdu du d 1u u udx d dx

(4.4)

and S Sa Sbu u u . The internal virtual work then is

int x x S S0 A 0 A

1

S S S S0 0

1

Sa Sa Sa Sb Sb Sa Sb Sb0

1 1 1 1Sa Sa Sa Sb Sb Sa Sb Sb0 0 0 0

W dA dx u E u dA dx

EA u u dx EA u u d

EA u u u u u u u u d

EA u u u u u u u u

Sa Sa Sa Sb Sb Sa Sb SbEA u u u u u u u u (4.5)

Using a similar procedure for the term with the load qx of the external virtual work in Eq. (4.2), the following can be obtained:

1 112 2

S x x S x Sa Sb0 00 0

Sa Sb x

1 1u q dx q u d q u u2 2

u u q 2 (4.6)

The result of Eq. (4.6) is shown in Figure 4.4. Both nodes of the beam element are assumed to be fixed there. As an equivalent replacement of qx, concentrated loads qx /2 act at both nodes.


Figure 4.4 Replacement of qx at a beam element by concentrated loads at both nodes

With Formulas (4.5) and (4.6), the virtual work, Eq. (4.2), can be written as

Sa a Sb b Sa x Sb x

Sa Sa Sa Sb Sb Sa Sb Sb

W u N u N u q 2 u q 2EA u u u u u u u u 0 (4.7)

If the virtual and actual nodal displacements are combined in a row and a column vector,

Te Sa Sbv u u and Sa

eSb

uv

u, (4.8)

then Formula (4.7) results, in matrix notation, to T T Te e e e e e ev s v p v K v 0 (4.9)

Compared to Eq. (4.1), this formulation shows that with the complete notation, all terms depend on the vector of the virtual nodal displacements. This is referred to as complete stiffness relationship. Since T

ev 0 is a trivial solution, this term is often neglected. This leads to the so-called incomplete stiffness relationship:

Te e e e ev s p K v 0 e e e es p K v 0 (4.10)

The vector of the virtual deformations and thus the “complete” relationship (4.9) is definitely of importance for understanding the situation. With it, the consideration of the boundary conditions through dropping columns and rows is more understandable (see also Section 4.5.4). Eq. (4.10) can be written like Eq. (4.1), i.e. in solution for the internal forces and moments. The expressions placed at the beginning of the rows “ uSa:” and “ uSb:” are a reminder that all terms of the rows belong to these virtual nodal deformations:

a Sa xSa

b Sb xSb

Te e e e e

N EA EA u q 2u :N EA EA u q 2u :

v : s K v p (4.11)


Eq. (4.11) is the desired stiffness relationship for a member under pure axial force; it connects the internal forces and moments to the corresponding deformations at the beam ends. With the load vector ep , the uniformly distributed load of the beam element is captured by equivalent concentrated loads at the beam ends.

The derivations above are shown in detail here to serve as a general example. For the further stress cases they will be greatly shortened.

4.2.3 Bending

Similarly to the axial force in Section 4.2.2, now the stiffness relationships for bend-ing about both principal axes y and z are now derived. For bending about the y-axisone obtains, with Figures 4.2 and 4.3 and qz = const.:

Ma za ya ya Mb zb yb yb

M z int0

W w V M w V M

w q dx W 0 (4.12)

With the relations

x yE z (4.13)

x yz (4.14)

from Eqs (1.5) or (1.4a), respectively, the internal virtual work is:

2int x x y y

0 A 0 A1

y y y y M M0 0

W dA dx E z dA dx

EI dx EI ( w ) ( w ) d(4.15)

Here y M( w ) and y M( w ) have been used. According to Section 3.5.2, the basic approach for the displacements is:

2 3 2 3M Ma ya

2 3 2 3Mb yb

w 1 3 2 w 2

3 2 w (4.16)

The fifth term of Eq. (3.31), containing qz, is not used in Eq. (4.16) since for that the internal virtual work results to zero. Because the internal virtual work according to Eq. (4.15) contains the second derivation of the displacement wM to x M( w ) , the displacement approach (4.16) must be differentiated with the help of the chain rule (compare Eq. (4.4) for axial force stresses).


With Eq. (4.16), the part of the uniformly distributed load in the virtual work results in:

1

M z z M0 0

2 2Ma z ya z Mb z yb z

w q dx q w d

w q 2 q 12 w q 2 q 12 (4.17)

The conversion from qz into equivalent nodal loads is clearly shown in Figure 4.5.Here one can obtain the shown concentrated loads and single moments since a beam fully restrained on both sides is a geometrically determinate principal system. Their directions of action do not result from the equilibrium, but from the equivalence principle.

Figure 4.5 Replacement of qz at the beam element by loads in the nodes

After the execution of the integrations in Eq. (4.15), the following stiffness relation-ship for a beam element in bending about the y-axis can be obtained:

za Ma zMa2 2 2

ya ya zya y3

zb Mb zMb2 2 2

yb yb zyb

V 12 6 12 6 w q 2w :M 6 4 6 2 q 12: EIV 12 6 12 6 w q 2w :M 6 2 6 4 q 12:

Te e e e ev : s K v p (4.18)

Since the element is of fundamental importance for bending, Table 4.1 gives an over-view of the derivation of the stiffness relationship (4.18).

The stiffness relationship for bending about the z-axis can be formulated through a comparison to bending about the y-axis:

ya Ma yMa2 2 2

za za yza z3

yb Mb yMb2 2 2

zb zb yzb

V 12 6 12 6 v q 2v :M 6 4 6 2 q 12: EIV 12 6 12 6 v q 2v :M 6 2 6 4 q 12:

(4.19)


Table 4.1 Derivation of the stiffness relationship for a beam element in bending about the y-axis (linear theory)

Beam element, internal forces/moments and deformations:

Sign definition II

Virtual work (equilibrium at beam element):

ext int

ext Za Ma ya ya zb Mb yb yb z M0

int x x y M M0 A 0

W W W 0

W V w M V w M q w dx

W dA dx EI w w dx

Deformation functions: 2 3 2 3

M Ma ya

2 3 2 3Mb yb

w ( ) 1 3 2 w 2

3 2 w2 3 2 3

M Ma ya

2 3 2 3Mb yb

w ( ) 1 3 2 w 2

3 2 w

Stiffness relationship for a beam element: 2

M Ma ya Mb ybw ( ) 6 12 w 4 6 6 12 w 2 6 2

M Ma ya Mb ybw ( ) 6 12 w 4 6 6 12 w 2 6

For W = 0 with dx = d and integration the following is obtained:

za Ma z2 2 2

ya ya zy3

zb Mb z2 2 2

yb yb z

V 12 6 12 6 w q 2M 6 4 6 2 q 12EIV 12 6 12 6 w q 2M 6 2 6 4 q 12

The virtual deformations are here omitted; see Eq. 4.18.


As for qz in Figure 4.5, the determination of the equivalent loads at the nodes is illustrated with Figure 4.6 for qy here.

Figure 4.6 Replacement of qy at the beam element by loads at the nodes

4.2.4 Torsion

For beams subjected to torsion the corresponding loads are carried by two completely different effects. Internal moments arise, which are allocated as follows (see also Table 3.3):

St Venant’s torsion with the primary torsional moment Mxp and the torsionalstiffness GITwarping torsion with the secondary torsional moment Mxs and the warping bi-moment M as well as the warping stiffness EI

For beam structures with steel cross sections the two bearing effects usually occur in combination. In practical applications for cross sections which are free of warping or which show very small warping deformations it is sufficient to consider the special case of St Venant’s torsion. Detailed explanations concerning this matter can be taken from [25], also see at the end of this section. Using Figure 4.2 and 4.3, the virtual work for a beam element subjected to torsional loadings reduce to:

a xa a a b xb b b

x int0

W M M M M

m dx W 0 (4.20)

Because the derivative of the angle of twist and the warping bimoment M act in opposite directions, the displacements at the beam ends a and b or a

and b , respectively, are used. For the internal virtual work from Eqs (1.5) and (1.4a) we have

x E (4.21)

x (4.22)

and by using the component for St Venant’s torsion according to [25], we obtain:


1 1

int T0 0

W EI d GI d (4.23)

In Section 3.5.3 it is shown that the exact solution for the rotations ( ) within the element contains the hyperbolic functions sinh and cosh:

0 1 2 T 3 T xc c c sinh c cosh f m (4.24)

The final term in Eq. (4.24), the one with mx, is not considered here since its internal virtual work is zero. In Eq. (4.24), T is the member characteristic for torsion:

TT

GIEI (4.25)

With the solution function for ( ) in Eq. (4.24), the integrations of Eqs (4.23) and (4.20) can be performed. If the integration constants c0, c1, c2 and c3 are replaced by deformations a, a , b and b , the following stiffness relation is obtained:

symmetric !

xa T T T T a xa2 2 2

a T T T a x Ta3

xb T T b xb2 2

b T b x Tb

M m 2:M m 2: EIM m 2:M m 2:

(4.26)

The parameters in the element stiffness matrix give:

T T T TT

T T T

sinh cosh2 cosh 1 sinh T T T

T T TT

T T T

sinh2 cosh 1 sinh

3T T

TT T T

sinh2 cosh 1 sinh

(4.27a-d)

Figure 4.7 Replacement of mx at beam element by torsional moments and warping bimoments at the nodes


For the conversion of mx into equivalent nodal loads Figure 4.7 is used with Eq. (4.20). Both torsional rotation (twist) and the derivative at the beam ends are restraint. Because u , the condition a b 0 means that the warping of the cross section at the beam ends is completely prevented. Torsional and warping moments result as equivalent nodal loads. For the limit case T 0 then 2 T = 12, so that the results are comparable to the loads of bending (see Figures 4.5 and Figure 4.6).

The element stiffness matrix for torsion of Eq. (4.26) is usually not used, but divided into two matrices. As shown in [93], the parameters T, T, T and T can, by ap-proximation, be replaced as follows:

T

2T

2415 T

2T

1610

T

2T

1230 T

2T

6125

(4.28a-d)

With these equations, the following matrices are obtained:

2 2 2 2T

e 3

2 2

12 6 12 6 36 3 36 34 6 2 4 3EI GIK

3012 6 36 3sym. 4 sym. 4

(4.29)

Figure 4.8 Percentage error concerning the approximations for T, T, T and T

As can be seen, the matrix for warping torsion (parameter EI ) is formally in accor-dance with the element stiffness matrices (4.18) and (4.19) for bending about the y- or z-axes, respectively. The solution can also be determined if a polynomial function


comparable to equation (3.31) is used, as in Eq. (4.76), and inserted into the virtual work according to Eq. (4.23). The approximation with both submatrices implies that a fine FE modelling is carried out.

The errors which occur with the usage of the approximations for the matrix parameters is indicated in Figure 4.8. For T 1.0 minor errors between 0.041% and

0.051% occur and therefore the parameters lead to adequate calculation results. Therefore the element lengths should be chosen so that T 1.0 is maintained, and this leads to the condition:

T T

EI I1.61GI I (4.30)

In order to get an overview, Eq. (4.30) has been evaluated for some common rolled I-sections in Figure 4.9.

Figure 4.9 Maximum element lengths for warping torsion

Special case: Pure St Venant’s torsion

For cross sections that are free of warping, I = 0 and torsional loadings are exclu-sively carried by St Venant’s torsion. If, for this load case, only the second matrix of Eq. (4.29) is used, warping bimoments will result numerically, and these are not allowed to occur according to the assumption. Furthermore, due to T numeric difficulties occur with the exact element stiffness matrix according to Eq. (4.26) when calculating the values of the matrix elements.

In Section 3.5.2, it is shown that a polynomial with two terms is able to exactly cap-ture the functional course of ( ) with regard to St Venant’s torsion. Since this applies for axial forces as well, the stiffness relationship for St Venant’s torsion can directly be formulated with the help of Eq. (4.11):


xa a xa T

xb b xb

M 1 1 m 2: GIM 1 1 m 2:

(4.31)

4.2.5 Arbitrary Stresses

For beam elements with arbitrary loadings seven deformations and internal forces/moments occur at each beam end; these are defined in Figures 4.2 and 4.3. If one combines the element stiffness matrices for the single load cases, i.e. the matrices of Eqs (4.11), (4.18), (4.19) and (4.26) or (4.29), respectively, and in doing so orders with respect to the nodes a and b, the element stiffness matrix as shown in Table 4.2 is obtained.

Table 4.2 Stiffness relation for arbitrarily stressed beam elements (without element load vector)

In the matrix in Table 4.2, the numbers in brackets indicate the numbering of the ele-ments (row, column). The value “0” means that for the matrix element, ki,j = 0. The following list is a summary of all the matrix elements that are not equal to zero and which are to be considered for the linear beam theory. Just like the overview in Table 4.3, the summary also includes components resulting from shear diaphragms S and distributed rotational springs c as well as distributed springs cv and cw (“elastic foundation”). The derivation of these terms is dealt with in Section 4.10. Concerningwarping torsion, the approximation according to Eq. (4.29) containing the submatri-ces EI and GIT is included. Alternatively, the exact solution according to Eq. (4.26) can be used, but this may lead to numeric difficulties for T 0 and T . For the limit case T Formula (4.31) is the exact solution, and for T 0 the matrix with EI of Eq. (4.29).


The Matrix elements for the element stiffness matrix Ke of beams with arbitrary load-ings according to the first order theory (linear beam theory) are as follows: k(1,1) = k(8,8) = EA/ k(1,8) = – EA/ k(2,2) = k(9,9) = 12 EIz/ 3 + 13/35 cv + 1.2 S/ k(2,3) = 6 EIz/ 2 + 11/210 cv

2 + 0.1 S k(2,6) = 13/35 cv z1 – 1.2 S/ z2 k(2,7) = – 11/210 cv

2 z1 + 0.1 S z2 k(2,9) = – 12 EIz/ 3 + 9/70 cv – 1.2 S/ k(2,10) = 6 EIz/ 2 – 13/420 cv

2 + 0.1 S k(2,13) = 9/70 cv z1 + 1.2 S/ z2 k(2,14) = 13/420 cv

2 z1 + 0.1 S z2 k(3,3) = k(10,10) = 4 EIz/ + 1/105 cv

3 + 2/15 S k(3,6) = 11/210 cv

2 z1 – 0.1 S z2 k(3,7) = – 1/105 cv

3 z1 + 2/15 S z2 k(3,9) = – 6 EIz/ 2 + 13/420 cv

2 – 0.1 S k(3,10) = 2 EIz/ – 1/140 cv

3 – 1/30 S k(3,13) = 13/420 cv

2 z1 + 0.1 S z2 k(3,14) = 1/140 cv

3 z1 – 1/30 S z2 k(4,4) = k(11,11) = 12 EIy/ 3 + 13/35 cw k(4,5) = – 6 EIy/ 2 – 11/210 cw

2

k(4,11) = – 12 EIy/ 3 + 9/70 cw k(4,12) = – 6 EIy/ 2 + 13/420 cw

2

k(5,5) = k(12,12) = 4 EIy/ + 1/105 cw3

k(5,11) = 6 EIy/ 2 – 13/420 cw2

k(5,12) = 2 EIy/ – 1/140 cw3

k(6,6) = k(13,13) = 12 EI / 3 + 1.2 GIT/ + 13/35 c + 13/35 cv z12

+ 1.2 S/ z22

k(6,7) = – 6 EI / 2 – 0.1 GIT – 11/210 c 2 – 11/210 cv2 z1

2 – 0.1 S z22

k(6,9) = 9/70 cv z1 + 1.2 S/ z2 k(6,10) = – 13/420 cv

2 z1 – 0.1 S z2 k(6,13) = – 12 EI / 3 – 1.2 GIT/ + 9/70 c + 9/70 cv z1

2 – 1.2 S/ z22

k(6,14) = – 6 EI / 2 – 0.1 GIT + 13/420 c 2 + 13/420 cv2 z1

2 – 0.1 S z22

k(7,7) = k(14,14) = 4 EI / + 2/15 GIT + 1/105 c 3 + 1/105 cv3 z1

2

+ 2/15 S z22

k(7,9) = – 13/420 cv2 z1 – 0.1 S z2

k(7,10) = 1/140 cv3 z1 – 1/30 S z2

k(7,13) = 6 EI / 2 + 0.1 GIT – 13/420 c 2 – 13/420 cv2 z1

2 + 0.1 S z22

k(7,14) = 2 EI / – 1/30 GIT – 1/140 c 3 – 1/140 cv3 z1

2 – 1/30 S z22

k(9,10) = – 6 EIz/ 2 – 11/210 cv2 – 0.1 S

k(9,13) = 13/35 cv z1 – 1.2 S/ z2 k(9,14) = 11/210 cv

2 z1 – 0.1 S z2 k(10,13) = – 11/210 cv

2 z1 + 0.1 S z2 k(10,14) = – 1/105 cv

3 z1 + 2/15 S z2 k(11,12) = 6 EIy/ 2 + 11/210 cw

2

k(13,14) = 6 EI / 2 + 0.1 GIT + 11/210 c 2 + 11/210 cv2 z1

2 + 0.1 S z22

with: z1 = zM zcv and z2 = zS zM (for distributed springs cv and shear diaphragms S)


Table 4.3 Allocation of the element stiffness matrix (first order theory) in principle

A load vector ep with 2 7 = 14 vector elements belongs to the element stiffness ma-trix of Table 4.2. Since the uniformly distributed loads qy and qz often do not act at the shear centre, but at an arbitrary point of application (yq, zq), additional corre-sponding components are considered with reference to the torsional moment:

x,ges x z q M y q Mm m q y y q z z (4.32)

Load vector ep of beam elements with 14 degrees of freedom (subscript “e” dropped):

p(1) = + qx /2p(2) = + qy /2p(3) = + qy

2/12p(4) = + qz /2p(5) = - qz

2/12p(6) = + mx,ges /2p(7) = - mx,ges

2/12 bzw. - mx,ges2/(2 T)

(4.33)p(8) = + qx /2p(9) = + qy /2p(10) = - qy

2/12p(11) = + qz /2p(12) = + qz

2/12p(13) = + mx,ges /2p(14) = + mx,ges

2/12 bzw. + mx,ges2/(2 T)

4.3 Nodal Equilibrium in the Global Coordinate System 123

4.3 Nodal Equilibrium in the Global Coordinate System

As already explained in Section 3.2, in FEM the equilibrium conditions are for-mulated at the nodes, i.e. at the joints of the beam elements. The nodal equilibrium is therefore the origin of the equation system in Section 4.5, with which structural systems are idealised for numerical analysis.

To illustrate the procedure, Figure 4.10 shows a straight beam segment, with three nodes and two elements, which is examined. Since general stress cases would lead to a confusing representation, only an axial force and the equilibrium at node k in the x-direction are examined in Figure 4.10. It is important here that all variables u, F and N are defined as positive to the right – the positive x-direction. For the internal force N this is valid for both beam ends (sign definition II) and for equilibrium reasons, the internal forces at the nodes act in the opposite direction. After a virtual displacement uK of the node k, the virtual work is:

k k xk i,k i 1,kW u F N N 0 (4.34)

Figure 4.10 Equilibrium at node k in x-direction

A central idea of FEM and the displacement method is to replace the internal forces Ni,k and Ni+1,k using the internal forces at the ends of the beams and, in doing so, to introduce the deformations as unknowns. Since the beam elements i and i+1 are connected to each other at node k, with the help of the element stiffness matrix of Eq. (4.11) we obtain:

i,k i i k 1 i i kN EA u EA u (4.35)

i 1,k i 1 i 1 k i 1 i 1 k 1N EA u EA u (4.36)


With these relations the axial forces can be substituted and Eq. (4.34) can be written as follows:

k i i k 1 i i i 1 i 1 k i 1 i 1 k 1

k xk

u EA u EA EA u EA u

u F (4.37)

Eq. (4.37) is one row of the above-mentioned system of equations in Section 4.5. With the round brackets it shows that at node k, the elements of the stiffness matrices related to uk of all adjacent beam elements are to be added. In Section 4.5.2, this is explained in detail and is clearly shown in Figures 4.35 and 4.36. The procedure shown in Figure 4.10 can be similarly carried out for different load cases and like Eq. (4.37), further equations can be formulated for all nodes of a structural system. They can be summed up to the system of equations

K v = p (4.38)

in which the number of rows and columns results from the number of nodes and the degrees of freedom at each node. For example, for the beam in Figure 4.37 the system of equations has 4 2 = 8 rows and columns which result from four nodes with two degrees of freedom (w, y) in each.

Figure 4.11 On the formulation of the equilibrium relations at the node k for uniaxial bending with axial force

In addition to Figure 4.10, uniaxial bending with axial force is considered in Figure 4.11. At node k three equilibrium relations can be formulated which correspond to thevirtual displacements u, w und y and one obtains the following conditions:

(Fxk – Ni,k – Nj,k) uk = 0 (4.39 a c)

4.3 Nodal Equilibrium in the Global Coordinate System 125

(Fzk – Vzi,k – Vzj,k) wk = 0

(MyLk – Myi,k – Myj,k) yk = 0

The procedure described so far is only correct as long as the structural system is a straight beam without any sharp bends. If a structural system includes beams with different directions, additional transformations into the global coordinate systemmust be carried out. For a general approach the system of equations is

K v p , (4.40)

in which the overbar indicates the reference to the global coordinate system. To ex-plain the issue, Figure 4.12 is given, where the stress case “bending with axial force” on the x-z plane is considered.

Figure 4.12 On the formulation of the nodal equilibrium in the global X-Z COS

The deformations in vector v and the loads in vector p always refer to the fixed directions of the global coordinate system. Therefore, the loads at node k in Figure 4.12 correspond to the directions X and Z. As can be seen, the adjacent beam elements with the local axes x and z have different directions and the internal forces/moments, which result from the stiffness relationship in Section 4.2, do not comply with the directions of the loads. On the right-hand side, Figure 4.12 shows that the local internal forces/moments have to be transformed into the global coordinate system for the formulation of equilibrium relations. The figure also con-tains the corresponding transformation relationships for the internal forces/moments. In Section 4.4, further transformation matrices are given and it shows how the


stiffness relationship Eq. (4.1) for beam elements has to be transformed into the global coordinate system.

The preceding remarks can be stated and extended to other load cases. For biaxial bending with axial force and warping torsion, the following equilibrium relations result at each node:

Xk k kF N u 0

Yk Yk kF V v 0

ZLk Zk ZkM M 0

Zk Zk kF V w 0

YLk Yk YkM M 0

XLk Xk kM M 0

Lk k kM M 0

(4.41a g)

The equations (4.41a g) capture the equilibrium at the node k of the loads acting and the internal forces/moments (with reference to the global coordinate system) of all beams connected at the node k. However, generally speaking, for each unknown nodal displacement an equilibrium condition has to be formulated. If nodes includes hinged connections of beams, corresponding equations have to be added. The consideration of hinges is covered in Section 4.11, partial hinges in conjunction with warping in Section 4.4.5. As shown at the beginning of this chapter, the internal forces/moments can be replaced by the displacements of the nodes using the element stiffness matrices.

4.4 Reference Systems and Transformations

4.4.1 Problem

In FEM, beam structures are discretised using finite beam elements. According to Section 3.2, the stiffness relationships for the beam elements are formulated with ref-erence to the local x-y-z coordinate system for that purpose; here the x-axis is the longitudinal axis of the beam elements and y and z are the principal axes of the cross section. To formulate the equation system in Section 4.5, the single beam ele-ments have to be transformed into a global X-Y-Z coordinate system. Since the transformation of the deformations is a very important task, it is shown in Figure 4.13, where the global coordinate system is shown on the left and the local coordinate

4.4 Reference Systems and Transformations 127

system for a beam element on the right. The figure shows a local COS where the direction of the y-axis is in agreement with the global Y-axis and the x-z plane is rotated in comparison to the X-Z plane. For straight beams (without sharp bends and steps), for which all beam elements can be described using an x-y-z COS, transformations are not necessary. The local COS can then be used as global COS as well, and the displacements of beams connected in a node are equal (for example wM,left = wM,right).

Figure 4.13 Displacements in global and local coordinate systems

The cross sections of a beam structure are of vital importance with regard to the re-quired transformations. Since the displacement in the longitudinal direction of a beam element refers to the centre of gravity S, and the displacements vM and wM cor-respond to the shear centre M, Figure 4.14 differs for the cross sections type A and B:

Type A: The shear centre is located at the centre of gravity. Since all variables refer to one point, the transformation relationships are simpler.

Type B: The shear centre is not located at the centre of gravity. This case re-quires more extensive transformations.

Also, the orientation of the principal axes has to be considered when different cross sections are combined.

For further explanations, consider the single-span beam of Figure 4.15, which consists of two doubly-symmetric I-sections, connected in such a way that the upper edges are flush.


Figure 4.14 On the position of the principal axes, the centre of gravity and the shear centre for standard cross sections

Figure 4.15 Single-span beam with an erratically nonuniform cross section


Usually, an idealisation as shown for case A is applied when the stress due to the bending of the beam in Figure 4.15 is to be analysed. In doing so, the mismatch of the neutral axes is neglected since it does not have an influence on the deflection and the internal forces/moments. However, if an additional longitudinal force Fx is acting, the mismatch has to be considered and then the beam has to be regarded as a beam structure. With idealisation B the mismatch of the neutral axes as well as the actual location of the bearings and the position of the load application are considered. This is of special importance for the analysis of lateral torsional buckling. For that case, information concerning the bearing and stiffness related to torsion are, of course, missing in Figure 4.15.

Figure 4.16 Hollow cross sections with openings

Figure 4.16 shows beams consisting of hollow cross sections. In some situations, beams are often supplied with openings in order to have access to the hollow space. Examples are poles of wind turbines or of tall illumination equipment. Due to the opening, the centre of gravity of the cross section shifts a little to the left and the shear centre is now located outside the cross-section. The load-bearing behaviour of the tube, which as it generally known is free of warping, changes markedly. In the area of the opening, warping torsion has to be taken into account.

Figure 4.17 Beam with U- and L-section


A further example, which ties in with the beam of Figure 4.15, will put the focus on different directions of the principal axes. The straight beam of Figure 4.17 consists of a U-profile for which the upper flange had to be removed in the right third of the beam length for structural reasons. As shown with the sketches of the cross section, this leads to a strong shift of the centre of gravity and of the shear centre. Also, the principal axes y and z show different orientations for the two cross sections. The change of cross sections leads to a deformation which is not only composed of the displacement w and the torsional rotation , but also of a displacement v per-pendicularly to the drawing plane.

Figure 4.18 On the transformation of displacements at joints of frames

Beam elements with different local x-directions and reference axes through S and M are shown in the context of the plane frame of Figure 4.18. Figure 4.18a shows a frame for which all beams consist of I-sections. According to Figure 4.14, the shear centre and the centre of gravity are then located in the same spot. If the displacements at the left joint (beam to column connection) for the adjacent beam elements are considered for this case, the deformations in the global COS are 2 1u u w and

2 1w w u (see sketch top right). These are deformations with respect to S and M, respectively. Compared to that, Figure 4.18b shows the frame with U-profiles where displacements of the centre of gravity and the shear centre have to be differentiated. With regard to the global deformations, Figure 4.18b leads to S2 M1u u w and

M2w w S1u . It should be noted that the displacements with respect to S and M are equalised, even though they are not linked to each other in the stiffness relationship of the elements. The connection in the nodes has to be dealt with by using transformation relationships.


With the previous explanations we have looked at cases that are of importance for different reference systems and the corresponding transformations. If problems like these are to be analysed using FE-programs, we have to ask the question whether the program is suitable for this task. A general solution for structural problems in civil engineering is rarely required. If necessary, we can proceed as follows:

Finite elements are used which refer to arbitrary reference points and arbitary coordinate systems. This approach is discussed in more detail in Section 4.4.6.

In the cross section plane, transformations are executed, which provide a link between the different reference points (S, M) and the principal axes. This approach is, for example, suggested in [74].

In Sections 4.4.2 and 4.4.3 plane and special transformation relationships for beam elements are set up. In Section 4.4.4 the transformation of loads is discussed, and Section 4.4.5 deals with the derivative of the angle of twist of the local x-axis for which particular advice is needed in order to consider it in the global coordinate system. The topic of transformation also includes the coverage of the boundary conditions which apply to the local x-direction or eccentric points regarding S and M. Further comments related to this can be found in Section 4.5.4 on geometric boundary conditions.

4.4.2 Beam Elements in the X-Z Plane

Besides straight beams and columns, plane structures often appear in engineering practice. Generally, the X-Z plane is chosen for the acquisition of their geometry and dimensions. Figure 4.19 shows a beam element with the local coordinates x and z and its position in the X-Z coordinate system. With the measurements X and Z, the element length as well as sin and cos can be calculated. In this Section, only beam elements are considered for which the local x-z coordinates comply with the X-Z plane, meaning that they are in that plane.

Figure 4.19 Beam element in the X-Z plane


Therefore, the direction of the local y-axis corresponds to the direction of the global Y-axis. If the principal axes y and z have different directions (rotation about the x-axis), as shown in Figure 4.17, the transformation relationships have to be determined according to Section 4.4.3.

As explained in Section 3.2, we have to distinguish between internal forces/moments and deformations relating to local and global coordinate systems. Figure 4.20 contains a compilation of the necessary transformation relations, which can be formulated in matrix notation as follows:

v T vs T s

cos sinwith : T

sin cos(4.42a, b, c)

T

T

v T v

s T sT cos sin

with : Tsin cos

(4.43a, b, c)

The inverse of T would actually have to be used instead of the transposed transforma-tion matrix TT (if converted formally), which is T -1. However, as is known, it is

T -1 = TT (4.44)

since the transformation matrix is orthogonal.

Figure 4.20 Transformation relationships for displacements and internal forces/moments in the X-Z plane

Transformation relationships for beam elements in the X-Z plane

Using beam elements, the displacements and internal forces/moments at both nodes must be transformed. The corresponding relationships result directly from Eqs 4.42 and 4.43. For the internal forces of uniaxial bending with axial force we obtain:


aa

zaza

yaya

bb

zbzb

ybyb

e e e

NN cos sinVV sin cosMM 1NN cos sinVV sin cosMM 1

s T s (4.45)

In this equation the local internal forces are transformed into the global coordinate system. Because the directions of the internal forces/moments correspond to those of the deformations, matrix Te is also valid for the deformations. The transpose T

eT has to be used as transformation matrix for the transformation of global variables into local ones. Figure 3.4 in Section 3.2 contains the relationship

Tv T v (4.46)

with all the necessary details.

For biaxial bending with axial force and torsion the elements of the transformation matrix are arranged slightly differently in comparison to Eq. (4.45). If the internal forces and deformations are arranged in the order shown in Table 4.2 (Section 4.2.5), the upper left quarter of the transformation matrix is:

(4.47)

This equation contains a question mark, to point out that the derivative of the angle of twist and the warping bimoment are not vectors and hence cannot be transformed in a formal manner. Section 4.4.5 will deal with the transformation of these variables, since the construction design of corners and sharp bends is of great importance in this context.


Transformation of element stiffness relationships

According to Section 4.2.2, the stiffness relation for a beam element is: T T Te e e e e e ev s v K v v p (4.48)

This results from the demand for equilibrium at a beam element with the use of virtual work, i.e. is We = 0. All variables of Eq. (4.48) refer to the local x-z co-ordinate system of the considered element and must be transformed into the global X-Z coordinate system as explained in Section 3.2.

If Eq. (4.43a) is used for the transformation of the element displacements, we obtain T

e e ev T v (4.49)

and for the virtual displacements: T Te e ev v T (4.50)

ve and Tev can be inserted into Eq. (4.48) and the following stiffness relationship

results:T T Te e e e e e ev s v K v v p (4.51)

with: e e es T sT

e e e eK T K T

e e ep T p

Now all variables of Eq. (4.51) refer to the global coordinate system, i.e. es and epcontain internal forces/moments and nodal element loads in the X- and Z-direction.

eK is the element stiffness matrix, which can, as a result of the transformations, be directly added to the total stiffness matrix of the system, also see Sections 3.2 and 4.5.2.

4.4.3 Beam Elements in a Three-Dimensional X-Y-Z COS

The position of beam elements is described by the coordinates of the starting node and the end node. In doing so, the x-axis, i.e. the beam axis or the longitudinal axis, is distinctly defined and the element length can be determined, as shown at the top of Figure 4.21. Moreover, the position of the x-axis can also be described using the an-gles and .


Figure 4.21 Beam element in the X-Y-Z coordinate system

For the clear definition of a beam element the direction of the principal axes have to be defined in addition to the x-axis. For that purpose, an additional point or angle can be used. In [93], a point H (XH, YH, ZH) is chosen which is located in the x-y plane, but not on the x-axis itself. Here, the direction of the principal axes is defined with the help of the angle explained in Figure 4.21 at the bottom. The projection of the beam element on the X-Y plane is notionally used to obtain the identified position

XY , as shown in Figure 4.21 (top). If the principal axis z of the beam element is in compliance with the global Z-axis in the projection, it is = 0. Other directions can be described with the angle as shown in Figure 4.21 at the bottom. As reference axis, y* is defined, which is located in the global X-Y plane.


The axial force N and the displacement u are vectors whose direction is in compli-ance with the local x-axis. Due to the proportionality, they can directly be decomposed regarding X, Y and Z:

N N N X , yV N N Y , zV N N Zu u u X , v u u Y , w u u Z

(4.52a)

Figure 4.22

For the determination of the global displacements, which result from the displace-

4.22, the following relationships can be stated:

u v v cos sin sin sin cosv v v cos cos sin sin sinw v v sin cos

(3.52b)

u w w sin sin cos sin cosv w w sin cos cos sin sinw w w cos cos

(3.52c)

The transformations for the rotations about the X-, Y- and Z-axis can be formulated in an analogous manner. Since formally identical relationships result, these are not stated explicitly here.

ments v and w, more extensive geometric examinations are necessary. With Figure

Transformation of the displacements v and w


The result is the following transformation relationship for the displacements and ro-tations in a node:

z z

y y

k k k

u u0 0 0v v T 0 0 0w w0 0 0

0 0 00 0 0 T0 0 0

v T v (4.53)

The transformation matrix T in Formula (4.53) is given with Eq. (4.54), where the trigonometric functions have been replaced by the following relations:

XY

Xcos ,XY

Ysin , XYcos , Zsin

XY XY XY XY

XY XY XY XY

XY XY

X Y X Z Y X Zcos sin sin cos

Y X Y Z X Y ZT cos sin sin cos

Z sin cos

(4.54)

The transformation matrix of Eq. (4.54) can also be derived, if for each of the spatial planes a transformation matrix comparable to Eq. (4.45) is formulated and afterwards combined by multiplication – see [93] for example. In doing so, the sequence of the multiplication and the definition of the angles have to be considered.

It is obvious that the transformation matrix cannot be determined for beams which show a longitudinal axis parallel to the Z-axis ( = ± /2) since in that case XY = 0. If, in this case, it is assumed that for = 0 the y-axis of the beam is in compliance with the global Y-axis, the following transformation matrices can be stated:

0 sin cos: T 0 cos sin

21 0 0

, 0 sin cos

: T 0 cos sin2

1 0 0(4.55)

With the help of the matrix TK according to Formula (4.53), the transformation relationship Te for the variables at the beam element can be arranged in an analogous manner, as shown in Section 4.4.2, here the order of the variables has to be con-


sidered. All relationships in matrix notation are similarly valid for beam elements in plane and space and can therefore be used as shown in Section 4.4.2. Section 4.4.5 deals with the transformation concerning the warping bimoment and the derivative of the angle of twist.

4.4.4 Loads

With FEM, the loads are assigned to the coordinate systems as follows:

According to the agreement, single (concentrated) loads act at nodes (see Section 4.9.1 as well) and their directions of action have to comply with the global X-Y-Z COS. Distributed loads are handled as “element loads” and they therefore refer to the local x-y-z COS of the beam elements on which they act.

As an example, Figure 4.23 shows a plane frame. The concentrated loads Fx and Fzact in the X- and Y-directions as defined above, and the distributed loads in the local x-z COS. If that is not the case, corresponding transformations have to be executed – see Figure 4.23 on the right.

Figure 4.23 Example of a plane frame with global and local loads and decomposition of loads

For calculations using FEM not only do the directions of the loads have to correspond to the axes of the coordinate systems, but they also have to act at the reference point, which is the centre of gravity S or the shear centre M. Since internal forces/moments correspond to the loads, the assignment to the reference points can be taken from Figure 4.3 of Section 4.2.1. The transformation of eccentric concentrated loads FYund FZ is shown in Figure 4.24. Since these loads are transferred to the shear centre, an additional torsional moment results as loading. When relocating an eccentricconcentrated force FX, three additional loads occur (see Figure 4.25). Besides the


moments MYL and MZL as loads, a warping bimoment M L arises as well, if the warping ordinate is not equal to zero at the point of load application. Such a warping bimoment as load also has to be considered when FX acts at the centre of gravity where S 0. This applies for example to Z-profiles.

Figure 4.24 Relocation of eccentric concentrated loads FY and FZ to the shear centre and the resulting torsional loading moment

The transformations described have to be applied analogously for eccentric distrib-uted loads. Moreover, the influence of eccentric loads has to be considered for second order theory see Section 5.5.

Figure 4.25 Additional load moments due to the transfer of an eccentric concentrated load FX to the centre of gravity

4.4.5 Warping Moment and Derivative of the Angle of Twist

Translations and rotations as well as internal forces and internal moments are vecto-rial variables which can be divided into components and can therefore be transformed into different reference systems. For the warping bimoment and the derivative of the angle of twist, this does not apply. For that reason, additional thoughts are necessary with regard to the nodal equilibrium, a transformation into the global COS and the inclusion in the total stiffness matrix.


)

Figure 4.26 Stresses in I-cross sections due to Mxs and M (pure warping torsion), [25]

As will be shown later, the constructional design of the nodes, i.e. the nature of the connections, is of crucial importance. However, first of all, some general explana-tions will be given.

Figure 4.27 Explanation of the warping torsion of a cantilever, [25]


Figure 4.26 taken from [25] shows a doubly symmetric I-cross section with internal forces M and Mxs as well as the resulting stresses. The secondary torsional moment can be replaced by local shear forces Vg = Mxs/ag at the flanges, so that the division of shear stresses shown is quite plausible. For the warping bimoment a division of shear stresses emerges which complies with the impact of local transverse bending moments Mg = M /ag in the flanges. A similar circumstance is shown in Figure 4.27, but, here, we have to explain how the warping bimoment arises in a structural system. If the cantilever, which is rigidly fixed on the left side, is loaded by a torsional moment at the free end, it can be replaced by a couple F = MxL/ag at the flanges. With this approach, the structural condition of pure warping torsion (GIT = 0) is covered. Due to couple F, a transverse bending emerges, which acts in opposite directions in the flanges and leads to local shear forces Vg and bending moments Mg. It is easy to recognise that for the warping torsion, in addition to shear stresses (as a result of Mxs)axial stresses must occur. They are combined as the internal force “warping bimoment M ”. With Figure 4.26, the correlation Mxs = M is easily understandable since relationships such as y zM V and z yM V are generally known: the shear forces are the first derivatives of the bending moments.

Loading the cantilever through the couple leads to a mutual displacement of the flanges in the transverse direction. This is clarified in Figure 4.28 through the displacements vo and vu, but would lead to a change of the cross section shape. With the basic assumption of the beam theory that the cross section shape does not change the cross-section rotation (twist) results. In addition to that, Figure 4.28 outlines the warping of the cross section at the end of the cantilever. It results from the local bending in the flanges according to Figure 4.27 (v at the top to the left, at the bottom to the right). With the preceding explanations, the following findings result for beam ends (with and without bearing):

Torsional restraint 0 ; Mx and M 0

The derivative of the angle of twist is the alteration of the rotation in the longitudinal direction of the beam. It is equal to zero if the warping according to Figure 4.28b is completely prevented.

Free beam ends Mx = M = 0, provided that the loads MxL and M L do not act there.

and 0

Fork bearing = M = 0 and Mx 0


Figure 4.28 Rotation and warping of the cross section at the free end of the cantilever in Figure 4.27

Figure 4.29 Warping springs C due to end plates, flat stiffeners, hollow stiffeners and beam extensions

The constructional form of beam ends has a major influence on the internal forces and displacements of the torsion. With the end plate in Figure 4.29a, the warping of the end cross section is constrained. Thus, with regard to the warping bimoment, it functions like a partial restraint. For calculations with FEM the plate can be replaced by a warping spring, which is defined as follows:


M C C M (4.56)

Flat and hollow stiffeners between the flanges have an effect which is similar to the effect of the end plate. Beam extensions likewise lead to a constraint of the warping – also see [56]. At the bearing, warping bimoments occur, which are reduced towards the beam end.

In the following, several nodes are considered in which beams are connected with each other. At each beam end local variables M and occur, which have to be coupled together appropriately at the node. Figure 4.30 shows three variants of a beam intersection, which are constructionly not practicable, but they are helpful for the understanding. For variant A the flanges of the beam cross sections end in front of the node and only the webs are completed and connected to each other. These are practically free beam ends, and all four warping bimoments are equal to zero. The four derivatives of the angle of twists are independent of each other, so that they have to be considered separately in the total stiffness matrix. However, overall, the node is very weak and there will hardly be cases where it is conducted this way.

For variant B in Figure 4.30 the flanges are connected to each other and the local bending moments as a result of M can compensate each other and are immediately in equilibrium. In this case, it is sufficient to assume two derivatives of the angle of twist for the nodes and the values for the total stiffness matrix can be added up at the corresponding positions. The web of the interrupted beam is, naturally, to be added and both webs act, similar to the flat stiffeners in Figure 4.29b, like weak warping springs.

Figure 4.30 Three variants of a beam intersection (top view, I-cross sections)


In variant C the flanges are connected to each other as in variant B, but the node is additionally stiffened. The stiffeners are arranged in such a way that, as in Figure 4.29c, hollow stiffeners emerge. If they are designed so that they are strong enough, a warping restraint can be assumed in the node for all four beam ends. Then, it is suffi-cient only to assume one derivative of the angle of twist for the node. At the corresponding position in the total stiffness matrix the boundary condition 0 can be considered or the values from the element stiffness matrices and the warping spring C can be added up. In connection with Figure 4.30, refer to [93] where the node of a girder grid is examined in detail.

Another design variation is shown in Figure 4.31. Here, a transverse beam is con-nected flexibly to the continuous longitudinal beam, according to variant A in Figure 4.30; this complies with a warping hinge. In the total stiffness matrix, two independ-ent derivatives of the angle of twist are then to be considered for the node: for the transverse beam and for both ends of the longitudinal beam.

In construction practice, often frame corners occur in which I-profiles or welded I-sections are connected to each other, as in Figure 4.32. The frame corner without stiffening, i.e. welded via bevel cut, is relatively weak and can only transfer small bending moments. The indicated warping bimoments, again, dispersed into flange bending moments, cannot compensate in the corner. They act at the joints of the flanges (beam to column) as local torsional moments and cause cross section defor-mations. However, the bending stiffness of the webs is usually so small that it can be neglected. The outcome of this is a warping hinge and M = 0 at both beam ends.

Figure 4.31 FEM idealisation of a hinged joint (transverse beam to longitudinal beam)


Figure 4.32 On the transfer of warping bimoments in frame corners

For the frame corner with an inclined plate two effects result: the plate functions, like the end plate in Figure 4.29, as warping spring and the resulting flange moments can equalise in the inclined plate between top and bottom flange. Therefore, M is not equal to zero and it is sufficient to only arrange one unknown derivative of the angle of twist for this node in the total stiffness matrix.

The frame corner with the extended flanges complies with the common construc-tional design, but the height of the frame corners is frequently increased by the arrangement of haunches. For the frame corner design in Figure 4.32 the extended flanges function like flat steel stiffeners or end plates which constrain warping. A possible model for the determination of warping springs C is shown in Figure 4.32 on the right. Here, the M of the waler is carried by the perpendicular flanges and the M of the strut by the horizontal ones. Because waler and strut are considered as beam ends being independent of each other, two derivatives of the angle of twist – for waler and strut – are to be determined. Provided that the beam ends are very stiff, a warping restraint can also be assumed and one unknown derivative of the angle of twist in the node is sufficient. Also, an inclined plate can be arranged as an addition.

To broaden the topic, it is referred to [57] and [74]. There, frame corners are covered by taking into account cross section deformations.


4.4.6 Finite Elements for Arbitrary Reference Points

For the examination of beams it is common practice to distinguish the following four subproblems:

axial force bending about the y-axis bending about the z-axis torsion

This is possible through the standardisation of the cross section properties, which leads to reference points S and M as well as to the principal axes (y, z) and standard-ised warping ordinates , as shown in Chapter 2. Since FE calculations can also be carried out using arbitrary reference points and axes, this is not necessary. Figure 4.33 shows a corresponding approach.

Point B is the zero-point of the y-z coordinate system, here, y and z are not princi-pal axes of the cross section. The warping ordinate can also refer to the arbitrary reference point B as centre. The approach shown complies with the starting point of the standardisation of the cross section properties described – see Chapter 2. However, the standardisation is not carried out and only cross section properties are calculated which depend on y , z and . On the basis of the arbitrary reference system in Figure 4.33, stiffness relationships for beam elements can be formulated. Detailed explanations on this can be found in [71]. There, element stiffness matrices according to the first and second order theory are explicitly compiled too, which are not covered here for reasons of spaces. In comparison with the matrices in Section 4.2.5 (and Table 4.3), they are complete and the subproblems mentioned above are linked. In [71] and [44] the stiffness matrices have been used for beams in terms of the expansion of plastic zones. They can also be used for the solution of the problems stated in Section 4.4.1, because for all cross sections one reference point B and oneposition of the y-z COS can be retained.

Figure 4.33 Arbitrary reference system for cross sections

)

4.5 Systems of Equations 147

4.5 Systems of Equations

4.5.1 Aim

Using the FEM and a calculation with the displacement method, the equilibrium rela-tions in the nodes are formulated according to Section 4.3 and structural systems are replaced by equation systems. According to first order theory (linear beam theory), the following equation system arises:

K v p (4.57)with: K Total stiffness matrix

v Vector of all displacements in the nodes p Vector of the loads of the system

The overbar indicates that all variables refer to a global coordinate system. The required transformations can be executed with the calculation operations given in Section 4.4. For straight members (beams, columns) the transformations are unnecessary and the designation with the overbar can be renounced. The aim of the following Sections is the set-up of the equation system according to Formula (4.57).

4.5.2 Total Stiffness Matrix

In order to formulate the total stiffness matrix, the stiffness matrices of each beam element have to be determined according to Section 4.2. The necessary element type depends on the stress to be analysed. With regard to the arrangement of the matrix elements in the matrix K , transformations of the local element coordinate systems into the global coordinate systems according to Section 4.4 might be necessary. To understand what is happening it is also helpful to subdivide the element stiffness matrices into submatrices assigned to the nodes – see Figure 4.34.

Figure 4.34 Submatrices of the element stiffness matrix

As an example, an elementation of a structural system with seven nodes is considered in Figure 4.35 and a beam element with the node a = 3 at the beginning and node b = 6 at the end is integrated into the total stiffness matrix. As can be seen, only a simple arithmetic statement is needed and the loads can be assigned without any problem.


Figure 4.35 Allocation of a beam element in the total stiffness matrix and load vector

Since element stiffness matrices are always symmetric to the main diagonal, this also applies to the total stiffness matrix. For that reason it is sufficient to assign the matrix elements located on and above the main diagonal. In addition, for many structural systems a band structure arises, which is distinct for continuous beams where the nodes are numbered from left to right throughout the beam. Figure 4.36 shows a beam with nine nodes as an example (without bearings or loadings). The matrix elements of the finite elements i and j, which are connected at the node k, are as-signed. The hatching shows that at the node k the matrix elements of the two beam elements are added – also see Section 4.3. The band width measured from the main diagonal to the right border of the band corresponds to the number of degrees of free-dom related to the chosen beam element.

Figure 4.36 Band structure of a total stiffness matrix for beams with serially numbered nodes


4.5.3 Total Load Vector

Distributed loads qx, qy, qz and mx are always assigned to the beam elements and they are therefore first of all considered in the local COS. In order to arrange a total load vector, they are then transformed into the global COS according to Formula (4.51). The positioning of the vector elements of ep can be taken from Figures 4.34 4.36.

Single (concentrated) loads, as for instance Fx, Fy and Fz, and single load moments always refer to the global COS. It is assumed here that they only act at nodes. For that reason, it is possible to directly integrate them into the total load vector. The corre-sponding assignment can be seen in Figure 1.4, for example. Loads which do not act in the nodes are discussed in Section 4.9.

Figure 4.37 Beam and FE modelling

Figure 4.37 shows a beam as an example of the assembly of the total stiffness matrix and the load vector. It is divided into three beam elements with four nodes and eight degrees of freedom (deformations w and y at each node). It must be mentioned that for practical engineering assignments the beam would usually be divided using a finer elementing since deformations and internal forces and moments are needed at closer intervals.

The stiffness relationships for the beam elements can be determined with the help of the explanations in Section 4.2.3. For simplification purposes, EIy = 64 106 kNcm2

is assumed. For that reason, EIy/ 3 = 1 kN/cm for element two. With Eq. (4.18), the stiffness relationship for element two leads to:


z2,2 2 z

y2,2 y2 z

z2,3 3 z

y2,3 y3 z

V w q 20012 2400 12 2400M q 40000 3640000 2400 320000V w q 20012 2400M q 40000 3symmetric 640000

2 2 2 2s K v p (4.58)

Here, the subscript “2” designates the number of the beam element. The element stiffness matrices K1 and K3 can be determined analogously since only different lengths have to be inserted into Eq. (4.18) and because of qz = 0, we have 1p = 3p = 0 for the element load vectors. The element stiffness matrices and the element load vectors have to be transformed into a global reference system according to Section 3.4 and Formula (4.51). Since the beam of Figure 4.37 is straight, the local COS can also be used in a global manner, and transformations do not have to be applied. As shown in Figure 4.36, the arrangement of the element stiffness matrices leads to a matrix with a distinct band structure:

6 6

6 6

6 6

6

96 9600 96 9600 0 0 0 01.28 10 9600 0.64 10 0 0 0 0

108 7200 12 2400 0 01.92 10 2400 0.32 10 0 0

Ksym. 108 7200 96 9600

1.92 10 9600 0.64 1096 9600

1.28 10

(4.59)

The load vector p of the system can be set up with the help of Figure 4.36 as well. Load elements of qz occur at nodes 2 and 3. The concentrated load acts at node 4 and therefore corresponds to the displacement w4. This leads to the following load vector:

z

z

z

z

z

00

q 200q 40000 3

pq 200

q 40000 3F0

(4.60)


The calculations for the beam of Figure 4.37 are continued in Sections 4.5.4, 4.6 and 4.7.

4.5.4 Geometric Boundary Conditions

After the insertion of the beam elements into the total stiffness matrix and the total load vector, the equation system (4.57) is singular since the bearings of the structural system have not been considered yet. In terms of the FEM and the displacement method, the bearings are the geometrical boundary conditions. At first, the beam of Figure 4.37 is taken as an example; it shows three geometric boundary conditions. At the restraint (left) w1 = y,1 = 0 and at node 3 w3 = 0. This means that the values in the first, second and fifth column of the total stiffness matrix are equal to zero. The corresponding columns and rows are dropped, as shown in Section 3.2 and Figure 3.5. Through the consideration of the geometric boundary conditions, a 5 5 matrix and five unknown degrees of freedom remain. A reduction like this is usually not executed for computer-oriented calculations since it causes interference with the course of calculation and demands rearrangements and a new numbering. If, alternatively, the elements of the rows and columns with the exception of the main diagonal are set to zero, the original size of the equation system can be maintained. This procedure is illustrated in Figure 4.38 by the zeros in the dashed rows and columns 1, 2 and 5. Figure 4.38 also shows the placement of the element matrices and their superposition.

Figure 4.38 Equation system for the beam in Figure 4.37 after taking into account the boundary conditions


Figure 4.39 The formulation of geometrical boundary conditions

For the formulation and consideration of the geometric boundary conditions it has to be observed that they always correspond to the global X-Y-Z COS since the dis-placements in the vector v are defined for these directions. According to Figure 4.39, three translations and three rotations can be set to zero in one node. Due to the fact that the derivative of the angle of twist is not a vector variable, it cannot be transformed with the help of trigonometric functions. It has to be analysed separately, as shown in Section 4.4. If the warping is completely prevented at a beam end, the boundary condition 0 can be applied.

Conditions for bearings in the local coordinate system

In structural systems bearings are often used for which the displacements are not dis-abled with respect to the direction of a global axis, but in a different direction. Figure 4.40 shows an example with the bearing on the right side. The condition uk = 0 at the node k cannot be considered in the equation system directly since the direction nei-ther corresponds to ku nor to kw . Using the transformation relationship of Figure 4.20 (Section 4.4) a side condition can be formulated, which is shown in Figure 4.40.

Figure 4.40 Condition for a bearing in the local COS

4.6 Determination of the Deformations 153

Undisplaceable points of support in the cross section

For beams, it must be pointed out that the longitudinal displacement uS corresponds to the centre of gravity and the lateral displacements vM and wM refer to the shear centre. With the conditions for bearings uS = 0, vM = 0 and wM = 0, the named points are therefore fixed unmovably. This circumstance is of importance for the cross sec-tion and the loading in Figure 4.41, for example. The boundary condition wM = 0 means that the shear centre cannot be displaced downwards. Constructionally, this can only be realised in exceptional cases though. It is more likely that the support is located in the area of the web. Since the load does not act in the shear centre, torsion occurs and a calculation with wM = 0 leads to wrong results. For that reason an ap-propriate side condition has to be formulated. From Formula (1.3), with w = wM + (y – yM) , the side condition given in Figure 4.41 can be obtained. A different ap-proach is to first of all set wM = 0 as boundary condition for the calculation and then to apply a torsional moment “reaction force (yweb – yM)” for the purpose of correc-tion.

Figure 4.41 Boundary conditions wM = 0 and wweb = 0 for torsionally loaded U-sections

Actually supported points in the global COS

With the transformation of beams in a global COS, as for instance for plane frames, stiffness components are added which correspond to local displacements uS and wM.If the points S and M of the cross section are not located at the same spot, it is unclear what the boundary conditions ku 0 or kw 0 are supposed to mean (also see Section 4.4.1). The question is, which point of the section is held unmovably?

4.6

tions in Figure 4.38 can be solved. Formally, the unknown deformations at the nodes come from the following matrix equation:

After considering the geometric boundary conditions (bearings), the system of equa-

Calculation of the Deformations


1v K p (4.61)

However, in many applications, the determination of the inverse of K is computationally intensive, so that usually time-saving methods are used for the solution. Generally known are the Gaussian algorithm, and for systems of equations with symmetric matrices the Cholesky method. The topic of the solution of the equation systems is dealt with in Chapter 6. There, the GAUCHO method is proposed for the calculation of deformations in the vector v . The name originates from the combination of Gauss and Cholesky. It is perfectly suitable for the solution of symmetric systems of equations with band structure and, furthermore, also for the determination of eigenvalues. For the beam of Figure 4.37 and the corresponding system of equations in Figure 4.38 we obtain the following deformations apart from w1 = y1 = w3 = 0:

2 z zw 5.864 q 0.02083 Fy2 z z0.03935 q 0.0001563 F

y3 z z0.0625 q 0.0004688 F

4 z zw 12.50 q 0.1354 Fy4 z z0.0625 q 0.0007813 F

(4.62)

With that, the deformations at the nodes are known. For the determination of the deformations within the element, Eq. (3.31) – the solution of the differential equation for the bending about the y-axis – can be used. With the nodal deformations, the displacement wM can directly be calculated at an arbitrary point f the element. In the context of linear beam theory with infinite shear stiffness, also for other problems the variables within the element can similarly be determined with the functions of the deformations, e.g. for axial force, bending about the y-axis or torsion (see Section 3.5).

4.7 Determination of the Internal Forces and Moments

After solving the equation system (4.57), the deformations at the nodes are available as a result of the calculation explained in Section 4.6. In order to determine the internal forces and moments, all beam elements are now considered again and Eq. (4.1) is applied:

se = Ke ve ep (4.63)Since the deformations at the nodes are now known, they can be assigned to the beam elements, i.e. ve, and the internal forces and moments at the beam ends se can easily be determined by matrix multiplication. However, in doing so we have to pay attention to the fact that the deformations in v of the structural system refer to the

4.7 Determination of the Internal Forces and Moments 155

global coordinates, but the deformations in ve must be inserted with respect to the local element coordinates. According to Eq. (4.46), the required transformation relationship is:

ve = TeT ev (4.64)

For the beam of Figure 4.37 the deformations v v have been determined in Section 4.6. As an example, the bending moment at the right-hand end of beam element 2 is here calculated with Eq. (4.58) and the deformations in Eq. (4.62):

y2,3 2 y2 3 y3 z

z z

M 2400 w 320000 2400 w 640000 q 40000 3

0 q 200 F(4.65)

The other internal forces at the beam ends can be calculated in a similar manner and we obtain with qz in kN/cm and Fz in kN, Vz in kN and My in kNcm: Element 1: Vz1 = –185.2 qz + 0.5 FzMy1 = 31111 qz – 100 FzVz2 = 185.2 qz – 0.5 FzMy2 = 5926 qz

Element 2: Vz2 = –1.85 qz + 0.5 FzMy2 = –5926 qzVz3 = –214.8 qz – 0.5 FzMy3 = –200 Fz

Element 3: Vz3 = – 1.0 FzMy3 = 200 FzVz4 = 1.0 FzMy4 = 0

(4.66)

The signs of these internal variables refer to sign definition II. In order to convert them into the usual sign convention of structural engineering, they have to be multiplied by 1 at the left end of the elements. Figure 4.42 shows the result of the calculations – the diagrams between the four nodes have been added.

Figure 4.42 My and Vz for the beam of Figure 4.37

The internal forces and moments within the elements can be determined as shown in Section 4.6 for the deformations using solutions of the differential equations, i.e. with the functions which describe the deformations; see Section 3.5. The correlation be-


tween the internal forces/moments and the deformations becomes apparent with Table 3.3. To determine the course of the bending moment y y MM EI w for in-stance, the displacement function for wM according to Eq. (3.31) has to be differentiated twice with respect to x. This succeeds with the chain rule dwM/d · d /dx. By using the nodal deformations at the element ends, the diagram for Mydepending on can now be specified.

4.8 Determination of Support Reactions

The reactions of the supports are the corresponding variables for the geometric boundary conditions and they can be calculated after the determination of the deformations v (Section 4.6). The procedure is described in Section 3.2 on the basis of the fundamental example for the understanding of FEM and it can directly be adopted from Figure 3.5. Since now the deformations in vector v are known, the product K v can be calculated. For this purpose, the vector p is divided into two parts

L Rp p p (4.67)

where the first part contains the loads and the second one the reactions at the supports. The support reactions can then be calculated as follows:

R Lp K v p (4.68)

In Eq. (4.68), it is sufficient to only include the rows of the calculation, which contain the support reactions, i.e. those which correspond to the geometric boundary condi-tions.With the deformations of Section 4.6 for the beam of Figure 4.37, we obtain:

Fz,R,1 = 185.2 qz – 0.5 Fz

My,R,1 = -31111 qz + 100 Fz

Fz,R,3 = 214.8 qz + 1.5 Fz

(4.69a to c)

The calculation is shown for the moment at the restraint on the left end of the girder. With the second row of the total stiffness matrix, Eq. (4.59), and the deformations w2and y2 according to Section 4.6, we obtain:

6 6y,R,1 1 y1 2 y2

z 2 z z

z z

M 9600 w 1.28 10 9600 w 0.64 10

0 0 56294 q 199.97 F 25184 q 100.03 F31110 q 99.94 F

(4.70)

The result matches Eq. (4.69b), except for a small inaccuracy due to rounding.

4.9 Loadings 157

4.9 Loadings

4.9.1 Concentrated Loads

In Section 3.2, it is assumed that concentrated loads always act at the nodes or that nodes are arranged at these places. Sometimes, it is advisable to deviate from this convention and to allow concentrated loads between the nodes. The corresponding loads are to be determined at the beam being restrained at both ends and compiled in Table 4.4 for selected load cases. These are local loads, which relate, like the distributed loads, to the beam elements – also see Figure 4.5.

Table 4.4 Equivalent loads for concentrated loads between two nodes a and b

Sign definition II

Load case Va Ma Vb Mb

F2

F8

F2

F8

2

3dF 3 2d

2

2F c d

aF - V2

2F c d

L3

6 M c d L2

M d3c L

3

6 M c d L2

M c 3d

adN F b

cN F

Distributed loads are related to the beam elements and always refer to the local COS of the element. Since only uniformly distributed loads ranging over the whole ele-ment length have been covered in Section 4.2, loads for further distributed loads are compiled in Table 4.5. These are the values for the vector of the loads ep in Eq. (4.1).

4.9.2 Distributed Loads


Table 4.5 Equivalent loads for distributed loads for the element load vector pe

Sign definition II

Load case Va Ma Vb Mb

q2

2q12

q2

2q12

3

3q d 2 d2

3

2q d 4 3d

12aq d V

3

2

2

q d

12

6 8 3dd

3 q20

2q30

7 q20

2q20

aqN2 b

qN2

4.9.3 Settlements

Settlements are scheduled or unscheduled displacements of the supports. First of all, a single settlement is considered.

For capturing settlements, one acts on the set-up of the total stiffness matrix of the structural system and inserts the numerical value for the corresponding support dis-placement. For reasons of clarification, the plane frame in Figure 3.1 (Section 3.2) is considered and it is assumed that the right support is displaced 3 cm downwards. Therefore 5w 3 cm . Since 5w corresponds to the last but one column of the total stiffness matrix (Figure 3.5), this column is to be multiplied by the value “3”. After-wards, the column is taken to the right side of the equation system as a load vector. After deleting the last but one row, which belongs to 5w , the system, reduced by one unknown, can be solved and we obtain the other deformations as solution. After that, the internal forces and support reactions can be calculated as usual; see Sections 4.7 and 4.8.

4.10 Springs and Shear Diaphragms 159

4.9.4 Influences of Temperature

The heating of building elements leads to an expansion and the cooling to a contrac-tion. Due to influences of temperature, deformations result in bearing structures and, moreover, internal forces are set-up if the deformations are constrained, which is the case for statically indeterminate structures. An important characteristic in this context is the coefficient of thermal expansion T, which is also called temperature coefficient of expansion. For structural steel it is T = 12 10-6/K, i.e. a 1 m steel beam expands by 0.012 mm for a heating of 1 kelvin.

Usually, only constant temperature ratios TN and, across the component height or width, linearly varying temperature ratios TM are examined numerically in con-structional engineering. For calculations with FEM the temperature influences are translated into equivalent loads (forces and moments). Figure 4.43 contains a compilation of the loads being needed for a beam element with bending about the y-axis and axial force. They are determined at the geometrically determinate main system, i.e. the beam element being retrained on both sides.

Figure 4.43 Temperature influences TN and TM as well as resulting loads for the element load vector ep

4.10 Springs and Shear Diaphragms

Beam structures are often supported by adjacent constructions, laterally supported or prevented in their rotations. One can differentiate:

point (concentrated) springs C distributed springs cshear diaphragms S


Point springs

In Section 3.4.2, the virtual work for concentrated loads is derived. With Figure 4.44, it can be formulated for point springs in a comparable manner. As a result, we obtain:

CW v C v (4.71)

In Eq. (4.71), v is a displacement at an arbitrary location of the structure and C is the corresponding spring constant.

Figure 4.44 Virtual work of a point spring

The point springs are first of all assigned to the single beams and hence they are captured in a local x-y-z COS. Figure 4.45 shows the three point springs Cu, Cv and Cw, which are considered to act at an arbitrary point of the cross section. With (4.71) for the virtual work, we obtain:

WC = uC Cu uC+ vC Cv vC + wC Cw wC (4.72)The displacements uc, vc and wc must now be replaced by displacements of the beam axes. This is done using the linearised relations for uc, vc and wc given in Figure 4.45, which are based on Eqs (1.1) (1.3). As an example, a point spring Cv is considered. The virtual work is:

Cv M C M v M C M

M v M M v C M v C M M2

v C M

W v z z C v z z

v C v v C z z C z z v

C z z

(4.73)

As can be seen, the eccentric spring Cv leads to virtual work components, which are connected to the displacement vM and the rotation (see also Figure 4.47). Since, according to the agreement, point springs only act at nodes (see Section 3.2), vM and

are nodal deformations the following matrix formulation constitutes the connection to the beam elements:

v v C M MCv M 2

v C M v C M

TCv Cv

C C z z vW v

C z z C z z

W v K v

(4.74)


Figure 4.45 Point springs Cu, Cv and Cw as well as corresponding displacements

This procedure can be applied in a similar manner for other point springs. Table 4.6 contains a compilation for the point springs Cu, Cv, Cw, C , C y, C z and C . Cu and Cw have only been considered with a point of application at the centre of gravity or the shear centre, respectively. C , C y and C z are rotational point springs, which cor-respond to the rotations , y and z. The warping point spring C belongs to the derivative of the angle of twist and is explained in Section 4.4.5.

For the consideration of the point springs in the total stiffness matrix, they must be transformed into the global X-Y-Z COS. In many cases, the assignment of the local springs to the global directions is directly recognisable. In a generalised procedure, the local displacements have to be transferred into global ones as shown in Section 4.4.

Table 4.6 Virtual work W as a result of point springs, distributed springs and shear diaphragms in the local COS

Point springs:S u S M w M y y y z z z

2v M M M c M c M M c M

W u C u w C w C C C C

C v v v z z z z v z z

Distributed springs:

M w M

2v M M M c M c M M c M

W w c w c

c v v v z z z z v z z dx

Shear diaphragm S: 2

z z z S M S M z S MW S z z z z z z dx


Distributed springs

The following distributed springs are considered:

Distributed translational spring cvThis acts in the direction of the local displacement v, is located eccentric to the shear centre with respect to the z-direction and is constant in the x-direction. Distributed rotational spring cThis is assumed to be constant in the x-direction and it prevents the torsional rotations (twist). c is designated as rotational bedding in DIN 18800 part 2. Distributed translational spring cwThis is assumed to be constant in the x-direction and it is considered to act at the shear centre in the direction of the local displacement wM.

Figure 4.46 Beam element with a distributed rotational spring c

The virtual work for the distributed springs is given in Table 4.6. Because these springs are assigned to the beam elements, the element spring matrices can be derived in the same way that the stiffness matrices in Section 4.2 were derived. In Section 4.2.5, there is a compilation of the matrix elements k(i,j), which already contains the springs cv, cw and c . Moreover, Table 4.3 illustrates at which positions the springs are to be considered in the element matrix. The derivation of the corresponding ma-trices is here shown for the example of a distributed rotational spring c = const. In the virtual work

cW c dx (4.75)

the rotations (x) are replaced by the polynomial function2 3 2 3

a a

2 3 2 3b b

( ) 1 3 2 2

3 2 (4.76)

see Sections 3.5.2, 3.5.3 and 4.2.4.


With = x/ and dx = d , and with virtual rotations ( ), also according to Eq. (4.76), but using the virtual nodal degrees of freedom a, a , b and b for the virtual work, we have:

1

c0

W c ( ) ( ) d (4.77)

The integration of the product ( ) ( ) can be accomplished term by term. After some intermediate calculations this leads to the following result in matrix notation:

aa xa2 2

aa a

bb xb2

bb b

Te e c ,e e

: M 156 22 54 13: M 4 13 3c

420: M 156 22: M sym. 4

v : s K v (4.78)

In this equation, the virtual nodal deformations are placed “as a reminder” in front of the rows of the four equations, as was done in Section 4.2, and the vector of the nodal internal moments is taken from Section 4.2.4; see Eq. (4.26). The transformation of the element spring matrix in order to assemble it in the total stiffness matrix is carried out as described in Section 4.4.2.

Shear diaphragms

Shear diaphragms originate for example, if trapezoidal steel sheeting or similar com-ponents are continuously connected with the construction underneath at all fouredges. Often, the compressed upper flange of girders is considered to be supported laterally in this way. The supporting effect does not rely on a disability of displacement as with translational springs, but on a decrease of the rotations.

Figure 4.47 Eccentric shear diaphragm S and translational spring Cv

According to Figure 4.47, eccentric shear diaphragms S in the z-direction relative to the shear centre are considered since this covers practically relevant cases. Therefore, in the virtual work the rotations z and the derivative of the angle of twist occur as


shown in Table 4.6. The element shear diaphragm matrix can be obtained as for a distributed spring cv. All matrix elements due to shear diaphragm S are in the compilation of the k(i,j) in Section 4.2.5 and the correlation to the deformations and internal moments can be seen in Table 4.3. The transformation into the global coordinate system (for the assembly in the total stiffness matrix) is done as for distributed springs in correlation with the element stiffness matrices – see Section 4.4.2.

When considering point and distributed springs as well as shear diaphragms, forces are associated which are of interest for the design of the structural bracings. As already shown in Figure 4.44, spring forces are connected with the deformations by spring stiffnesses. In Table 4.7, the relationships are compiled with which the spring and shear diaphragm forces can be determined using the displacements, which are determined for the nodes after solving the system of equations (4.57).

Table 4.7 Calculation of spring forces

Point springs C:

Cu u SF C u= ⋅ Cw w MF C w= ⋅ eccentric: ( )Cv v M Cv MF C v z z⎡ ⎤= ⋅ − − ⋅ ϑ⎣ ⎦

CM Cϑ ϑ= ⋅ ϑ C y y yM Cϕ ϕ= ⋅ ϕ C z z zM Cϕ ϕ= ⋅ ϕ CM Cω ω ′= ⋅ ϑ

Distributed springs c:

ϑ ϑ= ⋅ ϑcm c = ⋅cw w Mq c w eccentric: ( )cv v M cv Mq c v z z⎡ ⎤= ⋅ − − ⋅ ϑ⎣ ⎦

Shear diaphragm S: eccentric: ( ) ′⎡ ⎤= ⋅ ϕ − − ⋅ ϑ⎣ ⎦S z S MV S z z

4.11 Hinges

In beam structures, hinges often have to be considered. For plane structures these involve axial force, shear force and moment hinges. For three-dimensional structures it may also involve torsion and warping hinges. In principle, the finite element method offers two possibilities for considering these hinges:

• introduction of additional nodal degrees of freedom• reduction of the element stiffness matrix

Introduction of additional nodal degrees of freedom

If several beams or finite elements act at one node, the deformations of the beam ends are linked to each other at the nodes by transition conditions. The introduction of a hinge at a beam end has the effect that the deformation of the beam, which corresponds to the hinge, is independent of the equivalent deformation of the other beam ends acting at the node. For calculation with finite elements this can be realised through the introduction of an additional degree of freedom at the node. Figure 4.48

4.11 Hinges 165

clarifies this circumstance giving the example of a moment hinge. On the left of the figure, three beams are rigidly linked to each other at the node k. All three beam ends rotate according to node rotation. This is described with the help of the nodal deformation Ykϕ . On the right element 3 is joined with a hinge to the other beams. Thus, the beam end must be able to rotate independently of the other beams. This is achieved through the introduction of an additional degree of freedom at the node. While the rotation of the beam ends of element 1 and 2 is captured by the nodal rotation Ykϕ , an independent nodal rotation Yk,3ϕ is applied for element 3. Accord-

ing to Figure 4.48, an additional degree of freedom has to therefore be considered at node k. The nodal equilibrium according to Section 4.3 is thus extended correspond-ingly, and for the setup of the equation system according to Section 4.5 a further equilibrium relationship with the corresponding row and column is added. Therefore an additional nodal load at the hinge can be included in the calculation as well.

Figure 4.48 Transition condition and degrees of freedom of the node k with and without considering a beam end hinge

Reduction of the element stiffness matrix

As already explained, the consideration of a beam end hinge leads to the fact that the corresponding deformation can adjust independently. Instead of defining an additional nodal degree of freedom to do so, the stiffness relationship of the element that is to be joined as a hinge is changed in a way that it is independent of the nodal deformation belonging to that hinge. To put it another way, the degree of freedom corresponding to the hinge is eliminated from the stiffness relationship. This process of eliminating degrees of freedom is also called static condensation. As an example,


the reduction of the element stiffness relationship when considering a moment hinge at the beginning of the beam element is shown below. On the basis of Eq. (4.18) for the bending about the y-axis with the condition Mya = 0, the following is the relationship for the deformation which is to be eliminated (second row of the stiffness relationship):

2y 2 2

ya Ma ya Mb yb z3

EIM 6 w 4 6 w 2 q 0

123

zya Ma Mb yb

y

3 3 1 qw w2 2 2 48 EI (4.79)

With this relationship, the rotation of the node at the beginning can be described with the other degrees of freedom of the element. In matrix notation, the following equa-tions of the stiffness relationship in Eq. (4.18) remain:

MaMa za z

yayMb zb z3

Mb2 2 2yb yb z

yb

ww : V 12 6 12 6 q 2

EIw : V 12 6 12 6 q 2

w: M 6 2 6 4 q 12

(4.80)

If in Eq. (4.80) the relationship (4.79) is introduced for the deformation ya of the displacement vector, the rotation of the node at the beginning of the element is de-scribed with the help of the other degrees of freedom. The following stiffness relationship, which is now independent of ya, results after corresponding transfor-mations:

Ma za Ma zy

Mb zb Mb z32 2

yb yb yb z

Te e e e e

w : V 3 3 3 w 3 q 8EI

w : V 3 3 3 w 5 q 8: M 3 3 3 q 8

v : s K v p

(4.81)

In order to clarify that this is a matter of a reduced stiffness relationship, a “*” was added for the designation of the matrix and the vectors. This transformation can be generally formulated with the help of the following relationships for programming, so the elements ijk of the reduced stiffness matrix and ip of the reduced element load vector can be determined directly. For the reduction of row r due to a corresponding hinge we have:

iri i r

rr rr

kp p p

k C (4.82)

4.11 Hinges 167

irij ij rj

rr rr

kk k kk C (4.83)

Equations (4.82) and (4.83) also include the hinge stiffness C, so that semi rigid con-nections can be considered through hinge springs.

Besides the reduction of the element stiffness matrix, the geometric element stiffness matrix has to be reduced as well for calculations according to second order theory. Although nonlinear calculations are not covered until Chapter 5, it seems good to present the corresponding relationships for the consideration of hinges in a con-tinuous form. Based on the stiffness relationship of Eq. (5.36)

e e e e e 0,es K G v p p (4.84)

the following relationships for the determination of the reduced stiffness matrix and the reduced load vector can be stated with Eqs (4.82) and (4.83); this is also based on the reduction of row r:

ir iri i r

rr rr rr

k gp p pk C g (4.85)

ir irij ij ij ij rj rj

rr rr rr

k gk g k g k gk C g (4.86)

With the help of relationship

rr

rr rr rr rr rr rr rr rr

g1 1 1k C g k C k C g (4.87)

Eq. (4.86) can be transformed and the following equation for the reduction of the geo-metric element stiffness matrix can be stated:

rr ir irij ij ir rj ir ir rj rj rj

rr rr rr rr rr

g k g1g g g k k g g k gk C k C g

(4.88)

However, Eq. (4.83) is valid for elements ijk , so that for the determination of the reduced element stiffness matrix, independent of linear or nonlinear theory, the same relationship can still be used. Moreover, the division into the matrices Ke and Ge or Kand G, respectively, can be kept, which is of advantage for the calculation procedure.

5 FEM for Nonlinear Calculations of Beam Structures

5.1 General

Chapter 4 deals with FEM for linear calculations (first order theory) of beam structures. Since, according to Section 1.4, there is a difference between physical and geometric nonlinearities, “linear” refers to the following:

A linear elastic material behaviour (Hooke’s law) is assumed and plastic re-serves of the structural system are not utilised. For linear calculations the verification method Elastic-Elastic and Elastic-Plastic according to Table 1.1 can therefore be applied. Linear calculation methods are based on the assumption that the displacements occurring are very small and that the equilibrium may be formulated using the undeformed system as an approximation.

In contrast to linear calculations, the plastic reserves of a system are utilised and/or the influence of the deformations on the equilibrium is considered for nonlinearcalculations. The utilisation of the plastic reserves of the system is possible with the verification method Plastic-Plastic (see Table 1.1), which requires a calculation according to the plastic hinge or the plastic zones theory, see Sections 5.11 and 5.12. The influence of the deformations on the equilibrium affects all the verification procedures stated in Table 1.1, if this influence is to be considered in terms of structural safety. According to DIN 18800 Part 2, this, for example, applies to the buckling of beams and frameworks and especially affects the stability problems of flexural buckling and lateral torsional buckling. In this process, the geometric nonlinearity is linearised, which is referred to as “second order theory”, and this is the basis for the determination of critical loads and calculations according to second order theory. In the following Sections, second order theory is therefore deepened. Geometric nonlinear calculations are only covered in conjunction with the plastic zones theory in Section 5.12.

5.2 Equilibrium at the Deformed System

The column shown in Figure 5.1 is examined to give an explanation of the difference between first and second order theory. It is fixed elastically at the base point with a rotational point spring C and it has a very large bending stiffness EI . Similar conditions apply for the extensional stiffness EA, so that the deformations due to axial forces may be neglected. Because of these assumptions, the deformation of the column will only contain an inclination and no curvature. For that reason, all relationships can directly be stated without using differential equations or similar formulations.


5.2 Equilibrium at the Deformed System 169

Figure 5.1 Equilibrium of a column in the undeformed and deformed state

As is generally known, for linear beam theory the equilibrium conditions are postulated at the undeformed system, i.e. in the initial position. These can be taken from Figure 5.1. According to first order theory, we obtain for the fixing moment:

IaM H h (5.1)

However, with the rotational spring CI IaM C , (5.2)

so that: IC H h (5.3)

This equation can be compared with the equation system of an FE computation: K v p (5.4)

As can be seen, a single equation is adequate for the column since only one unknown deformation has to be determined. C corresponds to the total stiffness matrix, I to the vector of the nodal deformations and H h to the total load vector. Here, the rotation can be immediately determined with:

I H hC (5.5)

With the matrix equation (5.4), an equation solver (Gauss, Cholesky, GAUCHO) is needed for that purpose – see Chapter 6.

5 FEM for Nonlinear Calculations of Beam Structures 170

The equilibrium of the deformed system can also be taken from Figure 5.1. With the spring law according to Eq. (5.2), the following formula is obtained when using an arrangement comparable to Eq. (5.3):

C P h sin H h cos (5.6)

Since Eq. (5.6) contains the variables , sin and cos , it is a nonlinear conditional equation for , which can only be solved iteratively. It will now be linearised with the justification that the deformations are not too large in terms of systems of engineering and their serviceability. Using series expansions for the trigonometric functions, they are therefore set to sin and cos 1. With this formula, (5.6) can be written as:

C P h H h (5.7)

In comparison to the linear theory according to Eq. (5.3), Eq. (5.7) contains the addi-tional term “ P h ”. If, for the purpose of generalisation, Eq. (5.7) is now written as matrix equation

K G v p , (5.8)

the additional part, P h, complies with the geometric stiffness matrix G of a sys-tem according to second order theory, which will be derived in the following sections. Figure 5.2 can be used for an evaluation of the conducted linearisation. On the right side of the figure, the exactness of sin = and cos = 1 is shown. As can be seen, the approximations are larger angles, but up to 20° are sufficiently accurate for practical applications.

Figure 5.2 Rotation of the column in Figure 5.1 and the accuracy of the linearisations sin = and cos = 1

Whereas, just as for linear theory, for Eq. (5.8) an equation solver is required, the unknown column rotation of Eq. (5.7) can directly be determined:

H hC P h (5.9)

5.3 Extension of the Virtual Work 171

Here, the load P leads to the nonlinear increase of the rotation shown in Figure 5.2, which becomes infinitely large if the denominator of Eq. 5.9 approaches the value zero. This l value designates the stability problem, and with

P = Pcr = C /h (5.10)

it leads to the critical load Pcr. This is independent of the horizontal load. If a criticalload factor cr is used and the critical load is replaced by

Pcr = cr P, (5.11)

for the column we have

(C cr P h) = 0 (5.12)

as the generalised condition for the determination of the critical load factor. With reference to Eq. (5.8) for the displacement method we obtain in a similar way:

crK G v 0 (5.13)

Therefore, in FEM, the point of origin for the determination of the critical load factor, which is the lowest eigenvalue of Eq. (5.13), is the system of equations (5.13). Methods for the solution, which are also applicable for the determination of the eigenvector v, are dealt with in Chapter 6. Since the denominator was set to zero in Eq. (5.9) for the column example, then, for clarification, it should be stated here that the condition “determinant equal to zero”, which is

crdet K G 0 , (5.14)

can be used for the determination of cr. In this context, it is often referred to as the denominator determinant of the equation system.

Figure 5.1 also shows the familiar correlation between bending moment and shear force M V . With M(x) = Ma + V(x) x, we have

dM xM V x

dx(5.15)

This should clarify that this relation is valid for the internal forces referring to the de-formed state and not for a shear force in horizontal direction; also see Figure 5.9.

5.3 Extension of the Virtual Work

For calculations according to second order theory and for stability calculations, the virtual work according to Section 3.4 has to be supplemented, as explained in the previous Section. Table 5.1 contains a compilation which is used to set up the geometric stiffness matrix. Here, work components are distinguished, which are


caused in the deformed system by eccentrically applied loads and as a consequence of the internal forces needed for equilibrium. In Table 5.1, N is defined as positive as axial tension force. N, My, Mz and M are the internal forces and moments listed in Table 1.2 as resultants of the stresses. Mrr is an abbreviation which includes a tor-sional moment Mx( x) = Mrr due to the derivative of the angle of twist.

Table 5.1 Extension of the virtual work for second order theory and stability

a) Loads

ext y q M z q M

y F M z F M

z x F x F z y x F x F y

W q (y y ) q (z z ) dx

F y y F z z

F z F z F y F yb) Internal forces and moments

int M M M M

M M M M M M M M

M y y M M z z M rr

W N v v w w dx

N v z z v w y y w dx

( v M M v w M M w M ) dx

with:A

zyyz2M

2M

2Mxrr rMrMrMiNdA)yy()zz(M

2M

2M

2p

2M zyii

Ai zy2p

II

A

22 dA)zy(1r I

I2 2

y Mz A

1r y (y z ) dA 2 yA

M22

yz z2dA)zy(z1r I

Loads

For the derivation of the nonlinear work components due to the loads, Figure 5.3 is examined and the virtual work of eccentric concentrated loads is formulated, as explained in Section 5.2. The transformation of eccentric concentrated loads into the shear centre or centre of gravity, respectively, is also dealt with in Section 4.4.3. However, there it is a matter of additional loads for linear beam theory, whereas here, work components for second order theory as approximation for the geometric non-linear theory are added.

Within the scope of beam theory, the virtual displacements uF, vF and wF of the loading point in Figure 5.3 must be replaced by the displacements of the beam axis. Since the exact derivations are very extensive, please refer to [42]. Here the essentials are explained clearly and in a shortened manner.


Figure 5.3 Eccentric concentrated loads Fx, Fy and Fz as well as virtual displacements uF, vF and wF

Figure 5.4 Position of loading point F after rotation of the beam axis

The longitudinal displacement is given in Formula (1.1), so that the displacement of the loading point can be stated as

F S F z F y Fu u y z (5.16)

This approximation in terms of linear beam theory is complemented now with the help of Figure 5.4. For this purpose, the cross section is shifted and rotated by an an-gle . Since the y-z coordinate system rotates with the cross section, the loading point still corresponds to the coordinates yF and zF after the rotation. However, the rotations

y and z still refer to the horizontal and vertical directions of action. By approximation, the following lever arms result:

F

*F Fy y z (5.17)


F*

F Fz z y (5.18)

With regard to the rotation , Eq. (5.16) can be formulated more precisely now and this gives:

F S F F z F F y Fu u y z z y (5.19)

Since the virtual displacements comply with the first variation (cf. Section 3.4.3) and for variational calculus the same rules are valid as for differential calculus, it follows with the product rule:

F S F z F y F

F z z F y y

u u y z

z y (5.20)

The virtual work due to concentrated load Fx can be obtained with Wext (Fx) = FxuF. As can be seen, the work components shown in Table 5.1 arise when the nonlin-

ear terms of Eq. (5.20) are multiplied by Fx. Basically, the approximations according to second order theory are always formulated such that double products of the displacements emerge at maximum, e.g. y . This is also the reason why only the first term of the series expansion is considered for the trigonometric functions in Figure 5.4.

Figure 5.5 Determination of displacements vF and wF of loading point F

The virtual work for Fy and Fz can be complemented in a comparable way to that shown for Fx. Since both of these concentrated loads are to be applied at the shear centre M, the displacements vF and wF of the loading point F are determined with the help of Figure 5.5. For reasons of clarity, a singly symmetric cross section with zM = 0 is assumed and vF and wF are formulated based on vM, wM and . This gives the specified relations given in Figure 5.5, which can directly be taken from the figure


and which can be simplified in terms of small angles . If, for the purpose of generalisation, the ordinate zM of the shear centre is added and sin is substituted by

as well as cos by 1 ½ 2 (provision for two series terms), at maximum, double products of the deformations result and, we get as with the detailed derivation in [42]:

2F M F M F Mv v z z 1 2 y y (5.21)

2F M F M F Mw w y y 1 2 z z (5.22)

The virtual displacements are then (first variation):

F M F M F Mv v z z y y (5.23)

F M F M F Mw w y y z z (5.24)

With Wext (Fy, Fz) = Fy vF + Fz wF and the terms of Eqs (5.23) and (5.24) de-pending on , the work components in Table 5.1 for calculations according to second order theory and for stability analyses result. Distributed loads qx, qy and qzcan be included in a similar manner. Table 5.1 contains corresponding work compo-nents for qy and qz. The distributed load qx often acts at the centre of gravity, so that it is not included in Table 5.1. But this can be added in a similar was, without any problem.

All loads in Table 5.1 refer to local coordinate systems which are assigned to the beam elements. For the concentrated loads, this does not comply with the procedure described in Sections 3.2 and 4.4.4, but for the nonlinear work components it is more convenient. For the set-up of the geometric stiffness matrix of a system, they have to be transformed into the global coordinate system; see Sections 4.4 and 5.5.

Internal forces and moments

Table 5.1 contains work components depending on the internal forces and moments N, My, Mz and Mrr. The derivation of these terms is complex and not the subject of this book. For better comprehension, the principles and correlations are explained now. Full details can be found in [70] and [42].

In Figure 5.6, a cantilever is shown which is loaded at its end by concentrated loads Fx and Fz. As explained in detail in Section 5.2, if the equilibrium is formulated at the undeformed system, Fx does not have an influence on the bending moment M(x).


Figure 5.6 Example for the influence of deformations on the bending moments

Contrary to that, an increase of the bending moment due to Fx emerges when the de-formations are considered. One goal of the calculations with second order theory is to capture this effect. In the FEM, this is not achieved with the direct use of the loads, but with the internal forces acting in a beam element. As an example, the column of Figure 5.1 can be taken, for which the following equation has been derived with Formula (5.7).

C P h H h (5.25)

This equation contains the load P in the brackets. It can be replaced by NI according to Figure 5.1:

IC N h H h (5.26)

When transferred to arbitrary systems, the term, NI h, corresponds to the geomet-ric stiffness matrix, which contains the axial compression force according to first order theory – an internal force.

To generalise, consider Figure 5.7, which shows a fibre of a differential element with the length dx, which is taken to be a component of a beam. If the beam is loaded, the fibre is displaced and elongates. Stresses x and strains x refer to the deformed state. The designation du is used to show that it is a differential displacement du in the direction of x.

Applying Pythagoras’s Theorem to the triangle generated by the dashed line in Figure 5.7, we get

2 2 2dx du dx du dw (5.27)

and the strain x = du /dx = u gives:

2 2x 1 2u u w 1 (5.28)


Figure 5.7 Displacement and elongation of a fibre

With a series expansion for the root, this relation for the elongation can be linearised. If the deformation v(x) is considered analogously, we get:

2 2x u 1 2 v w (5.29)

According to custom, the displacements u, v and w are now related to the displace-ments of the centroid and the shear centre. Equations (1.1) (1.3) of the linear beam theory are the starting point, and they are completed in terms of second order theory.If one proceeds as for the loads, Equations (5.19), (5.21) and (5.22) can be adopted analogously and the following approximations can be made:

S M Mu u y z v z y w (5.30)

2M M Mv v z z 1 2 y y (5.31)

2M M Mv v y y 1 2 z z (5.32)

The internal virtual work (see Table 3.2)

int x xx A

W dA dx (5.33)

can be determined if derivations of the displacements are inserted into Eq. (5.33). After determining the virtual elongations x, we get:

int x x S M Mx A

W ( ) u y v z w

x M M M M M M M M M M M Mv v w w v z z v w y y w

x M M M Mv z z v w y y w2 2

x M My y z z dA dx (5.34)


The first row belongs to linear beam theory and yields the results in Table 3.2. The remainder is the supplement of the internal virtual work in Table 5.1 for second order theory and stability with the following classification:

second row in Eq. (5.34): work components due to N third row in Eq. (5.34): work components due to My and Mzfourth row in Eq. (5.34): work components due to Mrr

5.4 Nodal Equilibrium with Consideration of the Deformations

In Section 3.2, it was stated that the formulation of the equilibrium conditions at the nodes, i.e. at the connections of the beam elements, is an important basic concept of FEM. This was explained with the help of Figure 3.2 and dealt with in detail for the linear beam theory in Section 4.3. With nonlinear beam theory, the equilibrium hasto be stated at the deformed system. This is why the influence of the deformations on the nodal equilibrium is treated here.

Figure 5.8 Equilibrium at the node k under consideration of the deformations

Figure 5.8 is an addition to Figure 4.12 (see Section 4.3), in which, for reasons of clarity, only the concentrated load FZ,k and the corresponding equilibrium internal forces are considered in the global COS. It can be seen that the nodal deformations

Ku and Kw do no influence the nodal equilibrium for FZ. Even the nodal rotation y

does not affect the equilibrium condition, because the internal forces Z,i,kV and

Z, j,kV are by definition to be applied in the global Z-direction (which is vertically). The influence of the nodal rotation will be acquired in the next step, when the global equilibrium internal forces are replaced by the internal forces of the element ends, as it was done in Section 4.3.

5.4 Nodal Equilibrium with Consideration of the Deformations 179

Figure 5.9 Beam elements with internal forces in three different directions

Figure 5.9 shows the beam element of Figure 5.8, which is connected to node k on the right-hand side. With nonlinear beam theory, it is useful to distinguish three different directions for the internal forces:

Verification internal forces These are, as explained in Section 5.2 with the help of Figure 5.1, the internal forces (see Table 1.2) which are needed for the determination of stresses and the verification of the cross section bearing capacity using stresses or interac-tion conditions. Their directions refer to the deformed beam axis.

Global equilibrium internal forces (with “-”) Their directions refer to the axes of the global COS. They are required for the formulation of nodal equilibrium and therefore for setting up the equation system for the calculative capturing of the structural system.

Local equilibrium internal forces (with “ ”)These internal forces refer to the axes of the local COS of the beam elements. For linear beam theory it is assumed that the deformations, and particularly the angles of rotation, are small and hence that the verification internal forces are equal to the local equilibrium internal forces by approximation. For nonlinear beam theory, which is acquired here within the scope of second order theory, this not the case, and, due to this difference, the alignment of geometricstiffness matrices is treated in the next Section. The transformation relation-ships (local/global) in Section 4.4 are valid for the transformation of the local equilibrium internal forces into the global (not for the “verification internal forces”!).


5.5 Geometric Stiffness Matrix

Geometric element stiffness matrices eG

In Section 4.2, stiffness matrices were derived for beam elements using the virtual work of Table 3.2 and the definitions of the deformations and internal forces according to Figures 4.2 and 4.3. Figure 4.3 is replaced by Figure 5.10 and the local equilibrium internal forces are indicated (with “ ”, see Section 5.4).

Figure 5.10 Beam element with definition of the local equilibrium internal forces and moments at the element ends

While for linear beam theory using Eq. (4.1) we have

e e e e eˆs K v p s , (5.35)

for the local equilibrium internal forces according to second order theory, we have:

e e e e e 0,es K G v p p

with: es vector of the local equilibrium internal forces at the element ends (also see Figure 5.10)

eK element stiffness matrix (Section 4.2)

eG geometric element stiffness matrix

ev vector of the nodal deformations of the beam element ep vector of the loads due to qx, qy, qz and mx

0,ep loads due to initial deformations (see Section 5.7)

(5.36)

5.5 Geometric Stiffness Matrix 181

Just as in the case of the virtual work due to concentrated loads in Section 3.4, the virtual work due to internal forces and moments at the ends of beam elements results in a similar manner through the products “local equilibrium internal force times cor-responding virtual displacement”. With Figures 4.2 and 5.10, we obtain:

ext a Sa ya Ma za za

za Ma ya ya xa a a a

b Sb yb Mb zb zb

zb Mb yb yb xb b b b

ˆ ˆ ˆW N u V v Mˆ ˆ ˆ ˆV w M M Mˆ ˆ ˆN u V v Mˆ ˆ ˆ ˆV w M M M

(5.37)

The element stiffness matrix for arbitrary stresses can be taken from Section 4.2.5 and the link of the internal forces to the deformations follows from Table 4.1. According to Eq. (5.36), the geometric stiffness matrix Ge is required for second order theory. The basis for its determination is the virtual work in Table 5.1 with the supplement for second order theory and stability. Just as in Section 4.2, the following polynomials are chosen as basis functions for the description of the deformations:

S Sa Sbu ( ) 1 u u (5.38)

M 1 Ma 2 za 3 Mb 4 zbv ( ) f v f f v f (5.39)

M 1 Ma 2 ya 3 Mb 4 ybw ( ) f w f f w f (5.40)

1 a 2 a 3 b 4 b( ) f f f f (5.41)

Loads qx, qy, qz and mx are not considered in the basis functions in Eqs (5.38) (5.41). As mentioned in Chapter 4, the corresponding internal virtual work for the determination of the element stiffness matrix is zero. Regarding the geometric element stiffness matrix and the load vector, it has to be noted that the polynomial approaches are approximations. Section 5.6 includes derivations with the exact defor-mation functions for selected cases.

The shape functions f1 to f4 are defined and shown in Figure 3.10. With the polyno-mial approach of Eq. (4.76), the warping torsion is acquired by approximation, as explained in Section 4.2.4 – see Eq. (4.29) as well. Since the calculation of the matrix elements is done in the same manner as shown in Section 4.2 for the stiffness-de-pendent elements and the load case, flexural buckling is dealt with in detail in Section 5.6, further derivations are not needed here. The allocation of the matrix elements in principle can be seen in Table 5.2, where empty spaces mean that these matrix elements are equal to zero.

The calculation of the matrix elements for the geometric element stiffness matrix Gecan be taken from the following compilation:


g(2,2) = g(4,4) = g(9,9) = g(11,11) = 1.2 N/ g(2,3) = g(2,10) = g(5,11) = g(11,12) = 0.1 N g(2,6) = –1.1 Mya/ – 0.1 Myb/ – qz 9/140 + 1.2/ zM N g(2,7) = 0.1 Mya + qz

2/140 – 0.1 zM N g(2,9) = g(4,11) = – 1.2 N/ g(2,13) = 0.1 Mya/ + 1.1 Myb/ + qz 9/140 – 1.2/ zM N g(2,14) = 0.1 Myb + qz

2/140 – 0.1 zM N g(3,3) = g(5,5) = g(10,10) = g(12,12) = 2/15 N g(3,6) = – 0.9 Mya – 0.2 Myb – qz

2 31/420 + 0.1 zM N g(3,7) = 0.1 Mya + 1/30 Myb + qz

3/84 – 2/15 zM N g(3,9) = g(4,5) = g(4,12) = g(9,10) = –0.1 N g(3,10) = g(5,12) = –1/30 N g(3,13) = –0.1 Mya + 0.2 Myb – qz

2/105 – 0.1 zM N g(3,14) = –1/30 Mya – qz

3/210 + 1/30 zM N g(4,6) = –1.1 Mza/ – 0.1 Mzb/ – 1.2/ yM N g(4,7) = 0.1 Mza + 0.1 yM N g(4,13) = 0.1 Mza/ + 1.1 Mzb/ + 1.2/ yM N g(4,14) = 0.1 Mzb + 0.1 yM N g(5,6) = 0.9 Mza + 0.2 Mzb + 0.1 yM N g(5,7) = – 0.1 Mza – 1/30 Mzb – 2/15 yM N g(5,13) = 0.1 Mza – 0.2 Mzb – 0.1 yM N g(5,14) = 1/30 Mza +1/30 yM N g(6,6) = g(13,13) = 1.2 Mrr/ + mq 13/35 g(6,7) = – 0.1 Mrr – mq 11/210 2

g(6,9) = 1.1 Mya/ + 0.1 Myb/ + qz 9/140 – 1.2/ zM N g(6,10) = – 0.2 Mya + 0.1 Myb + qz

2/105 + 0.1 zM N g(6,11) = 1.1 Mza/ + 0.1 Mzb/ + 1.2/ yM N g(6,12) = 0.2 Mza – 0.1 Mzb + 0.1 yM N g(6,13) = – 1.2 Mrr/ + mq 9/70 g(6,14) = – 0.1 Mrr + mq

2 13/420 g(7,7) = g(14,14) = /7.5 Mrr + mq

3/105 g(7,9) = – 0.1 Mya – qz

2/140 + 0.1 zM N g(7,10) = – 1/30 Myb – qz

3/210 + 1/30 zM N g(7,11) = – 0.1 Mza – 0.1 yM N g(7,12) = 1/30 Mzb + 1/30 yM N g(7,13) = 0.1 Mrr – mq

2 13/420 g(7,14) = – 1/30 Mrr – mq

3/140 g(9,13) = – 0.1 Mya/ – 1.1 Myb/ – qz 9/140 + 1.2/ zM N g(9,14) = – 0.1 Myb – qz

2/140 + 0.1 zM N g(10,13) = 0.2 Mya + 0.9 Myb + qz

2 31/420 – 0.1 zM N g(10,14) = 1/30 Mya + 0.1 Myb + qz

3/84 – 2/15 zM N g(11,13) = – 0.1 Mza/ – 1.1 Mzb/ – 1.2/ yM N g(11,14) = – 0.1 Mzb – 0.1 yM N g(12,13) = – 0.2 Mza – 0.9 Mzb – 0.1 yM N g(12,14) = – 1/30 Mza – 0.1 Mzb – 2/15 yM g(13,14) = 0.1 Mrr + mq 11/210 2

with: mq = qy (yq yM) + qz (zq zM)

5.5 Geometric Stiffness Matrix 183

Table 5.2 Allocation of the geometric element stiffness matrix in principle (additional components for second order theory)

For a detailing of Eq. (5.36), Table 5.3 shows the link between the local equilibrium internal forces at the beam ends and the corresponding deformations according to second order theory.

Table 5.3 Link between the local equilibrium internal forces at the element ends and the corresponding deformations


The matrix elements g(i,j) compiled above were determined on the assumption that the bending moments My and Mz have a linearly varying course within the beam elements, which is covered by the Mya and Myb or Mza and Mzb. For the axial force N (tension positive) and Mrr defined in Table 5.1, a constant distribution is assumed. The signs of the internal forces are valid for the usual sign definition shown in Section 1.5. In principle, they are verification internal forces, which are, however, usually calculated according to first order theory, i.e. using Eq. (5.70) according to Section 4.7. Further details for the determination of the internal forces, which are needed for the geometric element stiffness matrix, are given in Section 5.8.

The element load vector 0ep due to initial deformations or equivalent geometric imperfections is derived in Section 5.7. For the element load vector ep , values according to first order theory are usually used, i.e. according to Eq. (4.33), since a correspondingly fine FE modelling is required. However, it is perfectly possible to use more precise load vectors according to Eq. (5.44) for bending with axial compression force or for bending with tension force, as explained in Section 5.6. Instead of dividing by 12, it is then divided by 2 D or 2 Z for the moments at the element ends. With this, the approximation for 1.0 and the corresponding FE modelling can be checked. Since the parameter for D = 1.0 is D = 5.89277, the moment resulting from bending with axial compression force using the approximation (12!) is 1.7% too small, i.e. a discrepancy which is tolerable from a structural point of view.

Geometric total stiffness matrix G

With Eq. (4.57), the system of equations for linear beam theory is

K v p (5.42)

For second order theory it is completed as follows:

0K G v p p (5.43)

G is the geometric total stiffness matrix, which is set up with the element matrices Ge. The procedure is in accordance with the set-up of the total stiffness matrix K , so that the same transformations into the global COS and an equal arranging of the ma-trix elements must be carried out. For that purpose, one can again refer to Sections 4.4.1 and 4.5.

5.6 Special Case: Bending with Compression or Tension Force 185

5.6 Special Case: Bending with Compression or Tension Force

In Section 5.5, the geometric element stiffness matrix is derived for the general stress state with the help of polynomial functions. This leads to an approximation of second order theory, so that a sufficiently fine FE modelling is required. As mentioned in Section 3.5.3, in special cases, the exact deformation functions may also be used, which contain the trigonometric functions sin x and cos x or hyperbolic functions sinh x and cosh x.

Bending with axial compression force

An important field of application of beam structures is flexural buckling, i.e. the bending stress with axial compression force. In Figure 5.11, a beam element with bending about the strong axis is shown. Bending about the weak axis is not covered here since it can be dealt with in a similar manner without any problems.

Figure 5.11 Beam element for flexural buckling about the strong axis

In Figure 5.11, the axial compression force is not shown with the internal forces, be-cause it has to be determined previously using a first order theory calculation and hence it is assumed to be known here. The purpose of the calculation according to second order theory is the determination of the deformations and internal forces shown. The basis for the stiffness relationship of the beam elements is the virtual work according to Figure 5.11, where the exact solution for the deformation function wM( , D) of Eq. (3.36) in Section 3.5.3 is inserted now. After solving the integrations, which is quite extensive, we get the following stiffness relationship for flexural buckling:


za D D D D zMaMa2 2 2

ya yaya D D D z Dy3

D D Mb zMb zb2 2

ybyb D z Dyb

V q 2ww :M: q 2EIˆ w q 2w : V

: ˆ sym. q 2M

(5.44)

The parameters in the element stiffness matrix are:

D D D DD

D D D

sin cos2 1 cos sin D D D

D D DD

D D D

sin2 1 cos sin

3D D

DD D D

sin2 1 cos sin

(5.45a d)

For small member characteristics D (see Figure 5.11) the parameters D, D, D and D can be replaced by approximations as follows:

2D D

2415

2D D

1610

2D D

1230

2D D

6125

(5.46a d)

Using these approximations, the stiffness matrix according to second order theory of Eq. (5.44) can be divided into two partial matrices:

2 2 2 2y D

e e 3

2 2

12 6 12 6 36 3 36 3

EI 4 6 2 4 3NK G12 6 36 330

sym. 4 sym. 4

(5.47)

The first matrix corresponds to Eq. (4.18) being the stiffness matrix according to first order theory. The second matrix is the geometric stiffness matrix, whose elements are already contained in the compilation of Section 5.5. The approximation with the partial matrices requires a correspondingly fine FE modelling. Figure 5.12 shows the percentage error arising from the use of the approximations for D, D, D and D. For

D 1.0 they are between 0.052% and 0.046%. For this reason, the approximation leads to accurate enough results for member characteristics smaller than 1.0. The element lengths can be defined with condition D 1.0 as follows:

y

D

EIN

(5.48)

5.6 Special Case: Bending with Compression or Tension Force 187

Figure 5.12 Percentage errors concerning the approximations for D, D, D and D

Apart from exceptional cases, ND is usually smaller than 50% of the perfectly plastic axial force in structural systems. As a guideline, one can apply ND = 0.5 Npl,d for steel grade S 235 in Eq. (5.48) and evaluate it for common rolled sections. The results are presented in Figure 5.13, where an evaluation for bending about the weak axis has also been added.

Figure 5.13 Maximum element lengths of rolled sections for flexural buckling with D = 1.0 and ND = 0.5 Npl,d (S 235)

Bending with tension force

While the deformation function wM(x or ) contains trigonometric functions for a compression force, it includes hyperbolic functions for a tension force, as shown in


Section 3.5.3. For the load case bending with tension force according to second order theory, the following stiffness matrix yields with an analogous procedure:

Z Z Z Z2 2

Z Z Z

Z Z2

Zsym.

(5.49)

This stiffness matrix complies with the matrix in the stiffness relationship of Eq. (5.44). Since here an axial tension force is considered to act in the beam element, the member characteristic

ZZ

y

NEI (5.50)

has to be considered, and the parameters of the matrix are

Z Z Z ZZ

Z Z Z

sinh cosh2 cosh 1 sinh Z Z Z (5.51a-d)

Z Z ZZ

Z Z Z

sinh2 cosh 1 sinh

3Z Z

ZZ Z Z

sinh2 cosh 1 sinh

The matrix and the parameters correspond to Eqs (4.26) and (4.27), if the indices “Z” and “T” are exchanged; this is due to an analogy between the “bending with axial tension force according to second order theory” and the “warping torsion”. This is of course also valid for the approximations

2Z Z

2415

2Z Z

1610

2Z Z

1230

2Z Z

6125

(5.52a-d)

and for the division into two partial matrices. Since in Eq. (4.29) EI has to only be replaced by EIy and GIT by NZ, an explicit formulation is not necessary here. If the geometric element matrices for compression and tension are compared, we see that they are identical except for the sign. This is a great advantage in comparison to the exact matrices since the geometric element matrix can generally be used with the factor N. Any value may be inserted for the axial force N without a distinction of cases: N = 0, tension force N and compression force N.

The error which occurs with the use of the approximations for Z, Z, Z and Z are shown Figure 4.8. The basis is the condition Z 1.0 again and for the element length it is:

5.7 Initial Deformations and Equivalent Geometric Imperfections 189

y

Z

EIN

(5.53)

Apart from exceptional cases, NZ will not exceed 50% of the perfectly plastic axial force in structural systems. As a guideline, one can apply NZ = 0.5 Npl,d for steel grade S 235 in Eq. (5.53) and evaluate it for rolled sections. The results are equal to the ones shown in Figure 5.13.

5.7 Initial Deformations and Equivalent Geometric Imperfections

With the introductory example in Figure 5.1 it has been shown that the consideration of the deformations of a beam leads to a change of the internal forces and moments. As a result, and also due to the equilibrium at the deformed system, other deformations yield beyond that in contrast to linear beam theory. If now, in addition to Figure 5.1, the initially deformed column in Figure 5.14 is considered, it can be easily seen that internal forces and deformations yield due to initial deformations.

Figure 5.14 Column with initial rotation 0 (initial sway imperfection)

In addition to the rotation in Figure 5.1, the initial rotation 0 shown in Figure 5.14 can be taken into consideration and the equilibrium at the initially deformed anddeformed system can be established. Using the relationships in Section 5.2, we obtain for the moment at the bottom of the column:

a 0 0M H h cos P h sin (5.54)

The linearisation in terms of second order theory leads to:


a 0M H h P h (5.55)

This equation can be written using the spring law, aM C , like Eq. (5.7):

0C P h H h P h (5.56)

In comparison to Eq. (5.7), the additional term P h 0 appears on the right-hand side. The matrix Eq. (5.8) can now also be expanded analogously:

0K G v p p (5.57)

In Eq. (5.57)

0 0p G v (5.58)

is an additional load vector as a result of initial deformations, i.e. the initial defor-mations are replaced by an equivalent load vector using the geometric stiffness matrix G. Since only one unknown deformation has to be determined, Eq. (5.56) can be solved directly and this gives:

0H h P hC P h (5.59)

The initial deformation 0 leads to an increase of the rotation and of course also to a change of the internal forces and moments, especially to an increase of the bending moment. The moment at the bottom of the column results to:

0a

Ki

H h P hM

1 P P (5.60)

The preceding information aims to explain clearly that initial deformations have an influence on the deformations and internal forces, and are also that they are always considered in the matrix equation of FEM. For buckling, however, not only effectively possible initial deformations are captured, but equivalent geometric imperfections are to be applied to cover further influences. These are residual stresses acting unfavourably and the spread of plastic zones, which can only directly be considered using calculations according to the plastic zones theory. They cannot be covered using customary calculations, where the stiffness is not changed. More detailed explanations on the plastic zones theory can be found in Section 5.12 and specifications concerning equivalent geometric imperfections in Section 9.6.

For calculations according to the FEM equivalent geometric imperfections, v0(x),w0(x) and 0(x) in the form of straight lines ( initial rotations) as well as quadratic pa-rabolas or sine half waves (initial deflections) have to be considered. To describe the methodology, the initially deformed (v0(x)) beam section A E in Figure 5.15 is con-sidered.

5.7 Initial Deformations and Equivalent Geometric Imperfections 191

Figure 5.15 Capturing the initial deformations v0(x) in beam section A E

According to Figure 5.15, the initial deformation can be divided into three parts: the displacement at the point of origin, the rotation and the deflection. The function is then:

2

0 0,A 0,E 0,A 0,m 0,m20 0 00

x x x xv x v v v v 4 v sin (5.61)

With the help of this equation the initial deformation ordinates can be calculated in every node of the beam section A E. It is often the case that the beam section is divided into n beam elements of equal length. With 0 = n e and x = (k – 1) e we then get:

0 0,A 0,E 0,A

2

0,m 0,m

k 1v x v v vn

k 1k 1 k 1v 4 v sinn n n

(5.62)

Besides the displacements, the rotations of the initial deformation function are also necessary at the nodes, because these are also nodal degrees of freedom of the beam elements. The first derivation of Eq. (5.61) gives

0,E 0,A 0,m 0,m0,z 0

0 0 0 0 0

v v 4 v vx xx v x 1 2 cos (5.63)

Provided that it is divided into beam elements of equal length, x/ 0 can be replaced by (k 1)/n in Eq. (5.63). As the calculations depicted here in connection with Figure 5.15 show, the initial deformation function v0(x) is replaced by v0,k and 0,zk at the nodes. This can be achieved similarly for the initial deformation functions w0(x) and

0(x), so that w0,k, 0,yk, 0,k and 0,k are obtained. Generally, it is advisable to de-scribe initial deformations in the local COS which relate to the beam elements. For every beam element an additional element load vector


0,e e 0,ep G v (5.64)can be determined. In Eq. (5.64), Ge is the geometric element stiffness matrix and the vector v0,e contains the initial deformation ordinates at the element nodes calculated above. The element vector 0,ep is the equivalent load for the consideration of the initial deformations of a beam element. Since, according to Table 5.1, the internal forces and moments N, My, Mz und Mrr are also included in Ge besides qy and qz, the load vector 0,ep cannot be determined before a system calculation according to first order theory – also see Sections 3.3 and 5.8.

The element load vectors according to Eq. (5.64) are determined for all beam ele-ments with initial deformations and, after transformations into the global COS, considered in the total load vector 0,ep . The transformations have to be conducted in the same manner as for the “normal” load vector ep , so that with Eq. (4.51) in Section 4.4.2 the following transformation relationship results:

0,e e 0,ep T p (5.65)

A look at Table 5.1 (extension of the virtual work) shows that the nonlinear load components as a result of Fx, Fy and Fz are still missing. Since these loads act at the nodes by convention, the corresponding work components can directly be calculated in the global COS. For this approach the work components are:

x F z 0 0,z x F y 0 0,y

y F M z F M 0

F z F y

F y y F z z(5.66)

These load components relate to the nodes at which the loads Fx, Fy or Fz act excen-trically. Accordingly, they can be arranged in the total load vector 0p .

The approach described leads to the fact that the initial deformations or equivalent geometric imperfections in connection with the internal forces and loads, which result in the nonlinear work components according to Table 5.1, are converted into equivalent nodal loads. Since the matrix K is not considered in this conversion, the equivalent nodal loads comply with equivalent loads in terms of DIN 18800 Part 2. Instead of the initially deformed beam, a straight beam with corresponding equivalent loads is therefore considered.

The sum of the deformations and the initial deformations yields the total defor-mations:

vM,tot = vM + v0,m

wM,tot = wM + w0,m

tot = + 0

(5.67)

5.8 Second Order Theory Calculations and Verification Internal Forces 193

A calculation of the total deformation is usually not reasonable, since for practical calculations instead of initial deformations, equivalent geometric imperfections are applied, which only partly include real initial deformations, i.e. geometric imper-fections.

5.8 Second Order Theory Calculations and Verification Internal Forces

System calculations

For calculations according to second order theory the equation system becomes:

0K G v p p (5.68)

It can not be directly solved in this form, because internal forces join the geometric total stiffness matrix G , and these are not known and represent an essential aim of the calculations. Furthermore, the total load vector 0p as a result of initial defor-mations can not be determined for the time being since, according to Section 5.7, internal forces are also needed to do so.

However, as starting point for the calculations according to second order theory the equation system

K v p (5.69)

can be set up for the linear beam theory as in Section 4.5 and the deformations in vector v can be calculated according to Section 4.6. If it is designated Iv , because it deals with deformations of first order theory, the internal forces in Section 4.7 produce, for every beam element:

I Ie e e es K v p (5.70)

The procedure for calculations with first order theory is listed in Table 3.1 (Section 3.3) where single calculation steps are summarised. With the internal forces according to first order theory, the calculation approach can now be repeated in an extended manner. To do so, the internal forces of first order theory are used to calculate G and 0p , so that Eq. (5.68) can be solved now. The results are the defor-

mations according to second order theory IIv , but in the following the designation with “II” is omitted.


Equilibrium internal forces

As explained in Section 5.4 and Figure 5.9, for calculations according to second order theory, global and local equilibrium internal forces and verification internal forces are distinguished. The local equilibrium internal forces refer to the local COS of the beam elements and can be, according to Eq. (5.36) in Section 5.5, calculated as:

e e e e e 0,es K G v p p (5.71)

Since the directions of action of these internal forces comply with the element-related distributed loads qx, qy and qz, they have been used in Section 5.4 to formulate the equililibrium. After the completion of the system calculations, they are needed for determination of the verification internal forces and for doing equilibrium checks.

Verification internal forces and moments

For the verification of sufficient cross section bearing capacity, internal forces are needed to determine the stresses of the cross sections. According to Figure 5.9, these are the verification internal forces since their directions of action relate to the deformed beam axis. They serve the determination of stresses or to verify sufficient cross section bearing capacity with internal forces (verification method Elastic-Plastic – see Table 1.1). For clarification, it should be mentioned that the verification internal forces comply with the internal forces defined in Table 1.2 (resultants of the stresses) and that only for these internal forces do the common methods for the determination of stresses apply.

Generally, the verification internal forces can be calculated with

e e e es K v p (5.72)

since this part of Eq. (5.71) contains the verification internal forces. However, matrix multiplication means that some relatively imprecise verification internal forces may result. This can lead to erratic changes from element to element. This effect is ad-dressed in [93], including the torsional internal forces.

More detailed verification internal forces result if the local equilibrium internal forceses from Eq. (5.71) are assumed, and the boundary conditions are consulted for the

formulation of the necessary transformation relationships. If it is preceded as in Section 2.11 in [25], and if in addition the influence of second order theory is considered, we obtain the following boundary conditions with the virtual work ac-cording to Tables 3.2 and 5.1:

S x 0u F N 0 (5.73a)

M y y z z M y 0v F V V N z M 0 (5.73b)


M z z y y M z0

w F V V N y M 0 (5.73c)

xL xp xs rr M z M y0

M M M M N z y 0 (5.73d)

y yL y z 0M M M 0 (5.73e)

z zL z y 0M M M 0 (5.73f)

L 0M M 0 (5.73g)

The boundary conditions mentioned above can be used to calculate the verification internal forces. To do so, the loads

Fx, Fy, Fz, MxL, MyL, MzL and M L

are replaced by the corresponding equilibrium internal forces:

y z x y zˆ ˆ ˆ ˆ ˆ ˆ ˆN, V , V , M , M , M and M

For clarification, it should be mentioned here that only shear stresses of the primary torsion are part of the virtual work according to Table 3.2 with:

xp TM GI (5.74)

All other parts are based on axial stresses x. However, the boundary conditions also contain verification internal forces, which are resultants of shear stresses (see Table 1.2). This is due to the fact that they emerged through the equilibrium conditions

y zV M , z yV M and xsM M (5.75a c)

from x internal forces.

Critical examination of Eq. (5.73) shows that the equation is a relatively rough approximation for the axial force, from which N N results. The reason for this is the linearisation in terms of second order theory compared to the precise formulations of the geometric nonlinear problem. Using Figure 5.16, the connection with the shear forces therefore has to be added. With the help of Figure 5.16, the verification internal forces N, Vz and My can be tranformed into the equilibrium internal forces z y

ˆ ˆ ˆN,V and M . In doing so, we have to take into account that the axial forces act at the centre of gravity S and the transverse forces at the shear centre M and that with y My and y different angles occur at these points. We then obtain the transformation relationships:


y yM M (5.76a)

y M z yN N cos y V sin (5.76b)

z z y y MV V cos N sin y (5.76c)

Figure 5.16 Equilibrium and verification internal forces for bending with axial force in the y-z plane

If the transverce force Vy is added analogously in Eq. (5.76b) and small rotations are assumed (sin , cos 1), we get the approximation:

y z z yN N V V (5.77)

Equations (5.73b g), (5.74) and (5.77) are now used to determine the verification internal forces. The following relationships result by approximation:

y z z yˆ ˆ ˆN N V V (5.77a)

y y z z M yˆ ˆV V V N z M (5.77b)

z z y y M zˆ ˆV V V N y M (5.77c)

xp TM GI (5.77d)

xs x xp rr M z M yˆM M M M N z y

with 2rr M z y y zM N i M r M r M r (also see Table 5.1)

(5.77e)

y y zˆ ˆM M M (5.77f)


z z yˆ ˆM M M (5.77g)

ˆM M (5.77h)

Figure 5.17 Calculation of bending moments My and Mz

(verification internal moments)

As an example, the division of the bending moments is clearly explained with the help of Figure 5.17. Here, as the sketches show, only the rotation about the longitudi-nal axis is taken into account. Rotations z and y are ignored.

As already mentioned, the verification internal forces and moments serve the analysis of the cross section bearing capacity using the theory of elasticity and the theory of plasticity. Therefore, they are the starting point for the stress analysis and the stress verification as well as for verifications with interaction conditions or with the partial internal forces method. Moreover, the verification internal forces N, My, Mz and M are used in the geometric stiffness matrix. Since they are not known at the be-ginning of the calculations and are only available by approximation after the calculation according to first order theory, the calculation according to second order theory can be repeated if required, and the verification internal forces, calculated with Eq. (5.77a, f, g and h), can be inserted in the equation system (5.68). Thus, in many practical cases, the accuracy can be increased.

For calculation of the internal forces with the element stiffness matrices it has to be taken into account that they result according to sign definition II and that a trans-formation into the common sign convention is necessary. Positive internal forces act at the positive cross section as shown in Figure 1.7 or in Figure 5.10 at element end b.At the negative intersection, the opposite directions of action apply to positive inter-nal forces, so that the direction of the arrows at the element end a in Figure 5.10 has to be switched.


Limitation of rotation

For calculation of the displacements and internal forces within the scope of the beam theory used here it is assumed that the rotations z , y and are small. This approximation is not only used for derivation of the virtual work and formulation of the stiffness matrix, but also for the calculation of internal forces (see, for example, Figure 5.17). Therefore, the rotations have to be limited in view of the ap-proximations sin and cos 1. This is unproblematic for the rotations z and

y , because they are usually small for practical calculations. The rotation , how-ever, can – especially with scheduled torsion – have large angles. Thus, the limitation

0.3 (5.78)

should be used and the calculation should be aborted if the limit is exceeded.

Calculation example

As an example of the use of FEM and of calculations according to first order theory, a bending beam (see Figure 4.37) is dealt with in Section 4.5.3. This example is picked up again at this point and, as shown in Figure 5.18, complemented with a concentrated load of 400 kN at the right-hand end of the beam in the longitudinal direction of the beam. Thus, a constant axial compression force occurs in the beam and the beam is susceptible to buckling. Since the beam is to be held nondisplaceably perpendicularly to the plane of projection, the flexural buckling in the x-z plane is examined.

Figure 5.18 Example beam with constant compression force

The FE modelling is taken over from Figure 4.37 and the beam is divided into three beam elements. Since beam element 2 is the longest, the highest member characteris-tic emerges for this element:

D400400 1.0

64000000


Because of D 1.0 for all beam elements, the approximation with both of the partial matrices of Eq. (5.47) can be used. For example, the geometric stiffness matrix for beam element 2 is:

2

1.2 40 1.2 4064000 3 40 16000 3

G1.2 40

sym. 64000 3

(5.79)

Eq. (5.79) complements Eq. (4.58) in Section 4.5.3, which contains the stiffness ma-trix for beam element 2. The geometric element matrices G1 and G3 result analoguously, and after arranging the three element matrices, the following geometric total stiffness matrix for the structural system in Figure 5.18 is obtained:

2.4 40 2.4 40 0 0 0 032000 3 40 8000 3 0 0 0 0

3.6 0 1.2 40 0 032000 40 16000 3 0 0

G3.6 0 2.4 40

32000 40 8000 32.4 40

sym. 32000 3

(5.80)

Since the stiffness matrix K is given in Eq. (4.59), the sum K + G can now be de-termined according to second order theory. As load vector, Eq. (4.60) is retained, although this is only an approximation. The approximation affects the load moments due to qz. With Eq. (5.44) for beam element 2, we get:

2 D = 12.144

Compared to that, with Eq. (4.60), which is based on Eq. (4.18), we get the value 12. At 1.2%, the difference is small.

As described in Section 4.5.4, only the geometric boundary conditions are considered and the deformations can be calculated according to Section 4.6 if the matrix K is replaced by K + G. As deformations from second order theory for qz in kN/cm and Fzin kN we get:

2 z zw 7.436 q 0.03334 Fy2 z z0.05080 q 0.0002499 F

y3 z z0.08924 q 0.0007125 F

4 z zw 19.50 q 0.2020 Fy4 z z0.1017 q 0.001161 F


Comparison with Eq. (4.62) shows, that according to second order theory, considerably larger deformations result.

The internal forces according to first order theory have been calculated for the example in Section 4.7 see Eq. (4.66) and Figure 4.42. According to second order theory, the local equilibrium internal forces can be determined element by element with Eq. (5.71), and the verification internal forces from Figure 5.9 are gained with the transformation relationships of Eqs (5.77a, c and f). The results are compiled in Table 5.4 for qz = 1 kN/cm and Fz = 1 kN and can be compared to Figure 4.42. As can be seen, significant differences occur between the internal forces according to first and second order theory.

Table 5.4 Internal forces and moments in kN and cm for the beam in Figure 5.18

Local equilibrium internal forces Verification internal forces qz Fz qz Fz

z1V 210.4 -0.732 z1V z1V z1V

y1M -38445 158.1 y1M y1M y1M

z2V 210.4 -0.732 z2V 230.7 -0.832

y2M 6611 -1.513 y2M y2M y2M

z3,V -189.6 -0.732 z3,V -225.3 -0.447

z3,rV 0 1.000 z3,rV -35.70 1.285

y3M 7800 -280.8 y3M y3M y3M

z4V 0 1.000 z4V -40.68 1.464

y4M 0 0 y4M y4M y4M

Note: This example only serves as explanation of FEM. For a design in practice equivalent geometric imperfections have to be applied and the stiffness EIy has to be reduced with M = 1.1. Using the exact stiffness relationships from Eq. (5.44), is not beneficial for this example, since for the determination of all values, that are relevant for the design, the beam would be divided into more than three elements. Furthermore, the separate examination of load cases qz and Fz is only acceptable because the axial compression force is the same in both cases (limited superposition). For a repeated calculation according to second order theory minor differences result.

5.9 Stability Analysis / Critical Loads 201

5.9 Stability Analysis / Critical Loads

For the buckling of beam structures the stability cases of flexural buckling and lateral torsional buckling occur. Important parameters which characterise the stability danger of a structural system are the eigenvalues of the system, which are called critical loads in steel construction. Amongst others, they are needed for a design with the or

M method of DIN 18800-2 (Eurocode 3: method) since Ncr or Mcr,y are used in the related slenderness ratios:

plK

cr

NN

or pl,yM

cr,y

MM

(5.81)

For the exclusive acting of axial compression forces N or bending moments My, Ncrand Mcr,y are the lowest critical loads according to the elastic theory. Mathe-matically expressed, the lowest critical load of a system is the lowest positive eigenvalue of the related equation system. However, higher eigenvalues are also needed sometimes or critical loads with simultaneous action of multiple internal forces.

For the solution of an eigenvalue problem with the finite element method the homo-genous equation system

crK G v 0 (5.82)

is examined and the critical load factor cr of the structural system or equation system is determined. In a general load case, the internal forces N, My, Mz and Mrr (see Table 5.1) enter the geometric stiffness matrix G as well loads acting excentrically (Fx, Fy,Fz, qx, qy and qz). Since internal forces are necessary for the matrix G, first of all, a calculation according to first order theory has to be executed, as they are for calculations using second order theory. For this calculation it is advisable to assume a load for which a sufficient bearing capacity is to be verified. The critical load factor

cr that has been determined with Eq. (5.82) then relates to this load factor.

For the further understanding of eigenvalue determination, the introductory example examined in Section 5.2 can be used. Since for the bending-resistant and elastically restrained column instead of the equation system (5.82), one equation for the determi-nation of

Pcr = cr P = C /h (5.83)

is sufficient, the connection is immediately obvious. From a general perpective, the eigenvalues of critical loads Pcr should not be determined, but the eigenvalues of internal forces such as Ncr or Mcr,y. For explanation, see the homogeneous equation system (5.82), since there the right-hand side of the equation system, i.e. the load vector, has been set to zero. Moreover, in the term cr G, the internal forces are


mainly increased with cr. Loads are only entered there if they act excentrically and then they cause an additional effect.

Figure 5.19 Flexural buckling of a restrained column

As an addition to the introductory example in Section 5.2, the flexural buckling of the restrained column in Figure 5.19 is examined. Since the profile HEB 200 can deform freely (no lateral support), the flexural buckling about the weak axis is critical. For a calculation with FEM, cr = 1.4315 results and with it Ncr,z = 1.4315 290 = 415.14 kN. For checking purposes, this value can be verified by

2 2z

cr,z 2 2EI E 2003N 415.14 kN

4 4 500

since this is the critical load of the column (= Euler case I).

The methods for the solution of the eigenvalue problem are dealt with in detail in Chapter 6. The matrix decomposition method and the inverse vector iteration have proven themselves especially useful. For the matrix decomposition method the condition “determinant equal to zero” is used instead of Eq. (5.82):

det (K + cr G) = 0 (5.84)

Since cr is not known, first of all a reasonable value is assumed and then iteratively improved until sufficient accuracy is attained. The determinant is not actually being calculated, but it is only ascertained with the matrix decomposition method whether the assumed value of cr is lower or larger than the searched eigenvalue. For the second method, the inverse vector iteration, the following equation is applied:

(K + start G) vi+1 = G vi (5.85)

At the beginning of the calculation with this method, an initial value for cr and an initial vector v for the eigenvector have to be assumed. Provided that the initial value and the initial vector are chosen sensibly, the iteration converges using Eq. (5.85) and the desired eigenvalue results. Futher details are given in Chapter 6. In Section 5.10,

5.10 Eigenmodes / Buckling Shapes 203

the example of Figure 5.18 is continued and the first eigenvalue as well as the related buckling shape are determined.

5.10 Eigenmodes / Buckling Shapes

The equation system (5.82) for the solution of the eigenvalue problem with FEM is the starting point not only for the determination of cr, but also for the determination of the eigenvector v. This vector describes, mathematically, the eigenmodes of structural systems. Engineers also refer to them as “buckling shapes”.

For the determination of eigenvectors numerable methods are available. We could, for example, use the inverse vector iteration according to Eq. (5.85), which leads to the eigenvector in the case of a corresponding convergence. The determination and estimation of eigenmodes requires in-depth knowledge. Therefore, the topic is covered in Chapter 6 in connection with the equation systems and eigenvalues. Eigenmodes or buckling shapes are needed for the following reasons:

For the calculation of eigenvalues according to Section 5.9 we can (with the relevant experience) check whether the searched eigenvalue has in fact been determined and whether the structural system has been entered correctly. Fur-ther, we can ascertain which stability case (flexural buckling, lateral torsional buckling, torsional buckling) is decisive or whether there are, in some case, decoupled subsystems (see Section 6.2.2).

For calculations according to second order theory (Section 5.8) “the equivalent geometric imperfections are to be scheduled in a way that they adjust as well as possible to the deformation figure belonging to the lowest buckling eigenvalue”, see DIN 18800-2. If the buckling shape is not known, it has to be determined in advance of the calculation according to second order theory.

Calculation example

The example in Figure 5.18 is continued and the lowest critical load and the corresponding buckling shape are determined. Since a concentrated load

xF 400kN is acting at the right-hand beam end, a constant compression ND = 400 kN is acting in the beam. The critical load factor is cr = 2.9728, so that Ncr = 2.9728 400 = 1189.1 kN results. This value can, for example, be determined with the partial matrices in Section 5.6 if the beam is divided into 20 beam elements. For the FE-modelling with three beam elements shown in Figure 4.37, we get the approximation:

cr = 2.9959 or Ncr = 1198.4 kN (100.78%)

The values have been determined in 16 iteration steps and the arithmetical accuracy is 0.0001. Figure 5.20 shows the course of the iterations.


Figure 5.20 Iterations for the determination of cr for the example in Figure 5.18.

The calculation of eigenvalues cannot be presented totally because of the numerical complexity of the iterations. Thus, the third iteration step in Figure 5.20, in which

= 3, is here examined as an example. The compression force in the beam is then N = 3 400 = 1200 kN. The analysis of the matrix (K + 3 G) with the matrix decomposition method GAUCHO in accordance with Section 6.1.5 leads to the following diagonal matrix:

97.2 0 0 0 00 1290667 0 0 0

D 0 0 1573014 0 00 0 0 31.66738 00 0 0 0 319.4213

The diagonal matrix contains one element with a negative sign. This means that for = 3 the first eigenvalue is exceeded and must be decreased. Figure 5.20 shows how the searched eigenvalue is nested through decrease and increase from to the desired accuracy. This method is discussed in more detail in Chapter 6.

There are also several methods for the determination of the buckling shape (i.e. of the eigenmode) belonging to the lowest positive eigenvalue. However, the structural sys-tem in Figure 5.18 is so simple that the buckling shape can in terms of quality be drawn without calculations.

Figure 5.21 Buckling shapes for the beam in Figure 5.18

In Figure 5.21, two buckling shapes are depicted which only differ in the directions and signs. But it should be emphasised that for eigenvalue problems the homogenous equation system (5.82) is examined, i.e. the loads qz and Fz have no influence on the

5.10 Eigenmodes / Buckling Shapes 205

eigenvalue and the eigenmode. Further, eigenmodes or buckling shapes can always only be determined in terms of quality and not in terms of quantity. The ordinates in Figure 5.21 can therefore be chosen arbitrarily.

A clear and simple method for the determination of the buckling shape can be shown with the help of Figure 5.21. The starting point has to be the FE modelling with three beam elements of Figure 4.37. After consideration of the boundary conditions, five unknown nodal degrees of freedom remain: w2, y2, y3, w4 and y4.For the determination of the buckling shape an arbitrary deformation can be assumed, it is only important that the buckling shape is not equal to zero at the chosen point. A reasonable choice suggesting itself according Figure 5.21 is w4 = 1. Since the eigen-value (critical load factor) is already known, the matrix (K + cr G) can be calculated. K is the stiffness matrix, Eq. (4.59), and G the geometric stiffness matrix, Eq. (5.80). The calculation is carried out with cr = 2.995, a value being a bit smaller than the eigenvalue. By doing so, we ensure that the first eigenmode results and not the second.

The column containing w4 is brought to the right-hand side as load vector and the following equation system results for w4 = 1:

2

y2

y3

y4

w97.218 7200 2280.2 0 01824160 335973 0 0

1824160 647987 9480.2sym. 1248053 9480.2

(5.86)

For reasons of clarity, the columns and rows containing the boundary conditions have been omitted here. For computer-oriented calculations it is more beneficial to keep the full extent of the equation system and, as described in connection with Figure 4.38, to consider the boundary conditions. Also, with this approach, the column and the row belonging to w4 are assigned with zeros. Exceptions are the main diagonal element and the element on the right-hand side, which are both assigned as equal to one, so that the calculations result in w4 = 1.

After the solution of equation system (5.86), we obtain

w2 = 0.1932 y2 = 0.001441 y3= 0.003685 y4 = 0.005683

These nodal deformation values describe the eigenmode of the system. Since only values for w2 and w4 are given, the sketching is difficult. From this point of view, it is reasonable to choose a higher number of beam elements (for example 10). However, the methodology cannot be shown as clearly as before.

Calculations using FEM are usually fully automated and it is then not possible to select a nodal deformation to be set equal to one. In computer programs, it is


therefore sometimes the first or the last free deformation which is set equal to one. This may lead to problems since the selected variable can be equal to zero in the eigenvector and then a wrong eigenmode is obtained with this method. Thus, another method is preferred, as given in Chapter 6: inverse vector iteration. Since much deeper mathematical principles are necessary for this, it cannot be shown in a shortened version here.

5.11 Plastic Hinge Theory

For the plastic hinge theory it is assumed that point-shaped plastic hinges can be established at fully plasticised cross sections. With this assumption, the ultimate limit load can be determined if plastic hinges are established until the entire structural system or parts of it become kinematic. Thus, for an n-times statically indeterminate system a maximum of n + 1 plastic hinges can be arranged.

Figure 5.22 Examples for the plastic hinge theory

Figure 5.22 contains two basic examples for the application of plastic hinge theory. If for a two-span beam a cross section is assumed which is uniform over the entire length, a kinematic chain emerges under the ultimate limit load qult in the left field. The plastic hinges occur at the inner support and in the left span. Since local rotations are allowed through the establishment of the plastic hinges, the corresponding cross sections have to be applied as loads. In the left span this is a bending moment pair where the perfectly plastic bending moment Mpl is achieved. Besides the bending moment, the influence of the shear force V on the cross section bearing capacity has to be considered at the inner support. Thus, a bending moment pair with Mpl,V is applied there, the subscript V designating the reduction due to the shear force.

5.11 Plastic Hinge Theory 207

For the plane frame in Figure 5.22 a maximum of four plastic hinges can be estab-lished since it is triply statically indeterminate. The depicted kinematic chain – a so-called combined chain – is only one possibility since also partial systems can be de-termined kinematically. It depends on the parameter , the dimensions and the chosen cross sections, whether the depicted chain, the lateral displacement chain or the beam chain becomes decisive. Furthermore, besides the bending moments, not only the shear forces, but also the axial forces have to be considered for plastic hinges. Another influence complicating the application of plastic hinge theory is the stability and second order theory, which must be examined for the plane frame in Figure 5.22. In this context, it also has to be mentioned that the equivalent geometric imperfections must be applied and that sufficient lateral bracing has to be arranged in the regions of the plastic hinges.

Verification using the plastic hinge theory is related to the verification method Plas-tic-Plastic. According Table 1.1, two methods are distinguished:

calculations with kinematic chains

step by step elastic calculations

Calculations with kinematic chains are suitable for hand calculations when dealing with simple (manageable) systems and if calculations using first order theory are sufficient. For the determination of the ultimate limit load the procedure consists of

choice of adequate kinematic chain

determination of the ultimate limit load with the help of the virtual work principle

determination of the internal forces and checking whether the bearing capacity of the cross sections is sufficient in the entire system.

Since the decisive kinematic chain is usually not known, this approach is a test method, which was originally designed for hand calculations.

With the appropriate experience and manageable systems, beam structure programs can be used and the position of the plastic hinges can be determined by trial and error. In doing so, the load, for which a sufficient bearing capacity is to be verified, is entered and as many plastic hinges as do not yet form a kinematic chain. Since, as ex-plained with Figure 5.22, bending moment pairs have to be considered at the plastic hinges, computer programs are needed where corresponding entries are possible. Moreover, it is of course helpful if the bearing capacity is immediately checked by the program.


Figure 5.23 Example for iterative elastic calculation with plastic hinge theory

More universally applicable and more computer-oriented is the method “step by step elastic calculations”. It is explained with the help of Figure 5.23. For the two-span beam verification is given that the uniformly distributed load of 80 kN/m can be carried. Lateral torsional buckling is prevented through a supporting construction (lateral bracing, elastic rotational bedding).

In the first step, the two-span beam is examined using the elastic theory. For q = 80 kN/m the bending moment results at the central support in MB = 360 kNm > 285.2 kNm = Mpl,d. Since the perfectly plastic moment is exceeded, a plastic hinge and a bending moment pair are established there in a second step. The bending moment must not be larger than Mpl,d and has to be determined by consideration of the shear

5.11 Plastic Hinge Theory 209

force. If Table 16 of DIN 18800 Part 1 (see Table 8.5 in Section 8.3.5) is used as condition for the plastic hinge, we obtain MB = 243.7 kNm and VB = 280.6 kN. The decisive interaction condition is then equal to 1.000, see “verification at the mid support” in Figure 5.23. As can be seen, the bearing capacity of the cross section in the entire two-span beam is sufficient. Due to max MF = 248.5 kNm < Mpl,d = 285.2 kNm, a plastic hinge does not emerge yet in the spans. In addition, the uniformly distributed load could be increased up to max q = 87.77 kN/m.

The plastic hinge theory is an approximation method for the plastic zones theory (see Section 5.12) since plastic zones extended in longitudinal direction of the beam are replaced by point-shaped plastic hinges. Concerning calculations using first order theory, this has no influence on the bearing capacity, but has to be corrected appropriately for calculations according to second order theory, since the plastic zones lead to major deformations. In both DIN 18800 Part 2 and Eurocode 3 equivalent geometric imperfections are therefore applied, which by approximation cover unfavourable effects due to residual stresses, geometric imperfections and plastic zones.

In constructional practice, structural safety verifications are rarely carried out with the plastic hinge theory. If at all, only pure bending beams are analysed with this method or structural systems which have to be subsequently “saved” due to an increase of the load. The following reasons predominantly lead to the fact that the plastic hinge the-ory is rarely used:

The calculations are more complex than those of elastic theory.

The superposition principle, i.e. the combination of load cases, is not valid any longer.

The effort for stabilising components (bracing, elastic rotational bedding) is considerably larger.

Dispersions concerning the yield strength demand additional verifications and can amongst others have an unfavourable influence on the connections – also see DIN 18800 Part 2, element 759.


5.12 Plastic Zone Theory

5.12.1 Application Areas

In order to analyse the actual load-bearing behaviour of beams and frameworks, experiments or calculations using plastic zone theory have to be carried out – also see Section 1.4. For these calculations the load-bearing behaviour is shown realistically, so that, compared to other calculation methods, the most precise results can be achieved. However, complex calculation programs with an exceptional performance scope are necessary whose application demands extensive knowledge and experience. In addition, the time expenditure for the execution of the calculations according to plastic zone theory is very high since the preparation of the calculations requires careful consideration and the results have to be checked closely.

For these reasons, plastic zone theory is presently applied in the following cases:

recalculation of experiments

parameteric studies for checking or development of verification procedures

calculative analysis in the area of expert advice, for example to detect collapse causes

Stability checks for structural application are currently not carried out with the plastic zone theory because they are time-consuming and error-prone. This is partly due to the available computer programs, which have to be further developed for speedy and safe usage. Still, it can be assumed that in the next ten years calculations using the plastic zone theory will increase and will therefore also become increasingly important for structural practice. Initially, people will certainly focus exclusively on simple applications such as flexural buckling of beams. However, for beam structures susceptible to losing stability and for lateral torsional buckling of beams with and without scheduled torsion, a certain restraint is advisable because these are considerably more complex application cases.

5.12.2 Realistic Calculation Assumptions

If a test is to be checked again with plastic zone theory, certain input information about the characteristics of the test specimen (structural system) are required. Here, the following points are of fundamental significance:

geometric imperfections

size and distribution of residual stresses

elastic-plastic material behaviour

5.12 Plastic Zone Theory 211

yield strength and E-modulus including dispersion in the cross section and in longitudinal direction

dimensions of the cross sections (plate thickness, length, shape)

Figure 5.24 Geometric imperfections for members

When performing a calculation according to plastic zone theory, assumptions for the values have to be made, even though we are not performing a calculation related to an experiment. For geometric imperfections, which are to capture the deviations from the regular system line, usually the values given in Figure 5.24 are assumed. It is common usage to apply initial bow imperfections w0 or v0 = /1000 as sine half wave or quadratic parabola. Concerning initial sway imperfections, Lindner argues in [62] that because of new measurements, 1/400 instead of 1/300 is sufficient. Wolf discusses the approach of geometric imperfections in [92] and also makes a suggestion for the initial rotations 0 with reference to the torsional buckling of beams being held undisplaceably at both ends.

Residual stresses emerge in rolled sections, through rolling and cooling, as well as in welded sections through strong local heating during welding and the subsequent cooling. For rolled sections with I profile, the residual stresses in Table 5.5 can be applied. The parabolic distribution is certainly more realistic, and the linear distribu-tion a simplification or approximation.

In [92], Wolf shows that for flexural buckling higher ultimate limit loads result for the parabolically distributed residual stresses than for the linearly distributed. The reason is that the stress zero crossing is further outside and resulting tension forces emerge in the flanges.


Table 5.5 Size and distribution of residual stresses for rolled sections

A further essential assumption relates to material behaviour. The starting point is the stress-strain relationship for constructional steel in a tensile test outlined in Figure 1.10. Usually, a linear elastic-perfectly plastic behaviour is assumed for the calculations, see Figure 5.25a. Since due to E = 0, plasticised areas have no stiffness any longer here, numeric problems may occur. They are moderated with the assump-tion in Figure 5.25b, because the calculations with Ev = E/10000 or a similarly small value are more stable. According to DIN 18800 Part 2, a strain hardening is also al-lowed to be considered if it ranges over locally limited areas. The material behaviourwith strain hardening shown in Figure 5.25c goes back to the ECCS publication No. 33 [12] and has been supplemented by the horizontal line with = fu, as in [92].

Figure 5.25 Assumptions for material behaviour for calculations with plastic zone theory


5.12.3 Influence of Imperfections

A look at Figure 5.26 immediately shows that compared to the ideally straight beam additional bending moments result for an initially deformed beam. Since they have some effect on the bearing capacity, they can not be neglected. Additional internal forces and moments also result for bending beams when the lateral torsional buckling is examined and initial deformations v0(x) are applied for example. Here, among others, torsional and warping moments occur which tend to reduce the bearing capacity. The influences mentioned also affect calculations using elastic theory as a result of equivalent geometric imperfections. Therefore, they are not fundamentally new in terms of plastic zone theory.

Figure 5.26 Perfectly straight and initially deformed compression member

The influences as a result of residual stresses and plastic zones, which can be covered by approximation with equivalent geometric imperfections for calculations with the verification method Elastic-Elastic and Elastic-Plastic, can only be captured directly by plastic zone theory. What influence the residual stresses have in doing so is to be principally explained with the help of Figure 5.27.

Figure 5.27 Example of the plasticising of the flange ends of rolled profiles under consideration of the residual stresses


As an example, the compression member on the left of Figure 5.26 is considered. An I-shaped rolled section with h/b 1.2 is chosen and a rectilinear residual stress distri-bution with a compression stress of x 0,5 fy at the flange ends according to Table 5.5 applied. In addition, a constant compression stress as a result of the compression force N is added in Figure 5.27; however, for reasons of clarity, only the upper flange of the rolled section is considered. From the superposition with the re-sidual stresses, it follows that, with the assumption of the material behaviour in Figure 5.25a, 22.5% of the upper flange is plasticised. Since there E = 0, these areas do not contribute to the calculation of the bending stiffnesses EIy and EIz. This effect has an especially strong impact on the weak axis for flexural buckling, because the flange ends providing the most substantial contribution to the moment of inertia about the weak axis are affected. Thus, the critical load Ncr,z decreases strongly and for initially deformed beams large deflections emerge and, as a result of this, additional bending moments. Therefore, the bearing capacity is influenced unfavourably by residual stresses, which has been clearly shown by this example of flexural buckling about the weak axis, being generally the case for stability problems.

5.12.4 Calculation Example

Details, explanations and calculation examples concerning plastic zone theory can be, among others, found in [92] and [39]. Basic relationships will be explained here, using an example from [92], which is shown in Figure 5.28.

The testing beam BE-IPE 200-1 depicted in Figure 5.28a was experimentally exam-ined in Berlin in a research project carried out conjointly by the departments of steel construction in Aachen, Berlin and Bochum. The simply supported beam with fork bearings, is 2.80 m long and has extensions of 5 cm at both ends. As a result of the eccentric load Fz, it is methodically subjected to bending and torsion and is also strongly susceptible to lateral torsional buckling.

A direct stability danger is observable for the deflection w through the sloping part of the load deformation graph in Figure 5.28b. For deflection v and rotation similar graphs result. In the test, a load of Fz = 38.04 kN was reached and with the computer program ABAQUS an ultimate limit load of Fz = 38.72 kN was calculated.

For the calculation with ABAQUS (plastic zone theory) a beam element with seven degrees of freedom per node is used, as described in Section 4.2.1, and the beam is divided into 200 elements. Residual stresses and geometric imperfections in the form of initial bow imperfections are applied. Details can be taken from [39]. The load was increased incrementally. Bending moments My and Mz as well as warping bimoment M are decisive for the bearing capacity of the beam.


Figure 5.28 Calculation results for the testing beam BE-IPE 200-1


Therefore, they are shown in Figure 5.28c on the left-hand side in related form (my = My/Mpl,y and so on). In an additional examination with the partial internal force method [25] and a strain-oriented computer calculation it was found that the bearing capacity of the cross section is utilised, when reaching max. Fz = 38.72 kN, by only 89.1% .

This result is confirmed in qualitative terms by Figure 5.28d since only the upper chord is plasticised area by area. Thus, the bearing capacity of the beam is not limited through the bearing capacity of the cross section, but through the eigenvalue failure of the partly plasticised system. With the help of Figure 5.28c on the right, it also becomes clear that the ABAQUS program cancels the calculation if at the end of the sloping branch the cross section bearing capacity is utilised by 100%. In this state, it gives Fz = 30.13 kN.

6 Solution of Equation Systems and Eigenvalue Problems

6.1 Equation Systems

6.1.1 Problem

For the analysis of structural systems with the finite element method linear equation systems are of the following form:

K v p (6.1)

The stiffness matrix is symmetric and has n rows and columns. Here, n is the number of unknown nodal deformations. For a structure with four unknown nodal de-formations the following equation system is obtained:

11 12 13 14 1 1

22 23 24 2 2

33 34 3 3

44 4 4

k k k k v sk k k v s

k k v ssym. k v s

(6.2)

Due to its symmetry, all elements being laterally reversed to the main diagonal are pairwise equal and we have kij = kji. This characteristic can also be expressed as KT = K, where KT is the transposed matrix to K which emerges by writing the rows as columns. Frequently, the stiffness matrix has a distinctive band structure, i.e. all elements of the matrix above – and because of the symmetry also below – certain secondary diagonals are equal to zero. For symmetric matrices the main and secondary diagonals containing nonzero values are counted to determine the bandwidth. The consideration of the band structure leads, depending on the band width, to a considerable saving of calculating time and memory capacity – see [60] and [79].

If, as in the following example, only one main diagonal and the directly adjacent sec-ondary diagonal show nonzero values , the band width is m = 2:

11 12

21 22 23

32 33 34

43 44

k k 0 0k k k 00 k k k0 0 k k

(6.3)

For calculations using second order theory linear equation systems also result:


6 Solution of Equation Systems and Eigenvalue Problems 218

K G v p (6.4)

Since the geometric stiffness matrix G is symmetric and has the same band structure as K, there are no differences for Eqs (6.1) and (6.4) concerning this matter. However, it has to be observed that for stability problems (K + G) is less well conditioned than matrix K, which unfavourably affects the numeric solvability of the equation system. As long as matrix K is always positive definite, this does not apply for (K + G) close to eigenvalues – also see Section 6.2.

6.1.2 Solution Methods

Equation systems (6.1) or (6.4) serve for the calculation of unknown deformations in the vector v. For the solution, the equation systems on the left are multiplied by the inverse matrices and for Eq. (6.1), for example, we get:

1K K v E v 1v = K p (6.5)

The calculation of inverse K 1 in Eq. (6.5) is extremely complex for large equation systems and is usually unnecessary for the solution of practical problems since other methods need shorter calculating times. Using the inverse is only then of advantage when there are many load vectors and their number is larger than the number of equations of the equation system. This practically never occurs.

There a numerable solution methods for solving linear equation systems. Here, a dis-tinction is made between direct and indirect solution methods. An essential topic in technical literature is the improvement of the conditioning through corresponding reorganisation and calculation techniques. Examinations in [60] have shown that problems treated within the context of this book can be reliably and efficiently solved with direct methods. The following topics are therefore covered in the following Sections:

the Gaussian algorithm

the Cholesky method

the Gaucho method

Summing up, they are called “matrix decomposition methods” since this is a basic characteristic of these methods. In technical literature, the expression “elimination methods” is frequently used – see for example [79]. For application purposes, the Gaucho method is recommended and it is thus also explained in more detail here. The method is not only suitable for the solution of Eqs (6.1) and (6.4) but also for the determination of eigenvalues and eigenmodes – also see Section 6.2. The choice suggests itself for the following reasons:

6.1 Equation Systems 219

reliable solution method

limited use of memory capacity

short calculating times

Note: Microsoft Excel contains a procedure for the formation of the inverse (“MINV”). This procedure is very powerful and can be advantageously used for up to 30 unknowns. The unknown nodal deformations can then be calculated with the help of a matrix multiplication (procedure “MMULT”).

6.1.3 Gaussian Algorithm

The best-known method for the solution of linear equation systems is the Gaussianalgorithm. Here, the matrix K is divided into a left and a right triangular matrix:

K L R (6.6)

For the left triangular matrix all values on the main diagonal are equal to one. The following example shows the decomposition and the chosen designations:

11 12 13

21 22 23

31 32 33

1 0 0 r r rL R 1 0 0 r r

1 0 0 r(6.7)

An advantage of the method is that it can also be used for the solution of asymmetricequation systems. Nevertheless, in the context of the symmetric systems given here, because of the memory space required and the long computing times, this is a disadvantage. In connection with the Gaussian algorithm, it is worth mentioning its applicability where the matrix is not positive definite – see [14]. The matrix elements in Eq. (6.7) can be calculated with the calculation formulas for the Gaucho method in Section 6.1.5. For the transformation the following formulas are valid:

TL Z (6.8)

R D Z (6.9)

After the decomposition of matrix K into the triangular matrices first of all a additional vector u is determined via a so-called forward substitution and then the unknowns in vector v are calculated via back substitution. The necessary arithmetic operations can be conducted as described in Section 6.1.5.


6.1.4 Cholesky Method

The Cholesky method is widely used for the solution of linear equation systems with FEM. It is similar to the Gaussian algorithm, but it utilises the symmetry properties. The decomposition

TK C C (6.10)

into triangular matrices is effected in the following form:

11 11 12 13T

12 22 22 23

31 32 33 33

c 0 0 c c cC C c c 0 0 c c

c c c 0 0 c(6.11)

An important precondition of the method is that the matrix K is positive definite. This is always the case for Eq. (6.1), but for Eq. (6.4) not always so. This characteristic becomes obvious in the course of the decomposition, since for the calculation of the main diagonal elements cii roots have to be extracted and therefore all radicands must be larger than zero. The elements of the triangular matrix C can be calculated as follows using Section 6.1.5:

C D Z (6.12)

The calculation of the unknown nodal deformations in vector v is effected as for the Gaussian algorithm via forward and back substitution.

6.1.5 Gaucho Method

The Gaucho method is a development of the Gaussian method for symmetric matri-ces. It has considerable similarities with the Cholesky method, so that here, as in [77], the name “Gaucho” is used (Gauss, Cholesky). The method is also explained in [79], where no new name is coined, rather it is treated in connection with the Gaussianalgorithm.

For the Gaucho method the matrix K is decomposed into three matrices TK Z D Z, (6.13)

which are defined as follows:

11 12 13

12 22 23

13 23 33

T

1 0 0 d 0 0 1 z zz 1 0 0 d 0 0Z 1 zz z 1 0 0 d 0 0 1

D Z (6.14)


Through the division into the matrices Z and ZT the symmetry of K is utilised. As for the matrix L for the Gaussian algorithm the main diagonal elements in Z are equal to one. D is a pure diagonal matrix containing only values on the main diagonal. The decomposition according Eq. (6.14) can even be carried out if the matrix K is not positive definite. Thus, the method is especially suitable for calculations using second order theory and for the determination of critical loads (eigenvalues) – also see Eq. (6.4). An important distinguishing feature here are the signs of the main diagonal elements dii: if all values are larger than zero, we are below the first eigenvalue. If for example matrix (K + G) in Eq. (6.4) is decomposed and two elements of matrix D are negative, this means that we are between the second and third eigenvalue. Details on this are given in Section 6.2.3.

The solution of Eq. (6.1) with the Gaucho method, which can also be used for (6.4), if K is replaced by K + G, can be classified as follows:

equation system TK v Z D Z v p (6.15)

decomposition of KTK Z D Z (6.16)

forward substitution set TD Z v u Z u p u (6.17)

considering matrix D set Z v w D w u w (6.18)

back substitution Z v w v (6.19)

Below, the calculations for the determination of matrix elements dii and zic are com-piled:

For i = 1 to n dii = kii For j = 1 to i – 1

dii = dii – djj z2ji (6.20)

zii = 1 (6.21)

For c = i + 1 to n zic = 0 For j = 1 to i – 1

zic = zic – djj zji zjczic = (kic + zic)/dii (6.22)


The instructions mentioned above for the calculation of the values dii and zic have been compiled with regard to an understandable identification of the calculation steps. For a concrete realisation it is reasonable to use a vector that includes the main diagonal element dii instead of the matrix D. Moreover, the values dii and zic can also be directly saved in matrix K for saving memory capacity.

After the decomposition of matrix K, the determination of vector v is effected in three steps. Here, first of all the additional vector u is determined via the so-called forward substitution, starting in the first row. After that, the supporting vector is determined by dividing the elements of u by the main diagonal elements of D. In the final step, the back substitution, started in the last row, the determination of the required vector v is conducted. The required calculation operations are compiled in Eqs. (6.23) (6.25).

For i = 1 to n ui = pi For j = 1 to i 1

ui = ui – zji uj (6.23)

For i = 1 to n wi= ui/dii (6.24)

For i = n to 1 step ( 1) vi = wi For j = i + 1 to n

vi = vi – zij vj (6.25)

The Gaucho method is ideally suited to computer programs. Reference [36] contains a compilation of the necessary procedures, programmed in Visual Basic. Access to the RUBSTAHL report 1-2004 is possible at: www.rub.de/stahlbau/publikationen.

6.1.6 Calculation Example

An application of the Gaucho method is shown for the cantilever with two concen-trated loads depicted in Figure 6.1. For calculation using FEM, the beam is divided into two beam elements with lengths 1 = 3 m and 2 = 1 m.

Figure 6.1 Cantilever with two concentrated loads


Figure 6.2 Arrangement of the equation system in principle for the example in Figure 6.1

Since only uniaxial bending using first order theory is examined, six nodal degrees of freedom result: w1, y1, w2, y2, w3 and y3. Figure 6.2 shows the arrangement of the equation system K v p in principle after adding the geometric boundary conditions (hatched lines).

Due to the boundary conditions – restraint at node 1 – the first two rows and columns can be dropped, so that a linear equation system for the determination of the four un-known nodal deformations w2, y2, w3 and y3 remains. Then, the stiffness matrix Kbecomes:

4772.731 204545.6 4602.276 230113.820454560 230113.8 7670460

K4602.276 230113.8

sym. 15340920

(6.26)

With Eqs. (6.20) (6.22) we obtain the following matrices Z and D for the decompo-sition of K into ZT D Z:

1 42.85713 0.9642854 48.214270 1 0.0028125 0.187500

Z0 0 1 199.99820 0 0 1

(6.27)

4772.732 0 0 00 11688322 0 0

D0 0 71.91151 00 0 0 958824.4

(6.28)

The additional vectors u and w and the required vector v, with Eqs (6.23) (6.25)produce:


20857.1426

u46.87508249.916

3

5

1

3

4.191 107.333 10

w6.51843 10

8.604 10

3

3

1.5252858.213 10

v2.3726678.604 10

(6.29)

6.1.7 Additional Notes

In this chapter, the solution of linear equation systems has been covered with the Gaussian algorithm, the Cholesky method and the Gaucho method, a combination of the first two methods. Despite there being numerous methods in technical literature, only a limited selection are practical and sufficient for the problems addressed in this book. Since the topic is only treated briefly here, it is worth referring to the relevant literature; here [79], [96], [78] and [2] are of special interest.

In contrast to the iterative solution method, the abovementioned methods are directsolution methods. They are suitable for linear equation systems with band structure, which are usually symmetric for practical problems in civil engineering. The Gaus-sian algorithm, also suitable for asymmetric equation systems, is often called the Gaussian elimination method in technical literature. The name characterises the solution method where the unknown values are eliminated step by step by corresponding calculation operations. In this book, the expression “matrix decom-position” is used since a clear separation between the decomposition of a matrix and the following calculation of the unknown nodal deformations is con-ducted. The basis of the method is the decomposition of the matrix into triangular matrices.

6.2 Eigenvalue Problems

6.2.1 Problem

The starting point for the solution of eigenvalue problems is always the basic homogeneous equations. In FEM, formulated with unknown displacements, this is the homogeneous matrix equation:

cr,r rK G v 0 (6.30)

Here, cr,r are eigenvalues and vr eigenvectors which describe the eigenmodes (modal shapes). A system of equations with n equations, i.e. n degrees of freedom, always has n eigenvalues and associated eigenvectors, designated by “r” as subscript. They do not all have to be different. It often happens that an equation system has two (or even more) equal eigenvalues. Eigenvalues and eigenmodes can only be obtained

6.2 Eigenvalue Problems 225

correctly if the FE model allows it and the eigenmode can be described exactly with the degrees of freedom included in vr. Even if the eigenmodes corresponding to low eigenvalues are covered precisely with the chosen model, the accuracy for higher eigenvalues will decrease since, in general, the shapes include a higher number of maxima, minima, points of contraflexure or zero crossings. This is not problematic though, because often only the determination of the lowest eigenvalue and, where required, the corresponding eigenmode, is necessary. Sometimes the second and third eigenvalues have to be calculated. Higher eigenvalues and eigenmodes are only occasionally necessary for plate buckling or oscillation problems. The subscript “r” in Formula (6.30) is often omitted, which leads to:

crK G v 0 (6.31)

The critical load factor cr refers to the lowest eigenvalue, and v is the associated eigenvector. The terms eigenvalue, eigenvector and eigenmode are standard mathematical language. Instead of eigenvalue, the terms critical load and critical stress are used by engineers. Eigenmodes being described through their eigenvectors are called “buckling shapes” for member and plate buckling. If Eq. (6.31) – which is, to be more precise, actually a condition – is formulated in words, the following task can be stated for buckling: the lowest critical load factor cr and all values of the eigenvector v describing the buckling shape have to be determined so that the value zero occurs in each row after the execution of the calculation operations of Eq. (6.31). As far as the determination of the critical load is concerned, we often don’t bother solving the condition in formula (6.31) but use the following condition instead:

crdet K G 0 (6.32)

The determination of eigenvalues and eigenmodes is an extremely challenging task, both mathematically and from the engineering viewpoint. A look at the relevant literature shows that many solution methods exist. In many cases, it is not directly noticeable which method is appropriate or practical for a given task. Since they are all iterative procedures, the convergence behaviour is of vital importance. After a calculation, we always need to ask whether the required eigenvalue and associated eigenmode really have been found. The evaluation is fairly difficult and therefore it requires some experience. In the following Section, some explanations will be given and then two methods, the matrix decomposition procedure and the inverse vector iteration, will be dealt with in detail.

6.2.2 Explanations for Understanding

Often one must judge whether the calculation results of computer programs are cor-rect or not. Maybe a eigenmode has been determined which does not seem to be ap-propriate or even has a negative eigenvalue.


Positive and negative eigenvalues It depends on the way of thinking of the programmer and on the solution method used whether negative eigenvalues are calculated as well as positive ones or not. Figures 6.3 and 6.4 show two examples.

Figure 6.3 Eigenvalues for compression and tension members

For the compression member at the top of Figure 6.3 an engineer of course knows that the member is susceptible to buckling and that therefore eigenvalues will appear. If a mathematician complements “all eigenvalues are positive”, presumably the engineer hardly takes notice of this specification. For the member in tension the mathematician will certainly come up with the result “all eigenvalues are negative”. It could very well be that the engineer replies indignantly: “Eigenvalues do not occur, the tension member is not at risk of buckling”. Certainly, the answers are connected with different ways of thinking and they are of little importance for the tension members. Engineers usually calculate positive eigenvalues and if they want to analyse the lateral torsional buckling of the beam in Figure 6.4 exposed to a suction loading, they apply the load as positive acting upwards. From the mathematician’s point of view, positive and negative eigenvalues emerge for the structure in Figure 6.4.

This is, for example, of interest if the beam in Figure 6.4 is a doubly symmetric I-pro-file and the superimposed load acts at the lower flange. A computer program could then determine a negative eigenvalue, because it is the lowest in absolute value.This result corresponds to the eigenvalue of a suction loading, even though a load acting downwards (superimposed load) was supposed to be analysed. In this context, the solution method used is of major importance. The lowest absolute eigenvalue is always determined with the inverse vector iteration in Section 6.2.4. If the lowest positive eigenvalue is to be determined, as seen in the example above, the solution method has to be programmed in such a way that a spectral displacement is carried out.

Figure 6.4 Lateral torsional buckling of a beam with a distributed load qz


Structural systems with several independent eigenmodes

If the critical load factor and the eigenmode are determined for the asymmetric frame with a hinged column in Figure 6.5, the solutions shown can emerge. Either the buckling of the single-haunched frame occurs, which is also referred to as system failure, or the flexural buckling of the hinged column is identified as the decisive sta-bility failure mode. The case determined with a computer program for the stability analysis depends on the stiffnesses EI of the frame and the column. Since both cases are important for the design, the second eigenvalue has to also be determined here and, provided that the verification method with equivalent geometric imperfections is to be applied, also the corresponding second eigenmode. While the portrayed cir-cumstance is immediately obvious with the simple system in Figure 6.5, for more complex systems the question of decoupled partial systems has to be addressed sys-tematically.

Figure 6.5 Decoupled stability behaviour of a single-haunched frame with a hinged column

In conjunction with independent eigenmodes of decoupled partial systems, the differ-ent forms of stability failure occurring for beam structures, which are explained in detail in Chapter 9, are also very important. A differentiation is made between:

flexural buckling about the weak axis (eigenmode: v(x))

flexural buckling about the strong axis (eigenmode: w(x))

torsional buckling (eigenmode: (x))

lateral torsional buckling (eigenmode: v(x) and (x) coupled)

The four cases are shown in Figure 6.6 for a single-span beam. If the lowest eigenvalue is calculated with an FE program, only one solution can be determined. This is either case 1 with Ncr,z = 2595 kN or case 4 with qcr,z = 239 kN/m depending on the values inserted for N and qz, since the program calculates the lowest critical load factor cr. Cases 2 and 3 are the higher eigenvalues and eigenmodes that also arise if, for example, we take the conditions v = = 0 or v = 0 into account at mid-span.


The example of Figure 6.6 is supposed to show that different stability cases can occur and that the result of an FE calculation depends on the given loading, the geometric boundary conditions and the stiffness relationships. For the design, each of the four cases can be of importance if they possess eigenvalues close together. In that case, higher eigenvalues and eigenmodes are to be calculated in addition to the lowest one.

qz acts on the shear centre, N in the centroid.

Case 1: Flexural buckling about the weak axis

Ncr,z = 2

z2EI 2595kN

Case 2: Flexural buckling about the strong axis

Ncr,y = 2

y2

EI7379kN

Case 3: Torsional buckling (special case of lateral torsional buckling)

Ncr, = 2

T2z y

EI AGI 7118kNI I

Case 4: Lateral torsional buckling

22z

cr,z T2 2 2EIEI8q 1.12 GI

239kN m

Figure 6.6 Differentiation of the buckling cases for beam structures

In Figure 6.7, the flexural buckling of a symmetric two-span girder with an intermediate hinge is analysed. Since both spans are decoupled from each other due to the hinge, for both of them Euler case II is decisive and the first eigenvalue is equal to the second one. The figure shows that different eigenmodes can be determined with an FE program and that all of them are correct solutions. With regard to the design, the first and second solution are slightly problematic since in each case the eigenmodes are equal to zero in one field, even though equivalent geometric im-perfections are also to be applied there using the equivalent geometric imperfections


method. This circumstance is immediately obvious for the simple system in Figure 6.7, but one should have it in mind if, for instance, the spans in both fields differ.

Figure 6.7 Eigenmodes for the flexural buckling of a symmetric two-span girder with intermediate hinge

It is much more difficult to evaluate the lateral torsional buckling of the system in Figure 6.8 than it is for the flexural buckling in Figure 6.7. Using a commercial FE program, the eigenmode of b) has been determined regarding a distributed torsional spring of c = 10 kNm/m. It cannot be completely correct, because it has to be sym-metric or antisymmetric. With the program FE-beams (see Section 1.7), one obtains the expected result and realises that the first and second eigenvalue are practically equal. However, if a very strong distributed torsional spring stiffness is regarded, numeric difficulties result for programs, shown with the eigenmodes determined by ABAQUS and KSTAB with c = 200 kNm/m. The strong torsional stiffness leads to a decoupling of the areas at the beam ends which are critical in terms of lateral torsional buckling, and the inner field acts like a restraint. Independent eigenmodes also occur in these areas, which have different ordinates and are qualitatively undetermined, which is, by the way, the case for eigenvalue problems in general. The example is supposed to make the interpretation of comparable calculation results easier and it should already have shown that the determination of eigenmodes is a challenging task for a program.


Figure 6.8 Eigenvalues and eigenmodes (x) for the lateral torsional buckling of a beam with weak and strong distributed torsional springs

6.2.3 Matrix Decomposition Method

In Section 6.1, the solution of linear equation systems with the help of the matrix de-composition method was dealt with. The Gaucho method, which has been described in Section 6.1.5, is used here for the determination of the eigenvalues. In contrast to the other methods, it allows the consideration of the symmetry (Cholesky method) and the calculation of higher eigenvalues (Gaussian algorithm).

The starting point for the determination of eigenvalues with the matrix decomposition method is the condition:

crdet K G 0 (6.33)


If the decomposition of the matrix has been performed using the Gaucho method, the determinant can be calculated with the elements of the diagonal matrix D:

n

ii 11 22 33 nni 1

det K G d d d d d (6.34)

Here, is an appropriate estimated value for the required eigenvalue cr.

Figure 6.9 Eigenvalues 1 3 and associated eigenmodes w(x) for the flexural buckling of a two-span beam

As an example of the execution of the calculation, the flexural buckling of a two-span beam is analysed in Figure 6.9. In order to ensure a clear illustration, we set EI/ 2 = 1 and the beam is (only) split up into two finite elements. Under consideration of the boundary conditions, we obtain the following homogeneous system of equations:

ya

cr yb

yc

K G v

4 2 0 2 15 1 30 0 02 8 2 1 30 4 15 1 30 00 2 4 0 1 30 2 15 0

(6.35)

If the estimated value is altered from 0 to 60, the value of the determinant can be calculated with the help of Eq. (6.34) with the result shown in Figure 6.10. The com-parison with the exact values of Figure 6.9 shows that the determined zero crossings 12, 30 and 60 are inaccurate. The first eigenvalue is 21.6% too large and the others 48.5% or 52%. This is certainly due to the beam being only divided into two elements. However, at this point, the exactness is not of importance; we are just illustrating the solution method.


Figure 6.10 Value of the determinant for the two-span beam of Figure 6.9

Note: Microsoft Excel includes a powerful procedure (“MDET”), which is able to calculate the value of determinants. With this, the zero crossings and the changes of sign can be analysed as shown in Figure 6.10 for example. However, with this methodology two equal eigenvalues cannot be identified since the determinant does not show a change of sign. An absolutely reliable method for the determination of eigenvalues is described below.

In FE programs, the value of the determinant is not calculated because it is not actually needed and because the information about the change of sign of the determinant is sufficient for the determination of the eigenvalues. In Figure 6.10, the areas with positive and negative signs are designated for that purpose. For the ele-ments d11, d22 and d33 of the matrix D the following applies:

0 < < 12: All values are positive.

12 < < 30: One value is negative.

30 < < 60: Two values are negative.

A generally applicable methodology can be developed from the circumstance shown with this example, and two aims for the search can be distinguished:

At the beginning, an interval is defined through a specific variation of , where the required eigenvalue is located, so that bottom cr top is valid. As nec-essary and sufficient criterion, the number of changes of sign of det (K - G)is used, i.e. the number of negative diagonal elements dii is counted. If the r-th eigenvalue is to be determined, it is located in the interval stated above, as long as negative diagonal elements for bottom (r 1) and for top (r) can be determined. Furthermore, it is completely clear that no further eigenvalues apply to this interval.


After an interval has been found in which cr is located, it can be reduced itera-tively, for example by bisection. The iteration is conducted until a prede-termined exactness is achieved. It has to be noted that it is a calculation exact-ness referring to the cr computable with FE modelling. In the example of Figure 6.9, this is the value 12 for the first eigenvalue and not the exact solu-tion 9.87. As an example for the realisation of the iteration, the first eigenvalue is calculated for the two-span beam of Figure 6.9. For the search of the interval it is increased by a factor of 5 and afterwards the interval is repeatedly halved. The iteration steps are shown in Figure 6.11.

Figure 6.11 Iteration for the determination of cr = 12 for the two-span beam of Figure 6.9

Advantageously, the iteration for the determination of the first positive eigenvalue is started with 1 = 1.0, because this is the loading which is to be carried. All diagonal elements of D are positive, so that 1 is smaller than cr. 1 is now increased by the factor 5 until at least one diagonal element is negative. This is the case for 3 = 25 and the following interval boundaries occur: bottom = 5 cr 25 = top. The following matrices can be used for a better comprehension of the calculations:

1 = 1:

3.86667 0 0D 0 6.66408 0

0 0 3.24626

3 = 25:

0.66667 0 0D 0 0

0 0 1.4163410.7083

19 = 11.999817 12:

2.40002 0 0D 0 2.40009 0

0 0 0.00012


The iteration now proceeds by using an interval bisection, starting with 4 = (5 + 25)/2 = 15. Since the decomposition leads to a negative diagonal element, 4 is used as the upper boundary again and the estimated value is then 5 = (5 + 15)/2 = 10. This localisation is continued until the eigenvalue is included in

the interval to the desired exactness. For a deviation of not more than 10 4 the value of 19 = 11.999817 is determined after 19 iterations steps; this is close to cr = 12. If the exact eigenvalue of cr = 12 is inserted, the matrix becomes singular since the last diagonal element d33 is zero.

The calculation of the higher eigenvalues is performed in a similar manner. If the second eigenvalue K,2 is sought for example, the estimated value r is increased until two negative diagonal elements emerge. If the iteration starts with 1 = 1 again and increased by a factor of 5, this is the case for 4 = 125. A further approximation of the eigenvalue is executed with the interval bisection [25, 125] and then, after 21 steps, the eigenvalue 21 = 29.9995 cr,2 20 = 30.0003 is localised with an exactness of 10 4.

The following matrices D result:

4 = 125: 21 = 29.9995 30 12.667 0 0

0 22.331 00 0 10.964

0.00006 0 00 146737.6 00 0 0.00012

Table 6.1 Decomposition of the matrix F (n n) and determination of the main diagonal elements dii 0

p = 0 For i = 1 to n dii = fii For j = 1 to i – 1

dii = dii – djj z2ji

If dii 0, then p = p + 1 Where required: if p = k, then leave slope i (for the determination of the kth eigenvalue) zii = 1 For c = i + 1 to n

zic = 0 For j = 1 to i – 1

zic = zic – djj zji zjc

zic = (fic + zic)/dii

The values zic and dii can also be written into the matrix F.


The main focus of an eigenvalue search with the matrix decomposition method Gaucho are the arithmetic operations of Table 6.1 compiled in a computer-oriented notation. The matrix

TF K G Z D Z (6.36)

is decomposed into the matrices D and Z as well as ZT, as shown in Section 6.1.5, and the number of main diagonal elements dii being smaller than (negative!) or equal to zero is determined. The result is the value p, which leads to the information that is smaller than or equal to the (p + 1)th eigenvalue:

cr,p+1, if p diagonal elements dii 0 (6.37)

Because normally a certain eigenvalue is sought, as for instance the kth, the calcu-lations can be aborted according to Table 6.1, if p = k. The row designated with “if necessary” shortens the computing time.

Determination of the eigenmodes

If the required eigenvalue is determined with the Gaucho procedure, as explained above, it is appropriate to determine the eigenmode according to the method stated in Section 5.10. In this process, a deformation variable in the vector v is chosen and set to one and then the corresponding column of the matrix (K – cr G) is put on the right-hand side representing a load vector. A system of equations emerges that can be described as follows:

cr iK G v q v 1 (6.38)

The formulation with Eq. (6.38) implies that the size of the equation system does not change. In the matrix, all elements of the rows and columns corresponding to the cho-sen deformation are set to zero with the exception of the element of the main diagonal, which is set to one just like the value of the load vector. The system of equations (6.38) can be solved using the Gaucho method and as a result we obtain the nodal deformations of the vector v describing the eigenmode.

With the method described, the eigenmode can in many cases easily be calculated. However, for a general use in computer programs running fully automatically, the method poses certain difficulties, since an appropriate variable vi = 1 has to be set. In computer programs, the first or the last deformation variable is often set to one. This approach may lead to a wrong eigenmode for the structural systems in Figures 6.5 to 6.8, because many independent eigenmodes can emerge, as described in Section 6.2.2. For the single-haunched frame with a hinged column of Figure 6.5 this is an obvious difficulty since the eigenmode has to be determined for the decisive case. This, of course, also applies to the four stability cases of Figure 6.6: if for torsional buckling (case 3) the decisive (i.e. lowest) eigenvalue occurs, the correspondingeigenmode (x) has to be determined. Choosing a nodal deformation variable as mentioned above, only a value of or may be set to one. Otherwise, the wrong


eigenmode will be determined. An additional difficulty can be seen with the help of Figure 6.9. If the second eigenmode is supposed to be determined, for instance, ybmay not be set to one, because yb = 0 for that eigenmode. In general, a deformation variable may not be set to one which is equal to zero in the sought eigenmode. This is a strong restriction since the eigenmode is not known and usually it cannot be deter-mined as easily as in Figure 6.9.

6.2.4 Inverse Vector Iteration

With the classic vector iteration according to von Mises, the specific eigenvalue problem

(A - E) x = 0 or A x = x (6.39)

can be solved. If the iteration

xi+1 = A xi (6.40)Ti i 1

Ti i

x xx x

(6.41)

converges, the largest absolute eigenvalue of the matrix A is obtained.

However, for stability problems, the lowest positive eigenvalue is usually sought and not the largest absolute value. The general eigenvalue problem

(K + cr G) v = 0 (6.42)

can be transferred to a specific eigenvalue problem with the help of matrix operations in order to determine the lowest absolute eigenvalue cr. After the multiplication of Eq. (6.42) from the left with the inverse of K and the division by cr, we obtain:

1

cr

1K G E v 0 (6.43)

This complies with Eq. (6.39). With it, the largest absolute eigenvalue = 1/ cr can be calculated and for the lowest absolute eigenvalue cr = 1/ . However, Eq. (6.43) has only limited applicability since for many structural applications convergence problems occur. This problem can be solved with an adequate spectral displacement

0 using:

cr = 0 + cr (6.44)

With this, the general eigenvalue problem can be formulated and solved as shown in Table 6.2.


Table 6.2 Modified vector iteration for the solution of stability problems

General eigenvalue problem: 0 crK G G v 0

Choose suitable spectral displacement 0.Choose adequate start vector v0.Conduct modified vector iteration:

ii

i

vvdomv

1i 1 i0v K G G v

Ti i

cr,i Ti i 1

v vv v

Result, if the iteration converges: Lowest absolute eigenvalue cr of 0 crK G G v 0

Eigenvalue cr = 0 + cr of (K + cr G) v = 0

The numerically reliable use of the vector iteration requires a further method in addi-tion to the spectral displacement. If a large number of iterations are needed and the numerical values increase excessively or decrease rapidly, the available range of numbers of the computer is often insufficient. The eigenvectors must therefore be kept within admissible limits through standardisation or other methods. In Table 6.2, this is achieved by dividing the elements of the eigenvectors by the largest absolute element, dom vi:

ii

i

vv

dom v (6.45)

Table 6.2 contains the inverse of the matrix 0K G for the calculation of

1i 1 i0v K G G v (6.46)

Its calculation is only reasonable for small equation systems if one wants to use an existing function or procedure, e.g. MINV of Microsoft Excel. For banded matrices the solution of the system of equation

i 1 i0K G v G v (6.47)

using a matrix decomposition method is more advantageous (shorter calculating time). The decomposition of 0K G is incidentally only necessary once and not at each iteration step. An appropriate matrix decomposition method is dealt with in detail in Section 6.1.5.


Notes on the application of the modified vector iteration

For the application of the modified vector iteration three aspects are of particular importance:

selection of the spectral displacement 0

selection of an appropriate starting vector 0v

checking whether with cr, the sought eigenvalue, has been found or not

Basically, the following correlation is valid: the closer 0 is located to cr and the more the starting vector and eigenvector are alike, the faster the vector iteration converges.

Figure 6.12 Eigenvalues and eigenmodes for the lateral torsional buckling of a beam

For the explanation of the selection of the spectral displacement 0 the lateral tor-sional buckling of the beam in Figure 6.12 is considered. For the case zp = 0 (bottom of the figure) the positive and negative eigenvalues are equal in size. With an execution of the vector iteration without a spectral displacement, which is 0 = 0, the procedure diverges, because the spans to the first positive and first negative eigen-value are equal. For that reason, a spectral displacement 0 0 being close to the


required eigenvalue has to be chosen, but in any case it has to be located far enough away from the others. This is also valid for the determination of higher eigenvalues. It is always highly disadvantageous if the mean value of the two adjacent eigenvalues is chosen for 0. However, as mentioned above, the assumed starting vector also has a crucial influence on the convergence or divergence of the procedure.

In the case of Figure 6.12 with the load of qz acting at zp = h/2, the first positive eigenvalue can be determined without any problem using 0 = 0. It has to be taken into account though, that for many structural systems the first negative eigenvalue is closer to = 0 than the first positive eigenvalue (e.g. the structure in Figure 6.12 with zp = +h/2). In that case, the result of the vector iteration is the first negative eigenvalue. Depending on which eigenvalue is to be found, an appropriate spectral displacement then has to be applied.

Besides the spectral displacement, an appropriate starting vector has to be chosen for the execution of the vector iteration. The starting vectors are completely inappropriate if they are orthogonal to the sought eigenvector, i.e. if

Tstart eigenv v 0 (6.48)

Hence, a starting vector leading to symmetric deformations may not be chosen if the sought eigenmode is antisymmetric (and vice versa). In principle, the following procedures can be distinguished for the selection of starting vectors:

1. Allocate random numbers to the starting vector. 2. Set one element or some elements of the starting vector equal to one (the oth-

ers equal to zero). 3. Choose a deformation variable and set it equal to one – see Section 6.2.3. 4. Set “disturbance loads” (vector s ) and calculating deformations v with

0K G v s .

For the explanation of the four procedures the example of Figure 6.9 is picked up again. It is assumed that always an appropriate spectral displacement is carried out.

“Random numbers” Starting vectors including random numbers almost always lead to a good con-vergence of the vector iteration. Bad convergence behaviour or even diver-gences are very rare. In cases of doubt, one should repeat the vector iteration, if necessary, several times.

“Elements equal to one” This method often leads to the desired result. If all elements of the starting vector are set equal to one for the two-span girder of Figure 6.9, however, it is the third eigenvector T

3v 1 1 1 . With this starting vector, only the third eigenvalue can be determined.


“Deformation variable is set vi = 1”This method often leads to a good convergence behaviour. As explained in Section 6.2.3, a deformation variable may not be set equal to one if it is equal to zero in the required eigenvector. Difficulties also occur if decoupled partial systems emerge since it is not clear for which partial system a deformation value has then to be set equal to one.

“Disturbance loads” Rating this method is comparable to “deformation variables set to vi = 1”. For example, a disturbance load qz may not be applied to both fields of the two-span girder (Figure 6.9) if the first eigenvalue (antisymmetric) is desired to be calculated since, with that, symmetric deformations emerge. For decoupled partial systems the load has to be applied to the decisive partial system in any case. Also, the disturbance load must lead to deformations corresponding to the proper failure case. No flexural buckling about the weak axis can be generated with a load qz for example, as for instance in case 1 of Figure 6.6.

RecommendationsThe previous explanations show that the application of the vector iteration requires deepened knowledge concerning the selection of the starting vector and the spectraldisplacement. If the first positive eigenvalue is sought, the choice of 0 = 1 is nor-mally a practical assumption since in the stability verification for this loading level an adequate bearing capacity is supposed to be proven. For the starting vector the allo-cation with random numbers is a promising approach in many cases. If there are any doubts whether the correct eigenvalues are determined, they can mostly be clarified by multiple repetitions using new random numbers each time. However, absolute certainty can only be achieved with a check using the so-called Sylvester test, i.e. with the decomposition of the matrices. In Section 6.2.5, the combination of “inversevector iteration” with the “matrix decomposition method Gaucho” is therefore treated.

Calculation example

In Section 6.2.3, the flexural buckling of a two-span girder is examined in detail with the matrix decomposition method Gaucho. The eigenvalues for the chosen FE model are: 12, 30 and 60. The corresponding eigenmodes can be taken from Figure 6.9 as follows:

T1v 1 1 1 ; T

2v 1 0 1 ; T3v 1 1 1

With the inverse vector iteration now the first eigenvalue and the first eigenmode are to be determined here. A spectral displacement does not have to be considered, because the first eigenvalue is sought and only positive eigenvalues can emerge. Therefore, the assumption of 0 = 0 leads to the lowest positive eigenvalue since it is


the lowest absolute value. With that, we have for the modified vector iteration of Table 6.2

1i 1 iv K G v (6.49)

where the subscript “i” indicates the iterations. No indication of the number of the eigenvalue is given here. The values of the starting vectors are determined with the help of random numbers. For that purpose the function “random number( )” in Microsoft Excel can be used, for example. Since values between 0 and 1 occur and negative values are required as well, the range can be extended to values between 1and +1, and with z = 2 (random number 0.5) it can be converted. “By sheer chance”, this calculation here leads to

0v 0.4906 0.3329 0.6984

The vector iteration can now be started. Since K and G are only 3 3 matrices, the calculations can be conducted with the functions MINV (inverse) and MMULT (ma-trix multiplication) in MS Excel. In each step of the iteration, the Rayleigh quotient

Ti i

cr,i Ti i 1

v vv v

(6.50)

can then be calculated. As stated in Table 6.2, the eigenvectors are divided by the largest absolute element idom v in each iteration step. For i = 1 we get:

1v 0.0976 0.5569 1.0000

cr,1 24.9476

The iteration steps 2 to 7 lead to the following approximations for the first eigenvalue: 16.2769; 12.8129; 12.1315; 12.0206; 12.0032; 12.005. Although the starting vector is relatively unfavourable, the eigenvalue is determined very precisely in the seventh step with an error of only 0.4 %. In contrast, the eigenvector

7v 0.9911 0.9956 1.0000

does not have this exactness. However, it can be improved very fast with further iterations.

6.2.5 Combination of the Solution Methods

In Sections 6.2.3 and 6.2.4, different methods are treated for the determination of eigenvalues and eigenmodes. With the matrix decomposition method Gaucho,arbitrary eigenvalues can be calculated and it is absolutely certain that the result is correct. However, it is only conditionally suitable for the determination of


eigenmodes and the calculating times are usually definitely higher than with compa-rable vector iterations. Moreover, Sylvester test is needed for the inverse vector iteration anyway; using a matrix decomposition, it determines whether the vector iteration has led to the required eigenvalue and eigenmode or not. Therefore, it seems reasonable to combine both methods – this is also suggested in [60] and [35].

Combination of the matrix decomposition method Gaucho and the inverse vector iteration

It is practical to classify the iterative calculation into four steps:

1. Search of interval (matrix decomposition)

At the beginning of the calculations, an interval bottom cr top is determined in which the required eigenvalue is located. This can be carried out with the matrixdecomposition method Gaucho, as described in Section 6.2.3.

2. Decrease of the interval (matrix decomposition)

After an interval has been found, it is decreased with the Gaucho method until as approximation of cr is available with a relatively rough exactness, e.g. within about 1%. If there could be many eigenvalues close together, we need to improve the approximation a bit.

3. Inverse vector iteration for cr and v

We now switch to the inverse vector iteration (see Section 6.2.4), and the approxi-mation of the Gaucho method from step 2 is used as spectral displacement, i.e. we have:

0 = cr,Gaucho = ( top + bottom)/2

Of course, an appropriate starting vector v1 for the vector iteration is needed. Many comparative calculations have shown that it is best to assign the starting vector with random numbers. The vector iteration is then carried out until the eigenvalue is calculated with the desired exactness. Generally, 10 4 is adequate, whereas for certain cases (see e.g. Figure 6.8) a higher accuracy is required.

4. Check (matrix decomposition)

Using a matrix decomposition according to the Gaucho method, we check whether the calculated eigenvalue under point 3 is the one we want. With this procedure, the eigenvector then also contains, with a high probability, the corresponding eigenmode. Absolute certainty is only obtained for special problematic cases if crof point 2 and 3 is calculated with a high exactness.

Calculation example

The combination of the solution methods is shown for the flexural buckling of the two-span girder in Figure 6.9. The first eigenvalue and the corresponding eigenmode are determined.


1. Search of interval

The calculations in Section 6.2.3 with the matrix decomposition method show that the first eigenvalue is located between 5 and 25. This result can be discovered after the third matrix decomposition, as shown in Figure 6.11.

2. Decrease of the interval

Starting with the fourth matrix decomposition, the interval [5, 25] is reduced. In the iteration steps i 4, the following values result for when bisecting: 15; 10; 12.5; 11.25; … The interval in which cr is located is getting smaller and smaller. A criterion is needed to decide when the decrease of the interval is considered to be finished. A conclusive answer to this question can not be given; it depends on the complexity of the problem. If, for example, it is assumed that the eigenvalue is to be determined with an exactness of about 10%, this can be discovered with the use of the interval boundaries. Since the mean value of the last interval is used as the new approximation for , the following criterion leads to the discontinuation of the interval reduction:

top

bottom

1 1 0.12 (6.51)

In the example, this criterion is reached for i = 7. With Figure 6.11, we have

1 12.5 1 0.055 0.12 11.25 (6.52)

and it is now known that the eigenvalue is located between 11.25 and 12.5.

3. Inverse vector iteration

The interval boundaries for cr mentioned above are the origin of the vector itera-tion of Section 6.2.4. As starting value

0 = (11.25 + 12.5)/2 = 11.875 is used. If the starting vector of the calculation example in Section 6.2.4 is chosen, among others, we obtain the following results:

i = 1 cr = 12.727913 Tv 0.963097 0.984225 1.000000

i = 2 cr = 11.999973 Tv 0.999741 0.999878 1.000000

i = 3 cr = 12.000000 Tv 0.999998 0.999999 1.000000

i = 4 cr = 12.000000 Tv 1.000000 1.000000 1.000000

As can be seen, the vector iteration leads to the exact solution very fast (within the scope of FE modelling!). Incidentally, it is also considerably faster than the consistently conducted matrix decomposition procedure in Section 6.2.3 – see Fig-ure 6.11 as well. The vector iteration can be aborted if cr has reached the desired


exactness, e.g. 10 4. Complex tasks (e.g. beams with distributed torsional springs, as in Figure 6.8) demand a higher exactness for cr if the eigenvector is to be determined precisely. It has to be “iterated cleanly” then, so further iteration steps are often required.

4. Check In order to check whether the desired eigenvalue and its eigenmode have actually been calculated with the vector iteration, the matrix decomposition method is used again. For cr we have d33 = 0 at the decomposition and both the other main diagonal elements of D are positive. Therefore, the result is approved. For structural problems it is rather unusual for the exact eigenvalue determined as shown in this example. Since usually approximation values are calculated, it is more suitable to conduct the checks u = 0.999 cr and o = 1.001 cr. With that, you can ascertain whether the eigenvalue is located in this interval and that the determined eigenmode belongs to this eigenvalue. For this example, we get, as expected, the following results:

u = 0.999 12: d11, d22 and d33 are positive.

o = 1.001 12: d11 and d22 are positive, d33 is negative.

7 Stresses According to the Theory of Elasticity

7.1 Preliminary Remarks

Within the scope of the beam theory, cross sections can be stressed by up to eight internal forces (see Chapters 4 and 5). It is assumed here that the internal forces and moments have already been calculated and that now the stresses need to be deter-mined.

The compilation of internal forces as “resultants of the stresses” in Table 1.2 (Section 1.6) shows the relationship between the internal forces and the axial and shear stresses. According to this, the following stressing results:

N, My, Mz and M x

Vy, Vz, Mxp and Mxs

Structure

The structure of this chapter is based on the different calculation methods. The following list provides a rough overview of the contents of the chapter:

x due to N, My and Mz

y zx

y z

M MN z yA I I

due to Vy and Vz (thin-walled cross sections)

open cross sections: z y y zxs

y z

V S (s) V S (s)I t(s) I t(s)

due to Mx = Mxp + Mxs (thin-walled cross sections)

open cross sections: xpxs

T

Mt(s)

I; xs

xsM A (s)

I t(s)

single-celled hollow cross sections: xpxs

m

M2 A t(s)

x due to M

IM

x

section includes hollow cells.


This list indicates that it is of significance for some internal forces whether the cross

7 Stresses According to the Theory of Elasticity 246

Equilibrium between internal forces and stresses

Internal forces and stresses in the cross section have to fulfil the equilibrium condi-tions. The equilibrium has been formulated in Section 3.4.2 with the help of the vir-tual work principle. If one, theoretically, starts with the internal forces, these are replaced by equivalent stresses. For generating the equilibrium Figure 7.1 is helpful. At the positive intersection the internal forces and moments are given and at the negative intersection, the stresses in the cross section.

Figure 7.1 Equilibrium between the internal forces (right) and stresses (left)

Figure 7.1 is also to make clear that either the internal forces or the stresses act in a beam intersection. If both internal forces and stresses are sketched in, this is not really correct. However, sometimes it can be useful to aid clearity. The sign or the direction of action of the stresses then results from the “replacement” of the internal forces by the stresses.

For the common case of uniaxial bending with axial force Figure 7.2 shows the cor-rect approach. In a beam section, the internal forces N, My and Vz are replaced by positive stresses x and xz at the positive intersection. When using the equilibrium conditions (Figure 7.2 at the bottom), dx 0 is assumed for the length of the beam section, since the internal forces or the stresses, respectively, are considered at onespecific position x.

7.2 Axial Stresses due to Biaxial Bending and Axial Force 247

Figure 7.2 Stresses and xz due to N, My and Vz

Material behaviour

In this chapter, the determination of the stresses is performed with the theory of elas-ticity, i.e the universal validity of Hooke’s law is assumed. According to Eqs. (1.5) (1.7), the relationship of strains or shearing strains, respectively, and stresses is then:

xx E (7.1a)

G (7.1b)

For details on Hooke’s law refer to [25].

7.2 Axial Stresses due to Biaxial Bending and Axial Force

If it is assumed that the cross section does not rotate about its longitudinal axis ( 0 ), according to Section 1.6, Eq. (1.1), we have

S z yu(x,y,z) u x y x z x (7.2)

for the displacements in the longitudinal direction of the beam. Since y and z only occur linearly for the rotations y and z, a plane area is described by Eq. (7.2). It complies with the Bernoulli hypothesis about the cross sections remaining plane. With xx E and ux it follows that

x S z yE u y z (7.3)


The stress is described by the elongation in the centre of gravity Su , the derivatives of the rotations z and y , the ordinates y and z in the principal coordinate system and the modulus of elasticity. The stress distribution also corresponds to a plane area, as can be seen from Eq. (7.3).

With the internal forces as resultants of the stresses in Figure 7.3, we obtain with regard to Eq. (7.3):

x SA

N dA EA u SNu

EA (7.4a)

y x y yA

M z dA EI yy

y

MEI (7.4b)

z x z zA

M y dA EI zz

z

MEI (7.4c)

For the integrations it was taken into account that in the y-z principal coordinate system

Ay = Az = Ayz = 0 (7.5)

is valid and that Ayy = Iz and Azz = Iy. If now Su , z and y from Eq. (7.4a c) are inserted in Eq. (7.3), we get the familiar formula for the determination of

yI

MzI

MAN

z

z

y

yx (7.6)

The formula is valid for completely arbitrary cross section shapes:

thin-walled, thick-walled or solid cross sections

with or without hollow cells

Figure 7.3 N, My and Mz as stress resultants of x

7.2 Axial Stresses due to Biaxial Bending and Axial Force 249

Figure 7.4 Distribution of the x stresses as plane area over the cross section with individual portions due to N, My and Mz

The distribution of x across the cross section, as already mentioned, corresponds to a plane area: a constant portion and linearly varying portions overlap in y and z. For the depiction in Figure 7.4 a rectangular cross section was chosen, so that the distribution of x can be observed without problems.

For the verification the form of a stress check, the largest magnitude of stress is always needed. The critical cross section points are at the edges of the cross section, i.e. points that are far away from the centre of gravity S.

Example: max x for a rolled profile IPE 330

Internal forces: N = 150 kN IPE 330: A = 62.6 cm2

My = 80 kNm Wy = 713 cm3

Mz = 6 kNm Wz = 98.5 cm3

Stress: 2x

150 8000 600max 2.40 11.22 6.09 19.71 kN cm62.6 713 98.5

The maximum stress occurs for y = –8.0 cm and z = +16.5 cm, i.e. at the bottom of the lower flange at the right edge. Here, the section moduli Wy and Wz have been used since they can be taken from profile tables (see [29]).

Note: As is common practice, the determination of x due to N, My and Mz has here been carried out in the y-z principal coordinate system. In Section 4.4.6, it is pointed out that an arbitrary y-z reference system can also be used. The computer-oriented calculations being required to do so are covered in more detail in Section 8.4.

In some application cases, it may be reasonable to carry out the stress distribution in the y-z centre of gravity system. The different reference systems are shown in Figure 2.5 and here in Figure 7.5. In the y-z centre of gravity system, the condi-


tions 0AA z~y~ are fulfilled. However, z~y~A is not equal to zero, leading to the following formula:

2z~y~z~z~y~y~

z~y~z~z~z~z~y~y~y~y~x AAA

z~Ay~AMy~Az~AMAN

The formula enables the determination of x without knowledge of the y-z prin-cipal coordinate system since the bending moments relate to the axes y~ and z~ .

y~y~A , z~z~A and z~y~A are the associated second degree area moments, which is covered in detail in Section 2.3.

Figure 7.5 Bending moments yM and zM in the y z centre of gravity system

7.3 Shear Stresses due to Shear Forces

7.3.1 Basics

The shear forces Vy and Vz generate shear stresses in a cross section. If this is looked at the other way around, for equilibrium reasons, the integration of the shear stresses over the cross section area must result in the shear forces – see Figure 7.6. This relationship has already been discussed in Section 7.1.

Figure 7.6 Shear forces Vy and Vz as stress resultants of xy and xz

7.3 Shear Stresses due to Shear Forces 251

Clear definition of the shear stress directions and utilisation of the symmetry properties

For later calculations it is advantageous to define the direction of the shear stresses by viewing. The shear forces can only be introduced into the cross section parts which can carry the relevant shear stress components. For carrying Vy, shear stresses xy inthe direction of Vy are required and, accordingly, xz in the direction of Vz. Figure 7.7 shows four thin-walled cross sections being loaded by vertical shear forces. Sketch b shows the cross section portions carrying the shear forces and the direction of action. Sketch c will be explained later.

All cross sections in Figure 7.7 are symmetric with respect to the z-axis. For cross sections 2 and 4 it is immediately obvious by viewing that V is carried by both webs in equal amounts. For cross section 3 this is true for the vertical components of the resulting shear forces in the inclined webs.

From the directions of the shear stresses in Figure 7.7b it can also be concluded that it must betrue that = 0 on the axis of symmetry. This becomes even clearer with the sketches under c, showing the horizontal directions of the shear stresses.

Figure 7.7 a) Four cross sections and shear forces V b) Direction of the shear stresses of the absorbing cross section parts c) Additional shear stresses needed for equilibrium reasons


Figure 7.8 Utilisation of the symmetry for shear stresses and partial internal forces due to shear forces

The following may serve as a standard rule:

For symmetric cross sections a symmetric shear stress distribution results due to a shear force acting in the direction of the axis of symmetry. At the intersection points of the profile centre line and the axis of symmetry, the shear stress is equal to zero. Provided that the profile centre line and the axis of symmetry coincide, as in case of the doubly symmetric I-cross section in Figure 7.8 top, the cross section can be theoretically separated along the axis of symmetry, so that two U-cross sections emerge. For the hat cross section shown at the bottom of Figure 7.8, cutting open the cross section is not necessary, since profile centre line and axis of symmetry do not coincide. A symmetric shear stress distribution results for a zero-crossing at the intersection point of the profile centre line and the axis of symmetry.

Figure 7.9 Shear stresses xs in thin-walled cross section parts, shear flow Txs,profile ordinate s and boundary conditions


Shear flow T = · t and = constant over t

Shear stresses cannot “go beyond” the edges of the cross section. Therefore, they must run tangentially at the edges or be equal to zero – see Figure 7.9. For thin-walled cross section parts the obvious assumption resulting from this is that the shear stresses are constant over the plate thickness, the resulting shear flow is Tx = xs · t (unit: kN/cm).

Frequently, the cross section is also only represented by its profile centre line, and the shear flow Txs is indicated – also see Figure 7.10.

Figure 7.10 Equilibrium of the shear flows in the cross section corners and directions of Txs in the cross section

Equilibium at the cross section nodes

The shear stresses indicated in Figure 7.7c can only be determined if the longitudinal direction of the beam (x-direction) is taken into consideration. According to [25], the following relationship between shear stresses in the cross section and in the longitudinal direction is valid due to the moment equilibrium:

xy = yx , xz = zx , xs = sx (7.7)

Also, the positive directions of the shear stresses in the cross section are needed. The moment equilibrium is only possible if the arrowheads or the arrow ends meet in the element corners. Based on this knowledge, the equilibrium at the transitions between the cross section parts can be formulated. With Figure 7.10, for a cross section corner we have:

22,sx11,sx2,sx1,sxx ttTT:0F (7.8)

With sx = xs, for the shear stresses and shear flows in the cross section, we get:


xs,1·t1 = xs,2 ·t2 and Txs,1 = Txs,2 (7.9)

From examination of the equilibrium it can be concluded that if one shear flow at a cross section node is known in terms of size and direction, the other can be determined directly.

The most application cases it is not required to formulate the equilibrium conditions explicitly. We can think of it as “flowing”, as if water were led into a pipe with a tube and flowed around the corner. The sketches in Figure 7.11 show the principle for cross section corners and T nodes. As can be immediately seen, the directions in Figure 7.7c starting from chart 7.7b have been referenced appropriately.

Figure 7.11 Shear flow T at cross section nodes

7.3.2 Calculation Formula for

Shear stresses due to Vy and Vz are usually calculated by formulating the equilibrium in the longitudinal direction. For that purpose, a segment of a beam is considered as shown in Figure 7.12, where the intersections are at the position of a particular shear stress xs = sx to be determined. It is assumed that the segment has a constant thickness t(s) and that the shear stress shows constant distribution across the plate thickness. This leads to the following equilibrium condition:

x sx sx,AA(s)

d dA(s) (s) t(s) dx 0

xsx sx,A

A(s)

d1(s) dA(s)t(s) dx

(7.10)

Regarding Eq. (7.6) and Table (1.3) the derivative of the normal stress can be expressed via shear forces:


y zx

y z

M Mz yI I

yzx

y z

VV z yI I (7.11)

The introduction of Eq. (7.11) into Formula (7.10) leads to

yzsx sx,A

y zA(s) A(s)

VV(s) z dA(s) y dA(s)I t(s) I t(s)

(7.12)

since Vy, Vz, Iy and Iz are independent of the profile ordinate s and therefore constant factors for the integration. sx,A is the shear stress in the chosen integration starting point A (s = 0).

Figure 7.12 Equilibrium in the longitudinal direction of a beam segment

7.3.3 Open Cross Sections

For open cross sections, the conditions sketched in Figure 7.9 are valid:

xs is always tangential to the edges

perpendicular to the edges xs = 0

With that, the shear stresses can be determined for open cross sections using Eq. (7.12) since the determination of the sx,A, which can also be considered as an integration constant, poses no problem. Advantageously, the edge of a cross section is chosen as starting point A, for which sx,A = 0, leading to

z y y zsx

y z

V S (s) V S (s)(s)

I t(s) I t(s) (7.13)

where Sy (s) and Sz (s) are the static moments taking the cross section into account until the regarded cross section point at the position s is reached:


yA(s)

S (s) z dA(s) and zA(s)

S (s) y dA(s) (7.14)

Because xs = constant across t(s), the shear flow results to:

st)s()s(T xsxs (7.15)

Figure 7.13 shows the calculation method for a rectangular cross section subjected to a shear force Vz. The shear stresses have to be determined at the position z = z1. It is reasonable to start with the profile ordinate s at the upper edge since then the directions of s and Vz coincide. It is obvious that xz (= xs) must then have a positive value. The static moment is

2z8hb2z2hz2hbzS 21

2111y (7.16)

With that and t(s) = b, the shear stress is

2z

8h

IVz

21

2

y

z1xz (7.17)

Of course, we can also start at the lower edge. The static moment is then positive and xz(z1) negative. From the negative xz it follows that it acts in the opposite direction

to the chosen s direction. Thus, the result complies with the one determined above.

Figure 7.13 Determination of xz due to Vz for a rectangular cross section

Example: Static moments and weld thicknesses for a T-cross section according to Figure 7.14

The shear centre M is at the intersection point of the profile centre lines. It is also directly obvious that this is where Vy and Vz act, since Vy must be carried by the up-per flange and Vz by the web.

The static moment Sz (to Vy) has a parabolic progress at the upper flange. It is nega-tive since it started with the integration on the right end of the upper flange (where y


is negative). The arrows indicate the positive direction of the profile ordinate s. It complies with the actual direction of action of the shear stresses. This results from the illustration and with the calculation formula due to the negative static moment. In the web, Sz = 0, since it is on the z-axis (y = 0). From this it follows that the welds due to Vy are not stressed. Therefore, it is sufficient to choose the weld thickness con-structively in this case.

Figure 7.14 Static moments Sy and Sz for a T-cross section

The static moment Sy (to Vz) varies linearly and symmetrically in the upper flange. The total static moment of the flange is connected to the web by the welds. It progresses parabolically in the web and has its maximum at the centre of gravity:

2y w 2max S t h 2 . The shear stresses at the top end of the web are:

y,UFzxz

y w

SVI t

with: Sy,UF = f 1A h (7.18)

For the determination of the stresses in the welds tw must be replaced by 2a. Due to xz = zx, the shear stress also acts in the longitudinal direction of the welds, or, to put

it differently, parallel to the weld length. Thus, it is identified by ||.

||y,UFz z f 1

y y

SV V A hI 2 a I 2 a (7.19)


As an alternative, the method “using difference forces” will be explained here. For a single-span beam with a concentrated load at midspan we have:

4FMmax z

y (7.20)

In the midspan region, the force of the upper flange is:

yf f f 1 f

y

max Mmax N A h A

I (7.21)

At the supports, Nf = 0. The maximum force of the upper flange, max Nf, is therefore introduced into the upper flange via half of the beam length by the welds. Because of the fact that max Nf is a compression force, we have

||f z f 1

y

max N F A h12 2 a 2 I 2 a (7.22)

The result with Eq. (7.19) is identical because Vz = Fz/2.

Figure 7.15 Static moments Sz and Sy for a hat-like cross section


Note: In Figure 7.14 on the left, the Sy progress in the flange is shown in dashed lines because this is the decisive point for the verification of the flange’s shear stresses due to Vz. Using the shear stresses in the middle of the flange (intersection with the web axis) is not appropriate because the shear-stiff connection of the flange to the web is not there, but at the web edges via the welds.

Example: Shear flows due to Vy and Vz for the hat-like cross section in Figure 7.15

The cross section is symmetric to the z-axis. Through choosing the cross section pa-rameters differently, several cross section variants can be captured. Figure 7.15 con-tains the progress of Sz and Sy as well as the calculation formulas for max Sz and max Sy. The single ordinates are also recognisable. The shear flows are obtained with:

)s(SIV)s(S

IV

)s(T yy

zz

z

yxs (7.23)

Example: For the bridge cross section in Figure 7.16 with three main girders, it is to be determined what proportion of Vz they carry.

Since the webs only carry vertical shear forces, then, for equilibrium reasons,

V1 + V2 + V3 = V (7.24)

Because of the cross section symmetry we have:

V1 = V3 (7.25)

Due to the fact that all three lower flanges and also the three webs are identical, the shear flow in the webs is identical as well – see sketch Figure 7.16 on the right. Because of identical shear flows then

V1 = V2 = V3 = V/3 (7.26)

Figure 7.16 Bridge cross section with three webs (schematic diagram)


Since sufficiently large flanges are provided, here the middle shear stress is used for the verification:

mw w w

V V3 A 3 h t (7.27)

Notes and example: If the resulting shear force is known in an individual part, the medial shear stress can be calculated with

i

ii,m A

V

For I-shaped cross sections it is then, for example

zm

w

VA

DIN 18800 Part 1 [8], element (752), allows the design with m for cross sections with distinctive flanges since then the difference to max is minor. Figure 7.17 shows the comparison between approximation and exact solution for an IPE 450. The cross section properties are taken from the tables in [29].

Figure 7.17 Shear stresses due to Vz = 100 kN in an IPE 450

7.3.4 Closed Cross Sections

If shear stresses due to Vy and Vz are to be determined for hollow (closed) cross sec-tions, there is a general problem in applying Formula (7.12):

Contrary to open cross sections, there is no starting point A for which xs,A = 0 is valid, and when formulating the equilibrium at a point where partial plates are con-nected there may be two or three unknown shear flows.

7.4 Stresses due to Torsion 261

Since the equilibrium conditions are not sufficient to determine the shear stresses, this can be referred to as a statically indeterminate problem. The order depends on the number of hollow cells that a cross section shows. However, by considering symmetry conditions of cross sections, the number of unknowns can be reduced. Details on this issue and the application of the force method can be found in [25], for example.

Nevertheless, the statically indeterminate calculations take a lot of effort, especially if a cross section consists of several hollow parts. For that reason, numerical procedures are recommended in that case. Details can be found in Chapter 11.

7.4 Stresses due to Torsion

7.4.1 General

If beams and frameworks are loaded by torsion, the cross sections rotate about the longitudinal axis by the angle x . Torsional moments Mx occur in the cross sec-tions and these lead to shear stresses. Then again, under consideration of its lever arms, the integration of the shear stresses has to result in the internal force Mx – see Table 7.1.

The sketch on the left of Table 7.1 shows the shear stresses xy and xz with their lever arms z–zM and y–yM regarding the shear centre. On the right, xs is the shear stress in the direction of the profile centre line of thin-walled cross sections; rt is the lever arm belonging to xs. Shear stresses can also occur perpendicularly to xs. These are often equal to zero and are not considered in this overview.

The shear stresses are carried differently by the cross sections. The torsional moment is therefore divided into two portions:

Mx = Mxp + Mxs (7.28)

Mxp is the St Venant’s or primary torsional moment and Mxs the secondary torsional moment (additionally occurring in warping torsion).

The starting point for the division is the stress equilibrium, which is not shown here in detail – see [25]. Only the gist is given in Table 7.1: if a secondary torsional mo-ment exists in a beam, nonuniform axial stresses x occur in the longitudinal di-rection of the beam. These are summed up to a resultant in the internal force “warping bimoment”. The relation between warping bimoment and secondary tor-sional moment can be compared to the bending moment and the shear force:

xsMM like zy VM (7.29)


The warping bimoment is explained in Section 7.4.2. Which internal moments – Mxp,Mxs and M – occur in a structural system must be determined together with the re-spective system calculation (see Chapters 4 and 5). However, it is of basic interest here whether a calculation according to the warping torsion is required at all. In addi-tion to the type of bearing and the load, it is the cross section shape which is impor-tant in the first place. One distinguishes as follows:

Table 7.1 Shear stresses and torsional moments (overview)

Torsional moment:

Division into primary and secondary shear stresses s,xzp,xzxzs,xyp,xyxy ;

xsxpx MMM

Conditions for the division ( Mxp and Mxs)

a) Txpp,xzp,xy IGM0

zy

b) IEMxzy xsxs,xzs,xy with: MMxs

Warping bimoment

Ax dAIEM


Warping-free cross sections The warping ordinate is equal to zero in the entire cross section. The examina-tion of the warping torsion is omitted since

M = Mxs = 0 and Mx = Mxp (7.30)

i.e., the torsion is exclusively carried via the shear stresses of the primary torsion. Figure 7.18 contains a compilation of some warping-free cross sections.

Cross sections with minor warping These are cross sections where 0, but the influence of the warping torsion is very minor, so that

M 0 and Mxs 0 (7.31)

can be assumed as an approximation. Cross sections with minor torsion are similar to warping-free cross sections. These are often cross sections with relatively large hollow cells. The member characteristic T is an indicator of torsion:

EIGIT

T ( : member length) (7.32)

If T > 10, it can usually be assumed that the influence of the warping torsion is minor.

Non-warping-free cross sections These are all open thin-walled cross sections consisting of more than two plates whose profile centre lines do not meet at one point. For these cross sections, the St Venant’s torsion (Mxp) occurs together with the secondary torsion (Mxs and M ) in combination as warping torsion.

Figure 7.18 Cross sections free of warping


Section 7.3 has covered the shear stresses due to shear forces. The explanations in Section 7.3.1 are in many ways also valid for the shear stresses due to torsion. But there is a fundamental difference concerning the shear stress distribution across the plate thickness for thin-walled cross section parts. Table 7.2 shows that besides the constant (as for the shear forces) also a linearly varying shear stress distribution across the plate thickness can occur, so the classification is simplified.

Table 7.2 Stresses in thin-walled cross section parts due to Mxp and Mxs

In the following sections, the stress determination due to Mxp, Mxs and M is covered in detail. Since this is directly connected to the warping ordinate , the torsional constant IT, the warping constant I and the shear centre M, this relates to Sections 2.6, 2.7 and 4.4.5. In addition, many parts of Chapter 11 are also useful.

7.4.2 Arbitrary Open Cross Sections

Shear stresses due to Mxp

For the determination of shear stresses the primary torsional moment Mxp is distrib-uted across the individual parts. The cross section in Figure 7.19 is taken as an example.

Figure 7.19 Distribution of the torsional moment Mxp across individual parts


It consists of four parts. The assumption that the cross section does not change its shape when stressed leads to the conclusion that the twist for each cross section part has to be identical. The torsional moment Mxp can be split up, as shown in Figure 7.19, and for each part we have

Mxp,i = G IT,i i (7.33)

The torsional constants can be calculated by formulas stated in Tables as for instance 7.3 and 11.13 or by means of numerical methods (see Chapter 11). Due to the preser-vation of the cross section shape, we have:

i = (7.34)

The conditions lead to

xpT

i,Ti,xp M

II

M (7.35)

From Mxp = n

1ii,xpM , we have

n

1ii,TT II (7.36)

With the knowledge of the single torsional moments, the shear stresses in the individ-ual parts can be calculated. For thin-walled plates (see Figure 11.13) we have

max ii,T

i,xpi t

IM

here: i = 2, 3 and 4 (7.37)

and for the full circle according to [25]:

max i = 2d

IM i

i,T

i,xp here: i = 1 (7.38)

Since different cross section shapes appear as partial cross sections, a formula like the one used for the I-cross sections

max = )s(tI

M

T

xp (7.39)

is only reasonable if t(s) = d/2 is defined for the full circle. The method, which is shown here for a warping-free cross section, for the shear stress determination due to Mxp is valid for arbitrary open cross sections.

For the calculation of IT with Eq. (7.36) the torsional constants of the individual parts are required. For welded plate cross sections and rolled profiles they can be calcu-lated as indicated in Table 7.3 – also see Table 11.13.


Table 7.3 Torsion constant IT for thin-walled open cross sections

Welded cross sections: Approximation for rolled profiles:

n3

T i ii 1

1I c b t3

c = 1.18 – 1.37 (IPE)

c = 1.14 – 1.40 (HEA) c = 1.11 – 1.29 (HEB) c = 1.06 – 1.10 (HEM)

c = 1.06 – 1.13

c = 1.0 – 1.14

More precise solution for rolled I-profiles according to [41] 3 3 3

T f w2 1I t b a h 2 d t d a3 3

2f w w

f

t r t r t / 4d

2 r t2

wf

r 4a td t 2

20.46 0.5 d / a 1.15

More precise solution for rolled angles according to [32]

3 4TI a b 2 c t / 3 0.237 c

1c t 0.4 r

Shear stresses due to Mxs

Next, the shear stress distribution due to the secondary torsional moment Mxs is treated. To do so, the stress equilibrium is assumed, as it was done for the shear stresses due to shear forces. With

xsxx xs

M M, M M andI x I (7.40)

Section 7.3.2 can be used completely analogously and it follows that


stsA

IMs xs

A,sxsx (7.41)

In that

A(s)

A (s) dA(s) (7.42)

is the first degree moment of area of the warping ordinate. However, Eq. (7.41) is only valid for thin-walled cross sections (cf. Section 7.3.2). sx,A is the shear stress at the chosen starting point of the integration A. Because of the fact that, for open cross sections one always begins at the free edge, then sx,A = 0.

To contribute to the understanding, and for checking, it is advisable to look at the resulting shear forces in the individual components. Concerning the partial internal forces, these are local lateral forces Vi. For equilibrium reasons, they have to fulfil the following conditions:

n

1itiixs rVM (7.43a)

Vy = n

1iii 0cosV (7.43b)

Vz = 0sinVn

1iii (7.43c)

The angle i indicates the position of the individual component – see Figure 2.39.

Example: Shear stress distribution due to Mxs for the gutter cross section in Figure 7.20

The cross section is symmetric to the z-axis, and on the axis of symmetry = 0. The position of the shear centre has been determined beforehand (10.91 cm below the floor panel). In the single points the warping ordinate results to

1 = 0 2 = 0 + 10.91 15 = 163.65 cm2

3 = 163.65 – 15 25 = – 211.35 cm2

4 = 211.35 + 35.91 10 = 147.75 cm2

Since the warping ordinate varies linearly in all parts, A ,e can be determined with

A ,e = A ,a + Aae ( a + e)/2


Figure 7.20 Shear stresses due to Mxs for a gutter cross section

If a and e are the ends of the plates, A (s) within the plates results in 2

e a,a ae a 2

ae ae

s sA (s) A A2

For the cross section in Figure 7.20 we begin at point 4. Since the integration is not performed in the direction of the profile ordinate, a negative sign must be considered in all three plates. We get:

A ,4 = 0 max A ,43 = 304 cm4 A ,3 = +318 cm4

max A ,32 = 1 807 cm4 A ,2 = 914 cm4 A ,1 = 313 cm4

A ,1 in detail: A ,1 = 914 – 15 1 (163.65 + 0)/2 = 313 cm4

Due to the constant plate thickness of t = 1 cm, the shear stress is equal to the shear flow:

)s(AI

MT xs

The progress is shown in Figure 7.20 on the right. This shows that Mxs is mainly carried by the two webs.

The warping constant is:

I = 2 2 21 10 147.75 147.75 211.35 211.353

+ 25 2 2 2211.35 211.35 163.65 163.65 15 163.65

= 1 117 000 cm6

For Mxs = 1000 kNcm (example) the maximum shear stress is:


max 1 000 1807 1.621 117 000

kN/cm2

In Figure 7.20, also the resulting shear forces V1, V2 and V3 have been charted. With the equilibrium of moment at point 1, we get:

Mxs = 2 V2 15 + 2 V1 25 = 30 V2 + 50 V1

With the approximation V1 = 0, V2 and the central shear stress in the web can be de-termined:

xsm

M 1000 1.3330 25 1.0 750

kN/cm2 (82% of max )

The result confirms that Mxs is predominantly carried by the two webs. In many applications, minor shear stresses result due to Mxs. For the design then, a cleverly determined approximation is mostly sufficient. If in the given case only the webs are applied for carrying Mxs (and 1.5 m), one is on the safe side.

Axial stresses due to M

As a third and final internal force for the torsional stresses, the warping bimoment M is now considered. As explained in Section 7.1, axial stresses

IM

x (7.44)

result from this. The derivation of this equation can be directly developed with the method of Section 7.2, if in Eq. (7.2) the term u is considered. With

xx E and ux , it follows:

x E (7.45)

Analogously to Eq. (7.4), can be replaced by

MEI (7.46)

leading to Formula (7.44). According to the equation, I and the warping ordinate are required for the calculation of axial stresses. The determination of both values has already been mentioned in detail (also see Sections 2.5, 2.6 and 11). The distribution of x in the cross section complies with the progress of the warping ordinate. For the gutter cross section in Figure 7.20 the maximum stress occurs in point 3:

xMmax 211.35

1117 000

For practical applications, the axial stresses due to M can definitely be of crucial importance for the design.


7.4.3 Closed Sections

The calculation of axial stresses due to M can be performed as shown for open sec-tions in Section 7.4.2. However, according to Eq. (7.44), the knowledge of the stan-dardised warping ordinate is necessary. If the cross sections consist of hollow cells,

has to be determined using the force method. This leads to a high calculation effort and numerical methods can provide advantages (see Chapter 11).

In addition, as mentioned in Section 7.3.4 in conjunction with shear stresses due to Vyand Vz, the same problems and difficulties occur with stresses due to Mxs. The cal-culation of xs due to Mxp also involves high effort, especially if the cross section consists of several hollow parts. For these reasons, numerical methods presented in Chapter 11 are advisable.

7.5 Interaction of All Internal Forces and Verifications

The given internal forces can be carried by the cross section if the equivalent stress does not cross the yield strength at any point in the cross section. Consequently, the following condition must be fulfilled:

y2xz

2xy

2xv f3 (7.47)

If all internal forces act in an arbitrary combination, it is not directly obvious at which cross section point the maximum equivalent stress occurs. With the stress distribu-tions stated in the previous sections, an calculation for arbitrary cases can be con-ducted. Usually, max v occurs in a corner or on the axes of symmetry. As an aid for I-profiles, Figure 7.22 shows the individual stress distributions due to all possible in-ternal forces.

Figure 7.21 Stress curves for the combined influence of Mel and Vel

7.5 Interaction of All Internal Forces and Verifications 271

An interesting application is the simultaneous acting of the internal forces M and V for a rectangular cross section. Figure 7.21 shows the stress distribution for Mel and Vel (see Section 7.6), that is for the respective exclusive influence of one internal force. The equivalent stress can be calculated with Eq. (7.47) for the entire cross sec-tion. We see that v = fy is obtained at the upper and lower edge as well as at the centre of gravity. In between, the cross section remains elastic and reaches its minimum utilisation with 86.6% at z = 0.354 h. The conclusion is that the internal forces Mel and Vel can both be carried simultaneously.

Figure 7.22 Stresses in I-cross sections


7.6 Limit Internal Forces and Moments on the Basis of the Theory of Elasticity

If the stresses are calculated according to the theory of elasticity and the maximum value of the equivalent stress v is equal to the yield stress fy, the corresponding in-ternal forces are the limit internal forces in an elastic state. For the special case that only one internal force is involved (all other internal forces equal to zero), this limit internal force is indicated by “el” (elastic) in the subscript. Figure 7.23 shows the stress distributions corresponding to the limit internal forces Nel, Mel,xp, Vel and Melfor a rectangular cross section. For Nel and Mel it is max x = fy and for Mel,xp and Vel

it is max = R = yf 3 .

(for c values see Figure 11.13: c1 = 1 / 3, c2 = 1 / 2)

Figure 7.23 Stress distributions for Nel, Mel, Mel,xp and Vel of a rectangular cross section

8 Plastic Cross Section Bearing Capacity

8.1 Effect of Single Internal Forces

If the cross section is only loaded by a single internal force (all other internal forces are equal to zero), the limit internal force is designated by the index “pl” on the basis of the theory of plasticity (plastic).

Figure 8.1 Strains and stresses at the transition from Mel to Mpl

The stress distributions due to plastic limit internal forces must fulfil the following conditions:

The equivalent stress must not go beyond the yield strength in any point of the cross section.


8 Plastic Cross Section Bearing Capacity 274

There must be equilibrium between the internal forces and the stresses, e.g. M =

A

dAz .

The resulting internal force must be the maximum possible single internal force (all other internal forces equal to zero). This leads to the cross section plasticis-ing.

As an example, the transition from the elastic to the plastic limit bending moment is shown in Figure 8.1. The strains and stresses in the first chart show the case M = Mel,where the maximum boundary strains just reach el = fy/E. If the strain line is “rotated further”, the rectangular cross section begins to plasticise in the boundary areas. For the case max = 2 el, it is M = 1.375 Mel. The limit state is reached if the strain line is horizontal and the cross section is completely plasticised. Then it is

M = Mpl = 1.5 Mel (8.1)

Thus, the plastic shape coefficient for the rectangular cross section with regard to the influence of a bending moment is

pl = Mpl/Mel = 1.5 (8.2)

Figure 8.2 Plastic reserves of the rectangular cross section depending on the maximum boundary strain

The calculation assumption max is taken to be for pl = 1.5. In fact, the strains are limited due to the ultimate strain u – also see Figure 1.11. Further information about the connection of strains and reachable bending moments is given in Figure 8.2. As can be seen, an increase factor of = M/Mel = 1.495 belongs to

10/max el . Almost pl = 1.5 is achieved. The maximum strain for a construction steel S 355 ( u 20%) with 10 is

8.2 Limit Load-Bearing Capacity of Cross Sections 275

max = 1.71%

This is a value which is far below the ultimate strain. Therefore, the calculation as-sumption max can be used without any concerns and restrictions.

For thin-walled rectangular cross sections the plastic limit internal forces Npl, Mpl,Mpl,xp and Vpl are of interest. The stress distributions and the respective calculation formulas are compiled in Figure 8.3. The limit internal force for the primary torsional moment is here calculated with an approximation according to the box girder modelsince this is of advantage for the combined effect with the shear force. A precise Mpl,xp with the hip roof model results to

th3t61M 2

Rxp,pl (8.3)

Figure 8.3 Plastic limit internal forces and stress distributions

8.2 Limit Load-Bearing Capacity of Cross Sections


The examinations in Section 8.1 show that the rectangular cross section has substan-tial plastic reserves. With the effect of single internal forces, according to Figure 8.3, these can be up to 50% and with combinations of internal forces even considerably higher. This is shown in Figure 8.4 using the interaction of N and M.

The rectangular cross section, considered in isolation, has no significance for steel construction. For that reason, this chapter covers cross sections occurring in steel construction and methods for the verification of sufficient load-bearing capacity of


the cross sections on the basis of plastic theory. In doing so, it is assumed that the internal forces are known. Generally, these are

N, My, Mz, Vy, Vz, M , Mxp and Mxs.

The internal forces are always valid for the principal system of the cross section. According to Figure 1.8, they thus refer to the centre of gravity and the shear centre as well as to the cross section principal axes y and z.

Figure 8.4 Plastic reserves for the N-M interaction of rectangular cross sections

Moreover, it is proceeded according to DIN 18800 [8]. This means:

observance of the b/t relations for the selected verification method Elastic-Plas-tic or Plastic-Plastic, see [29] (complete involvement of all cross section parts under compression load)

linear elastic-perfectly plastic material behaviour (see Figure 1.11)

plastification condition according to von Mises – see Eq. (7.47)

where required, observance of pl 1.25 – see Section 7.3.4 of [25] or DIN 18800

Further details on the use of DIN 18800 can be found in [25]. The consideration of other preconditions, as for example according to Eurocode 3 [10], does not mean fundamental problems.


Chapter 9 includes examples on the bearing capacity of structural systems for which the limit load-bearing capacity of the cross sections is captured according to plastic theory. Detailed explanations on the use of verification methods Elastic-Elastic, Elastic-Plastic and Plastic-Plastic with examples showing basic phenomena of system load-bearing behaviour and the cross section bearing capacity are also given in the Chapters 1, 5 and 7. In this chapter, the fundamental precondition being used is that the equilibrium of internal forces and stresses must be fulfilled.

8.2.2 Plastic Cross Section Reserves

There are two reasons for verifying the bearing capacity of cross sections on the basis of plastic theory:

Compared to elastic theory, the verification requires little effort.

Higher bearing capacities and thus smaller cross sections can be allowed. This is not only important for higher cost-effectiveness and improved competitiveness but also for the preservation of resources.

For minimum effort of verification, two aspects need to be mentioned. For rolled sections the perfectly plastic internal forces can be taken from tables or data sets – see for example [29]. If a single internal force (e.g. a bending moment) occurs, the verification on the basis of plastic theory is easier and also clearer since it is directly compared with the relevant limit internal force. For a combined occurrence of severalinternal forces the verification with elastic theory can be extensive. With the partial internal forces method (PIF-method), which can be applied for many common cross section shapes and combinations of internal forces, the verification can be simplified considerably (see Section 8.3). How advantageously the PIF-method can be used is shown by the RUBSTAHL program FE-Beams. Here, the utilisation Sd/Rd of the cross section bearing capacity is determined along the entire beam length and is graphically shown, which makes an immediate evaluation possible.

Figure 8.5 Stress distributions according to elastic theory and plastic reserves


For the user it is of great interest which bearing capacity can be expected to increase using plastic theory compared to elastic theory. The plastic cross section reservesdepend, of course, on the cross section shape and the occurring internal forces. The following always applies: if the cross section in only utilised in single points up to the yield stress (elastic theory), the plastic reserves are large. Figure 8.5 gives three examples on this.

Plastic shape coefficients pl / Effect of single internal forces

Usually, the plastic coefficient is used for the evaluation of the bending moment bearing capacity (see Section 8.1). The definition

pl,M = Mpl/Mel (8.4)

can also be generalised as follows:

plpl

el

S limit internal force according plastic theoryS limit internal force according elastic theory (8.5)

For the rectangular cross section and the effect of different internal forces, pl values have already been given in Figure 8.3. They are between 1.0 and 1.5.

Figure 8.6 Shape coefficients pl for selected cross sections and bending about the horizontal axis

Since the effect of an axial force leads to constant stresses ( x = N/A), for arbitrary cross sections we have


pl,N = Npl/Nel = 1.0 (8.6)

For the effect of bending moments the pl values depend on the cross section shape. The overview in Figure 8.6 shows values between 1.0 and 2.37. Theoretically, even larger values are possible. For this, large parts of the cross section area would have to be concentrated near the neutral axis. However, for practical applications this is counterproductive. The dashed line in Figure 8.6 indicates the achievement of 0.99 Mpl. It is recognisable that for cross sections with large pl values, larger boundary strains are required compared to cross sections with small pl values (also see Figure 8.2). Rolled I-profiles are of great importance for construction purposes. For the common profiles their plastic shape coefficients are in the following range:

pl,y = 1.09 to 1.24 (bending about the strong axis)

pl,z = 1.50 to 1.60 (bending about the weak axis)

As an approximation, pl,y = 1.14 can be applied.

Combination of internal forces

Figure 8.4 shows the plastic reserves of the N-M interaction of rectangular cross sec-tions. These are 67% (for N/Npl = 0.2) at maximum and may be larger than for the single effect of the internal forces ( pl,M = 1.5 or pl,N = 1.0).

Figure 8.7 N-My limit load-bearing capacity for I-cross sections according to plastic theory (DIN 18800) and elastic theory


More interesting for application purposes than the rectangular cross section are, for example, I-shaped cross sections. For comparison purposes, in Figure 8.7 the N-Myinteraction of DIN 18800 and the respective limit bearing capacity are charted according to elastic theory. Similar tendencies result as for the rectangular cross section, although plastic reserves here are, of course, smaller. The explanations and comparisons in Section 8.3.5 show that the parameter

= Aweb/Acomplete (8.7)

has a decisive effect on the plastic reserves. This can also be seen from the following interaction relation, which is valid as an acceptable approximation for N-My-Mzcombinations of internal forces:

121

z,pl

z

y,pl

y1

pl MM1

MM

NN for rolled profiles: 0.20 to 0.45 (8.8)

For sandwich cross sections (no web) = 0 and it becomes

z,pl

z

y,pl

y

pl MM1

MM

NN

(8.9)

If pl,y = 1.14 and pl,z = 1.5 are assumed by approximation, the limit bearing capacity can be determined with

y z

pl pl,y pl,z

M MN 1N M 1.14 M 1.5 (8.10)

The case = 0.4 (40% web area) and N = 0 are to be studied as an example here. From the comparison of Eqs (8.8) and (8.10) the plastic reserve under the effect of My and Mz can be determined by approximation. It is, as shown by Figure 8.8, be-tween 14 and 85%, depending on the combination of internal forces.

Figure 8.8 Maximum plastic cross section reserve under the effect of My and Mz


Summary: For the common cross sections and combinations of internal forces the plastic reserves are usually 10 25%. However, they can be significantly larger as the example “My and Mz for I cross sections” with a maximum of 85% shows.

8.2.3 Calculation Methods and Overview

For the determination of bearing capacity on the basis of plastic theory two different calculation methods exist, which will be more or less suitable for any given problem. The problem revolves around two questions:

Which cross section shape is being used?

What internal forces occur?

Moreover, the calculation methods can be distinguished according to whether they are predominantly suitable for

hand calculation

use in spreadsheet programs

use in computer programs.

Frequently, there is no clear-cut dividing line. This depends on the individual assess-ment.

Below, four calculation methods are explained, together with their basic features:

choose stress distribution

strain iteration

partial internal forces method

choose neutral stress axis

These methods are used and explained in detail in the following sections; Table 8.1 provides an overview.

Choose stress distribution

This is the classic method and predominantly the starting point for hand calculations regarding simple problems. The principle of this method can be summarised as fol-lows:

Choose the stress distribution, so that the corresponding internal force is maximal and all others are equal to zero.


Figure 8.9 T-cross section and x stress distributions for Npl and Mpl,y

As an example, Figure 8.9 shows a T-cross section providing the equal yield stress in the entire cross section. For determining Npl, this yield stress is assumed to act in the entire cross section. The determination of Mpl,y is not quite that simple. Since the condition N = 0 must be fulfilled, the change from +fy to –fy occurs at the height of the area bisectrix and two cases have to be distinguished.

Case 1: w oA A (stress change in the web)

w w oN 0 : A 1 A A 0

o w1 A A 2 1

2pl,y w y w w w w o y oM A f h h a 2 a A f a

(8.11)

Case 2: w oA A (stress change in the flange, here simplifying assumption: o = const. in the flange)

w y o oN 0 : A f A 0

o y w of A A

pl,y y w w o o o y w w oM f A a A a f A h t 2

(8.12)

To consider the combined effect of N and My for the T-cross section, we always proceed as for Mpl,y. The stress distributions for both case 1 and case 2 can then be used for the determination of

Ax dAN and

Axy dAzM (8.13)

The sign change of fy no longer occurs at the height of the area bisectrix.


In Figure 8.10, an I-cross section consisting of different steel grades is considered: The upper flange and web consist of S 235 and the lower flange consists of S 355. In all three parts, a stress distribution with fy = 21.82 kN/cm2 occurs for Npl. The cross section is not completely plasticised in this case. If it is fully plasticised, we obtain max N = 5158 kN > Npl = 4372 kN, but we also get a corresponding bending moment of My = 157 kNm. If an axial force larger than Npl is to be allowed, it must be verified in a system calculation that the corresponding bending moment can be carried. Provided that the upper flange also consists of S 355, a complete plasticising of the entire cross section gives Npl and Npl = 5943 kN. Due to the symmetric stress distribution, no bending moment occurs.

Figure 8.10 Welded I-cross section consisting of different steel grades: Npl and max N

The example in Figure 8.10 shows that for the achievement of the limit bearing ca-pacity the cross section does not have to be necessarily fully plasticised.

Strain iteration

The method being described as strain iteration here is also a long-known classic method. It is covered in detail in Section 8.4, but two variants are distinguished here:

Variant 1: The internal forces are directly applied in their full size.

Variant 2: The internal forces are increased step by step. More information on this can be found in Section 8.4.3 – see Table 8.10.

The principle of the first variant can be described as follows:

application of given internal force as cross section load calculation of strains in the entire cross section determination of the stresses due to the strain distribution (here only xinternal forces)

el: x = E or > el: x = sgn ( ) fy


calculation of internal forces from the stresses distribution by integration, and determination of the cross section properties of the elastic rest cross sectionif the given internal forces are different than the ones of the previous step, repeat the calculation (iteration)

Because of the numerical effort involved, this method is only suitable for realisation by computer programs. Here, the cross section is often separated into fibres and stripes (see, for example, Figure 2.18). This method is, for example, used byRoik/Kindmann in [71] and Kindmann in [44] for the determination of the cross section bearing capacity due to the x internal forces and for calculations of frames according to plastic zones theory (see Section 5.12). A detailed description with examples of the method as well as explanations of the determination of shear stresses can be found in Section 8.4. For the following comments the focus is on the principle of this method.

The starting point for the determination of the cross section strains is Section 1.6 and Eq. (1.4a):

MMSx wzvyuu (8.14)

Here, Su is the strain at the centre of gravity, Mv and Mw the two flexions as well as the second derivative of the twist. With the first three terms, a plane area is

described complying with the Bernoulli hypothesis about the cross sections remaining plane. The fourth term belongs to warping torsion and leads to a warped strain area.

Figure 8.11 T-cross section with two strain distributions for My

The strain iteration method is, besides its use in computer programs, also helpful for understanding. Figure 8.1 shows the transition from Mel to Mpl using the example of the rectangular cross section. For explanation, we again come back to the calculation of Mpl,y (case 1) from Figure 8.9 and different strain distributions are considered in Figure 8.11. Here, the strains are described by the linear equation


MSx wzuz (8.15)

For y,ely MM we have 0uS and M el w ww h 2 a . Larger bending moments result if Mw is increased (Figure 8.11). Due to the condition N = 0, the position of the neutral stress axis is displaced and 0uS . In the ultimate limit state My = Mpl,y, the strain line is horizontal and has its zero crossing at the height of the area bisectrix.

Partial internal forces method (PIF-method)

The PIF-method was introduced by Kindmann/Frickel in [27] for I-cross sections and afterwards in [26] for further cross section shapes. In principle, it is based on the method “choose stress distribution” and systematically uses the possible internal forces in the individual parts. The fundamental ideas of the PIF-method are

forming equilibrium between internal forces and partial internal forces (“static of cross sections”).

variation of partial internal forces in the possible boundaries, so that the limit bearing capacity is achieved.

For explanation, the calculation of Mpl,y for a T-cross section is covered again. From the stress distribution in Figure 8.9 it can be seen that the following partial internal forces occur: No, Nw and Mw (see Figure 8.12 as well). The respective equilibrium conditions, are regarding Figure 8.12:

N = Nw + No = 0 (8.16a)

My = Mw + Nw aw – No ao (8.16b)

In these equilibrium conditions three unknowns occur. Since, due to My = Mpl,y, the bending moment should become maximum and the partial internal forces are varied accordingly. In doing so, two cases for Mpl,y result, as can be seen in Figure 8.9.

Case 1: Npl,o Npl,w (corresponds to the stress change in the web)

chosen No = –Npl,o

from N = 0 Nw = –No = Npl,o

with the N-M interaction in the web 2w pl,o pl,w pl,wM 1 (N N ) M

insertion of No and Nw in the equilibrium condition (8.16b) Mpl,y = Mw + Npl,o (aw + ao) = w pl,o w oM N h t 2

Case 2: Npl,o > Npl,w (corresponds to the stress change in the flange)

chosen Nw = Npl,w


from N = 0 No = –Nw = –Npl,w

with the N-M interaction in the web Mw = 0

insertion of No, Nw and Mw in the equilibrium condition (8.16b) Mpl,y = Npl,w (aw + ao) = pl,w w oN h t 2

Figure 8.12 Basics of the PIF-method using the example of a T-cross section

As can be seen, the effort for the PIF-method and the method “choose stress distri-bution” is approximately the same. The clear classification of the PIF-method, how-ever, simplifies the determination of the limit bearing capacity and makes it actually possible to use it for more complex cross section shapes and internal force combinations, see e.g. [25].

Choose stress neutral axis

Basically, this method complies with the method “choose stress distribution”. With the choice of the stress neutral axis, however, an additional condition for the stress distribution is introduced.

Figure 8.13 Two examples for the choice of a straight neutral stress axis


For the description for the strains with Eqs (1.4) we find that the neutral axis of the strains for biaxial bending with axial force is a straight line. The same is true for the stress neutral axis. Two cases are considered for explanation in Figure 8.13. On the left side, a doubly symmetric I-cross section is shown and the stress neutral axis for the effect of My and Mz is sketched in. It runs through the centre of gravity and has different inclinations depending on the relation Mz/My. For the assumption of being fully plasticised ( fy in the cross section) we have N = M = 0 and we find the limit load-bearing capacity for the combined effect of My and Mz.

Table 8.1 Overview of the limit load-bearing capacity of cross sections

Reference Cross section Internal forces Method Remarks

Section8.3

arbitrary partial internal forces method(PIF-method)

suitable for hand calculation

[25] arbitrary choose stress distribution


[25] arbitrary choose stress distribution


[25] arbitrary PIF-method recommendation: spreadsheet program

[25]

2 or 3 plates

arbitrary PIF-method recommendation: spreadsheet program

[25] arbitrary, but thin-walled plates

arbitrary PIF-method without redistribution


Section8.4

arbitrary N, My, Mz, MMxp, Vy, Vz, Mxs

strain iteration partial internal forces according to elastic theory

use in computer programs

If an axial force N is added, the assumption in Figure 8.13 on the right is obvious. From the stress distribution, however, it can directly be seen that the flange moments


are different (Mo Mu). This leads to a warping bimoment M 0. If M = 0 is to be fulfilled, the cross section remains partly elastic. This case is treated in detail in Section 8.3.5.

The methods described above are partly used and explained in the following sections for the determination of the limit load-bearing capacity of cross sections. Table 8.1 gives an overview of this.

8.3 Limit Load-Bearing Capacity of Doubly-Symmetric I-Cross Sections

8.3.1 Description of the Cross Section

Doubly symmetric I-cross sections are very frequently used in steel construction. Therefore, this cross section type is treated here in detail. According to Figure 8.14, rolled profiles and welded cross sections can be distinguished. They are idealised via the division into three plates (rectangles): upper flange, web and lower flange. Since these three plates are assumed to be thin-walled, capturing by approximation with the straight line method shown in Figure 8.14 on the right is sufficient.

With the parameter af, the distance between the flange centres is described. Since the entire cross section height is designated with h for rolled profiles, we have af = h – tf.According to Figure 8.14, the web height hw results for the welded cross section in hw = h – 2 tf. For the rolled profile, the fillets can be captured by approximation via an increased web height. It is common practice to choose hw = af. With the parameter hw,however, cross sections of equal area can also be idealised for example.

Figure 8.14 Idealisation of doubly symmetric I-cross sections

8.3 Limit Load-Bearing Capacity of Doubly-Symmetric I-Cross Sections 289

Note: The cross sections of rolled profiles can, of course, also be captured precisely, i.e. with the direct consideration of the fillets. Calculation formulas and verifications though, would then be considerably more time-consuming – also see Table 8.3.

8.3.2 Perfectly Plastic Internal Forces Spl

The starting point for the determination of the perfectly plastic internal forces are the stress distributions according to elastic theory. Figure 7.22 shows the elastic stress distributions if in each case only one single internal force is acting. According to elastic theory, the maximum ordinates may reach x = fy or 3/fyR , respec-tively. The elastic stress distributions given in Figure 7.22 show where the cross section still has load-bearing capacities. Through a respective “filling” to stress blocks, the stress distributions for the perfectly plastic internal forces Spl in Figure 8.15 result. They fulfil the conditions that the regarded internal force must be maximum and all other internal forces equal to zero. Note that the stress distributions comply with the standards of DIN 18800 [8].

As can be seen, the cross section is not completely plasticised in all cases. This fact is explained in Section 8.2.3 and will not be expanded on here.

Figure 8.15 Stress distributions for the determination of perfectly plastic internal forces Spl


Table 8.2 Perfectly plastic internal forces for I-cross sections (line model)

pl f w w yN 2 t b t h f

2pl,y f f w w yM t b a t h 4 f

2pl,z y fM f t b 2

pl, pl,z fM M a 2

pl,y f yV 2 t b f 3

pl,z w w yV t h f 3

pl,xs pl,y fM V a 2

2 2pl,xp f f w w w y

1M 2 3 b t t 3 h t t f 36

For the determination of perfectly plastic internal forces on the basis of Figure 8.15 we go back to Section 8.1. Since the cross section consists of three rectangles, the limit internal forces Npl, Mpl, Vpl and Mpl,xp in the three individual parts can be determined with the equations in Figure 8.3 and Eq. (8.3). These are then inserted in the equilibrium relations (8.17) and (8.18) and lead directly to the perfectly plastic in-ternal forces compiled in Table 8.2. In this respect, it must be pointed out that the hip roof model corresponding to Eq. (8.3) has been used for all three components. Due to this approach, the cross section fully plasticises as a result of Mpl,xp. For concurrence with other internal forces a slightly different load transfer model is of advantage – see [25] with regard to the PIF-method for all internal forces of beam theory (including torsional stress).

Table 8.3 Perfectly plastic internal forces of rolled profiles

pl f w w r yN 2 t b t h A f

2pl,y f f w w r w r yM t b a t h 4 A h 2 a f

2 2pl,z f w w r w r yM t b 2 t h 4 A t 2 a f

with: f fa h t w fh h 2 t

r10 3a r

42

r r)4(A

Vpl,y, Vpl,z, Mpl, , Mpl,xs and Mpl,xp:See line model, Table 8.2


There are cases where a more accurate determination of the perfectly plastic internal forces for rolled profiles is reasonable. The respective values are compiled in Table 8.3. Here, the effect of the fillets is captured using Table 2.9. The formulas belonging to the torsional internal forces are here adopted from the line model by approxima-tion. As the examinations of Gruttmann/Wagner for Mpl,xp show in [20], the consid-eration of the fillets can definitely lead to slightly larger values. For an HEM 300 the difference is about 8%.

8.3.3 Equilibrium between Internal Forces and Partial Internal Forces

In [25], the relations between the internal forces and the partial internal forces are derived. These relations are now used for the doubly symmetric I-cross section, and here the partial internal forces are applied with the directions of action in Figure 8.16. The following relations between the internal forces and partial internal forces result:

a) internal forces

Vy = Vo + Vu (8.17a)

Vz = Vw (8.17b)

Mxs = Vo af/2 – Vu af/2 (8.17c)

Mxp = Mxp,o + Mxp,s + Mxp,u (8.17d)

b) internal forces

N = Nu + Nw + No (8.18a)

My = Nu af/2 + Mw – No af/2 (8.18b)

Mz = Mu + Mo (8.18c)

M = Mu af/2 – Mo af/2 (8.18d)

Figure 8.16 Internal and partial internal forces for a doubly symmetric I-cross section


Since for the verification of the load-bearing capacity of the cross section the internal forces are known from a calculation at the structural system, with Eq. (8.17) and (8.18), eight equations are available for the determination of the 12 unknown partial internal forces. Thus, a calculative determination is not directly possible. It should be mentioned here that the problem would be solved if the partial internal forces were known. In this case, one could verify for each component (upper flange, web, lower flange) with the help of corresponding conditions (see [25]) whether the load-bearing capacity of the single rectangular cross sections is sufficient. Since this is not directly possible this way, the following procedure is followed:

VwFrom Eq. (8.17b) we have

Vw = Vz (8.19)

Vo and VuEquations (8.17a) and (8.17c) lead to

Vo = Vy/2 + Mxs/af (8.20a)

Vu = Vy/2 – Mxs/af (8.20b)

Mo and MuEquations (8.18c) and (8.18d) give

Mo = Mz/2 – M /af (8.21a)

Mu = Mz/2 + M /af (8.21b)

Mxp,o, Mxp,w and Mxp,uThe primary torsional moment Mxp is partitioned corresponding to the moments of inertia of the plates:

Mxp,o = Mxp IT,o/IT with: IT,o = b tf3/3 (8.22a)

Mxp,w = Mxp IT,w/IT with: IT,w = hw tw3/3 (8.22b)

Mxp,u = Mxp IT,u/IT with: IT,u = b tf3/3 (8.22c)

IT = IT,o + IT,w + IT,u (8.22d)

The chosen partition corresponds to the approach of elastic theory. It results from the postulation that the cross section keeps its entire shape (same twist of all three plates). This partition does not lead to the full plastification of the whole cross section (for common dimensions the web stays elastic).

No, Nu, Nw and MwFrom Eqs (8.17) and (8.18) the Eq. (8.18a) and (8.18b) remain, which are repeated at this point:

N = Nu + Nw + No (8.23a)


My = Nu af/2 + Mw – No af/2 (8.23b)

They include the still unknown four partial internal forces Nu, Ns, No and Mw.These internal forces can exist only between certain limits in consideration of the other internal forces. Therefore, the problem will be solved within the following section by limit considerations, i.e. in terms of an optimisation.

8.3.4 Combined Internal Forces N, My, Mz, Vy and Vz

The combination of all internal forces and moments (including warping torsion) is, of course, quite time-consuming and therefore not considered at this point. However, a corresponding procedure is presented in [25]. Here, it is assumed that the torsional internal forces are equal to zero, Mxp = Mxs = M = 0. The remaining combination of internal forces covers many construction-relevant cases.

Since Mxp = 0, no primary torsional moments occur in the individual parts. With the assumed conditions Mxs = M = 0, the following results from Eqs (8.19) (8.21):

Vw = Vz (8.24a)

Vo = Vu = Vy/2 = Vf (8.24b)

Mo = Mu = Mz/2 = Mf (8.24c)

Of course, the result could have been directly determined: the shear force Vz is car-ried by the web and 50% of Vy as well as Mz are carried by the upper and lower flange. Irregular distributions in the flanges are not permitted since the occurrence of torsion would be connected to this.

The verification of sufficient cross section load-bearing capacity for the internal forces N, My, Mz, Vy and Vz is performed with the method shown in Figure 8.17. In doing so, three steps are carried out, one after another, and five single proofs are con-ducted.

The inclusion of Vy and Vz (first step) can be verified with the shear forces Vo, Vwand Vu in the partial cross sections. Equivalent to this are the verifications

Vy Vpl,y and Vz Vpl,z, (8.25)

i.e. directly using the shear forces Vy and Vz. If the verifications succeed, according to Figure 8.17 at the top right, the yield stress in the three individual parts can be reduced due to the acting of the shear stresses. The reduced yield stresses are then boundary stresses which can be used for carrying N, My and Mz.

In the second step the bending moment Mz is considered. It can be carried by the axial stresses at the flange ends according to Kindmann/Ding in [24] and here in Figure 8.17. With this very clear depiction, the determination of reduced flange


widths is possible – see [24]. Here (equivalent to that) an interaction relation of the rectangular cross section is used and after a short transformation the corresponding verification gives

2y,plyz,pl,z,plz VV1MMM (8.26)

The subscript “ ” for Mpl,z, points to the effect of shear stresses, here due to Vy.

Figure 8.17 Verification of the adequate cross section load-bearing capacity for the internal forces N, My ,Mz, Vy and Vz

In the third step, the internal forces N and My are considered (at the bottom of Figure 8.17). The corresponding partial internal forces cannot be calculated directly from the


equilibrium conditions (8.18a) and (8.18b). However, it is possible to determine limit internal forces in the single plates, which can still be carried after the consideration of Vy, Vz and Mz. For the web we have:

2gr,w w w y z pl,zN t h f 1 V V (for: Mw = 0) (8.27a)

gr,w gr,w wM N h 4 (for: Nw = 0) (8.27b)

The limit flange forces are of equal size due to Mo = Mu = Mz/2, i.e. Ngr,o = Ngr,u = Ngr,f.

2gr,f f y y pl,y z pl,z,N t b f 1 V V 1 M M (8.28)

The maximum possible axial force is then

Ngr = 2 Ngr,f + Ngr,w (8.29)

According to Figure 8.17, at the bottom right, the maximum bending moment max My can now be determined for two cases depending on the axial force acting. With this moment, a final verification

My max My (8.30)

is possible. The verification for the axial force is included in the case differentiation.

Case 1: Stress neutral axis in the web

chosen u gr,fN N and o gr,fN N

from u w oN N N N wN N

with the N-M interaction of the web 2

w gr,w gr,w wM 1 N N N h 4

Insertion of Nu and No into the equilibrium condition (8.18b) y gr,f f wmax M N a M

Case 2: Stress neutral axis in the flange

chosen u gr,fN N and w gr,wN N

from u w oN N N N o gr,f gr,wN N N N

with the N-M interaction of the web wM 0

Insertion of Nu and No into the equilibrium condition (8.18b)

y u o f gr fmax M N N a 2 N N a 2


Since five internal forces exist, overall five verifications have to be executed. Table 8.4 contains a compilation of the necessary verifications and calculation formulas. The internal forces always have to be inserted with their absolute values, i.e. with a positive sign. The method presented here goes back to the publication of Kind-mann/Frickel in [27]. It is called the partial internal forces method (PIF-method)since the solution is gained by a systematic examination of the partial internal forces.

Table 8.4 Verification of the limit load-bearing capacity of I-cross sections for the internal forces N, My, Mz, Vy and Vz

Verification conditions (partial internal forces method):

Shear force Vy: y pl,y f yV V 2 b t f / 3

Shear force Vz: z pl,z w w yV V h t f / 3Bending moment Mz:

22z pl,z, y f y pl,yM M f t b / 2 1 V V

Axial force N and bending moment My:gr,wN N : y gr,f f wM N a M

or

gr,w grN N N : y gr fM N N a 2

Insert all internal forces according to magnitude!

Calculation values:2

gr,f f y y pl,y z pl,z,N t b f 1 V V 1 M M gr gr,f gr,wN 2 N N

2gr,w w w y z pl,zN t h f 1 V V

2w gr,w gr,w wM 1 N N N h 4

Usually the design value of the yield strength fy,d is to be used!

8.3.5 Interaction Conditions of DIN 18800 and Comparison with the PIF-Method

Doubly symmetric I-cross sections with constant yield stress over the cross section may be verified for uniaxial bending, shear force and axial force with the conditions in Tables 8.5 and 8.6 in order to analyse that the boundary internal forces of the plas-tic state are not exceeded.


Besides the interaction relations of Tables 8.5 and 8.6, respectively, for uniaxial bending with normal and shear force about the strong or weak axis, also an in-teraction relation for biaxial bending with normal force is specified in DIN 18800 part 1. Since this does not always provide appropriate results for the verification method Elastic-Plastic and is, furthermore, unnecessarily complicated, it is not cov-ered here. Instead, the verification of sufficient load-bearing capacity using the PIF-method of Section 8.3.4 is recommended.

Table 8.5 Interaction conditions for doubly symmetric I-cross sections with N, My and Vz(bending about the strong axis) according to DIN 18800 Part 1, Table 16

Moments about y-axis

Range of validity pl,d

V 0.33V pl,d

V0.33 1.0V

pl,d

N 0.1N

1M

M

d,pl pl,d pl,d

M V0.88 0.37 1M V

pl,d

N0.1 1N pl,d pl,d

M N0.9 1M N pl,d pl,d pl,d

M N V0.8 0.89 0.33 1M N V

Table 8.6 Interaction conditions for doubly symmetric I-cross sections with N, Mz, and Vy (bending about the weak axis) according to DIN 18800 Part 1, Table 17

Moments about z-axis

Range of validity pl,d

V 0.25V pl,d

V0.25 0.9V

pl,d

N 0.3N

1M

M

d,pl

2

pl,d pl,d

M V0.95 0.82 1M V

pl,d

N0.3 1N

2

pl,d pl,d

M N0.91 1M N

2 2

pl,d pl,d pl,d

M N V0.87 0.95 0.75 1M N V

Summing up, it can be stated that DIN 18800 [8] contains interaction relations for I-cross sections with which the following internal forces can be captured:

N-My-Vz according to Table 8.5

N-Mz-Vy, according to Table 8.6

N-My-Mz, see Eqs (40) (42) and Figure 19 of DIN 18800 Part 1

The three mentioned internal force combinations are completely captured through the verifications derived in Section 8.3.4 for the N-My-Mz-Vy-Vz combination. Table 8.4 shows that the verifications on the basis of the partial internal forces method are con-siderably easier to use than the three interaction relations in DIN 18800. Also all five internal forces can be captured in any combination.


The verifications in Table 8.4 can, with little effort, be reduced to the three cases of the DIN. If the non existing internal forces in each case are set to zero, we obtain the interaction relations compiled in Table 8.7. They can be used for the direct com-parison with the interaction relations of DIN 18800 (cf. Tables 8.5 and 8.6). For practical usage, the authors would recommend Table 8.4, since here a larger range of application is covered and fewer case differentiations have to be performed.

Table 8.7 N-My-Vz, N-Mz-Vy and N-My-Mz interaction conditions for I-cross sections (partial internal forces method)

Internalforces

Verification conditions Calculation values

Vz z pl,z w w yV V h t f 3

N gr,wN N : y gr,f f wM N a M with: 2

gr,w w w y z pl,zN t h f 1 V V

My gr,w grN N N : y gr fM N N a 2 gr,f f yN t b f

Vy y pl,y f yV V 2 b t f 3

Mz2

y,plyz,pl,z,plz VV1MMM with: 2pl,z y fM f t b 2

N2

f y y pl,y z pl,z, w w yN 2 t b f 1 V V 1 M M t h f

Mz z,plz MM

N gr,wN N : y gr,g g sM N a M with: gr,w w w yN t h f

My gr,w grN N N : y gr fM N N a 2 gr,f f y z pl,zN t b f 1 M M

Insert all internal forces according to magnitude!gr gr,f gr,wN 2 N N

2w gr,w gr,w wM 1 N N N h 4

Combined effect of N, My and Vz

For the formulation with the PIF-method in Table 8.7 and for the conditions of DIN 18800, three verifications have to be carried out since with DIN 18800 also the examination of the extent of validity includes respective verifications for N and Vz.

Moreover, it stands out that in DIN 18800 always the perfectly plastic internal forces are used as reference values. For the PIF-method in Table 8.7 only Vpl,z is considered. One could naturally also use the reference values Npl and Mpl,y. However, this would extend the verification conditions unnecessarily.


Figure 8.18 N-My-Vz interaction relations (PIF-method)

Figure 8.19 a) N-My-Vz interaction relations according to DIN 18800 b) N-My interaction relations for different web area ratios

and comparison with DIN 18800

Figure 8.18 shows the N-My-Vz interaction relations (PIF-method) for two different web area ratios (20% and 40%) depending on Vz/Vpl,z. Compared to that, the respective relations according to DIN 18800 can be found in Figure 8.19a. We can see that in DIN 18800 the accurate interaction curves are approaching straight lines. Furthermore, the relations of DIN 18800 are independent of the ratio of the web area to the total area:

= Aweb/Atotal

Thus, they must cover the complete possible range of application. For the rolled pro-files the web area portions are between about 20% and 45% (examples: HEB 300 20.7%; IPE 600 44.7%). The relations of DIN 18800 are valid for this range, so that for the verification of welded I-cross sections this parameter area should be main-tained. Figure 8.19b shows the comparison for the case Vz = 0. The relations of DIN


18800 in broad areas comply with the case = 0.2 (20% web area portion), thus they cover the minimum web area of the rolled profiles. Between N/Npl = 0 and 0.2, this interaction is a bit on the unsafe side. For Vz 0 fairly clear differences occur on the safe side, as Figures 8.18 and 8.19 show. For large ratios of Vz/Vpl,z the relations of DIN 18800 capture cross sections with large web areas ( 45%) better than with small ones ( 20%). For Vz = Vpl,z the web is fully plasticised. Both of the flanges are then available for the inclusion of axial forces and bending moments.

Combined effect of N, Mz and Vy

As a matter of principle, the explanations concerning the N-My-Vz interaction are also valid for the N-Mz-Vy interaction.

Figure 8.20 N-Mz-Vy interaction relations (PIF-method)

Figure 8.21 N-Mz-Vy interaction relations according to DIN 18800

The influence of the parameter “web area portion ”, for example, leads to consider-able differences (Figure 8.20). The approximations of DIN 18800 (Figure 8.21) are


partly significantly on the safe side, but partly also a bit on the unsafe side. The verifications with the PIF-method according to Table 8.7 are so simple that one should give them priority, not only because of their higher accuracy.

Combined effect of N, My and Mz

The evaluation of the verification conditions for the N-My-Mz interaction according to Table 8.4 (or Table 8.7) can be found for selected cases in Figure 8.22. Mz/Mpl,z and cross sections with web area portions of 20% and 40% have been chosen as curve parameters. The curves are similar to the N-My-Vz interaction. The influence of Mz,however, is larger than that of Vz.

Figure 8.22 N-My-Mz interaction relations (PIF-method)

The N-My-Mz interaction according to DIN 18800 is not responded to here for various reasons. Compared to the verification conditions in Table 8.7, it is extraordi-narily complex and, for safe use, the supplementary clarifications in the explanations of DIN 18800 [62] are needed. Furthermore, for the user, it does not visibly contain a warping bimoment M , so that in reality it does not involve an N-My-Mz interaction.

The matter is made clear here with the example in Table 8.8 and Figure 8.23. In that respect, the case “PIF-method and M = 0” is considered first, with the achieved limit load-bearing capacity of 100 %.

If the internal forces N, My und Mz occur in a structural system, deformations u, v and w result. The rotation (torsion) is equal to zero. From this it is immediately clear that for a doubly symmetric I-cross section 50% of the bending moment Mzmust be carried by the upper flange and 50% by the lower flange. Thus, the flange moments are of equal size for the example, Mo = Mu = Mz/2 = 1310 kNcm. The cross section for the chosen N-My-Mz internal forces combination, however, is not yet plasticised. Figure 8.36b shows the respective stress distribution, which has been de-termined with the computer-oriented calculation in Section 8.4 (strain iteration).


Fully plasticising can be forced by adding a warping bimoment M of an appropriate size. This is the case “PIF-method and M = –26200 kNcm2” in Table 8.8, for which a larger load-bearing capacity by up to 19.6% results. The flange moments Mo = 2983 kNcm and Mu = 150 kNcm are very different now, so that torsional rota-tions occur in the structural system. Moreover, the warping bimoment also leads to torsional moments Mxp and Mxs not equal to zero, whose shear stresses may decrease the bearing capacity.

Figure 8.23 Comparison of the N-My-Mz interaction using an example

The bearing capacity gained by the N-My-Mz interaction of DIN 18800 is influenced by warping bimoments, which are not considered in the structural system. For that reason, it is actually not a pure N-My-Mz interaction. This does not mean that the lar-ger bearing capacities cannot be achieved by adding M . However, the precondition for that is that torsion (Mxp, Mxs, ) occurring in the structural system can be cap-tured and influences decreasing the bearing capacity are considered.

8.4 Computer-Oriented Methods 303

Table 8.8 Example of the N-My-Mz interaction with M = 0 and M 0

Case PIF-method and M 0 PIF-method and M 26200 kNcm2

N = 1000 kN 1196 kN My = 3450 kNcm 4126 kNcm Mz = 2620 kNcm 3133 kNcm M = 0 kNcm² 26 200 kNcm²

Comparison 100% 119.6%

Crosssection:

See Figure 8.23

Notes Upper flange elastic Rotation = 0 Torsional moments Mxp = Mxs = 0

Cross section completely plasticised Rotation 0 Torsional moments Mxp = Mxs 0

Note: The PIF-method including torsional loading, which is required for the example of Table 8.8 and Figure 8.23 regarding the case “PIF-method and M 0 ”, can be found be found in [25].

8.4 Computer-Oriented Methods

8.4.1 Problem Definition

In Section 8.3, the limit load-bearing capacity of I-cross section shapes was covered. Now, the question which follows is how sufficient load-bearing capacity on the basis of plastic theory can be verified for

arbitrary cross section shapes and internal forces combinations.

For the solution, the

strain iteration

calculation method is used, which is explained in Section 8.2. The main features of the method have already been explained there, and Figure 8.11 gives a first clarifying example. Here, the methodology is further explained and, using examples, the application illustrated. In doing so, only x internal forces and, later, internal forces as well as their combined effect are considered for the time being. Due to the required iteration and the numeric effort, the method is only suitable for use in computer programs.


8.4.2 Strain Iteration for a Simple Example

For the doubly-symmetric I-cross section in Figure 8.25 we need to determine whether the bending moment My = 16500 kNcm can be carried. Moreover, the corre-sponding stress distribution should be determined. For the perfectly plastic bending moment we obtain with Table 8.2

2

pl,y31.85 24M 1.15 16 31.85 0.75 12 786.3 4 149.9 16 936.2 kNcm

4 1.1

It is clear from this that My = 16500 kNcm can be carried by the cross section. For performing the strain iteration, the basic equations

x Mz w and yM

y

Mw

EI

are assumed – see Section 8.2, and Table 3.3 in Section 3.4.4. With that, we obtain for the completely elastic cross section,

5 1Mw 6.921 10 cm

and the maximum strain comes to

max = 1.10 ‰ > el = fy/E = 1.04 ‰

Thus, the strain el corresponding to the yield stress is exceeded in the flanges. The stress distribution in the cross section can now be determined with the constitutive equations in Figure 8.24.

stresses:

xxelx E:

yxelx f:

with: Efyel

Figure 8.24 Stress determination with the bilinear stress-strain relation

Figure 8.25 shows the strains and stresses in the cross section – see iteration step i = 1 there. As can be seen, both flanges and end areas of the web are plasticised. The height hel,1 can be calculated using


el,1 el Mh 2 w 30.022 cm

due to x M elz w .

Figure 8.25 Example of the strain iteration (method of the elastic residual stiffness)

For the further performance of the strain iteration, the partly plasticised web is considered and with the help of Figure 8.26, the values Iy,el, My,el and My,pt of the web are determined. Here, the indices “el” and “pt” identify elastic and plastic areas of the web.

The moment of inertia of the elastic rest cross section now gives

3 4y,el,1

1I 0.75 30.022 1 691 cm12

And this part of the cross section carries 2

y,el,1M 1 6 0.75 30.022 24 1.1 2 458 kNcm


3y,el w el

1I t h12

2y,el w el y

1M t h f6

2 2y,pt w w el y

1M t h h f4

Figure 8.26 Symmetric partly plasticised rectangular cross section (web)

The following belongs to the plasticised areas:2 2

y,pt,1M 12 786.3 0.75 4 31.85 30.022 24 1.1 13 249 kNcm

Since the sum of 2 458 + 13 249 = 15 707 is smaller than 16 500, now the remaining elastic cross section is loaded with the difference

y,1 y y,ptM vorh M M 16 500 13 249 3 251 kNcm

and the abovementioned calculation is continued iteratively. Thus, the bending in the iteration step i is

y y,pt,iM,i

y,el,i

vorh M Mw

E I

With Table 8.9, the course of the iteration can be followed. In the fifth iteration step, the equilibrium state between the existing bending moment and the stresses in the cross section is achieved. The results of Table 8.9 are shown in Figure 8.27. In the detailed representation on the right side, the iteration is clearly recognisable, steps 4 and 5 are very close here. Figure 8.27 shows the decrease of My = 16500 kNcm to the respective My which is actually carried in each iteration step, cf. Table 8.9 last column.

Table 8.9 Iteration course for the example in Figure 8.25

Step w max hel Iy,el My,el My,pt My

[1/cm2] [‰] [cm] [cm4] [kNcm] [kNcm] [kNcm] 1 -6.921 10-5 1.10 30.02 1 691 2 458 13 249 15 707 2 -9.154 10-5 1.46 22.70 731 1 405 14 828 16 234 3 -1.089 10-4 1.73 19.08 434 993 15 447 16 440 4 -1.155 10-4 1.84 17.99 364 883 15 612 16 495 5 -1.162 10-4 1.85 17.89 358 873 15 627 16 500


Figure 8.27 Relation between My and max for the iteration (example in Figure 8.25)

8.4.3 Strain Iteration for Internal Forces

It is assumed that the internal forces N, My, Mz and M are known from a system calculation. Using the strain iteration method, we want to determine whether the internal forces can be carried by the cross section.

The basic equations can be taken from Chapter 7, see Eqs (7.4) and (7.46). For a state in which the cross section is completely elastic, in matrix notation we have:

M

MMN

wvu

I000

0I0000I0000A

Ey

z

M

M

S

y

z(8.31)

or

SAEVSVAE 1

Since the four equations are decoupled, the unknowns MMS w,v,u and can be di-rectly determined and with that the strain

MMSx wzvyu (8.32)

at any point of the cross section as well. With the constitutive equations in Figure 8.24, the stresses x result as does an indication of whether the cross section parts are elastic or plastic. However, this approach is inexpedient in some cases, which can be directly recognised from the example in Figure 8.25. As can be seen, both flanges are fully plasticised in the first iteration step, so that the cross section has lost a large part


of its stiffness. For the remaining elastic cross section (part of the web) Iz = I = 0. For computer-oriented calculations using Eq. (8.31), this is extremely inauspicious since the equation system cannot be solved any more (the determinant of A is equal to zero).

Therefore, instead of Figure 8.24, the modified material behaviour in Figure 8.28 is used. For a long time it has proven useful for calculations according the plastic zones theory (Section 5.12) – see for example [71], [44] – and is in a similar form suggested in Eurocode 3 [10].

Figure 8.28 Modified material behaviour

The decisive difference affects the horizontal straight line (E = 0!) for which a slight slope of, for example, Ev = E/10000 is assumed. With this, the cross section properties in Eq. (8.31) can decrease to 1/10000 at maximum, so that numeric diffi-culties can be avoided. Additionally, the strain is limited by an arithmetic ultimate strain u of, for example, 20% and the yield stress fy is assigned to this value. Due to this, the yield stress in the cross section is not exceeded at any point, and for less ductile materials the strains can be limited accordingly.

To explain the further calculations, it is for the time being assumed that the strain state in the cross section is known – see Eq. (8.32). Thus, the cross section can be divided into elastic and plastic partial areas. As an example, this is shown in Figure 8.29. Due to the constitutive equations in Figure 8.28 and because of Ev = E/10000, the plastic partial area has a certain residual stiffness contributing to the cross section properties.


Figure 8.29 Partly plasticised cross section

With the iteration state assumed to be known, it can be calculated which internal forces are carried by the partly plasticised cross section. To do so, the internal force definitions in Table 1.2 are used. The bending moment My is regarded as an example, and the stresses are replaced by strains using Figure 8.28:

Axy dAzM

plel Aplx

Aelx dAzdAz

pt,yA

plxvA

elx MdAzEdAzEplel

(8.33)

with: plA

plxuvypt,y dAzsgnEfM

With My,pt (plasticised), the constant stress portions are captured in the plasticised cross section parts. Here, sgn ( x) takes into consideration the sign. The strains can now be replaced by Eq. (8.32) and this gives

pt,yzMzzMyzSzy MAwAvAuAEM (8.34)

Here, the cross section properties are defined as follows:

el plA Apl

velz dAz

EEdAzA (8.35a)

el plA Apl

velyz dAzy

EEdAzyA (8.35b)

etc.

This calculation shown for the bending moment My can be performed for N, Mz and M in an analogous manner, so that overall four equations result. In matrix notation, this is


E

MM

MN

y

z

pt,

pt,y

pt,z

pt

M

M

S

zy

zzzzyz

yyzyyy

zy

M

M

M

N

wvu

AAAAAAAAAAAAAAAA

(8.36)

or:

ptSVAES

Up to now, the derivations have been carried out assuming that the strain state, i.e. the vector V, is known, although this is not the case in reality. We could now attempt to choose the unknowns in the vector V by trying in such a way that the internal forces S from to Eq. (8.36) are equal to the existing internal forces. If this should suc-ceed, the problem would be solved.

As a matter of principle, a trial-and-error solution is certainly possible in terms of an optimisation task, but an iterative solution algorithm is given priority here. To do this, iteration step i is considered and it is assumed that from iteration step i 1, adequate approximations for A and Spt are present. With this assumption, the equation system (8.36) can be solved for V and this gives

1i,pt1

1ii SSAEV (8.37)

Since the matrix is symmetric, among others the method of Cholesky can be used for the solution (see Section 6.1). The vector S (without subscript!) includes the existing internal forces, which are to be carried by the cross section. The following can be calculated with knowledge of the vector Vi:

strains, Eq. (8.32)

stresses, Figure 8.28

cross section properties Ai , Eq. (8.35)

internal forces in the plasticised cross section parts Spt,i, Eq. (8.33)

Equation (8.36) now becomes

i,ptiii SVAES (8.38)

Vector iS contains the internal forces which are actually carried by the cross section in the iteration step i. Provided that an equilibrium state is possible, the method usu-ally converges relatively fast. This can be seen from the fact that the difference be-tween the given and carried internal forces gets smaller from iteration step to iteration step, 0SS i . Here, it can actually happen that the difference for single internal forces becomes intermediately larger. For terminating the iteration an appropriate cancel criterion is needed. A reasonable possibility is defined in Eq. (8.39):


6

,el

i,2

z,el

i,zz2

y,el

i,yy2

el

ii 10

MMM

MMM

MMM

NNN

(8.39)

The elastic limit internal forces are chosen as reference values (for exclusive effect) since these can be easily determined in a preliminary calculation. With this reference, all quotients are dimensionless, so that with it a comparable criterion is available for arbitrary cross sections and dimensions. Through changing the error bound, the accuracy can be increased or decreased if needed. As an alternative to Eq. (8.39), a change of the maximum strain can also be used as the cancel criterion. A corre-sponding condition is formulated in Eq. (8.40):

6

i

1ii 10max

maxmax (example) (8.40)

Further, it has, of course, to be checked in every iteration step whether the strains are still below the ultimate strain. If > u, the iteration can be cancelled, because the cross section obviously fails. Moreover, it is advisable to limit the number of iteration steps (e.g. max i = 100) in order to be able to end the calculation in case of a diverging calculation process.

For the iterative improvement of the strain state, the cross section properties Ai and the internal forces Si and Spt,i are required. For computer-oriented applications it is advisable to determine the values via numeric integration. As described in Section 2.4.5, the cross section is divided into fibres and stripes to do this. Since the cross sections are often made of plane plates, a single plate (rectangular cross section) is considered in Figure 8.30 and divided into fibres of equal size. It is assumed that the ordinates of the plate ends “a” and “e” are known.

Figure 8.30 shows the division of a plate into five fibres. The small number of fibres was chosen here only for reasons of a clear depiction. For calculations, one should choose 50 – 200 fibres (depending on the desired accuracy). In Figure 8.30, a fibre j is considered, and ordinates, strain and stress are determined at the centre of gravity of this fibre. For the determination of the internal forces it is assumed that the stress in the fibre is constant, which is justified for a sufficiently fine fibre division. For the determination of the cross section properties Ai (iteration step i), the formulas in Ta-ble 2.6 can be referred to and, due to the fine fibre division, only the Steiner parts are used here. The calculations must capture all fibres of the cross section parts in every load step i.

Alternatively, the cross section could also be modelled using a finite element mesh regarding polynomial functions for the determination of the cross section properties and stress resultants via numeric integration as for instance with the Gauss quadrature (compare Chapter 11).


Fibre j:

Ordinates: j a e a j a e aj 0.5 j 0.5y y y y ; z z z zmax j max j

j a e aj 0.5max j

Strain: x,j S j M j M ju y v z w

Stress: x,j x,jE for: x,j el

x,j x,j y v u v x,jsgn f E E for: x,j el

Internal forces: (example)

y,j j x,j jM z A

y,pt,jM 0 for: x,j el

y,pt,j x,j y v u j jM sgn f E z A for: x,j el

Cross section properties: (example)

yz,j j j jA y z A for: x,j el

vyz,j j j j

EA y z AE

for: x,j el

Figure 8.30 Division of a plate into fibres and examination of fibre j

Cross sections free of warping and cross sections with minor warping

For cross sections free of warping, the fourth equation in (8.31), (8.36) and (8.37) must be deleted due to I = M = 0. Also, the fourth column in the matrices is dropped as well.

Figure 8.31 Reduction for cross sections free of warping

For computer-oriented calculations, however, it is often more convenient to keep the number of equations and the size of the matrices depicted by Figure 8.31. In doing so,


a “1” is set on the principal diagonal of the matrix and all other affected elements are set equal to zero. The reduction described here can also be conducted for cross sections showing very small torsional warpings if the warping torsion is to be neglected.

Limit load-bearing capacity and incremental calculations

The method of strain iteration which has been described so far is a method of total steps, because the existing internal forces are overall applied in full – see, for example, Eq. (8.31). As a result, difficulties with the convergence may result in individual cases. With regard to the convergence behaviour, incremental methodswhere the internal forces are increased step by step are of advantage. With a slight modification of the methodology described so far, incremental calculation can also be carried out. It is described in Table 8.10 for the determination of the limit load-bearing capacity.

Table 8.10 Incremental calculation of the limit load-bearing capacity of cross sections with the strain iteration method

Given Internal forces combinations S ; M,M,M,NS yzT

Cross section dimensions Material properties: fy, E, Ev, u

Starti = 0

Internal forces close to the elastic limit load: 0SChoose internal force increment: e.g. 100S0 (1%) Cross section properties: 0A ; matrix in Eq. (8.31)

0S 0,pt

Iterationsi = 1, 2, ...

100SiSS 00i

1i,pti1

1ii SSAEV

Strains x,i iV Stresses x,i

Cross section properties iA Internal force portions i,ptS

Internal forces i,ptiiist SVAES

Continue iterative calculation until max > u

Choose last carried load step: jS

Where required, iterate in the load steps until 0SS i,isti or until Eq. (8.39) is fulfilled. This depends on the chosen increment.

Iterationsk = 1, 2, ...

Choose new internal forces increment: e.g 0001S0

0001SkSS 0jk

Repeat iterations as described for i = 1, 2,...; Termination of the calculation if the internal force increment is sufficiently small.


For comparison with the total step method in Section 8.4.2, here the incremental cal-culation according to Table 8.10 is performed for the example in Figure 8.25 (E = 21000 kN/cm2, Ev = 2.1 kN/cm2, u = 20%). Using a spreadsheet program, one ob-tains as limit internal force Mpl,y = 16 910 kNcm after 30 iterations. This value is about 0.15% smaller than the one calculated in Section 8.4.2 for Mpl,y. The reason for this are the modified constitutive equations with Ev = E/10000, so that the stress in the web does not everywhere reach the yield strength. Figure 8.32 shows the graph of the bending moment My depending on the maximum strain. For reasons of represen-tation, the graph is only shown until = 2% (so not until u = 20%).

Figure 8.32 Incremental determination of the limit load-bearing capacity for the example in Figure 8.25

8.4.4 Consideration of the Internal Forces

Shear stresses in cross sections are a result of the effect of the internal forces Vy, Vz,Mxp and Mxs. The principles of the classic determination of the shear stresses are discussed in Sections 7.3 and 7.4, where the stress equilibrium in the longitudinal direction of the beam is the decisive basis (see also Chapter 11, where the shear stress calculation it is dealt regarding numerical approaches). Since the shear stresses for elastic beam theory are calculated from the equilibrium with the x stresses, which are nonuniform in the longitudinal direction of the beam, the combined effect of the internal forces N, My, Mz and M has to be rconsidered.

The precise solution of the problem is only possible if the longitudinal direction of the beam is considered – see [25]. However, such calculations are very time-con-suming and are only worthwhile in exceptional cases. For structural calculations, the cross section plane is considered only by approximation. Basically, there are two methods:

priority x stresses

priority stresses


Figure 8.33 illustrates the fundamental differences between the two methods using the example of a doubly symmetric I-cross section being loaded by a bending moment My and a shear force Vz. The internal forces should be so large that the elastic stress distributions lead to a clear exceeding of the equivalent stress vaccording to von Mises – see Section 7.5. For the method “priority x stresses” first of all a stress state x is determined which is in balance with the existing bending moment. This can, for example, be performed with the method of strain iteration according to Section 8.4.3. As can be seen in Figure 8.33, the yield stress fy acts in the flanges and the boundary areas of the web only as a result of the bending moment. Further stresses cannot be carried in these areas, so that the shear stresses due to Vzhave to be limited to the remaining elastic area of the web. In a second step a shear stress course must now be determined which is in equilibrium with Vz.

Figure 8.33 Comparison of the qualitative stress courses for the methods “priority x” and “priority ”

According to the method “priority stresses”, we calculate the shear stress course first. Provided that only due to , the equivalent stress v or the limit shear stress R is already exceeded, the shear stress distribution must be determined iteratively. Subse-quent to this, a x stress state is determined – see above. Here, however, the entire yield stress fy is not available for the x stresses, but depending on the shear stress only a reduced yield stress. As a consequence of this approach, the relatively unusual

x stress distribution on the right of Figure 8.33 results. These methods, which are here described roughly, are frequently used in the relevant literature. Nevertheless, for the occurrence of arbitrary internal force combinations it is very time-consuming when it comes to practical application. Moreover, it shows that with shear stress distributions which are assumed to be constant, larger load-bearing capacities can be achieved.

Suggestion for capturing shear stresses

Since the cross sections almost exclusively consist of thin-walled rectangular plates or can be idealised with these, it comes back to the bearing models for the partial plates. As a matter of principle, the suggested approximation method uses the method


“priority internal forces” regarding a certain shear stress distribution. The funda-mental condition for this shear stress state is the assumption that the value of the stress is constant across the total partial plate – see, for example, [25]. With the existing internal forces Vy, Vz, Mxp and Mxs as starting point, the following calculations are performed:

Determine the shear stresses in the cross section according to elastic theory (Chapters 7 and 11).

Calculate the partial internal forces V and Mxp in every cross section part according to Figure 8.34.

Determine the constant shear stress with: 2

pl

2

xp,pl

xp

xp,pl

xp

R VV

M2M

M2M

with: htV;th2t41M Rpl

2Rxp,pl

If / R > 1, the internal forces cannot be carried. Otherwise, a reduced yield stress

2Ryy 1ffred

can be calculated for every partial cross section.

The inclusion of the x internal forces N, My, Mz and M is examined according to Section 8.4.3. In doing so, the reduced yield stress is applied in every cross section part. Thus, instead of Figure 8.28, the constitutive equa-tions according to Figure 8.35 are applied.

Figure 8.34 Calculation of partial internal forces using the elastic stress distribution


Figure 8.35 Modified constitutive equations for the consideration of the shear stresses

With this procedure, combining the internal forces according to step (no itera-tion) with the strain iteration according to Section 8.4.3 for the internal forces (good convergence), one obtains a very efficient method which can be used for arbi-trary thin-walled cross sections.

8.4.5 Examples / Benchmarks

Figure 8.36 includes two examples which should serve readers as benchmarks for the verification of their own calculations. The cross section idealisation is in all cases based on the straight line model with overlapping. With 500 fibres in each case, the cross section parts have been divided very finely. As further parameters, E = 21000 kN/cm2, Ev = 0 and u = 0.20 have been chosen. The choice of Ev = 0 has not led to numeric problems in any respect – also see Section 8.4.3.

The verification of the L-cross section in Figure 8.36a (web: 100 10; lower flange: 50 5) can also be performed with the partial internal forces method (see Section 8.3) and the RUBSTAHL program CSP-Three Plates. With My = 740 kNcm, the perfectly plastic bending moment Mpl,y is almost reached (utilisation: 99.98%). The stress distribution shows that cross sections do not always have to fully plasticise under the effect of Mpl.

For the doubly symmetric I-cross section in Figure 8.36b the cross section load-bearing capacity is 99.764% utilised for the given internal forces. This is about the example on the N-My-Mz interaction in Section 8.3.5 (dimensions as Figure 8.23) where the question of fully plasticising is discussed. Figure 8.36b shows which areas


of the cross section are still elastic and documents again the compliance between the PIF-method and the strain-based calculation.

Figure 8.36 Benchmarks for the determination of the load-bearing capacity on the basis of the strain iteration

9 Verifications for Stability and according to Second Order Theory

9.1 Introduction

The numeric background and solution methods for stability (eigenvalue) problems have already been discussed in detail in Chapters 5 and 6. In this Chapter, the focus is on the understanding of the different stability cases and also on the procedures and methods for the verification of sufficient bearing capacity.

In Chapters 5 and 6 with its illustrations and explanations, it has been clarified that the stability cases flexural buckling and lateral torsional buckling are caused by compression stresses. Furthermore, in the case of lateral torsional buckling, an eccentric load application leads to an increasing risk of losing stability. Stability case plate buckling is not dealt with in this Chapter (see Chapter 10), even though many of the following explanations apply to that stability case as well.

Figure 9.1 Pointer under tension (left) and compression (right)

A small experiment can demonstrate that compression forces are more critical than tension forces. For that purpose only a pointer is needed, which should be long and thin as they usually are. The material of the pointer is not of primary interest in this case. On the left of Figure 9.1, Mr Vette pulls with both hands at the ends of the pointer. Despite his best efforts, he cannot visibly elongate the pointer. If, in contrast, he pushes the pointer against the wall, as shown in Figure 9.1 on the right, causing deformations does not seem to be a problem. It has to be mentioned that a small deflection has to be applied to the pointer in case it is perfectly straight. Alternatively, a poiter with slight initial deflections, i.e. an “imperfect” pointer, can be used. The compression in the member causes a stability problem leading to the main issue of this chapter: The stability cases and the calculation of deformations and stresses according to second order theory under consideration of imperfections.


9 Verifications for Stability and according to Second Order Theory 320

The classic theory of stability has a long tradition. However, the calculation methods and the way of thinking have changed in the past 10 – 15 years. Figure 9.2 shows the differences. In case a, which is the classic theory of stability, a perfectly straight compression member with a force applied centrically is assumed. With the applica-tion and increase of the load, the member is compressed and it stays straight at first because it is in a state of stable equilibrium. For N = Ncr, which is the critical load, an indifferent equilibrium occurs, which is an undetermined state: the member is indecisive whether it should remain straight or whether it should buckle. More precisely in technical terms, the transition to the unstable equilibrium is called “indifferent equilibrium” and it is also referred to as “equilibrium bifurcation”. So much for the classic theory of stability.

Figure 9.2 Stability and bearing capacity of a compression member

The way of thinking has now changed and an imperfect (initially deformed) compres-sion member is assumed as in case b. The nonlinear load-deformation relationship presented is the result of this, and the compression member shows certain deflections from the outset. If the imperfection is small and an unlimited elastic material behaviour is assumed, the curve converges asymptotically to the horizontal line characterising Ncr. The curve also shows that the deflections increase disproportion-ately with a rising N. This is also true for bending moments as well as for shear forces.

9.2 Definition of Stability Cases 321

Due to the fact that the material is not indefinitely elastic, the maximum axial force is reached when a plastic hinge occurs at midspan due to N and M. With this approach, all imperfections and simplifications of the calculation which are of significance re-garding the bearing capacity have to be covered by initial deformations w0. Of cour-se, this is also the case when the stability problem is assumed, i.e. Ncr, and max N =

Npl,d is determined with the help of reduction factors – see Section 9.4.

The change in the way of thinking is closely connected to the old and new verifica-tion methods. Previously, i.e. according to the old German stability standard DIN 4114, the stability was almost always verified using the condition

zulSF

(9.1)

and, for the determination of the buckling values , the buckling length resulting from the critical load was used. The influence of imperfections and the consequences of second order theory were included in the buckling values . However, this was actually not definitely present in the minds of engineers, so that with the introduction of DIN 18800 [8] many believed that the second order theory was an invention of the people laying down the standards. With

pl,d

N 1N (9.2)

a verification comparable to Formula (9.1) is also included in DIN 18800 Part 2. In contrast to former active engineers, it is nowadays clear to everyone what the reduc-tion factors covers (comparable with 1/ ). Also verifications are now often exe-cuted where calculations according to second order theory can be noticed directly.

9.2 Definition of Stability Cases

Regarding the buckling of beam structures, a distinction is made between flexural buckling and lateral torsional buckling. These expressions refer to the deformations that occur in case the of stability failure.

For the restrained beam shown in Figure 9.3a an axial compression force N arises due to the action of the load Fx. The beam is compressed, which causes a displacement u in x-direction. There are no further regular displacements because the beam is assumed to be perfectly straight and the load to be acting centrically. As long as there is a stable equilibrium, this state does not change. However, at a certain load inten-sity, the stable equilibrium is lost and the stability problem of flexural buckling occurs. The transition to the unstable equilibrium occurs when the critical axial force N = Ncr is reached. For structural systems this must not happen and the loading has to always be smaller in order to ensure a stable equilibrium. Thus, Ncr is the upper limit


for N. The stability problem of flexural buckling is characterised by a displacement states, where cross section rotations do not occur. Two cases are distinguished:

flexural buckling about the y-axis, often also called flexural buckling about the strong axis, with displacements w(x), i.e. in the z-directionflexural buckling about the z-axis, often also called flexural buckling about the weak axis, with lateral displacements v(x) as in Figure 9.3a, i.e. in the y-direction

The stability problem of flexural buckling can only occur if axial compression forces exist.

Figure 9.3 Stability problems for the buckling of columns and beams

9.3 Verification according to Second Order Theory 323

In Figure 9.3b, a restrained beam is considered which is subjected to a transverse load Fz. Scheduled bending moments My occur and displacements w(x) downwards result. As shown for the compression member in Figure 9.3a, there is a transition from the stable to the unstable equilibrium, which is a result of the compression stresses in the cross section due to My, and which is influenced by the point of load application of Fz. The case shown in Figure 9.3b is a typical example of the stability problem of lateral torsional buckling where displacements v(x) occur in combination with rota-tions (x). The rotations (x) shown in Figure 9.3b are the characteristic feature of lateral torsional buckling. One special case of lateral torsional buckling is torsional buckling, which is characterised by deformations where only rotations (x) occur and no displacements. Figure 9.33 contains an example for torsional buckling of a column due to an axial compression force. In Section 6.2.2, the different stability cases for beam structures are also explained with Figure 6.6 in conjunction with eigenvalues and eigenmodes.

Note: The lateral torsional buckling of bending girders (without axial forces), as shown in Figure 9.3b, used to be identified as tilting. This expression is, for exam-ple, used in the old stability standard DIN 4114 and in many other old references.

9.3 Verification according to Second Order Theory

For the determination of the bearing capacity of structures all influences that increase the stresses or decrease the bearing capacity have to be considered. Those are:

variations of the load values and the material propertiesdeviations of the cross section geometry and the plate thicknessesimperfections due to fabrication and erection work, i.e. due to initial out-of-straightnesses as well as residual stresses due to rolling, welding or straighten-ing works structural deformations and hence the increase of internal forces and moments according to second order theory plastic zones as a result of the material behaviour in Figure 1.11 causing a de-crease of stiffness

The first two points, which include the loads, the material properties, the cross section geometry and the plate thickness do not need to be followed up. They are part of the safety concept included in the verification Sd/Rd 1, i.e. these variations and deviations are covered by the safety factors M and F. The remaining points – imper-fections, structural deformations and plastic zones – have to be covered for the verifi-cation in a convenient manner. The different possibilities of how verifications may be executed are illustrated in Table 9.1.


Figure 9.4 Explanations on the verification of sufficient bearing capacity using the example of a compression member

Using the example of a compression member, Figure 9.4 clearly shows the differences between the three verification possibilities. An ideal case is presented, for which all three methods lead to an equal bearing capacity. However, for structural systems of civil engineering this is something of an exception.

Note: According to DIN 18800 Part 1, element 728, the equilibrium conditions are to be set up for a deformed system (second order theory). The influence of the de-formations according to second order theory on the equilibrium may be neglected if the increase of the relevant internal forces and moments due to the deformations determined using first order theory is less than 10%. According to element 739, this condition can be considered as fulfilled if the axial forces N of the system are not larger than 10% of the axial forces Ncr,d belonging to the critical load of the system. This means that cr,d = Ncr,d/N > 10 for the critical load factor.

Use of the reduction factors With this method, all influences reducing the bearing capacity are captured by reduc-tion factors, and the verifications are executed with the internal forces and moments according to first order theory. For that purpose, eigenvalues are required, e.g. Ncr and Mcr. For the compression member in Figure 9.4, the critical load Ncr is determined for flexural buckling, so that the nondimensional slenderness K can be obtained. It serves the determination of the reduction factor that is required for the verification condition N Npl,d.


Table 9.1 Alternative methods for the verification of adequate structural safety for flexural buckling and lateral torsional buckling

Methodology Flexural buckling Lateral torsional buckling Verification with reduction factors Stresses for the verifications Internal forces and moments according to

first order theory Critical loads (eigenvalues) Ncr,y; Ncr,z Mcr,y; Ncr

Nondimensional slendernesses K M

Reduction factors M

Verification with equivalent imperfections Equivalent geometric

imperfectionsw0 or v0 v0

Stresses for the verifications Internal forces and moments according to second order theory

Verifications Stress verifications or verifications utilising plastic capacities

Plastic zone theory Imperfections Geometric imperfections and residual stresses

Stresses for the verifications Internal forces and moments according to second order theory under consideration of plastic zones

Verifications Utilisation of the plastic material behaviour according to Figure 1.11

Note Conditionally applicable!

Table 9.2 contains examples for flexural buckling and lateral torsional buckling. In this, the verification process of both stability cases can be followed and compared. The verifications are to be assigned to the verification method Elastic-Plastic in Table 1.1 since the plastic bearing capacity of the cross sections is taken into consideration. Verifications using reduction factors are the most common verifications for flexural buckling and lateral torsional buckling of simple beams.

In DIN 18800-2, these verifications are called “simplified verifications”, which might possibly be deceptive, because the verification approach with geometric imperfections (see Table 9.1) also contains simplifications in terms of approximations and it is not more extensive for many applications. They are only “simplified verifications” in the case where the critical loads can be determined without much complexity, as for example for the Euler cases – see Figure 9.5.


Table 9.2 Verifications with reduction factors for flexural and lateral torsional buckling Flexural buckling about the weak axis – see Section 9.4

2 2z

cr 2 2EI E 2003N 1153.2 kN

600, see Figure 9.5

K pl crN N 1704 1.1 1153.2 1.28

= 0.397 for curve c according to Table 9.5

Verification:pl,d

N 650 0.960 1N 0.397 1704

Lateral torsional buckling – see Section 9.62 2

cr cr,z p pM N c 0.25 z 0.5 z

231.7 kNm1.12 see Table 9.13

cr,zN 1153.2 kN (see above!)

22 T

z

I 0.039 IcI

22167060 0.039 600 59.59 501.1cm

2003

M pl,y cr,yM M 140.2 1.1 231.7 0.816

M 0.883 and LT,mod = 0.833 according to Table 9.82 2

y zmaxM q 8 26 6 8 117 kNm The verification is executed with LT,mod since with M it is on the unsafe side for this system – see Section 9.6.7:

y

M pl,y,d

M 117 1.002 1M 0.833 140.2

Verifications with the equivalent imperfections method For this method equivalent geometric imperfections are applied, and with these the internal forces and moments are calculated according to second order theory. For the compression member shown in Figure 9.4, a initial bow imperfection w0 is applied at midspan, so that bending moments result and the limit load-bearing capacity is reached if a plastic hinge occurs at midspan due to N and M. The equivalent geometric imperfections capture influences due to actual geometric imperfections, residual stresses and the expansion of plastic zones. The internal forces and moments are determined with the elastic system. We can also insert plastic hinges, so that the system stays elastic next to the plastic hinges. This method is an approximation for the plastic zone theory. The correlation to the verification procedures stated in Table 1.1 depends on the way the system is calculated and verified:


Table 9.3 Verifications with equivalent imperfections for flexural and lateral torsional buckling

Flexural buckling about the weak axis (see Section 9.8)Equivalent geometric imperfection: v0 = /200 = 600/200 = 3.0 cm Maximum bending moment according to second order theory:

z 0cr,d

1 1max M N v 650 3.01 N N 1 650 1.1 1153.2

5132 kNcmVerification with Table 8.6 for Vy/Vpl,y,d 0.25: Mpl,z,d must be limited to 1.25 Mel,z,d = 1.25 Wz fy,d

= 1.25 200.3 24/1.1 = 5463 kNcm. With N/Npl,d = 650/1704 = 0.381

it is: 251320.91 0.381 0.855 0.145 1.0005463

Lateral torsional buckling (see Section 9.8)

With a computer program, the following internal forces and moments can be calculated according to second order theory: My = 117 kNm, Mz = –10.92 kNm, M = 86.57 kNmcm The verification is executed with the help of the PIF-method – see Chapter 8 and [25]:

Mpl,g,d = 0.25 1.5 202 24/1.1 = 3273 kNcm

z

g

MM 1092 8657 1014 kNcm 3273 kNcm2 a 2 20 1.5

ob 20 1 78 3273 19.76 cm ; ub 20 1 1014 3273 16.62 cm

o18.5 19.76 16.62 1.5h 6.63 cm 0

2 2 0.9Since ho > 0, the following verification is decisive:

My = 11700 kNcm < (1.5 16.62 + 0.9 18.5/2) 18.5 24/1.1 – 0.9 6.632 24/1.1 = 12560 kNcm

Elastic-Elastic: Elastic system and stress verification Elastic-Plastic: Elastic system and verification with interaction conditions or other conditions according to Section 8.3 Plastic-Plastic: System with plastic hinges (see Section 5.11) and verification of the cross section bearing capacity as for the Elastic-Plastic method


Since about 1995, the equivalent imperfections method has become more and more accepted in the practical use of civil engineering. Predominantly, the Elastic-Plastic method is used in this context because, with adequate calculation complexity, economical results are achieved for the design. This is at least true for cross sections consisting of two or three plates for which the PIF-method can be applied (see Chapter 8 and [25]). If the cross section shape is more complex, stress verifications with the Elastic-Elastic method are reasonable. The Plastic-Plastic verification method is rarely used in civil engineering practice. Table 9.2 contains typical cases for the application of the Elastic-Plastic verification method. For comparison, the same structural systems are used as shown in Table 9.3.

Note: In the old stability standard DIN 4114, a comparable approach was identified with “verifications according to second order stress theory”. Since this expression can occasionally also be found in the relevant literature, it should be clarified that the internal forces and moments, and not the stresses, are calculated according to second order theory. The way of obtaining the stresses does not change. Incidentally, the verifications are usually conducted using internal forces and moments – see Table 9.3.

Plastic zone theory Calculations according to plastic zone theory cover the load-bearing behaviour most suitably. On the other hand, it cannot be done without a computer program for which, moreover, deepened knowledge and experience are required in order to be able to apply the program and interpret the results properly. Section 5.12 gives an overview of calculations according to plastic zone theory. However, for practical use, the application of plastic zone theory is inappropriate. Then again, it can be assumed that further developments of corresponding computer programs for flexural buckling willenable it to be used more widely in the near future.

For the compression member in Figure 9.4, a nonlinear load-displacement relationship occurs, which decreases when the bearing capacity is reached and which is typical for stability problems. Section 5.12.4 contains a calculation example for the application of plastic zone theory describing the load-bearing behaviour in detail.

Note: The technical standards demand stability verifications, and in DIN 18800-1 it says: “It has to be verified that the system is in a stable equilibrium.” Correspon-ding verifications are for example: cr,d > 1, N < Ncr,d and My < Mcr,d. Such verifications are usually not directly executed, but included using the -verifications and the determinations of the internal forces according to second order theory. In the case of plate buckling (see Chapter 10), the existing stresses may even exceed the critical buckling stresses since the values take into consideration postcritical reserves.

9.4 Verifications for Flexural Buckling with Reduction Factors 329

9.4 Verifications for Flexural Buckling with Reduction Factors


According to DIN 18800 Part 2, element 301, flexural buckling and lateral torsional buckling of members may be analysed separately. In this Section, the verifications for flexural buckling using the method are dealt with. DIN 18800 Part 2 distin-guishes verification conditions for three cases:

axial compression force N axial compression force N and bending moment My or Mzaxial compression force N and bending moments My and Mz

For uniaxial bending with axial compression force N, the verification condition of the DIN is valid for the flexural buckling in the plane of the bending moment. This case covering a regular moment My about the strong axis and a member deformation orthogonal to that, i.e. in the y-direction, is missing. In the commentaries of the DIN [62], this circumstance is mentioned, and it explaines that this case is not decisive for the design. If there are any doubts concerning this issue, it is recommended to analyse this case with the help of the procedure with equivalent geometric imperfections (see Section 9.8) or to use the verification condition for biaxial bending with axial compression force of [8] and set Mz = 0.

In DIN 18800, the verifications with the -procedure are identified with “simplifiedverifications”. In the relevant literature, the expression “effective length method” is often used, since the actual member length is replaced by the buckling length. How-ever, the verification conditions do not yet reveal this methodology directly.

The verification of compression members using the method assumes an ideal (straight) beam, and the influence of imperfections (initial out-of-straightnesses, residual stresses) is captured by the reduction factor . The imperfections lead to bending moments, and plastic zones emerge reducing stiffness (also see Section 5.12). These effects have to be covered by the values as well. Also the increase of the bending moments according to second order theory is taken into consideration. Therefore, internal forces and moments according to first order theory are inserted for verifications with the method, and imperfections are not applied. However, for systems including hinged columns, initial sway imperfections of the columns have to be considered. Since shear forces are not taken into consideration with the veri-fication conditions, it is assumed that their influence is negligible. If necessary, addi-tional verifications have to be performed and the increase of the shear forces accord-ing to second order theory has to be considered.


Below the verification conditions for single members are dealt first with. Veri-fications for flexural buckling according to DIN 18800 Part 2 are given priority here. In addition, the corresponding regulations of Eurocode 3 are mentioned. For certain applications, Section 9.4.4 contains references to the values and reduction factors of higher accuracy.

For the determination of the reduction factors , the buckling lengths sK or the critical axial compression forces Ncr are required. Their calculation is a central task of the verification using the method and, apart from simple standard systems, it requires deeper knowledge. The determination of cr and Ncr is therefore further discussed in Section 9.5.

9.4.2 Axial Compression

According to DIN 18800 Part 2, the bearing capacity verification with regard to regular centrically compressed members may be performed with the following condi-tion:

pl,d

N 1N (9.3)

The reduction factor has to be determined for the decisive buckling direction ( yor z) depending on the nondimensional slenderness K :

K 0.2 : 1 (9.4a)

K 2 2K

10.2 :k k

(9.4b)

as simplification for KK K

13.0 : (9.4c)

The parameters in Eq. (9.4) are: 2

K Kk 0.5 1 0.2 (9.5)

= 0.13 for buckling curve a0 = 0.21 for buckling curve a; = 0.34 for buckling curve b = 0.49 for buckling curve c; = 0.76 for buckling curve d

The cross sections of the compressed members are assigned to four different buckling curves. In doing so, not only is the shape of the cross section distinguished, but also buckling perpendicularly to the y- or z-axis. The allocation of the cross sections to the buckling curves can be taken from Table 9.4. At the bottom of the table, a direct classification for rolled I-profiles is included.


Table 9.4 Allocation of the cross sections to buckling curves

For members of steel grade S460, a0 instead of a, a instead of b, b instead of c and c instead of d may be used.


The nondimensional slenderness ratio is calculated with:

plK

cr

NN

(9.6)

Alternatively, the calculation formula

KK

a(9.7)

may be used. Here the slenderness K = cr /i is determined using the buckling length cr and the gyration radius i. The reference slenderness ratio

ay,k

Ef (9.8)

for common structural steels is:

a = 92.9 for fy,k = 24 kN/cm2 (S 235, t 40 mm) a = 75.9 for fy,k = 36 kN/cm2 (S 355, t 40 mm)

cr crFigure 9.5 for four compression members with constant compression force N and a constant bending stiffness EI.

Figure 9.5 cr cr for the four Euler cases

forces and no other internal forces or moments occur. Because the member axis is not straight in reality, bending moments will develop, reducing the bearing capacity. The reduction factors according to Eq. (9.4) contain all the influences reducing the

and the calculation of the lowest critical load N is shown in The buckling lengths

The section heading “axial compression” expresses that only regular compression

Buckling lengths and critical loads N


bearing capacity of compression members. These are geometric imperfections and re-sulting bending moments according to second order theory as well as the influences of residual stresses and plastic zones – also see Section 5.12.

Figure 9.6 Buckling curves a, b, c and d for the flexural buckling of compression members

The buckling curves a, b, c and d according to Eq. (9.4) are illustrated in Figure 9.6. They are bordered by two lines: = 1 and 2

cr K1 . The value = 1 means that N = Npl,d and x = fy,d. It represents the upper limit since the maximum axial stress is allowed to be equal to the yield strength. Then again, Euler’s buckling stress cr = Ncr/A corresponding to the critical load Ncr is an upper limit as well. As a nondimensional formulation, Euler’s buckling stress is obtained as follows:

22cr a

cr 2 2 2y,k K y,k K K

E 1f f (9.9)

As it can be seen, the strongest deviation of the buckling curves from the mentioned boundaries can be found in the central slenderness area (at about K 1.0 ). There-fore, the imperfections have the largest influence in that range. In Table 9.5, numerical values of for the buckling curves a, b and c are compiled as design aid for compression members.


Table 9.5 Reduction factors according to DIN 18800 Part 2 and of EC 3


Table 9.5 (continuation)


Calculation example: Base restraint column Figure 9.7 shows a column restraint at the base, so that the buckling length cr = 2.0 = 12 m can be determined with Figure 9.5. Because moment of inertia Iz = 2003

cm4 < Iy = 5696 cm4, the lowest critical load for flexural buckling results about the weak axis. This case is also significant because it is to be assigned to the buckling curve c and the flexural buckling about the strong axis according to Table 9.4 to the buckling curve b. With Figure 9.5, we have

2

cr,z 221000 2003N 288.3 kN1200

and taking Npl,d = 1704 kN from the tables in [29]

K1704 1.1 2.55

288.3

With that, the value z = 0.128 can be taken from Table 9.5 for the buckling curve c, and the bearing capacity can be verified with Condition (9.3):

pl,d

N 210 0.963 1N 0.128 1704

Figure 9.7 Base restraint column

Note: The verification using Condition (9.3) is to be assigned to the Elastic-Plastic verification method according to DIN 18800. Thus, the cross section parts may have a maximum proportion of dimensions of “limit (b/t)”, as stated in Table 15 of DIN 18800 Part 1. For the rolled profile HEB 200 of S 235 of the previous calcu-lation example, the conditions “existing (b/t) limit (b/t)” are met for the flanges and the web. This can be seen in the tables of [29].

Verification according to Eurocode 3: The verification according Eurocode 3 [10] is directly comparable to the one of DIN 18800 Part 2. However, in [10], the reduction factor is designated . The buckling curves and the corresponding reduction factors are identical. For the verification it


has to be considered that the value comparable to Npl,d has to be determined with the comparable yield strength and the partial safety factors according to [10].

Comparison of the method with the equivalent imperfections methodCompression members can be verified with Eq. (9.4) or alternatively with the equi-valent imperfections method. In doing so, differences occur which are definitely worth mentioning and which will be discussed here. For Euler case II and selected cross sections, Figure 9.8 shows the relation of the bearing capacities. Since the method is the more precise method for compression members, the load-bearing capacity N( ) is selected as base value. For w0 and v0, the values of Table 9.17 have been inserted.

Figure 9.8 Bearing capacity of compression members with the equivalent imperfections method compared to the -method

As can be seen, the load-bearing capacities max N(w0) and max N(v0), determined with the equivalent imperfections method, are on the safe side. For flexural buckling about the weak axis and K 0.7 , the results of both methods are almost equal. The largest deviations arise for the flexural buckling of an HEM 340 about the strong axis with K 0.9 . Here, the bearing capacity resulting from the equivalent imperfections method is almost 12% smaller than the one obtained with the method.


9.4.3 Uniaxial Bending with Compression Force

Where a bending moment occurs as well as the compression force, Condition (9.3) is extended and the bearing capacity has to be verified as follows, using the buckling curves:

m

pl,d pl,d

N M n 1N M

(9.10)

withreduction factor for buckling in the moment plane

M maximum absolute value of the bending moment according to elastic firstorder theory without application of imperfections

m moment factor for flexural buckling according to Table 9.6. Factors m < 1 are only allowed if the member is supported nondisplaceably at both member ends with a constant cross section and constant compressionforce without transverse loads.

n 2 2K

pl,d pl,d

N N1N N

, but n 0.1

As a simplification, n may either be set to 2 2K0.25 or 0.1.

Table 9.6 Moment coefficient m for flexural buckling

Moment diagram m ExplanationsEnd moments m, 0.66 0.44

but m,Ki

11

and m, 0.44 For most structural cases m = 1.0 as an approximation.

For members with end moments we have m > 1 for > 0.77.

Since m is the numerator of the moment amplification factor

m

Ki1 N N

more precise values can also be used – see [42].

Moments due to in-plane loading

m,Q 1.0

Moments due to in-plane loading and end moments m0.77 : 1.0

0.77 :

Q 1 m,m

Q 1

M MM M


For the calculation of Mpl,d, the limitation of pl = 1.25 has to be considered according to DIN 18800 Part 2, element 123. For doubly symmetric cross sections which have a web area larger than 18% of the total cross section area, Mpl,d in Condition (9.10) may be replaced by 1.1 Mpl,d provided that

pl,dN 0.2 N (9.11)

All rolled profiles of the series IPE, IPEa, IPEo, IPEv, HEAA, HEA, HEB and HEM have at least 18% web area.

Calculation example

Figure 9.9 Calculation example: Compression member with transverse loading

For the compression member of Figure 9.9 with transverse loading, the flexural buckling about the strong axis is analysed. The following values can be taken from [29]:

Npl,d = 742 kN, Mpl,d = 36.05 kNm, Iy = 864.4 cm4, buckling curve b With cr = 5 m, we have:

2

cr,d 221000 864.4N 651.5 kN500 1.1

K b742 651,5 1.067 0.555 (see Table 9.5)

The maximum bending moment occurs at midspan and is

max My = 3.6 (2.52 3/4 – 2.5 1.25) + 5.1 5/4 = 5.625 + 6.375 = 12.00 kNm

1.1 Mpl,d is introduced in Eq. (9.10) due to N > 0.2 Npl,d. With m = 1.0 and the ap-proximation for n, the verification is:

2 2250 12.00 0.25 0.555 1.0670.555 742 1.1 36.050.607 0.303 0.088 0.998 1


For the application of Condition (9.10), the elements 315 to 319 in DIN 18800-2 have to be considered. They contain additional regulations for the following aspects:

influence of transverse forces nonuniform cross sections and axial forces rigid connections beam sections without compression forces load cases of bearing displacements and temperature

Most important for structural applications is element 318, “beam sections without compression forces”. The column in Figure 9.10 is used as an example. In the upper column area N = 0, so that there a verification with Condition (9.10) cannot be conducted. However, additional stress occurs in the uncompressed part due to imperfections and second order theory since the bending moments at the transition to the compressed part are equal. The verification is given as shown in Figure 9.10. Here the value 1.15 (> 1) captures the influence of imperfections by approximation. Incidentally, the verification condition for parts free of compression contains the moment amplification factor 1 / (1 N / Ncr). The verification is only relevant for the design if the cross section of the uncompressed areas is weaker or if the scheduled bending moments there are larger than in the part loaded by compression force.

Figure 9.10 Column with parts free of compression force

9.4.4 Modified Reduction Factors

According to Table 9.4, the cross sections of the compression members with steel grade S235 and S355 are assigned to four different buckling curves. Since this assignment must lead to verifications being on the safe side, important bearing reserves are definitely present in some individual cases.


For explanation, the differences between the values and the buckling curves a to b, b to c and c to d are shown in Figure 9.11. The largest deviations occur at K 1.05with 11.5%, 10.6% and 15.6%. Between K 0.5 and 2.5, the differences are con-tinuously larger than 5%.

Figure 9.11 Relation of the reduction factors for the buckling curves a/b, b/c and c/d as well as a/ab and ab/b

The question is how these obvious bearing reserves may be utilised. Although with maximum 15.6%, the largest difference can be found with curves c and d, this case is not further followed here. Since cross sections to be assigned to curve d are relatively rarely used, this case is of minor significance for construction practice.

Due to their frequency of use, rolled I-profiles are particularly interesting, so that they are analysed in more detail here. According to Table 9.4 (at the bottom), these cross sections are assigned to buckling curves a, b and c, and the following cases are to be distinguished regarding S235 and S355:

flexural buckling about the strong axis and h/b > 1.2: curve a

flexural buckling about the weak axis and h/b > 1.2: curve b

flexural buckling about the strong axis and h/b 1.2: curve b

flexural buckling about the weak axis and h/b 1.2: curve c As can be seen, the four cases are assigned to three buckling curves. It is therefore quite likely that a case assigned to line b is comparatively clear on the safe side.


This is confirmed by the analyses in [40], where for the case of

flexural buckling about the strong axis and profiles with h/b 1.2,a buckling curve “ab” is suggested, which is between the curves a and b. The sugges-tion is substantiated in detail in [40] and = 0.26 is applied for the calculation of the

values with Eq. (9.5). Typical columns, for which h/b is frequently smaller than 1.2, can be designed more economically than with the use of curve b for the verifications. Figure 9.11 shows that the values of the new curve “ab” are only up to 6.6% larger than the values of curve b. Figure 9.12 gives the accuracy of the buckling curves compared to the exact calculations according to plastic zone theory (PZT). The figure contains the minimum values of all profiles for the four cases mentioned above. Curve “ab” for flexural buckling about the strong axis with profiles h/b > 1.2 captures the bearing capacity with much more accuracy than curve b of the buckling curves.

Figure 9.12 Bearing capacity of compression members with the method compared to the plastic zone theory (PZT)

9.5 Calculation of Critical Forces

9.5.1 Details for the Determination

Figure 9.5 has shown the determination of axial critical forces Ncr for the Euler cases I to IV. Since the axial compression force N is equal to the acting load and,

9.5 Calculation of Critical Forces 343

moreover, N does not change in the beam (N = const.), the Euler cases belong to the simple systems for an analysis of flexural buckling. For complex structural systems, not only is the calculation of Ncr more time-consuming, but also the methodology is intellectually more difficult. Here, some information is thus given on the approach and on the specifics. In doing so, it is assumed that the eigenvalue determination is optionally performed as follows:

use of computer programs, i.e. determination of cr (see Chapter 6)

set up of buckling conditions and determination of Ncr, cr, or cr

use of formulas or diagrams taken from the relevant literature, i.e. determina-tion of

Figure 9.13 Single-haunched frame with internal forces and moments according to first order theory

Principle: For a structural system first of all the axial forces are calculated ac-cording to first order theory, then this axial force distribution is used for an eigenvalue analysis. Conceptually, all other external loads are removed and only the axial forces applied. This always leads to a homogeneous problem (differential equa-tion, equation system). Possibly present bending moments have no effect on Ncr. As an example, the single-haunched frame in Figure 9.13 is considered and, first of all, the internal forces and moments are calculated according to first order theory. The bending moments do not affect Ncr, so that for the eigenvalue analysis we can have H = 0. Ncr only depends on the axial compression support force. With the approach in Section 9.5.2 (adding springs) and the derivations in Section 9.5.3, the buckling condition is:

s

s

Ctan

EI with: R

R

3EIC (9.12)


Critical load factor cr

Figure 9.14 shows structural systems with nonuniform axial forces. For systems of this kind it is advisable to use the critical load factor cr for the eigenvalue determination. This is so that the relations between the different axial forces are clearly defined. Theoretically, the existent axial force distribution is always assumed and increased until the first eigenvalue is reached. This factor is the critical load factor cr, so that

Ncr,i = cr Ni or Ncr(x) = cr N(x) (9.13)

is valid. cr applies for the structural system and depends on the following parame-ters:

axial force distribution

dimensions of the members

stiffnesses EI

boundary conditions Equation (9.13) shows that the axial critical forces are affine to the existent axial force distribution.

Figure 9.14 Compression members with nonuniform axial forces

Figure 9.14 clarifies that the critical load factor is determined for the existent axial forces. For the sketches cr = 2 has been assumed and the ordinates for Ncr(x) are thus twice as large as for N(x). For the verifications with the method (Section 9.4) it must be observed that Eq. (9.3) has to be fulfilled everywhere. Provided that mem-bers have uniform cross sections, it is sufficient to perform the verification at the point with the largest N.


Due to the importance of the critical load factor cr for the eigenvalue determination, another structural system is considered with the frame in Figure 9.15. From the loading an axial force distribution N(x) results, which leads to a critical load factor

cr. Now, the critical forces Ncr,i required for the verifications can be determined in the individual members.

Note: For the eigenvalue determination with computer programs, loads are combined regarding F (partial safety factor) and i (combination coefficients) – see Chapter 1. In doing so we always obtain the critical load factor cr as a result. If

M = 1.1 and this value is also used by the program for the reduction of the stiff-ness values EI, it is the design value cr,d which leads to Ncr,d. For this approach, the nondimensional slenderness K must be calculated according to Eq. (9.6) using Ncr = Ncr,d M and Npl or with Npl,d = Npl/ M and Ncr,d.

Figure 9.15 Frame with loading, N(x) and Ncr(x)

Independent buckling of partial systems For some structural systems, partial systems may deform independently from each other. If partial systems emerge, which are decoupled by hinges, all required buckling shapes (eigenmodes) have to be analysed and the corresponding eigenvalues to be de-termined. Figure 9.16 shows a simple example.


Through the hinge at the inner support, both compression members are decoupled and they buckle independently of each other. Since Euler case III occurs for the left and Euler case II for the right member, the critical forces can be determined with the help of Figure 9.5. If the calculation is performed with a computer program, the first and the second eigenvalues (where required also a higher value) must be determined. In doing so, we also need the respective buckling shapes since the eigenvalues have to be assigned to both compression members, i.e., in more general terms, to the partial systems. Thus, an appropriate capable computer program is required.

Figure 9.16 Example on the flexural buckling of decoupled partial systems

For the single-haunched frame with a pendulum column in Figure 9.17, two cases can also be distinguished. The buckling shapes show that for the flexural buckling of the single-haunched frame a lateral displacement occurs and the hinged column remains straight. Furthermore, the hinged column may buckle completely independently of this and the frame is not displaced. If this problem is presented to students, they tend to “mix” both buckling curves, i.e. they deform the frame and the hinged column (in one figure). However, at this point, as already mentioned above, a clear way of thinking is required: there is only an “either or ”. For the verifications, as for the example in Figure 9.16, both eigenvalues cr,1 and cr,2 are necessary.

Figure 9.17 Independent flexural buckling of the system and the hinged column


Buckling condition with parameter Buckling conditions (see Table 9.8) are frequently formulated with the member char-acteristic N EI . The evaluation of these buckling conditions leads to the eigenvalue cr. With that, the buckling length coefficient

= / cr (9.14)

or directly 2cr

cr 2EIN (9.15)

can be determined.

Eigenvalue determination with the relevant literature Numerous formulas and diagrams can be found in the relevant literature on the de-termination of eigenvalues. Here, the buckling length coefficient is usually stated. From cr = , we have

2

cr 2EIN (9.16)

For structural systems with nonuniform axial forces it is often not immediately clear which axial force the given solution refers to. Usually, this is the maximum axial compression force. For example, for the column in Figure 9.14 on the left, one can assume that the -value given in the relevant literature for the axial compression force N2 is valid, so that

2

cr,2 2lit

EIN (9.17)

applies. From that the critical load factor of the system

cr = Ncr,2/N2 (9.18)can of course be calculated. Provided that the cross section in the upper part of the column is less capable of bearing than the lower part, also

Ncr,1 = Ncr,2 N1/N2 (9.19)

is required. This transformation results from the circumstance shown in Figure 9.14:

cr = Ncr,1/N1 = Ncr,2/N2 or cr = Ncr(x)/N(x) (9.20)


9.5.2 Replacement of Structural Parts by Springs

If structural systems are a bit more complex than the Euler cases, we frequently try to simplify the systems and to reduce them to compression members with springs at the ends. If there are any symmetry properties, they are utilised for this.

Figure 9.18 Reduction of a two-hinged sway frame

Figure 9.18 shows a typical example. The depicted two-hinged frame is symmetric,and an antimetric buckling shape belongs to the lowest eigenvalue. It is sufficient to analyse one half of the system and to apply a flexible, vertically nondisplaceable support at the frame centre. The support directly results from the shown buckling shape, but it can also be taken from Table 9.7. In a second simplification step, half of the horizontal member can be replaced by a rotational spring C , because there is no axial compression force. According to Figure 9.18 at the bottom right, the spring stiffness can be calculated with the spring law Ml = C l if the moment Ml is applied and the corresponding rotation l determined with the energy theorem.

For simple systems, as for example members which are hinged on both sides, the deformations can also be determined with calculation formulas from the relevant literature, i.e. for this example, the rotation l due to the moment Ml. As a result of the simplifications, we obtain the equivalent system shown in Figure 9.18 on the right, which has exactly the same risk of buckling as the original system. It can be immediately seen that the buckling length coefficient must be between 2.0 and (C

or C = 0). The exact solution can be determined using Section 9.5.3 – also see Figure 9.24.


Details on the replacement by springs In general, arbitrary parts of a structural system may be replaced by springs. If axial compression forces occur in these parts though, nonlinear springs result, so that the solution of the eigenvalue problem cannot be simplified. From this, the following principle results:

For the flexural buckling stability problem, parts of the structural system without compression axial forces may be replaced by springs.

However, provided that axial tension forces occur, the relieving effect may be neglected and we can have N = 0. For the application of the principle above the fol-lowing limitation must be observed:

Point springs Cw (translation) and C (rotation) must be independent of each other.

The circumstance is explained using Figure 9.19. Both systems differ only in the hinge in the right corner of the frame (at the top) contrary to the restraint (at the bot-tom). For the determination of the springs, the part free of compression forces (right) is separated from the column loaded by compression forces (left) for both systems. Since the deformations have to be calculated with the energy theorem, H and M are applied as loads and the moment diagrams determined. For the upper system (with hinge!) a displacement w results due to H, and the corresponding rotation is equal to zero. As a result of M, we have 0 and w = 0, so that for these systems single springs being independent of each other can be determined for the equivalent system. When the lower system with the rigid frame corner (right) is analysed, we notice that displacements and rotations result due to H and M. At the intersection, both springs are thus not independent of each other and the reduction to the equivalent system with two single springs is not possible.

Figure 9.19 Frames with independent single springs and coupled springs


Note: The coupling of the springs in Figure 9.19 at the bottom can be seen from the moment diagrams due to H and M, since the superposition leads to 12 0 when applying the energy theorem. The safest way for the determination of the inde-pendence of springs is the idealisation with finite elements of the displacement method. The displacement and rotation at the intersection are always independent of each other if the secondary diagonal members of the matrices at the points con-cerned are equal to zero.

Symmetric systems If for structural systems the geometry, the rigidity and the axial compression forces are symmetric, they may be separated in the symmetry axis and bearings may be applied there. The respective bearings are compiled in Table 9.7, where symmetricand antisymmetric buckling shapes are shown.

Table 9.7 Bearings for the symmetry axis of systems

Symmetric buckling shape Antisymmetric buckling shape

uc = 0 w 0 (V = 0)

= 0 (M 0)

u 0 w = 0 (V 0)

0 (M = 0)

Figure 9.20 Equivalent system for a laterally nondisplaceable two-hinged frame


For the two-hinged frame in Figure 9.20 the horizontal beam is laterally rigidly sup-ported and a symmetric buckling shape belongs to the lowest eigenvalue. The points of contraflexure are in the columns and designated with an “x”. According to Table 9.7, a bearing with a rigid restraint, which is displaceable in vertical direction, can be applied in the axis of symmetry. For half of the horizontal beam then, the stated rota-tional spring C and the equivalent system shown on the right result. The buckling length can be determined with the help of Section 9.5.3. It must be between cr = h (for C = 0) and cr = 0.7 h (for C ). Within this range, it is then also verified that for the laterally displaceable two-hinged frame in Figure 9.18, the antisymmetricbuckling length provides the lowest eigenvalue due to 2 .

Figure 9.21 Compression member with support by a beam

A further example for using springs as replacement of parts of a structural system is shown in Figure 9.21. Here, the compression member is connected to a beam. In the plane of the structure, the compression member has the buckling length cr = (Euler case II). Perpendicular to this, the compression member is supported by the beam at the end. The displacement wF of the statically indeterminate beam is not calculated with the energy theorem here, but with the formula

2 3

F 3a bw F 4a 3b

12 a b EI(9.21)

taken from [88]. The spring then results as follows: 3

w 2 3F

12 a b EIFCw a b 4a 3b

(9.22)

The eigenvalue determination for the equivalent system in Figure 9.21 may be carried out according to Section 9.5.3. For a stiff spring with

2

w 3EIC (9.23)


Euler case II results, i.e. cr = . If the spring is weaker, it is compressed and the com-pression member remains straight going along with:

Ncr = Cw (9.24)

9.5.3 Compression Members with Springs

In Section 9.5.2, we looked at how structural systems can be reduced to the equivalent system “compression member with springs at the ends”. The analysis of systems of this kind is the consequent extension following the Euler cases I to IV and helps us to understand the eigenvalue determination of systems.

Compression members with one spring First of all, compression members with only one spring are considered, as in [59], and six systems are analysed. For systems 3 and 4 (Figure 9.22), two compression mem-bers are shown in each case. This means that it does not matter whether the rotational springs act at the top or at the bottom.

The buckling conditions can be derived using the homogeneous differential equation of Eq. (3.32) and corresponding boundary conditions of the systems, as shown in [42]. The emerging conditions are compiled in Table 9.8. These can be used for cal-culating the eigenvalues cr. Since the solution must be determined iteratively, it is advisable to replace cr by / and to conduct the evaluation in the respective area of validity for . The results are shown in Figures 9.23 and 9.24, from which the buck-ling length coefficients can be taken.

Figure 9.22 Compression members with one spring at the ends


Table 9.8 Buckling conditions for the compression members shown in Figure 9.22

System Buckling condition Range of validity

1 2cr cr wsin 1 C 0 1

2 2cr cr cr w crcos 1 C sin 0 0.7 2

3 2cr cr cr crcos 1 C sin 0 0.7 1

4 cr cr crcos C sin 0 2

5 cr cr crsin C cos 0 1 2

6 cr cr cr cr cr cr crC cos sin 2 1 cos sin 0 0.5 0.7

For iterative analysis of the buckling conditions it is advisable to replace cr by / and to determine .

Figure 9.23 Buckling length coefficients for systems 2, 3, 5 and 6 in Figure 9.22


Figure 9.24 Buckling length coefficients for systems 1 and 4 in Figure 9.22

For Figure 9.24, it should be noted that 1/ is used, i.e. the reciprocal of . This is advantageous because can take very large values if the spring stiffness values are small. For system 1, the exact solution can be determined with Table 9.8 without any problems. Since the buckling condition contains two factors, results as follows:

2w

w

C :C

2wC : 1 (9.25)

Using the parameters

ww

11 C

and 11 C (9.26)

the following approximations can be used for the other systems:

System 2 2 3

w w w0.7 3.8 4.3 1.8 (9.27)

System 3 20.7 0.6 0.3 (9.28)

System 4 1

0.5 0.45 but 0.8 (9.29)


System 5 21.0 1.2 0.2 (9.30)

System 6 20.5 0.5 0.3 (9.31)

Compression members with two or three springs Many systems which are relevant for practical construction may be reduced to com-pression members with two or three springs. The columns of frames in multi-storey buildings are for example compression members being non-rigidly restrained at both ends. Therefore, the system shown in Figure 9.25 is analysed.

Figure 9.25 Compression member with three springs at the ends

C1 and C2 are rotational springs and Cw is a translational spring. They are converted into nondimensional spring stiffness values referring to the member stiffness as fol-lows:

11

CCEI

; 22

CCEI

;3

ww

CCEI

(9.32)

According to [42], the buckling condition for the member in Figure 9.25 is:

w 1 2 1 2 w

3 4 5w 1 2 1 2

C C C 2 2 cos sin C C C sin cos

C C C sin C C cos sin 0(9.33)

From this formula, various special cases can be developed for compression members with two springs, for example the one for

wC 0 , wC ( nondisplaceable support), 1C ( restraint)

Solving condition (9.33) for selected special cases leads to their eigenvalues = cr.The corresponding buckling length coefficients are stated in the following figures.


Figure 9.26 Buckling length coefficients for compression members supported and with rotational springs at both ends

Figure 9.27 Buckling length coefficients for compression members displaceable at one end and rotational springs at both ends


Figure 9.28 Buckling length coefficients for compression members with a hinged end and Cw as well as C at the other end

Figure 9.29 Buckling length coefficients for compression members with a restrainedend and Cw as well as C at the other end


In principle, Figures 9.26 and 9.27 comply with Figures 27 and 29 in DIN 18800 Part 2 being given there for the determination of the critical load factor cr and the buckling lengths cr for columns of nondisplaceable and displaceable frames. For im-proving readability, the curves for are here shown completely.

With Figures 9.28 and 9.29, the buckling length coefficients of compression members being supported at the upper end by a translational and a rotational spring can be determined. The base point is hinged (Figure 9.28) or is restrained (Figure 9.29). The diagram in Figure 9.28 is also valid where the rotational spring acts at the base.

Calculation example: Buckling length of a two-hinged frame In Section 9.9.3, the internal forces and moments according to second order theory are calculated for a two-hinged frame and the ultimate limit state analysis is provided. Here, the buckling length of the frame columns is determined for this frame, but the following simplifications are made:

The haunches are neglected.

N = 0 is assumed in the rafter.

In both frame columns, equal axial compression forces are applied. Using Figure 9.18, the two-hinged frame can be reduced to half of the system, as shown in Figure 9.30. The equivalent system complies with system 4 in Figure 9.22, where the length of the half girder is to be applied for the calculation of the rotational spring stiffness.

Figure 9.30 Frame and equivalent system for the determination of the buckling length


For the nondimensional spring stiffness related to the bending stiffness of the column we obtain

C 3 16266 700C 1.897EI 18263 986

and as parameter according to Eq. (9.26):

1 0.3451 C

As approximation

1 2.900.5 0.45 0.345

results from Eq. (9.29). Alternatively, the buckling length coefficient can be taken from Figure 9.24, so that

1 0.34 and 2.94

follows for system 4. The exact solution can be determined with the buckling condi-tion in Table 9.8. After a short iteration, we obtain = 2.96. That gives

cr = = 2.96 7.00 = 20.72 m

as the buckling length and 2

cr 2E 18263N 882 kN

2072With this value, the nondimensional slenderness can be calculated and the ultimate limit state can be analysed with the method according to Section 9.4. However, we can also determine amplification factors (see [42]), calculate internal forces and moments according to second order theory and provide the ultimate limit state analyses with the equivalent imperfections method. No further calculations are given here, because the two-hinged frame is analysed in detail in Sections 9.9.3 and 9.9.4 and, furthermore, the critical load factor is large. Using N = –82.2 kN in Table 9.22 we find:

cr,d882 9.75 10

82.2 1.1


9.6 Verifications for Lateral Torsional Buckling with Reduction Factors


For stability verifications we generally proceed as stated in DIN 18800 Part 2, ele-ment 112: For the purpose of simplification, flexural buckling and lateral torsional buckling may be analysed separately. In doing so, the lateral torsional buckling has to be verified after the analysis of the flexural buckling for the single members thought to be detached from the complete system, which are stressed by the internal forces and moments at the member ends determined for the total system and the loads acting on the regarded member.

Moreover, element 303 has to be considered: The analysis for lateral torsional buckling shall be performed with the single mem-bers virtually detached from the framework. In this process, the internal moments at the member ends have to be determined according to second order theory if nec-essary.

As already explained in detail in Section 9.2, lateral displacements v(x) and rotations (x) about the x-axis occur for the stability problem lateral torsional buckling, as

shown in Figure 9.3. These rotations are usually caused by bending moments, so that, in general, members regularly loaded by bending moments are susceptible to lateral torsional buckling. However, the stability cases torional flexural buckling or torsional buckling can be caused by a compression force. In this Section, simplified verify-cations for lateral torsional buckling are presented, so the M according to DIN 18800 or the LT procedure according to Eurocode 3 is focused on here. This Section is in principle comparable to Section 9.4, where the verifications using the method for flexural buckling are compiled.

9.6.2 Beams Not Susceptible to Lateral Torsional Buckling

For some structural systems, lateral torsional buckling does not have an influence on the design since it is not decisive or only leads to small additional stresses. This is al-ways the case if the rotations (of the eigenmode!) are equal to zero or very small in comparison to the displacements. According to DIN 18800 Part 2, element 303, the analysis of lateral torsional buckling is not necessary for:

beams with hollow sections beams for which the rotation or the lateral displacement v is sufficiently pre-vented

9.6 Verifications for Lateral Torsional Buckling with Reduction Factors 361

beams with regular bending, if for the nondimensional slenderness, M 0.4

For beams with hollow sections, the torsional stiffness is usually large. For that rea-son, there is a correspondingly large resistance with respect to the torsional rotations

(of the eigenmode!). Moreover, the additional stress occurring is generally low.

Basically, there is no danger of lateral torsional buckling if compressed cross section parts are sufficiently supported by constructional provisions. Figure 9.31 shows a single-span beam loaded by a distributed load. Due to the positive bending moment, the upper flange and the upper half of the web are compressed. If the upper flange is laterally supported as sketched in Figure 9.31, the displacement v is zero there. The analysis of the eigenvalue and eigenmode shows that is zero and the beam is not susceptible to losing stability. This also applies if the beam is not continuously, but selectively supported laterally at sufficiently close distances at the upper flange.

Figure 9.31 Single-span beam laterally braced at the compression flange

A sufficient prevention of lateral displacements is also given for members where the compression flange is constantly braced by brickwork. According to Figure 9.32, the thickness of the brickwork may not be smaller than 0.3 × beam height.

Figure 9.32 Bracing with brickwork, [8]

Lateral displacements of compression flanges can be prevented with the help of con-structions in the form of planes or shear diaphragms. As condition for a nondis-placeable support, DIN 18800 Part 2 demands that


2 22

T z2 2 270S EI GI EI 0.25 hh

(9.34)

S is the portion of the shear stiffness of profiled sheetings or comparable structural elements regarding the analysed girder. The profiled sheetings are to be fastened to the beams in each corrugation and at each of the four edges. If the fastening is only applied at every second corrugation, so that there is no shear diaphragm, 20% of the shear stiffness may be taken into account.

Condition (9.34) is relatively far on the safe side. In [21], Heil shows that, with re-gard to an adequate structural safety, the shear stiffness

plMS 10.2

h(9.35)

is sufficient. In [63], Lindner discusses different possibilities for considering the shear stiffness and concludes that it is usually appropriate to apply the available value of S for the calculation of Mcr to a computer program.

9.6.3 Scheduled Centric Compression

For members regularly and centrically compressed, flexural buckling must be analysed and simplified verifications with the procedure according to Section 9.4.2 are to be executed. However, also torsional flexural buckling or torsional buckling may be relevant, although this is rather rare for practical applications.

Where it is relevant, the following verification has to be applied according to DIN 18800 Part 2: The bearing capacity must be verified with Condition (9.3), which is

pl,d

N 1N (9.36)

In doing so, members with arbitrary but nondisplaceable bearings at the ends, a constant cross section and a constant axial compression force are assumed. For the calculation of the nondimensional slenderness K , the axial force with regard to the critical load of torsional buckling has to be applied for Ncr. The reduction factor has to be determined for buckling perpendicularly to the z-axis, i.e. it can be determined using Table 9.4 and 9.5. The commentary [62] of the DIN includes information on when torsional buckling may be decisive. The lowest critical load in example 8.4 of [62], a column with an single-symmetric cross section, is the critical load for torsional buckling Ncr, .

The statement of DIN 18800 Part 2, which says that for rolled profiles with I-section and for I-girders with similar dimensions a verification for torsional buckling does


not have to be performed, has to be put into perspective. The Ncr for torsional buckling depends on the cross section properties, the column length and the boundary conditions (bearings). For explanation and as a calculation example, the column in Figure 9.33 is discussed. It is restrained at the column base and simply supported at the top. Furthermore, it is supported in the middle in y-direction.

Figure 9.33 Torsional buckling of a column

If the critical compression force is now determined using a computer program, we get

cr = 3.0774 Ncr = 3.0774 495 = 1523 kN

Since the corresponding eigenmode only includes rotations (x), the Ncr corresponds to torsional buckling. With Figure 9.33, it becomes obvious that v(x) = w(x) = 0. As explained above, the verification has to be performed as follows:

plK c

cr

N 685.4 1,1 0.704 0.723N 1523

495 0.999 10.723 685.4

To allow comparison, the second eigenvalue is determined as well:

cr = 3.3772 Ncr = 3.3772 495 = 1672 kN The corresponding eigenmode only shows displacements v(x). Since it is buckling about the weak axis, the verification with the -procedure is conducted as follows:

K c685.4 1.1 0.672 0.741

1672


495 0.975 10.741 685.4

The comparison with the verification of torsional buckling shows that the difference of 2.5% is low, even though Ncr is almost 10% larger. If the column in Figure 9.33 is supported laterally at several positions in field, torsional buckling is clearly decisive and the verification for flexural buckling may be far on the unsafe side. The example should clarify that the boundary conditions have a crucial influence on the design.

9.6.4 Uniaxial Bending without Compression Force

The verification is to be performed with the following conditions for beams with I, U and C-profiles which are not regularly loaded by torsion:

y

M pl,y,d

M1

M (9.37)

where My maximum absolute bending moment

M reduction factor for bending moments depending on the nondimensinal slenderness:

pl,yM

cr,y

MM (9.38)

M = 1 for M 0.41n

M 2nM

11

for M 0.4 (9.39)

Table 9.9 Section coefficient n for the determination of M

Rolled n = 2.5 Castellated n = 1.5

Welded n = 2.0 Notched n = 2.0

Haunchedmin hn 0.7 1.8max h

The coefficient n has to be additionally multiplied by 0.8 if the flanges are welded to the web.


Figure 9.34 Factor kn for the coefficient n

The coefficient n can be taken from Table 9.9. For beams with end moments, it has to be reduced by the factor kn according to Figure 9.34 if the ratio of the end moments is

> 0.5.

The reduction factors M are shown in Figure 9.35 for n = 2.5 and n = 2.0. For com-parison with flexural buckling, the buckling curves a, b and c are illustrated in the figure as well – also see Figure 9.6. Table 9.10 contains a compilation from which numeric values M for n = 2.5 can be directly taken. In addition, the reduction factors

LT,mod according to DIN EN 1993-1-1 for rolled I-profiles are also stated in Table 9.10. These values are given here since the M-values are, according to Section 9.6.7, up to 15% on the unsafe side – see also Figure 9.39.

Figure 9.35 Reduction factors M and comparison with the buckling curves a, b and c


Table 9.10 Reduction factors M according to DIN 18800 Part 2 and LT,mod of EC 3


Calculation example: Beam with distributed load and fork bearings at the ends

The beam of Figure 9.36 is verified with the M method. The maximum bending moment acts at midspan:

max My = 30 62/8 = 135 kNm The following values can be taken from the tables in [29]:

Mpl,y,d = 285.2 kNm Iz = 1318 cm4

IT = 50.41 cm4 I = 482890 cm4

According to Section 9.7, the critical moment is:

2 2cr,y cr,z p pM N c 0.25 z 0.5 z

2 2cr,zN 21000 1318 600 758.8 kN

22 2482890 0.039 600 50.41c 903.37 cm

13181.12 see Table 9.15

2cr,yM 1.12 758.8 903.37 0.25 20 0.5 20 18421 kNcm

With this, M according to Eq. (9.38) and the reduction factor according to Table 9.10 can be determined in order to perform the verification with Eq. (9.37):

pl,yM

cr,y

M 285.2 1.1 1.305M 184.21 M = 0.534 and LT,mod = 0.477

Because M is on the unsafe side according to Section 9.6.7 (also see Figure 9.39) the verification is executed with LT,mod:

135 0.992 10.477 285.2

Figure 9.36 Single-span beam with uniformly distributed load qz


9.6.5 Uniaxial Bending with Axial Compression Force

For beams not loaded by scheduled torsion but with constant axial compression and doubly or singly symmetric I-shaped cross sections and whose cross section dimen-sions comply with those of rolled profiles as well as for U- and C-profiles, the ultimate limit state analysis shall be provided with the following condition:

yy

z pl,d M pl,y,d

MN k 1N M (9.40)

Apart from the ones explained in Section 9.6.4, the following variables mean:

z reduction factor according to Section 9.4.2 with K,z for buckling perpendicular to the z-axis

plK,z

cr

NN

nondimensional slenderness for axial compression stress

Ncr axial force under the lowest critical load for buckling perpendicular to the z-axis or torsional buckling

Table 9.11 Moment coefficients M for lateral torsional buckling

Moment diagram M

Beam end moments

M, 1.8 – 0.7 ·

Moments from transverse loads

M,Q 1.3

M,Q 1.4

Moments from transverse loads and end moments

QM M, M,Q M,

MM

MQ = |max M| only from transverse loads maxM for transgressingmoment distributions

MmaxM + minM for transgressingmoment distributions

non


ky coefficient for the consideration of the moment diagram My and the non-dimensional slenderness K,z

ky yz pl,d

N1 aN

, but ky 1

ay K,z M,y0.15 0.15 , but ay 0.9

M,y moment coefficient M for lateral torsional buckling according to Table 9.11 for capturing the distribution of the bending moment My

Note: Especially for U- and C-profiles should be noted that scheduled torsion is not captured with this verification. T-cross sections are not included with these regulations. An approximation on the safe side is given with ky = 1. The torsional buckling load is, for example, important for a beam with bound axis of rotation.

9.6.6 Reduction Factors according to Eurocode 3

The stability verifications for structural members are regulated in Section 6.3 of part 1-1 of EC 3 [10]. For lateral torsional buckling, reduction factors LT are used which, in principle, comply with the M values in DIN 18800 Part 2. For uniform structural members with bending about the principal axis, the verification condition is

Ed

b,Rd

M 1,0M (9.41)

MEd is the design value of the acting bending moment Mb,Rd is the design value of the lateral torsional buckling resistance

The design value of the lateral torsional buckling resistance of a beam not supported laterally is usually to be determined as follows:

yb,Rd LT y

M1

fM W (9.42)

For the section modulus, the following are valid:

Wy = Wpl,y for cross sections of class 1 and 2 Wy = Wel,y for cross sections of class 3 Wy = Weff,y for cross sections of class 4

The verification condition (9.41) for cross sections of class 1 and 2 can be described as follows:

Ed

LT pl,y y M1

M 1.0W f (9.43)


A comparison with Condition (9.37) shows formal compliance with the verifications in DIN 18800 Part 2. The values of the reduction factors, however, are defined differ-ently in EC 3 and DIN 18800.

Buckling curves for lateral torsional buckling – general case

The formulas for the calculation of LT comply with Eqs (9.4b) and (9.5) in Section 9.4.2. Therefore these are the known buckling curves and the numeric values can thus be taken from Table 9.5. The allocation of the cross sections to the buckling curves is included in Table 9.12.

Table 9.12 Recommended buckling curves for lateral torsional buckling – general case

Cross section Limits Buckling curves

rolled I-profiles h/b 2 h/b > 2

ab

welded I-profiles h/b 2 h/b > 2

cd

other cross sections d

Lateral torsional buckling curves of rolled or similar welded cross sections

For rolled and similar welded cross sections under bending stress, the values LT are determined with the slenderness ratio LT from the decisive lateral torsional buckling curve according to the following equation:

LT 2 2LT LT LT

1 but LT 1.0 and 2LT

1(9.44)

with: 2LT LT LT LT,0 LT0.5 1

The national annex may define the parameters LT,0 and , but the following values are recommended for rolled or similar cross sections:

LT,0 0.4 (maximum value) and = 0.75 (minimum value)

The recommended allocation to the cross sections should be taken from Table 9.13.

Table 9.13 Recommended buckling curves for Eq. (9.44)

Cross section Limits Buckling curves

rolled I-profiles h/b 2 h/b > 2

b LT = 0.34 c LT = 0.49

welded I-profiles h/b 2 h/b > 2

c LT = 0.49 d LT = 0.76


Table 9.14 Recommended correction coefficients kc for f in Eq. (9.46)

Depending on the moment distribution between both supports of structural members, the reduction factor LT may be modified as follows:

LTLT,mod f

but LT,mod 1 and 2LT

1(9.45)

The national annex may define the values f. The following minimum values are rec-ommended:

2c LTf 1 0.5 1 k 1 2.0 0.8 but f 1.0 (9.46)

kc is a correction coefficient according to Table 9.14.

The modification values f are always less than or equal to 1, so that LT,mod LT and the lateral torsional buckling can be evaluated more favourably. With 1/f, the evaluation in Figure 9.37 directly shows the effects on the reduction factors. The largest differences result at about LT 0.8 .

While for lateral torsional buckling DIN 18800 Part 2 only distinguishes between rolled and welded girders (see Section 5.4.), two further parameters are included for the determination of the reduction factors in EC 3:

the cross section geometry with h/b 2 and > 2 the course of the bending moments

Because of its significance for construction practice, Figure 9.38 shows the reduction factors LT and LT,mod as well as M (see Section 9.6.4) for rolled I-profiles. The un-favourable values result for the general case, i.e. with the bending curves a and b ac-cording to Table 9.12. Therefore, the reduction factors LT,mod should be used for the


design since they allow a more economical construction. The curves in Figure 9.38 are valid for kc = 0.94, and thus, for a parabolic moment distribution. Other moment diagrams can be rated with the help of Figure 9.37. In the next Section, 9.6.7, comparisons are made and the accuracy of reduction fac-tors is discussed.

Figure 9.37 Values 1/f for the modification of LT

Figure 9.38 M, LT and LT,mod for lateral torsional buckling of rolled I-profiles


Uniform structural members loaded by bending and compression Structural members loaded by bending and compression (with doubly symmetric cross sections) must usually meet the following requirements:

y,Ed y,Ed z,Ed z,EdEdyy yz

y Rk y,Rk z,RkLT

M1M1 M1

M M M MN k k 1N M M (9.47)

y,Ed y,Ed z,Ed z,EdEdzy zz

z Rk y,Rk z,RkLT

M1 M1M1

M M M MN k k 1N M M (9.48)

In principle, Conditions (9.47) and (9.48) comply with a corresponding verification of DIN 18800 Part 2, but this is not shown here. According to EC 3, two verifications are to be provided compared to that. The performance of the verifications is time-consuming, unclear and error-prone. The reason for this is not only the mentioned double verification, but particularly parameters kyy, kyz, kzy and kzz as well as respective auxiliary values defined in appendices A and B of EC 3. Since the in-teraction coefficients kij have to be determined with two different methods, the two appendices comprise five (!) pages which have to be evaluated. It is hard to imagine that this verification management will prevail in construction practice. Therefore, it is not further covered here.

Note: It is to be expected that for complex load cases with N, My and Mz, the equivalent imperfections method will prevail, see Section 9.8.

9.6.7 Accuracy of Reduction Factors

In Figure 9.38, five lateral torsional buckling curves are shown for rolled I-profiles with very clear differences. Following, they are further determined quantitatively and, in doing so, the LT and LT,mod values are related to M for n = 2.5. Figure 9.39 contains four curves, namely for the following cases:

LT,mod for h/b 2 and kc = 0.94 (parabola) LT,mod for h/b > 2 and kc = 0.94 (parabola) LT for h/b 2 LT for h/b > 2

Since LT,mod/ M and LT/ M are shown in the figure, readings smaller than one mean that the M values are larger and thus more advantageous for the design. One could therefore draw the conclusion from Figure 9.39 that the M-values are almost continuously more advantageous and to be preferred for the design. With a deepened


view, however, we come to a different conclusion: the M values are on the unsafe side in many cases and require adjustments.

In Figure 9.39 some exemplary calculation results according to plastic zone theory are included which mark the actual arithmetic bearing capacity. Here, single-span beams with fork bearing on both sides have been analysed for which a uniformly dis-tributed load acts at the upper flange. With h/b = 0.95, the profile HEA 200 captures wide I-profiles and the IPE 600 with h/b = 2.73 narrow profiles.

Figure 9.39 On the accuracy of reduction factors for lateral torsional buckling

The bearing capacity of beams of IPE 600 profiles is up to 15% smaller than ac-cording to a design with the M method. Compared to that, the LT values (for h/b > 2) are on the safe side, but they are partly also very disadvantageous. The best conformance is given for the LT,mod values (for h/b > 2) since the arithmetic bearing capacity is relatively well-represented. What is striking is the deviation about 4% on the unsafe side for LT 0.4 .

For HEA 200 with the small h/b ratio, the effect of lateral torsional buckling is smaller than for the IPE 600. The M method is only up to 6% on the unsafe side and with LT,mod, the bearing capacity is met quite well. The insecurities described here for the M method have already been pointed out in the context of the research project [16]. The analyses for the correction of the M values are not yet finished though. There is no need to raise concerns, even with reference to the “mix interdiction”, when using the LT,mod values according to EC 3 for verifications according to DIN 18800 Part 2 with the M method until the matter is finally resolved.

9.7 Calculation of Critical Moments 375

Note: For lateral torsional buckling, I-profiles with large h/b-ratios are less advan-tageous than comparably wide profiles. For flexural buckling, the tendency is ex-actly the other way round. For profiles with h/b > 1.2, the values are larger (more advantageous) than for profiles with h/b 1.2. This is for flexural buckling the occurrence of residual stresses, which differ concerning their size (see Table 5.5). Obviously, their effect is smaller for lateral torsional buckling.

9.7 Calculation of Critical Moments

Basic Cases Similar to the Euler cases for the determination of Ncr regarding flexural buckling, there are also basic cases for lateral torsional buckling often occurring in practice for which formulas for the calculation of critical moments Mcr are provided. This critical moment for doubly symmetric I-sections with a uniform cross section shape may be calculated according to DIN 18800 with the following formula:

2 2cr,y cr,z p pM N 0.5 z 0.25 z c (9.49)

Table 9.15 Coefficients for four basic cases

The moment coefficient covers the course of the bending moment My across the beam. For four basic cases regarding beams with fork bearings at both ends, the


factor is stated in DIN 18800 and compiled in Table 9.15. The Mcr,y, calculated with Eq. (9.49), always refers to max My of the beam. In Eq. (9.49), zp is the ordinate of the point of load application. When loads acting downwards have their point of ap-plication above the shear centre, it is negative. In addition, we have:

2z

cr,z 2EIN and

22 T

z

I 0.039 IcI

(9.50a, b)

Beams with end moments Especially when structural systems are separated into partial systems, solutions for beams with moments at the ends are necessary. In [45], the beam with negative moments at the ends shown in Figure 9.40 has been analysed and a formula for the determination of Mcr,y has been derived:

2cr,y0 cr,z

2 20 cr,z 0 q 0 q

M q 8

N 0.4 z 0.4 z c (9.51)

Ncr,z and c2 are the same parameters as in Eq. (9.49). They are defined in Eqs (9.50a) and (9.50b). zq is the ordinate of the point of load application of qz as explained previously (compare zp).

Figure 9.40 Beam with fork bearings at both ends and moments as well as uniformly distributed load

The moment coefficient 0 can be compared to the value in Table 9.15. The sub-script 0 indicates that Mcr,y0 = 2

cr,zq 8 refers to My0 = qz2/8. Also, Mcr,y may be

calculated at every point of the beam. The 0 values determined in [45] are compiled in Table 9.16. They have been calculated with the computer program FE-Beams ac-cording to the finite element method (FEM). In Figure 9.41, the moment coefficients

0 are shown graphically.


Table 9.16 Moment coefficients 0 for the determination of Mcr,y,0 with Eq. (9.51)

or 1/ MyA = 0 MyA = MyB/2 MyA = MyB Eigenmodes

yB

y0

MM

0 1.12 1.12 1.12 Lateral torsional buckling due to positive bending moments: v(x) and (x) are single-wavedfunctions and have equal signs.

-0.1 1.19 1.22 1.26 -0.2 1.26 1.34 1.44 -0.3 1.34 1.49 1.67 -0.4 1.43 1.67 2.00 -0.5 1.53 1.90 2.46 -0.6 1.64 2.19 3.17 -0.7 1.76 2.57 4.30 -0.8 1.91 3.09 5.61 Transition area:

In the area of max 0 v(x) or (x) change the sign. The course of v(x) is partly multi-waved.

-0.9 2.06 3.78 5.15 -1.0 2.24 4.43 4.10

y0

yB

M1M

-0.9 2.42 4.19 3.12 -0.8 2.66 3.42 2.31

Lateral torsional buckling due to negative bending moments: v(x) and (x) are single-wavedfunctions and have unequal signs.

-0.7 2.78 2.63 1.68 -0.6 2.38 1.93 1.21 -0.5 1.80 1.35 0.87 -0.4 1.26 0.91 0.60 -0.3 0.82 0.58 0.40 -0.2 0.47 0.33 0.24 -0.1 0.20 0.14 0.11

My0 0: according to Table 9.15, system 4

Figure 9.41 Graphic depiction of the moment coefficient 0


When comparing with the calculation formula according to DIN 18800, Eq. (9.49), we see that the moment coefficient in Eq. (9.51) also occurs for the terms dealing with the point of load application. These formulations make it possible to get by with one parameter by approximation, the moment coefficient , which, on closer observation, also depends on the member characteristic T and the point of load application. If in Eq. (9.50b) IT is replaced by T according to Eq. (4.25) and I by Iz · (ag / 2)2, Eq. (9.51) can be written as

2 2g q q T

cr,y0 0 cr,z 0 0g g

a z zM N 0.8 0.8 1

2 a a (9.52)

which shows that the member characteristic T and the ratio zq/ag are further pa-rameters for the determination of the moment coefficient 0. The values in Table 9.16 have therefore been defined so that good approximations result for zq = –ag/2 as well as T from 1 to 30. This will cover the application cases relevant for practical construction, with few exceptions.

From = 0 to shortly before reaching the maximum 0 values in Figure 9.41, the ef-fect of parameters zq and T is minor. After that, at max 0 and to the right side, im-portant differences and larger moment coefficients result for T 0. In the area of the curve maxima, the eigenmodes change strongly and must be captured relatively precisely. This does not work using simple approximation approaches. Therefore, it is also not possible to state a continuously valid calculation formula for 0. In wide ar-eas, the following approximation is useful:

0

222

1 0.78 1 k 0.869 k 1 k 0.283

with: yB y0M M and yA yBk M M(9.53)

The value of 0 from Eq. (9.53) is sufficiently accurate for from 0 to –0.8 (MyA=MyB), 0 to –1.0 (MyA = MyB/2) and 0 to –1.3 (MyA = 0). The 0 values are then stated with a maximum of 5% larger than in Table 6.2.

Example: k = 0.5 and = –1

0

22

1 0.78 1.5 0.869 0.5 0.5 0.283 0.04725

01 0.04725 4.60 (Table 9.16: 4.43)

For the verifications in Section 9.6.4 with the M method, the largest magnitude bending moment is required. This is, for the beam in Figure 9.40, either the hogging moment MyB or the maximum sagging moment max MyF. It may be calculated with


My(x) and the stated point x in Figure 9.40. Figure 9.42 shows an evaluation simplifying the determination of ymax M .

Figure 9.42 Determination of the largest magnitude bending moment for the beam in Figure 9.40

For the determination of M , the critical moment belonging to ymax M is neces-sary. Since Mcr,y0, according to Eq. (9.51), relates to My0, it must be converted with

ycr,y cr,y0

y0

max Mmax M M

M(9.54)

Calculation example: three-span beam For the three-span beam in Figure 9.43, the lateral torsional buckling verification needs to be provided. The continuous beam is separated into three single-span beams with moments at the ends. The following bending moments result:

MyB = –0.100 48 62 = – 172.8 kNm max My1 = 0.080 48 62 = 138.2 kNm max My2 = 0.025 48 62 = 43.2 kNm


The largest magnitude bending moment occurs at the inner supports. The design has to therefore be performed with MyB, and the respective Mcr,yB has to be determined. With the reference value My0 = 48 62/8 = 216 kNm, for the boundary spans we get

= – 172.8/216 = –0.80, and from Table 6.2 it follows with MyA = 0 0 = 1.91. For the inner span also = –0.80, but, because 0 is 5.61 due to equal moments at the ends, the boundary spans are decisive.

Figure 9.43 Calculation example: three-span beam

With the help of Eq. (9.51), Mcr,y0 can be calculated for the boundary span. Using the numeric values of the calculation example in Section 9.6.4, we get:

Ncr,z = 758.8 kN c2 = 903.37 cm2

2cr,y0M 1.91 758.8 1.91 0.4 20 1.91 0.4 20 903.37

26721 kNcm

At the inner support

cr,yB cr,y0 yB y0M M M M 267.21 172.8 216 213.8 kNm

With that, it follows

M285.2 1.1 1.21

213.8

and from Table 9.10 M = 0.600 can be read for n = 2.5. For the verification with Eq. (9.37), the result is a slight excess:

172.8 1.01 1.00.600 285.2

For the eigenvalue analysis of the three-span beam with a computer program, we obtain cr = 1.373 and cr,yBM 1.373 172.8 237.3 kNm . The stabilisation of the boundary spans due to the inner span thus leads to a Mcr which is about 11% larger.

9.8 Verifications with Equivalent Imperfections 381

9.8 Verifications with Equivalent Imperfections

9.8.1 Verification Guidance

According to Table 9.1, the verification for structural safety for flexural bucklingand lateral torsional buckling may be carried out with the “equivalent imperfections method”. Furthermore, cases with axial tension force or scheduled torsional loadscan be analysed. The only requirement is that applicable equivalent geometric imper-fections are known which alternatively capture the influences of the plastic zones, residual stresses and initial geometric imperfections. The execution of the verification can be structured as follows:

1. Assumption of the equivalent geometric imperfections 2. Determination of the internal forces and moments according to second order

theory3. Verification of adequate cross section bearing capacity

The single steps of the verification procedure are explained in the following Sections.

9.8.2 Equivalent Geometric Imperfections

Besides real geometric imperfections, equivalent geometric imperfections cover in-fluences of the residual stresses and the expansion of the plastic zones.

Flexural bucklingAccording to DIN 18800-2, the equivalent imperfections given in Table 9.17 have to be applied to one-piece compression members. A distinction is made between initialbow imperpections and initial sway imperfections. Initial bow imperfections are to be assumed if both ends of the member are held nondisplaceably regarding displacements and initial sway imperfections if a rotation of the member is possible. If the member characteristic is > 1.6, both imperfections must be applied. This case occurs rarely for structural systems because here cr must be much larger than 1.6 and high compression forces must occur. This is possible, for example, for sway frames with restrained columns.

The equivalent geometric imperfections given in Table 9.17 are valid for the flexural buckling of simple compression members and for compression members in frameworks. The initial bow imperpections and initial sway imperfections are to be applied in such a way that they match the analysed direction of buckling. With regard to the initial bow imperpections, it should be noted that w0 and v0 have to be chosen depending on the relevant buckling curves. The allocation of the cross sections to the buckling curves is shown in Table 9.4. For rolled I-cross sections the allocation can also be taken from the tables in [29] or from Table 9.19.


Table 9.17 Equivalent geometric imperfections w0, v0 and 0, according to DIN 18800-2 for flexural buckling

Table 9.18 Initial sway imperfection 0 = r1 r2/200 for columns

Columnheigth

Number of columns n =1 2 3 4 5 6 8 10

5 m 1/200 1/234 1/254 1/267 1/276 1/284 1/296 1/3046 m 1/219 1/257 1/278 1/292 1/303 1/311 1/324 1/3337 m 1/237 1/277 1/300 1/316 1/327 1/336 1/350 1/3608 m 1/253 1/296 1/321 1/337 1/350 1/359 1/374 1/384

10 m 1/283 1/331 1/359 1/377 1/391 1/402 1/418 1/43012 m 1/310 1/363 1/393 1/413 1/428 1/440 1/458 1/47115 m 1/346 1/406 1/439 1/462 1/479 1/492 1/512 1/52620 m 1/400 1/469 1/507 1/533 1/553 1/568 1/591 1/608

In many cases, it is advisable to assume straight members without initial imperfections, i.e. in the initial position, and to take into account the equivalent geometric imperfections with the help of equivalent forces. The corresponding assumptions are presented in Table 9.17, and the procedure will be explained later using examples. It must be pointed out that the axial compression force N is required for the determination of the equivalent forces.


Table 9.19 Initial bow imperfections and buckling curves for rolled I-cross sections

Cross sections Bucklingabout axis

Bucklingcurve

Initial bow imperfections

all IPE, IPEa, IPEo, IPEv, HEAA 400 to 1000 HEA 400 to 1000 HEB 400 to 1000 HEM 340 to 1000

y – y

z – z

a

b

w0 = /300

v0 = /250

HEAA 100 to 360 HEA 100 to 360 HEB 100 to 360 HEM 100 to 320

y – y

z – z

b

c

w0 = /250

v0 = /200

According to DIN 18800-2, the following principles are valid for the assumption of equivalent geometric imperfections:

1. They will be applied in such a way that they are well adjusted to the eigenmode of the lowest buckling eigenvalue.

2. They will be applied in the unfavourable direction.3. It is not necessary that they are compatible with the geometric boundary condi-

tions of the system. From point 1 it follows that the equivalent geometric imperfections can only be ap-plied correctly if the buckling shape is known. This topic is treated in detail in Sections 5.10 and 6.2. If it is not possible to determine the modal shape with basic functions (for instance with Figure 9.5), respective calculations are necessary, which usually require suitable computer programs.

Figure 9.44 Three examples for the application of the equivalent imperfections


Figure 9.44 contains three examples for the application of the equivalent imperfec-tions. Since these are the Euler cases II, I and III, the buckling shapes can be taken from Figure 9.5. According to Table 9.19, they are converted into initial bow imperfections and initial sway imperfections. For the first example the solution is clear since the geometric imperfection as initial bow imperfection is affine to the buckling shape and w0 (downwards) increases the regular bending moments. The second system corresponds to Euler case I, where the right end of the beam is not supported. Thus, an initial sway imperfection 0 has to be used, which is not compatible with the restraint according to point 3 (see above). An additional initial bow imperfection is not necessary because we have cr = /2 < 1.6. The third system (Euler case III) is very similar to the first system. In this example, it is only supposed to be shown that the initial bow imperfection does not correlate with the restraint. The solutions on the right side are not correct because the equivalent imperfections applied upwards lead to a reduction of the regular bending moment.

Figure 9.45 Equivalent imperfections for a two-span beam

There are different opinions about the equivalent imperfections used in case of a two-span beam loaded by a uniformly distributed load and a compression force. Since the lowest eigenvalue according to the theory of elasticity corresponds to the antisymmetric buckling shape, use of the equivalent imperfection shown in Figure 9.45a is obvious. That will increase the moments in the spans and the hogging moment will not change. On the other hand, according to first order theory, the hogging moment is the largest bending moment and the symmetric equivalent imperfection, which complies with the buckling shape of the second eigenvalue, leads to an increase of the hogging moment. Of course, both cases can be analysed and considered for the design. In the opinion of the authors, it is sufficient only to analyse


the first case with the antisymmetric equivalent imperfection since the affinity to the buckling shape of the lowest eigenvalue has priority. In this context, the following conclusions should be emphasised:

The equivalent geometric imperfections only partly contain real geometric deformations, for example for w0 /1000. The differences to the values of Table 9.17 cover the residual stresses and the spread of plastic zones.

The equivalent imperfections must comply with the buckling shape of the lowest eigenvalue because the existing stability risk and additional loads of the system are to be covered that way.

The outcome of this is that an increase of the hogging moment due to the symmetric equivalent imperfections, shown in Figure 9.45b, is irrelevant. We can choose the Plastic-Plastic method and apply a plastic hinge at the support as shown in Figure 9.45c. Equivalent geometric imperfections do not lead to stresses there, no matter which direction the imperfections are in. For the analysis of flexural buckling, the system can be divided into two single span-beams, whereas the plastic hinge occurs with the acting of the regular bending as well as the axial force and therefore M, V and N have to be considered.

Figure 9.46 Single-span beam with cantilever under compression

There are naturally systems for which several possibilities can be analysed. Con-cerning this, Figure 9.46 shows a single-span beam with a cantilever. The buckling shape w(x) is independent of the uniformly distributed load and with a little experience it can immediately be sketched. Here, w(x) is the correct solution, even with a mirror-inverted development with respect to the x-axis. With the diagram for


the bending moment MI(x), it becomes obvious that Mb or max MF could become relevant for the design. This depends on the lengths. If the hogging moment Mb is to be analysed, an initial sway imperfection 0 downwards has to be applied to the cantilever in order to increase the moment at the support. Within the span, an ap-plication of equivalent geometric imperfections is not necessary, because they do not affect Mb. In the case of a shorter cantilever, the moment in the span will be decisive. In that case, w0 must be applied in the area a-b as presented. With an upward initial sway imperfection 0 in the area of the cantilever, the moment in the field will increase and the moment at the support will decrease, so that this assumption is on the safe side. At the transition from the initial sway imperfection to the initial bow imperfection, the equivalent imperfections have a sharp bend, which is acceptable on the basis of point 3 of the previously mentioned principles.

Figure 9.47 The influence of the equivalent geometric imperfections on the reactions at the supports

Further examples for the application of the initial sway imperfections and the initial bow imperfections can be found in Figures 4, 5 and 6 of DIN 18800-2 and in the


comment [62]. The calculation examples in Section 9.9 explain the use of the equivalent geometric imperfections for buckling.

For second order theory calculations the equivalent geometric imperfections have an influence not only on the internal forces and moments, but also on the reactions at the supports. As a result of conditions Fx = 0 and Fz = 0, the sum of the reactions at the supports does not change. The examples in Figure 9.47 show that single values and the moments at the supports change.

Note: The values given in the Table 9.17 are valid for the verification methods Elastic-Elastic and Elastic-Plastic. The limit bending moments of the plastic state are to be defined with the limiting elastic moments multiplied by the factor 1.25 here, i.e. the condition M 1.25 Mel has to be maintained. Using the Elastic-Elastic verify-cation method, only 2/3 of the values for the w0, v0 and 0 have to be considered.

Table 9.20 Initial bow imperfections for verification for the flexural buckling of rolled I-profiles in case of compression loads according to [92]

Cross sections Bucklingabout axis

Initial bow imperfections

all IPE HEAA 400 to 1000 y – y w0 = /500HEA 400 to 1000 HEB 400 to 1000 z – z v0 = /250HEM 340 to 1000 ( /200)HEAA 100 to 360 y – y w0 = /400 HEA 100 to 360 HEB 100 to 360 z – z v0 = /200

( /150)HEM 100 to 320

Precise studies according to plastic zone theory show that, to some extent, lower initial bow imperfections than the ones in Table 9.17 are sufficient. In Table 9.20, values are presented which were published in [92] and which are only valid for cases with axial compression force without transverse loads. The comparison with Table 9.17 using Table 9.19 shows compliance for the values of v0, whereas for w0, they are much smaller. The reason for that is basically that, according to DIN 18800-2, fourpossible cases for flexural buckling of rolled profiles are assigned to three buckling stress curves, i.e. according to Table 9.19, the buckling stress curves a, b (twice) and c; see Section 9.4.4.

The values v0 in Table 9.20 are also valid with regard to the condition Mz 1.25 Mel,z. Since it can be adequate to renounce this condition when calculating using computer programs, initial bow imperfections have also been determined which allow an unlimited utilisation of Mpl,z,d. The corresponding initial bow imperfections v0 = /200 or /150, respectively, are stated in brackets in Table 9.20.


Figure 9.48 Equivalent imperfections method compared to the plastic zone theory, flexural buckling about the strong axis

Figure 9.49 Equivalent imperfections method compared to the plastic zone theory, flexural buckling about the weak axis

The achievable accuracy can be evaluated with Figures 9.48 and 9.49. There, the load-bearing capacity according to the equivalent imperfections method with refe-


rence to the plastic zone theory for selected cross sections is presented, which are representative for rolled profiles of the types IPE, HEA, HEB and HEM. The stated lines are valid for Euler case II, but the studies in [92] show that they can be used for the other Euler cases as well. For structural systems with transverse loads the values given in Table 9.17 should remain in use.

Figure 9.48 contains the results for flexural buckling about the strong axis. Accord-ing to Table 9.20, w0 = /500 or /400, respectively, have been applied here. As can be seen, the determined limit loads are continuously on the safe side. The greatest de-viation of slightly more than 3% on the safe side occurs for K 0.4 .

For flexural buckling about the weak axis, two cases are distinguished: Mz 1.25 Mel,z (i.e. with limitation of pl) and Mz Mpl,z. With v0 = /250 or /200 and v0 = /200 or /150 according to Table 9.20, the method using equivalent geometric im-

perfections is safe except for a few cases. The maximum is 2.9% with K 0.8 . It should be noted that this applies for the equivalent geometric imperfections stated in DIN 18800 Part 2. In this context, Figure 9.12, which allows a judgement of the accu-racy of the method, is interesting as well.

Multi-part compression members For the verification of multi-part compression members, e.g. columns with lacings and battenings, the initial bow imperfection may be assumed to /500 according to DIN 18800-2 verifying the direction perpendicular to the axis free of material. This value is smaller than the one for one-piece members due to a smaller influence of residual stresses and plastic zones.

Lateral torsional bucklingUntil the version of DIN 18800 released in 2008, initial bow imperfections had to be applied according to Part 2 of DIN 18800 in case of lateral torsional buckling with a value half the size of the values for flexural buckling given in Table 9.17. For rolled I-profiles, the following values were obtained:

h/b 1.2: v0 = /400

h/b > 1.2: v0 = /500

These equivalent imperfections were used for any of the verification methods listed in Table 1.1, but according to element 123 of the DIN, the limitation of the plastic coefficient pl had to be considered here. This affects the buckling about the weak axis of rolled I-profiles and thus “the maximum bending moment of the perfectly plasticised cross section regarding the axial and shear forces acting at the same time has to be reduced using the factor 1.25/ pl”.


Eurocode 3 [10] refers to the national annex for corresponding values, which is cur-rently not available in its final version. The recommendations in [10] make a dis-tinction between elastic and plastic calculations of structures, equivalent geometric imperfections for flexural buckling are given and, as also stated in the previous DIN 18800 Part 2, for lateral torsional buckling the factor 0.5 is suggested. For elastic cal-culations of structures this leads to the same values as according to the previous DIN 18800 Part 2 – see above. In case of the plastic calculations, v0 = /300 (h/b 1.2)

and v0 = /400 (h/b > 1.2) have to be applied.

The studies in [16] have already showed that the above-mentioned equivalent geometric imperfections for lateral torsional buckling are partly too small. Thus, additional research and investigation is required. Here, the accuracy of selected cases will be evaluated with the help of ultimate limit state calculations (limit loads) and it will at first be tied in with Section 9.6.7 “accuracy of reduction factors”. In terms of accuracy, the following results can be taken from Figure 9.39:

For IPE 600 with h/b = 2.73, the M values are up to 15 % unsafe. For HEA 200 with h/b = 0.95, the deviations are smaller and the M values are up to 6 % unsafe. The LT values are partly far on the safe side. With the LT,mod values, the actual load-bearing behaviour is relatively well captured.

Table 9.21 Maximum load-bearing capacity and req v0 for selected application cases Maximum

bearing capacity y pl,ymaxM M

req v0 withlimitation

of pl

req v0 withoutlimitation

of pl Structuralsystem

MHEM200

IPE600

HEM200

IPE600

HEM200

IPE600

0.6 0.963 0.887 /199 /234 /166 /195

0.8 0.899 0.760 /161 /172 /134 /1430.9 0.855 0.698 /170 /175 /142 /146

1.0 0.801 0.642 /202 /194 /168 /161

1.1 0.741 0.584 /301 /219 /261 /183

1.2 0.679 0.530 <

/1000 /257

</1000

/214

1.4 0.513 0.432 <

/1000 /341

</1000

/284

1.6 0.393 0.353 <

/1000 /442

</1000

/368


In order to rate the equivalent geometric imperfections, selective results of [3] are compiled in Table 9.21 and Figure 9.50. As structural system for the studies, a single-span beam with two supports (fork bearings) at the ends and a distributed load qzacting at the upper flange is examined. The two profiles HEM 200 and IPE 600 are representing profiles with small and large h/b ratios. Compared to Figure 9.39, which contains the reduction ratio for profiles HEA 200 and IPE 600, the profile HEM 200 is selected here, since for this profile larger equivalent imperfections result than for an HEA 200 profile. For both profiles, the limit load-bearing capacity according to plastic zones theory, as defined in Section 5.12, has been calculated with M bet-ween 0.6 and 1.6. The maximum load-bearing capacities max My/Mpl,y are shown in Table 9.21 as the result. With that, the required equivalent geometric imperfections v0, which lead to the same load-bearing capacity as the ones obtained with plastic zone theory, can be determined. Table 9.21 contains the required values req v0 with a limitation of pl 1.25 (for Mz and M ) and without this limitation.

Figure 9.50 Required equivalent geometric imperfections v0

The illustration in Figure 9.50 shows that req v0 to a great extent depends on the non-dimensional slenderness. If M is between 0.6 and 1.1, very large equivalent imperfections between /134 and /301 are necessary. With increasing slenderness, v0

is much smaller and for large M , the values are very small because in this case the decisive influence for the limit load-bearing capacity is the danger of losing stability. With regard to both profiles, it can be noted that for the HEM 200 with slenderness up to M 0.9 , larger req v0 are required than for the IPE 600. In contrast to that, for beams with a higher slenderness the tendency reverses.


For short beams the shear force at the supports is relevant for the design. In that case, the limit load-bearing capacity is defined by Vz = Vpl,z,d. This case is significant for beams with IPE 600 profiles and M 0.65 .

The study presented here shows that the equivalent geometric imperfections given in the mentioned standards are too small for many application cases. For that reason and as a supplement to Table 9.21, Figure 9.51 shows the maximum load-bearing capacities using the equivalent imperfections max qz(v0) compared to the results of the plastic zone method max qz(PZT). As can be seen, the application of v0 = /400(HEM 200) or v0 = /500 (IPE 600) leads to bearing capacities that are up to 7.7% (HEM 200) or 22.6% (IPE 600) unsafe. Since pl is limited to 1.25, the calculations are in line with the standards.

Figure 9.51 Equivalent geometric imperfections in comparison to plastic zone theory for different v0 and limitation of pl

The load-bearing capacities according to the standards are partly so unsafe, that they cannot be covered by the safety concept with M = 1.1. Further studies are required for the definition of the equivalent geometric imperfections since different structural systems, loads (bending moment distributions) and cross sections need to be covered. Until this issue is finally clarified, it is best to use the following equivalent geometric imperfections:

v0 = /200 and limitation of pl 1.25

v0 = /150 without limitation of pl

9.9 Calculation Examples 393

With these values, the deviations are up to 3.6% or 1.8%, respectively, unsafe, as shown in Figure 9.51. The more precise v0 values compiled in Table 9.21 may be ap-plied as well. This requires the knowledge of the nondimensional slenderness M ,which is usually not determined when applying the equivalent imperfections method. The above-mentioned v0 values are valid for beams with a parabolic moment dis-tribution in Table 9.21, for which kc = 0.94 according to Table 9.14. With the help of Figure 9.37, it can be assessed qualitatively how different moment distributions affect the required equivalent geometric imperfections.

The new version of DIN 18800 released in 2008 picks up the conclusions of the research work presented in a shortened manner above. According to the new DIN 18800, regarding lateral torsional buckling the equivalent geometric imperfections may not be reduced by the factor of 0.5 anymore, when the ratio h / b > 2 and M is in the range of M0.7 1.3. However, further investigations have shown that this approach does not lead to a safe design in any case.

Lateral torsional buckling with scheduled torsion For beams without scheduled torsion, torsional and warping bimoments occur under consideration of equivalent geometric imperfections v0(x) regarding calculations ac-cording to second order theory – see Section 9.9.1. If regular torsional loading is added, the question is how equivalent imperfections are to be applied. This question is not answered in the standards and a conclusive clarification can also not be found in the relevant literature.

Experimental and numeric analyses in [16] and [3] show that the required equivalent geometric imperfections depend to a great extent on the case of application when regular torsion is involved, and that partly larger values have to be applied. The pre-sent state of research shows a maximum discrepancy of 5% if the recommended v0values of Figure 9.51b are applied and regular torsion is added. More precise specifi-cations cannot be made until the completion of the research.

9.9 Calculation Examples

9.9.1 Single-Span Beam with Cantilever

Figure 9.52 shows a single-span beam with cantilever and corresponding internal forces and moments Vz(x) and My(x) according to first order theory. The calculation has been performed using the program FE-Beams (see Section 1.7), which includes the verification of the bearing capacity according to the partial internal forces method (PIF-method). The solution is shown with Sd/Rd. The verification is decisive within the span with a utilisation of 65.5%.


The beam in Figure 9.52 is statically determinate. For that reason, a finite element program is actually not needed to determine the internal forces and moments. How-ever, the input only takes a few minutes so it is very convenient to use a computer program instead of performing a hand calculation, especially since the program verifies the beam over the total member length. For the calculation, finite elements are applied which have been derived for bending about the y-axis in Section 4.2.3 – see Eq. (4.18). With regard to the plot of the internal forces and moments dis-tributions of the cantilever beam, five elements have been chosen. However, for the verification, one element would be sufficient. The single-span beam is divided into 10 elements of equal length, since the maximum bending moment has to be determined with a sufficient accuracy within the span. To allow comparison, the exact value is determined here:

z42x 10 4.20 m (V 0!)

42 58max My = 42 4.20 – 10 4.20 2.10 = 88.20 kNm

The program calculation leads to 80 kNm due to the partition into 10 elements. This value is only 0.2 % smaller.

Figure 9.52 Single-span beam with cantilever

Lateral torsional buckling

The beam of Figure 9.52 is susceptible to a stability failure due to the compressed narrow flanges of the IPE 300 and the load application of qz at the upper flange. For


that reason, the failure mode lateral torsional buckling has to be examined, for which torsional rotations and lateral displacements v have to be considered. For the finite element calculation, beam elements providing biaxial bending and warping torsion according to second order theory are needed. Thus, the beam elements include the nodal degrees of freedom vM, z, wM, y, and - also see Sections 4.2.5 (linear beam theory) and 5.5 (geometric stiffness matrix). In addition to Figure 9.52, the following boundary conditions are assumed, which have to be realised by the con-structive design: vM = = 0 at both supports. Regarding these assumptions, the modal analysis leads to a critical load factor (first eigenvalue) of cr = 0.48, which means that the beam cannot be built that way. This is an expected result, since the span of the IPE 300 is relatively large and its flanges rather narrow. For the stabilisation pro-file sheeting is regarded, which is arranged at the upper flange in the perpendicular direction of the beam. It is assumed that it provides a distributed rotational spring stiffness of c = 9 kNm/m. FE-Beams now calculates a cr of 1.329 and the verification can be performed using the M-procedure of DIN 18800 (see Section 9.6). However, as explained in the previous sections, the verification is executed with

LT,mod instead of M for safety reasons:

max My = 8800 kNcm max Mcr,y,d = 1.329 8800 = 11695 kNcm

Mpl,y,d = 13710 kNcm from [29]

M LT,mod13710 11695 1.08 0.668

8800Verification : 0.96 10.668 13710

Figure 9.53 Modal shape of the beam of Figure 9.52

With regard to Section 9.8, the verification against lateral torsional buckling can also be executed by applying equivalent geometric imperfections. Due to the eigenmode in Figure 9.53, a parabola-shaped imperfection v0(x) is considered for the single-span beam with a stitch of l/200 = 5.0 cm. As shown in Section 9.8.2, an initial sway imperfection with 1/200 = 2.0 cm and an initial bow imperfection of l/200 = 2.0 cm is scheduled for the cantilever. The course of the corresponding equivalent imperfection is sketched in Figure 9.54. Due to these initial imperfections, internal forces and


moments Mz, Vy, M , Mxp and Mxs occur next to the scheduled ones My and Vz. For the finite element analysis, the program FE-Beams is again used. The bearing capacity is checked by the program using the PIF-method according to [25]. Fundamental results are compiled in Figure 9.54. The maximum cross section utilisation Sd/Rd is only 71%. The considered limiting of pl = 1.25 only has a small influence for this example. It should be pointed out again here that one verification against lateral torsional buckling is sufficient: either the M procedure or using equivalent geometric imperfections.

Figure 9.54 Verification using equivalent geometric imperfections

9.9.2 Beam with Scheduled Torsion

Figure 9.55 shows a single-span beam as a UPE-profile with fork bearings, which is loaded by a uniformly distributed load. The load acts at the upper flange due to an introduction of profile sheeting. For the calculation it is assumed that its path of ac-tion is in the middle of the web. In Figure 9.55b, the load is shifted into the shear centre with a determination of the torsional load moment mx. Because of the influence on the lateral torsional buckling, it has to be noted in the following calculation that the distributed load is still acting at the upper flange.

The beam is regularly loaded by bending about the y-axis and warping torsion. Due to the spatial carrying behaviour, bending about the z-axis according to second order theory occurs. For that reason, beam elements with vM, z, wM, y, and as de-grees of freedom are needed. The required matrices for these elements can be taken from Sections 4.2.5 (linear beam theory) and 5.5 (supplement for second order the-ory). For the beam in Figure 9.55, the stability case lateral torsional buckling is rele-vant. Since regular torsion occurs, it may not be verified using the M procedure – see DIN 18800-2, element 311. The bearing capacity is therefore verified using equiva-lent geometric imperfections. For the finite element analysis, the beam is divided into


20 elements of equal length, even though Condition (4.30) would allow a rougher elementing with:

T

EI 21000 11880 58.9 cmGI 8100 8.884

Figure 9.55 UPE beam with scheduled torsion

First, the eigenvalue and corresponding eigenmode are calculated. The critical load factor cr,d = 1.467 is then determined. The modal shape complying with lateral tor-sional buckling consists of single-waved functions for v(x) and (x). With regard to Section 9.8, the equivalent imperfection v0 must therefore be scheduled as single-waved as well as with a stitch of:

v0,m = –0.5 /200 = –1.0 cm

Note: The equivalent imperfection applied here corresponds to the regulations of the previous DIN 18800. According to new conclusions for I-sections (see Section 9.8.2), larger imperfections are presumably necessary for a safe verification here as well. However, in the context of this example, on which the general proceeding is supposed to be focused, the regulations of the older DIN 18800 are retained.

The equivalent imperfection is set negative, which is against the y-direction, since it leads to higher torsional moments and is therefore decisive. The system calculation is performed with the program FE-Beams, and some of the results are shown in Figure 9.56a. Since the procedure Elastic-Plastic is supposed to be applied for the verification, the PIF-method according to [25] is also needed in this example and the cross section is separated into three plates. The flanges are considered with dimensions of b = 80 mm and tf = 11.0 mm and the web with hw = 178 mm and tw =


6.0 mm. The fillets are therefore neglected. Since pl,z of this profile is 1.8, it is limited to 1.25 according to DIN 18800-2, element 123, and the verification is performed using a bending moment Mz increased by the factor pl,z /1.25. In addition, the warping bimoment M is also increased in a comparable way.

The utilisation in Figure 9.56a shows that the cross section capacity is not sufficient. However, it is not only the cross section at midspan with the bending moment My,which is loaded too high (110.5%), but also the ones at the beam ends with 106.8%. The primary torsional moments Mxp are responsible for this, which shows the large influence of the torsion. The torsional rotation at midspan with = 0.276 rad (15.8°) is very large as well.

Figure 9.56 Results for the UPE beam without and with beam extensions


Because a sufficient bearing capacity could not be verified with the calculation, the system is slightly changed, as shown in Figure 9.56b. In this structural system, the beam extensions of 10 cm at the beam ends, which are actually provided by the structure, are regarded. According to Figure 4.29, they act as warping springs, i.e. with regard to the warping torsion as a partial restraint. The increase of stiffness leads to a larger critical load factor (1.53 instead of 1.467) and smaller torsional rotations at midspan ( 0.238 instead of 0.276). Furthermore, the internal forces and moments change going along with a maximum utilisation of 100% and a sufficient bearing capacity.

9.9.3 Two-Hinged Frame – Calculation in the Frame Plane

In this Section, the calculation of a typical two-hinged frame is conducted with a veri-fication of sufficient bearing capacity. The frame is part of a storage hall with ground dimensions of about 30 m length and 20 m width – see Figure 9.57. The roof shows an inclination of 2° and the frames are arranged at a distance of 5 m in between. The construction site is assumed to be in Bochum.

Figure 9.57 Structural elements of the hall

For the verification, the Elastic-Plastic procedure according to DIN 18800 is chosen. The equivalent geometric imperfections are applied by using equivalent loads according to Section 9.6.


ConstructionThe two-hinged frame is shown in Figure 9.58. The columns are of rolled profiles HEA 300 (S235) and the rafter a rolled profile IPE 360 (S235). In the region of the frame corners, the rafter is strengthened by haunches. The haunches as well as the connections to the columns are designed according to [28] shown in Figure 9.59.

Figure 9.58 Two-hinged frame of the hall

Loads

Self-weight of roof Profile sheeting 0.12 kN/m2

Insulation 0.10 kN/m2

Bitumen roof membrane 0.15 kN/m2

0.37 kN/m2

Self-weight of steel structure Rafter: g 1.1 kN/m (including braces and installations) Column: g 0.9 kN/m

Snow according to DIN 1055 Part 5 The construction site in Bochum is less than 400 m above sea level and as-signed to snow load zone 1:

sk = 0.65 kN/m2

The roof inclination of = 2° < 30° leads to a coefficient 1 = 0.8. From this, it follows:

s1 = 0.8 0.65 = 0.52 kN/m2

Wind according to DIN 1055 Part 4 The construction site (Bochum) is assigned to wind load zone I. Since the height of the hall is less than 10 m, the velocity pressure is:

q = 0.5 kN/m2


According to DIN 1055, due to = 2° < 5°, the roof is classified as a flat roof. The wind load on the walls and the roof are caused by the wind action on the gable wall as well as on the longitudinal walls. The determination is relatively extensive and therefore not shown in detail here. The result is sketched in the following compilation of the load cases.

Figure 9.59 Frame corner and haunch (cut profile IPE 360) according to [28]

Load cases and load combinations Essential load cases for the two-hinged frame are shown in Figure 9.60 with regard to the frame spacing of 5 m. The upper extensions of the columns due to the attic are neglected for reasons of simplification and instead concentrated loads WA are considered in the frame corners.

The load cases in Figure 9.60 lead to two load case combinations (LCC) for which the partial safety factors of DIN 18800 are used.

Load case combination 1 1.35 g 0.9 1.5 s 0.9 1.5 w (gable wall) 0

Load case combination 2 1.35 g 0.9 1.5 s 0.9 1.5 w (longitudinal wall) 0

In these combinations, 0 emphasises the equivalent geometric imperfections to be used according to Section 9.6. Because the eigenmode belonging to the lowest eigenvalue shows an antisymmetric course, the initial sway imperfections are sche-duleed to the right for both columns, as sketched in Figure 9.58. Using Table 9.18, we have:

0 = 1/277


Figure 9.60 Load cases for the two-hinged frame

Determination of internal forces and moments Generally, it is common practice to determine the internal forces and moments of a two-hinged frame according to second order theory using a computer program. This is also reasonable since haunches have to be considered and the load case “wind at longitudinal wall” is difficult to deal with using calculation formulas. The results of the computer calculation decisive for the verification are compiled in Table 9.22.

Table 9.22 Internal forces and moments for the frame in Figure 9.58

LCC Node (position) Myin kNm

Vzin kN

Nin kN Node numbering

1

4 right (roof ridge) 159.50 0.71 –27.50

5 (beginning of haunch) –90.15 –63.97 –29.55

6 beam cut –218.62 –76.24 –36.68

6 column (frame corner) –234.47 26.26 –79.59

2

4 right (roof ridge) 143.86 –1.73 –32.06

5 (beginning of haunch) –125.09 –66.32 –34.01

6 beam cut –257.04 –78.08 –41.49

6 column (frame corner) –273.96 33.74 –82.17


VerificationsThe verifications for the bearing capacity are performed with the interaction condi-tions from DIN 18800 Part 1; see Table 8.5. The limit internal forces and moments needed are taken from tables in [29]:

IPE 360, S 235 Mpl,y,d = 222.4 kNm Vpl,z,d = 350 kN Npl,d = 1587 kN HEA 300, S 235 Mpl,y,d = 301.8 kNm Vpl,z,d = 295.5 kN Npl,d = 2455 kN

With this, the following verifications can be conducted:

Node 4 right (roof ridge)

pl,d

N 27.5 0.017 0.1N 1587

z

pl,z,d

V 0.71 0.002 0.33V 350

y

pl,y,d

M 159.5 0.717 1M 222.4

Node 5 (beginning of haunch)

pl,d

N 34.01 0.021 0.1N 1587

z

pl,z,d

V 66.32 0.189 0.33V 350

y

pl,y,d

M 125.09 0.562 1M 222.4

Node 6, column (frame corner)

pl,d

N 82.17 0.037 0.1N 2455

z

pl,z,d

V 33.74 0.114 0.33V 295.5

y

pl,y,d

M 273.96 0.908 1M 301.8


These show that the conditions are met. A check of the b/t ratios for the Elastic-Plastic procedure can be executed using the tables of [29]. The conditions are fulfilled for both profiles due to the small compression forces.

Note: For the cross section capacity the verifications show that the influence of axial and shear forces is low. The maximums are N/Npl,d = 0.037 and Vz/Vpl,z,d = 0.189, which are below the limits of 0.1 and 0.33 of the interaction conditions. For the usual two-hinged frames this is often the case and, generally, the condition My/Mpl,y,d 1 is decisive and sufficient. Contrary to that, at least the axial force has a reducing influence in terms of the bearing capacity using the Elastic-Elastic procedure when stresses are determined according to Chapter 7.

9.9.4 Two-Hinged Frame – Stability Perpendicular to the Frame Plane

With the calculation of the frame in the previous Section according to second order theory, the stability (flexural buckling) in the frame plane is covered. For further analysis (flexural buckling perpendicular to the frame plane and lateral torsional buckling), the frame is divided into the subsystems frame columns and rafter. For the definition of these substituted systems it is assumed that the frame corner is sup-ported perpendicular to the frame plane in a nondisplaceable manner. This has to be ensured by correspondingly stiff braces in the side walls (longitudinal direction). In addition, fork bearings are assumed at the frame corners since rotations are con-strained due to the bending stiffness of adjacent members.

Stability of the frame columns According to Table 9.22, load case combination 2 leads to the highest loading in the right column of the frame. For that reason, the equivalent system in Figure 9.61 is analysed, for which flexural buckling about the weak axis of the HEA 300 or lateral torsional buckling due to N and My may be relevant. As supplement to Table 9.22, the axial force N at the column top includes the self-weight of the column.

Using Euler case II, cr,d = 26.75 > 10 for flexural buckling about the weak axis. Therefore, this stability case does not have to be analysed in more detail. Lateral torsional buckling is decisive, for which cr,d = 3.07 is determined using the program FE-Beams. The fact that for N = 0 cr,d = 3.34 shows that the compression force has only a small influence on the lateral torsional buckling. The verification is performed with Condition 9.40 from Section 9.6.5. However, instead of using M, the reduction factor LT,mod from Section 9.6.6 is applied. The cross section at the column top is decisive because of the maximum moment (absolute value) – also see Figure 9.62.

Npl,d = 2455 kN Mpl,y,d = 301.8 kNm (from [29])


Ncr,z,d = 26.75 90.7 = 2426 kN K,z 1.006 and z = 0.536 for curve c

Mcr,y,d = 3.34 274.0 = 915.2 kNm M 0.574

With kc = 1/1.33 according to Table 9.14 it is LT,mod = 1.0.

ay 0.15 1.006 1.8 – 0.15 = 0.12 < 0,9

y90.7k 1 0.12 0.992

0.536 2455 < 1

Verification: yy

z pl,d LT,mod pl,y,d

MN k 0.069 0.901 0.970 1N M

The column can be successfully verified using Condition (9.40). An additional lateral support within the column is therefore not necessary. Since the bearing behaviour of the column can be captured in a more appropriate manner using the verification with equivalent geometric imperfections, this procedure is also applied below.

Figure 9.61 Right column of the frame in Figure 9.58

The geometric imperfection to be applied is discussed in detail in Section 9.8. Ac-cording to Section 9.8.2, an initial bow imperfection of v0 = /200 = 700/200 = 3.5 cm is selected and pl,z is limited for the verification of the cross section capacity. Figure 9.62 shows the solutions calculated with the program FE-Beam. The bending moment Mz about the weak axis and the warping bimoment M are a result of the geometric imperfections and the calculation according to second order theory. The utilisation of Sd/Rd = 0.976 < 1 shows that the verification is decisive at the column top using equivalent imperfections as well. In this example, the internal moments Mzand M do not lead to an utilisation Sd/Rd within the span being larger than the one at the column top.


Figure 9.62 Verification of the right column with equivalent geometric imperfections

Lateral torsional buckling of the rafter The analysis of the rafter is performed using the structural system of Figure 9.63a, and the internal forces and moments of the calculation in the frame plane according to the previous section are applied. The verifications are executed for the load case combinations (LCC) 1 and 2 with bending moments and H-forces stated in the table. The rafter is substituted as a straight member given the low roof inclination. In order to adjust the distribution of the bending moments to this substitution, a single force Fzacting upwards is applied at midspan. It can be determined with the condition:

zF H 0.34 m4

As an additional simplification, the loads acting at the rafter are not applied according to the sketches in Figure 9.60, but as distributed distributed load qz. They are ar-ranged so that the acting moments at midspan Mym are attained, as shown in Figure 9.63.

For the equivalent system of Figure 9.63, there is no reasonable chance of verifying the system against lateral torsional buckling. Without any kind of stabilisation, the program FE-Beams provides cr,d = 0.220 and cr,d = 0.237 for both systems and therefore no stable equilibrium. Due to their length of 19.7 m, the rafters IPE 360 are much too slender and have to be stabilised. This can be achieved by roof bracings (lateral support) and profile sheeting (distributed rotational spring stiffness). Applying only a horizontal bracing will not give sufficient stabilisation for highly utilised rafters as the one in Figure 9.63 since positive and negative bending moments occur, which will cause compression in regions of the upper flange (midspan) and the lower flange (rafter ends).


Figure 9.63 Equivalent systems for the rafter

Even if a continuous nondisplaceable lateral support were used for the rafter, additional stabilisations would be necessary. The critical load factors cr,d gained for different points of application of the lateral support can be seen in Figure 9.64. The corresponding parameter zsupport is displayed at the vertical axis. Through this, the level of the rafter support can directly be noticed. LCC 2 is analysed as an example and cr,d is determined with c = 0. Figure 9.64 shows that for all support levels, the critical load factor is below 1. For that reason, the system is not in a stable equilibrium, and additional provisions for the stabilisation are required. With respect to the distributed rotational spring stiffness gained from the profile sheeting, cr,dincreases significantly. For zsupport = –1 cm it is max cr,d = 4.645.

Note: The calculations of Figure 9.64 have been performed with the program FE-Beams. However, it is not possible to regard eccentric supports. Due to this fact, very stiff eccentric distributed springs cv have alternatively been considered.


Figure 9.64 Critical load factor cr,d for different levels of the continuous lateral support

In the structural system considered here, continuous support is not available for the rafter since the roof bracing is only connected at midspan (L/2) and the quarter lengths of the rafter (L/4, 3L/4), see Figure (9.57). Since one bracing stabilises half of the roof, Figure 9.65 is adopted for the verification and it is assumed that the wind compression on the gable wall is introduced into the bracings at the roof level. With respect to the rafter, the bracings show an eccentric point of application of zVB = –9 cm, which leads to torsional loadings for the rafter. These are relatively small in this case and therefore neglected.

Figure 9.65 Stabilisation of the rafter

The sketch in Figure 9.65 shows that rafters are stabilised by the profile sheeting and the roof bracing. The roof bracing carries wind loads and loads due to the stabilising of the rafters and transfers the wind loads to the vertical bracings in the side walls.

For the verification of the rafter, the roof cladding is considered, first determining the distributed rotational spring stiffness due to the profile sheeting. The stiffness results from three components: the bending stiffness of the profile sheeting c M, the flexibility of the connection c and the distortion of the profile contour c P. For the sheeting used here, these values are:

Mc 289 kNm m , Ac 6.60 kNm m , Pc 77.5 kNm m


Information on how the values may be determined is given in [42]. The resulting distributed rotational spring stiffness is then:

1 1 1 1 c 5.96 kNm mc 289 6.60 77.5

and ,dc 5.4 kNm m

Using this spring stiffness leads to cr,d = 0.662 for the system in Figure 9.63b and cr,d = 0.675 for the system in Figure 9.63c. As expected, this stiffness is not suf-

ficient. Therefore, additional roof bracings are considered for the stabilisation of the rafter. It consists of diagonal members 20 (rods) and post members 76.1 4.0 (circular tubes) and the rafters as chords of the framework. Figure 9.66 shows the structural system of the roof bracing. The bracing is replaced by single springs using the shear stiffness as described in [51]. From Figure 9.66, the equivalent shear stiffness is:

3

2

1S 20860 kN7.02 5

21000 9.06 4.92521000 3.14 5 4.925

Figure 9.66 Replacement of the roof bracing by single springs

According to Figure 9.57, two bracings have to stabilise five inner rafters and the two outer ones at the gable walls. Because of the fact that the outer rafters are only influ-enced by half of the load due to the width of load, the following equivalent shear stiffness is used for one rafter:

12 SS 6950 kN

0.5 5 0.5With this, the single spring stiffness values are:


Cy,1/2 = 6950/1970 2 = 7.06 kN/cm

Cy,1/4 = Cy,3/4 = 6950/1970 2.667 = 9.41 kN/cm

With regard to M = 1.1, the stiffness values are applied with 6.4 kN/cm and 8.5 kN/cm for the calculation, and the position of the bracing is assumed to be at zVB = –9 cm according to Figure 9.67 – also see Figure 9.65.

According to Figure 9.59, the rafter is connected to the columns using end plates and bolts. These end plates act as warping springs, as described in Section 4.4.5. Since the rafters are highly utilised, they are regarded as additional stabilisation. Using Figure 4.29, the spring stiffness may be determined for the IPE 360 and an end plate thickness of tp = 25 mm to give

C ,d = 17 2.53 (36.0 – 1.27) 8100/(3 1.1) = 22.643 106 kNcm3

Figure 9.67 Distributed rotational spring stiffness c , translational springs Cy and warping springs C for the stabilisation of the rafter

From the stabilisations compiled in Figure 9.67, the calculation with the program FE-Beams leads to cr,d = 1.563 for the system in Figure 9.63b and to cr,d = 1.489 for LCC 2. The verification with the M procedure is performed for the maximum span moment in Figure 9.63b (LCC 1). For this verification, Mcr,y and Ncr,z are required for the individual acting of My and N. Using FE-Beams, we get cr,M = 1.609 and cr,N = 30.03 for these loadings and the verification can be performed as follows:

Mcr,y,d = 1.609 159.5 = 256.6 kNm

M LT,mod222.4 0.931 0.716256.6

(with kc = 0.90, see Table 9.14)

Ncr,z,d = 30.03 26.3 = 789.8 kN

K,z z1587 1.42 0.373789.8

(see Table 9.5)


M,y391.61.11 1.3 1.11 1.30394

ay = 0.15 1.42 1.30 – 0.15 = 0.127 < 0.9

y26.3k 1 0.127 0.994 1

0.373 1587

Verification: 26.3 159.5 0.994 0.044 0.996 1.040 10.373 1587 0.716 222.4

Using Condition (9.40), a sufficient bearing capacity cannot be verified at midspan as shown above. On the other hand, the verification with LT,mod for kc = 0.90 is on the safe side since the kc value can be expected to be more favourable regarding the course of the moment distribution in Figure 9.63b. In addition, the eigenmode does not show a single wave, which is the basis of the correction and reduction factors of Section 9.6.6. Thus, it is convenient to apply the verification procedure using equivalent geometric imperfections. With this procedure, the sufficient bearing ca-pacity of the member can be verified. This is not shown in detail here since LCC 2 is analysed with that approach. With this load combination, large negative bending moments occur at the right end of the rafter, as shown in Figure 9.63c. According to Figure 9.59, haunches are at that position, but it can be expected that the cross section at the end of the haunch will be relevant for the verification.

Figure 9.68 Eigenmode of the system in Figure 9.37c and equivalent geometric imperfection v0

The critical load factor is cr,d = 1.489 (see above), and the corresponding eigenmode determined with FE-Beams is shown in Figure 9.68. The functions v(x) and (x)


show large amplitudes at the right rafter end with lateral torsional buckling being decisive due to the large negative bending moment. The displacement function v(x) has four waves, which strongly die out from the right to the left. At the position of the individual springs, the displacements are approximately zero, going along with a lateral torsional buckling in between the posts of the bracing as decisive failure. As sketched in Figure 9.68, a four-waved equivalent geometric imperfection is chosen and the stitch of the initial bow imperfection is scheduled according to Section 9.8.2 with v0 = i/200 = 492.5/200 = 2.47 cm.

Figure 9.69 Decisive internal forces and moments for the verification of the rafter in Figure 9.63c and utilisation of the cross section capacity Sd/Rd

For the verification, the internal forces are determined with the program FE-Beams using the stabilisations according to Figure 9.67 and the equivalent geometric im-perfections. Selected results are compiled in Figure 9.69, including the decisive inter-nal forces and moments relevant to the verification. At the bottom of the figure, the utilisation of the cross section bearing capacity Sd/Rd using the partial internal forces method is shown, for which a rolled profile IPE 360 without haunches is used. Since an exceeding of up to 42.1% can be noticed for the right rafter end, additional verifications for the haunch have to be performed, which are not shown here.


Note: The higher stiffness in the region of the haunches is neglected in the calcula-tions. In addition, a higher distributed rotational spring stiffness could be used according to the commentary to DIN 18800 [8], which is a result of a beneficial estimation of the connection stiffness.

Further verifications for the frame are not shown here. However, the following analy-ses are necessary as well:

capacity of the haunches bearing capacity of the stabilising components, see [42] bolted connection at the roof ridge and the frame corners, see [38] load introduction, load redirecting, column bases

9.9.5 Frame Considering Joint Stiffness

Figure 9.71 shows frame construction with two storeys and two bays. The columns are HEB 200 profiles. On the lower horizontal beam (IPE 300), a concrete slab is ar-ranged and the upper beam (IPE 240) carries a roof of purlins, insulation and profile sheeting.

Figure 9.70 Load figure for the load combination 1.35 · (g + s + w + p)

The load width of the frame is 4 metres, from which the loadings sketched in Figure 9.70 for the load case combination 1.35 · (g + s + w + p) with the loads for self-weight g, snow s, wind w, and life load p result.


Figure 9.71 Frame with two bays and two storeys

The beams are connected to the columns with extended end plate connections – see Figure 9.72 (left). With the models of the component method, which is regulated in Eurocode 3 Part 1-8 [10] on a normative basis, the moment-rotation-relationships shown in Figure 9.72 (right) can be derived for the description of the joint behaviour. For calculations of beam structures according to elastic theory, the relationships including the initial stiffness Sj,ini may by idealised by linear stiffness-relationships. In


that case, the stiffness values are to be regarded with Sj,ini /2 for the joints of this ex-ample – see Figure 9.72 (right). With the programs of the institute the stiffness of joints with flush or extended end plates can be calculated – see Section 1.10.

Dimensions in cm Values in brackets apply for the connection IPE 240 / HEB 200

Figure 9.72 Beam to column joints and moment-rotation relationships

Figure 9.73 FE-model for the frame


At first, the frame is analysed in the frame plane. For the finite element method, ele-ments with three degrees of freedom in each node (u, w, ) are necessary. They are composed of the matrices given with Eq. (4.11) and (4.18) as well as of the corre-sponding nonlinear components of Section 5.5. The elementing of the frame is shown in Figure 9.73. For the description of the joints, rotational springs are arranged with the corresponding spring stiffness. In order to capture the eccentricities of the con-nections to the reference axes of the columns, the rotational springs are not directly modelled at these axes, but slightly shifted according to Figure 9.73.

Figure 9.74 Internal forces and moments of the frame in Figure 9.71 utilisation

The calculation of the frame according to second order theory of elasticity with the program FE-frames leads to the internal forces and moments shown in Figure 9.74 (internal verification forces), whereas for the beams, the values refer to the forces at the connections. The maximum utilisation, determined with the partial internal forces method (PIF-method), is 62%.


Figure 9.75 Modal shape of the frame in Figure 9.71

A modal analysis leads to the lowest eigenvalue of cr,d = 11.73 and the eigenmode sketched in Figure 9.75. Since effects perpendicular to the frame plane are not covered with the two-dimensional analysis using second order theory, additional examinations and verifications are necessary for the members (lateral torsional buck-ling and flexural buckling about the weak axis). The upper beam is idealised for a cal-culation with the program FE-Beams as shown in Figure 9.76a. Finite elements with seven degrees of freedom for each node are needed to cover the spatial deformation behaviour. The stabilising effect of the purlins is neglected. The calculation according to second order theory taking into account equivalent geometric imperfections (l/250) and the verification using the PIF-method leads to a maximum beam utilisation of 52.0%. The imperfection has been applied independently of the actual failure mode as safe approximation for the verification of the cross section bearing capacity. The eigenvalue of the system is cr,d = 1.32.

The lower beam is not analysed here any further, the verification for the capacity of the columns will now follow. Just as for the upper beams, the program FE-Beams is used for that purpose, and the lower part of the middle as well as the right column is examined – see Figure 9.76b. The calculation (second order theory and PIF-method) shows a maximum utilisation of 43.4% for the outer column, while the middle column is not decisive. The critical load factors are cr,d = 5.28 for the middle column and cr,d = 9.67 for the outer column.

Note: The relatively low utilisations of the columns result from the demands of the serviceability limit state of the column. In order to meet the requirements, a corre-sponding lateral stiffness of the frame is needed. For the structural system here, it can be ensured by two provisions: strengthening of the joints or choice of stiffer columns. As shown with the following calculation regarding rigid connections, the strengthening of joints will lead to much higher bending moments being transferred by the end-plate connections. If the column profile is downsized at the same time, in order to obtain an economic solution, only a limited increase of the moment resistance of the joint can be achieved. Because the slender column flange


will be the decisive component at the force transfer, a much more extensive joint configuration would be necessary. For that reason, the strengthening of the joints using stiffeners for the columns is not considered for this example.

Figure 9.76 Equivalent systems for beams and columns

For comparison purposes, the frame is analysed regarding rigid joints and an addi-tional calculation with the program FE-frames is performed. The PIF-method shows a maximum utilisation of 87% see Figure 9.77. The modal analysis leads to cr,d = 13.02.


The rigid connections of the beams to the columns lead to clearly higher bending moments according to elastic theory being transferred by the connection, which the joint configurations of the frame would not be able to carry. The joints would have to be strengthened for that reason. The higher utilisation at the area of the columns lead to a load relief within the beam span, but these have clear reserves. For the example of the beams, there is a more constant utilisation across the member length when taking into account the calculative joint stiffness. However, for practical applications it should be mentioned that considering the joint stiffness may involve a large effort for the structural analysis. In addition, it strongly depends on the structure and the construction whether a design omitting stiffeners leads to economic advantages. For typical frames of halls this is not usually the case.

Figure 9.77 Utilisation of the frame assuming rigid joints

10 FEM for Plate Buckling

10.1 Plates with Lateral and In-Plane Loading

Plates are structural members with a proportionately small thickness compared to the length and width. Due to their thinness, it is sufficient to consider their midplane. This is comparable with the reduction of a beam member to its member axis. As shown in Figure 10.1, plates with different loadings are distinguished: plates with loads acting in-plane and those with loads acting laterally to the plane. Typical examples for plates loaded in-plane are walls and for plates laterally loaded slabs; however, walls and slabs are usually massive structural members and not steel constructions. Plates made of steel are dealt with in detail in Section 10.6.

Figure 10.1 Plates with in-plane and lateral loading

10.2

only the stresses at the positive intersection x = const. and y = const. are depicted. The figure shows merely the directions and designations without considering the equilibrium of the element.


Figure 10.2 contains the definition of the stresses for plates. For reasons of clarity,


Stresses and Internal Forces

10.2 Stresses and Internal Forces 421

Figure 10.2 Stresses of plates

Plates with in-plane loading

The stresses of plates loaded in-plane shown in Figure 10.1 are summarised to resul-tant longitudinal and shear forces per unit length. Figure 10.3 contains the corresponding definitions according to DIN 1080 Part 2. In addition, the commonly used constant stress distributions and the longitudinal and shear forces resulting from these are shown on the right side of the figure. Because we have xy = yx, also nxy = nyx. The internal forces are forces per unit length and therefore they are designated with the small letter n. nx and ny are comparable to the axial force N of a beam member and nxy to a shear force V.

Figure 10.3 Normal and shear forces of plates loaded in-plane

Plates with lateral loading

The internal forces of laterally loaded plates are also designated with a small letter since they are forces or moments per unit length (kN/m or kNm/m). Figure 10.4 shows the definition of the designations and the directions. However, in DIN 1080 Part 2, two different definitions are included. Selected and depicted here is the “ori-entation according to coordinates”. In many publications and computer programs, the “orientation according to a characterised side” is used. With regard to stiffened plates, i.e. the combination of plates and beams, the designations in Figure 10.4 are more

10 FEM for Plate Buckling 422

advantageous. mxy is comparable to the bending moment My for beams and mxx to the torsional moment Mx. In addition to DIN 1080 Part 2, the stress distributions result-ing from the internal plate forces according to the theory of elasticity are sketched in Figure 10.4 on the right side.

Figure 10.4 Shear forces, bending moments and torsional moments of laterally loaded plates as well as stress distributions according to the theory of elasticity

10.3 Displacements

According to Figure 1.6, seven displacements are distinguished for beams:

displacements u, v and w rotations x, y and z (or , w , v )twist of the x-axis

10.4 Constitutive Relationships 423

In Figure 10.5, these displacements are assigned to plates with lateral and in-plane loading and they are complemented with regard to the theory of plates. As already discussed in detail in Section 3.5.4, rectangular finite plate elements with four nodes and the nodal displacements w, w , w and w are recommended according to Figure 3.12. The displacement w describes the deflection of the plate, and the derivations w and w are the corresponding rotations about the y- or x-axis, re-spectively. The following connection to the displacement values of beams is valid:

y w and xw .

Additionally, w is included, which is the derivation of the deflection function w ac-cording to x and y. It corresponds to the twist of beams. Due to w w , it is also equal to y , i.e. the change of the rotation y in the y-direction.

Figure 10.5 Displacement values for plates loaded laterally and in-plane

The deformations of plates loaded in-plane are described by displacement functions u(x,y) and v(x,y). Besides these displacements, the rotations u u y and v v x are used. Generally, they correspond to a rotation z, for which, however, both components have to be distinguished due to the theory of plates. As for beams, the sum of the two angles leads to the shearing strain:

xy xy yxu vy x

. (10.1)

The sketch in Figure 10.5 gives a corresponding illustration of that.

10.4 Constitutive Relationships

In Section 10.5, the virtual work for plates with lateral and in-plane loading is formu-lated. Just as for beams, some basic relationships are needed for that purpose; these are compiled below.


Hooke’s law for the plane stress state

The assumption z = 0 leads to the following strains and stresses:

x x y1E

(10.2)

y y x1E

(10.3)

xy xy1

2G(10.4)

x x y2E

1 (10.5)

y y x2E

1 (10.6)

xy xyE

1 (10.7)

The following correlation exists for the material constants:

E = 2 (1 + ) G (10.8)

Description of the displacement state

According to Section 1.6, the displacements u, v and w of beams are described by the deformations of the beam axes through the centre of gravity S and the shear centre M. Since the “centre of gravity line” and the “shear centre line” of plates are located in the centre face (S = M), it is referred to this face. If the subscript “m” is used for labeling, for plates loaded in-plane we then have:

u = um (10.9)

v = vm (10.10)

For laterally loaded plates, the relationships of beams are valid in a similar manner:

u = um + z y um – z w (10.11)

v = vm z x vm – z w (10.12)

w = wm (10.13)

These correlations can directly be taken from Eqs (1.1) (1.3), since at this point only the dependency of z is to be considered. The Eqs (10.11) (10.13) are the starting point for Kirchhoff’s plate theory, i.e. for plates with infinite shear stiffness.

10.5 Principle of Virtual Work 425

Relationships between strains and displacement values

According to [25] and many other publications, the following formulas are valid for the linear theory, i.e. for small deformations:

x u (10.14)

y v (10.15)

xy yx 1 2 u v (10.16)

With regard to the buckling of plates, the geometric nonlinearity has to also be considered in terms of second order theory. If we proceed as shown in Section 5.3 for beams and transfer the result of Eq. (5.29) to plates, this gives:

2x u 1 2 w (10.17)

2y v 1 2 w (10.18)

xy 1 2 u v 1 2 w w (10.19)

Using Eqs (10.11) (10.13), we can refer to the displacements of the centre face and the following relationships result for the strains:

2x m m mu z w 1 2 w (10.20)

2y m m mv z w 1 2 w (10.21)

xy m m m m1 2 u z w v z w w w (10.22)

10.5 Principle of Virtual Work

Just as for beams, the following equation is also used for plates as equilibrium condition:

W = Wext + Wint = 0 (10.23)

Explanations on the virtual work principle can be found in Section 3.4.2.

Internal virtual work

According to [25], the following is generally valid:


intV

W dV (10.24)

For plates, the volume integral can be formulated as t 2

int x x y y xy xy yx yxA t 2

W dz dA (10.25)

As is common practice, shear stresses xz and yz, i.e. perpendicular to the centre face, are neglected. In Eq. (10.25), the virtual strains can be replaced using Eqs (10.20) (10.22) after formation of the variation. If the stresses are substituted using Eqs (10.5) (10.7) and (10.20) (10.22), the virtual work is a function of the displacement functions um(x,y), vm(x,y) and wm(x,y). Within the scope of second order theory, double products of the displacement functions are considered at maximum. Since

t 2

t 2z dz 0 (10.26)

the following work components result:

a) Plates loaded in-plane

int m m m m m m m mA

m m m m m m m m

W D u u u v v v v u

G t u u u v v u v v dA (10.27)

b) Plates laterally loaded

(10.28)

c) Coupling in-plane/laterally loaded plates (for plate buckling)

int m x m m y m xy m m m mmA

W w n w w n w n w w w w dA

(10.29)

In Eq. (10.27), D is the extensional stiffness of the plate:

2E tD

1 (10.30)

and in Eq. (10.28), B is the bending stiffness of the plate:


3

2E tB

12 1 (10.31)

Equation (10.29) contains the longitudinal and shear forces nx, ny and nxy of the plates loaded in-plane according to Figure 10.3, i.e. the stresses x, y and xy, which are linked to the plate deflection wm(x,y). Since A is the centre face of the plate, for rectangular plates we have:

dA = dx dy (10.32)

External virtual work

In Section 3.4.2, the formulation of the virtual work for beams is dealt with and Table 3.2 contains a compilation for concentrated loads and distributed loads. Using this, the external virtual work for plates can directly be stated. With the help of Figure 10.6, we can see that for the selected loads it would be

x b

ext z Fz z qzx a

W F w q w dx (10.33)

In Eq. (10.33), wFz and wqz are the virtual displacements in the direction of the acting loads. For a constant area load pz, the integration has to be performed via the loaded area, so that the external virtual work yields

pz

ext z pzA

W p w dA (10.34)

Figure 10.6 Loads Fz, qz and pz for plates


10.6 Plates in Steel Structures

Usually, steel constructions are almost exclusively idealised as beam structures. Re-garding profile-oriented structures, which are often realised with rolled cross sections, this is in the nature of things, since they are actually beam-shaped constructions. Other structures which seem planar at first glance are also treated as beam structuresfor the calculation of the deformations and stresses. Even large bridges are idealised using beams/girders, girder grid constructions or spatial beam structures and the engineering standards are adjusted to these calculations as well. This is for instance realised with the specification of an effective width for flanges.

If, by way of exception, plates are consulted for the calculation, it may for example be a case of load introduction problems, which are calculatively analysed with the help of plates. Nevertheless, analyses with FEM are rare since they are actually only suitable if the material fatigue is to be considered. Otherwise, i.e. if local plastifi-cations can be accepted, we usually use simplified design models as for instance for the load introduction into I-profiles without stiffeners as shown in Figure 10.7.

Figure 10.7 Design model for the transmission of forces into I-sections without stiffeners

Another possible case for the use of plate elements for steel structures may be the analysis for the load-bearing capacity of beams with thin-walled cross sections. Figure 10.8 shows different cross sections for which the different parts are idealised using plates. However, for engineering practice, this application is of no relevance – it is only of interest for scientific analyses. Sporadically, this methodology has been used for the examination of bridges. Whether this is reasonable or not has not gener-ally been clarified yet.

Figure 10.8 Idealisation of thin cross-sectional parts through plates (plane shells)

10.7 Stiffness Matrix for a Plate Element 429

A reasonable field of application for the use of plate elements is plate buckling, al-though the calculations are normally not executed with FEM, but conducted by using formulas or tables. Usually, plate buckling is about the analysis of plane load-bearing structures being components of beam structures. A typical example of that are bridges, where single or stiffened plates are extracted in order to analyse the influence of the buckling on the bearing capacity of the components. Figure 10.9 gives an explanatory example. Due to the significance of plate buckling for steel structures, it is discussed in the following section in detail and initially the basics for the appli-cation of FEM are derived.

Upper flange: field range of single-span and continuous beams (pos. My) Webs: end-bearing (shear force), field range (pos. My), supported areas

of continuous beams (neg. My and shear force) Lower flange: supported areas of continuous beams (neg. My)

Figure 10.9 Required verifications against plate buckling using the example of a pedestrian bridge

10.7 Stiffness Matrix for a Plate Element

In this section, the stiffness matrix for a plate element with infinite shear stiffness is derived on the basis of Kirchhoff’s plate theory. As explained in detail in Section 3.5.4, a rectangular plate element with four nodes and a total of 16 nodal degrees of freedom is chosen. At each node, w, w , w and w are the unknown displace-ments.

The bicubic polynomial function chosen with the help of Figure 3.14 leads to the following displacement function for the deflection of the plate element in Figure 3.16:

w(x,y) = c1 + c2 x + c3 y + c4 x2 + c5 xy + c6 y2 + c7 x3 + c8 x2 y + c9 xy2 +c10 y3 + c11 xy3 + c12 x3 y + c13 x2 y2

+ c14 x2 y3 + c15 x3 y2 + c16 x3 y3 (10.35)


Similar to beams, the dimensionless coordinates = x/ x and = y/ y are introduced and the unknowns c1 to c16 are replaced by the unknown displacements at the four nodes:

wa, aw , aw , aw , wb , bw , bw , bw , wc , cw , cw , cw , wd , dw , dw and dw .

This gives the displacement function w( , ) in the formulation of Eq. (3.58) – see also Figure 3.10. The subscript “m” for the designation of the plate centre face is omitted here for reasons of a simplified depiction and better readability.

Figure 10.10 Rectangular plate element with 16 degrees of freedom and deformations, internal forces and loads in the nodes

In Figure 10.10, the rectangular plate element with the chosen designations is repre-sented. The internal forces and loads correspond to the four deformation values at the nodes. They are values in kN or kNm, respectively, and are therefore designated with capital letters. The indices comply with the definitions of Figure 10.4 for the internal forces of the plate. The subscript “L” designates the load values, as shown in Figure 1.4 for beams.

The stiffness relationship for the plate element in Figure 10.10 can now be formulated in a similar manner as was shown in Section 4.2 for beam elements. In doing so, especially the comparison with Eq. (4.18) and Table 4.2 directly leads to the desired result. For the plate element, the stiffness relation is:

10.7 Stiffness Matrix for a Plate Element 431

Te e e e e v : s K v p (10.36)

The element stiffness matrix Ke for the bending of plates can be stated explicitly with the virtual work in Eq. (10.28), if we use the displacement function w( , ) according to Eq. (3.58) for the deflection of the plate. To do so, the relevant derivatives of the function have to be set up and the integration over the plate area has to be performed. Since the displacement function in Eq. (3.58) is formulated with dimensionless coordinates, dA = dx dy is replaced by d d x y in Eq. (10.28) and the integration is executed from = 0 to = 1 as well as = 0 to = 1. The result is the stiffness matrix stated in Table 10.1 for the plate element of Figure 10.10. Each com-ponent of the matrix consists of four terms which are to be added after the multiplication with the given factors. The shear modulus G in Eq. (10.28) was re-placed by the material constants E and with the help of Formula (10.8). In Eq. (10.36), the vectors T

ev and ve contain the element lengths x and y. These lengths have to be multiplied into the element stiffness matrix of Table 10.1 if plates are to be discretised using elements of different lengths.


Table 10.1 Stiffness matrix for the plate element in Figure 10.10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

12808280812961296

1404396648108

3961404

648108

198198

999

972-2808-1296 -1296

486-396 -648 -108

-2341404108108

-11719854

9

-2808972

-1296-1296

1404-234108108

-396486

-648-108

198-117

549

-972 -972 12961296

486234

-108 -108

234486

-108 -108

-117-117

99

2 936

72144144

198198549

9

132367212

486-396 -648 -108

324-72

-144 -144

-117198

549

-78361212

-1404234

-108-108

468-54-36-36

-198117-54

-9

66-27-18-3

-486 -234 108108

162543636

117117

-9-9

-39-27

-3-3

3 72

936144144

36132

7212

234-1404

-108 -108

117-198 -54

-9

-54468-36-36

-2766

-18-3

-396486

-648-108

198-117

549

-72324

-144-144

36-781212

-234 -486 108108

117117

-9-9

541623636

-27-39-3-3

4 24

241616

117-198 -54

-9

78-36-12-12

-2766

-18-3

-1812-4-4

-198117-54

-9

66-27-18-3

-3678

-12-12

12-18

-4-4

-117 -117

99

392733

273933

-9-911

5 2808

280812961296

1404396648108

-396-1404-648-108

-198-198-99-9

-972-97212961296

486234

-108-108

-234-486108108

117117

-9-9

-2808 972

-1296 -1296

1404-234 108108

396-486 648108

-198117-54

-9

6 936

72144144

-198-198-549

-9

-132-36-72-12

-486-234108108

162543636

-117-117

99

392733

-1404 234

-108 -108

468-54-36-36

198-117

549

-662718

3

7 72

936144144

361327212

234486

-108-108

-117-117

99

54162

3636

-27-39

-3-3

396-486 648108

-198 117-54

-9

-72324

-144 -144

36-781212

8 24

241616

117117

-9-9

-39-27

-3-3

273933

-9-911

198-117

549

-6627183

-3678

-12-12

12-18-4-4

9 2808

280812961296

-1404-396-648-108

3961404648108

-198-198-99

-9

972-2808 -1296 -1296

-486 396648108

-234 1404

108108

117-198-54-9

10 936

72144144

-198-198-549

-9

132367212

-486 396648108

324-72

-144 -144

117-198 -54-9

-78361212

11 72

936144144

-36-132-72-12

234-1404

-108 -108

-117 19854

9

-54468-36-36

27-66183

12 24

241616

-117 19854

9

78-36-12-12

27-66183

-1812-4-4

13 2808

280812961296

-1404 -396 -648 -108

-396 -1404 -648 -108

198198

999

14 936

72144144

198198549

9

-132-36-72-12

15 72

936144144

-36-132-72-12

16 24

241616

10.8 Geometric Stiffness Matrix for Plate Buckling

In the following, the stiffness relationship for the bending of plates according to Eq. (10.36) is expanded for the analysis of plate buckling. For this purpose, a geometric

Matrix is symmetric.

Factors for the four values of the matrix elements:

y3x

x3y

x y

x y

3

2

B630B

630

B450

(1 ) B450

E t with :B12 (1 )

10.8 Geometric Stiffness Matrix for Plate Buckling 433

element stiffness matrix Ge is added to the stiffness relationship as done in Section 5.5 for beams:

e e e e es K G v p (10.37)

The differentiation of the internal forces se and es (see Figure 5.9) is not explicitly required here and the vector of the load ep is not needed since only the eigenvalue problem “plate buckling” is to be solved.

Table 10.2 Geometric stiffness matrix for the plate element in Figure 10.10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

156165616900

468792

0

792468

0

6666

-36

1944-5616

0

162-792-180

-468468

0

-396636

-56161944

0

468-468

0

-792162

-180

66-3936

-1944 -1944

-900

162468180

468162180

-39-39-36

2 624

1440

666636

88120

162-792 180

216-144

0

-3966

-36

-52120

-468468

0

-156-108

0

-6639

-36

-22-96

-162 -468 -180

-5410830

393936

13-9-6

3 144

6240

12880

468-468

0

39-66-36

-108-156

0

-9-22

6

-792162180

66-39-36

-144216

0

12-52

0

-468 -162 -180

393936

108-5430

-913-6

4 16

160

39-6636

52-12

0

-9-22-6

-12-40

-663936

-22-9-6

-12520

-4-12

0

-39-39-36

-1396

9-13

6

33

-1

5 5616

5616-900

468792

0

-792-468

0

-66-66-36

-1944-1944

900

162468

-180

-468-162180

3939

-36

-5616 1944

0

468-468

0

792-162-180

-663936

6 624

1440

-66-6636

-88-12

0

-162-468180

-54108-30

-39-3936

-139

-6

-468 468

0

-156 -108

0

66-39-36

2296

7 144

6240

12880

468162

-180

-39-3936

108-54-30

-913

6

792-162 180

-6639

-36

-144216

0

12-52

0

8 16

160

3939

-36

13-96

9-13

-6

331

66-3936

229

-6

-1252

0

-4-12

0

9 5616

5616-900

-468-792

0

792468

0

-66-66-36

1944-5616

0

-162 792

-180

-468468

0

39-6636

10 624

1440

-66-6636

8812

0

-162 792180

216-144

0

39-66-36

-5212

0

11 144

6240

-12-88

0

468-468

0

-3966

-36

-108-156

0

922

6

12 16

160

-396636

52-12

0

922-6

-12-40

13 5616

5616900

-468 -792

0

-792-468

0

6666

-36

14 624

1440

666636

-88-12

0

15 144

6240

-12-88

0

16 16

160

The starting point for the set-up of the matrix Ge is the virtual work in Eq. (10.29). As for the flexural buckling of beam members, the internal forces nx = x t, ny = y t and nxy = xy t are assumed to be known. Moreover, it is assumed that they are constant within the element. In a similar manner as described in Section 10.7 for the plate bending, the necessary integrations can be carried out without any problems. The geometric element stiffness matrix shown in Table 10.2 is the result. Each

Matrix is symmetric.

Factors for the three values of the matrix elements:

y

x

x

y

x

y

xy

x y

t12600

t12600

t1800and : tension positive


element of the matrix consists of three terms which are to be added after multiplication with the given factors.

As an alternative to the assumption of constant in-plane internal forces within the plate elements also linearly or quadratically varying courses can be considered. The integrations are somewhat more extensive then and the geometric stiffness ma-trix is significantly more complex since the matrix elements have two or three times as many terms.

10.9 Plates with Longitudinal and Transverse Stiffeners

Plates are often strengthened in longitudinal and transverse directions. The stiffened plates can be analysed with a combination of plate and beam elements using FEM.

Figure 10.11 Plate with a longitudinal and a transverse stiffener

The plate of Figure 10.11, which is supposed to represent a portion of a larger plate, is stiffened with a longitudinal and a transverse stiffener. It is divided into finite plate elements, what also directly leads to the corresponding beam elements. For the set-up of the stiffness matrix K and the geometric stiffness matrix G, first the plate elements are considered. The allocation of the matrix elements is carried out as described in Section 4.5.2 for beam structures. In a second step, the beam elements are added. Forbeams in the x-direction, the element stiffness matrices can directly be taken from Sections 4.2.3, 4.2.4 and 5.6. Since the bending about the y-axis and the warping tor-sion are to be acquired, 8 8 element matrices ensue. The allocation of the degrees of freedom of the beams to the ones of the plates is also possible without any problem. With the assumption that the beam element begins at the point a and ends at point b, the following allocation results with Figures 4.2 and 10.12:

10.9 Plates with Longitudinal and Transverse Stiffeners 435

Ma a

ya a

a a

a a

Mb b

yb b

b b

b b

w ww

w

ww w

w

w

w

Figure 10.12 Beam elements in the x- and y-directions and allocation of the degrees of freedom to the plate elements

The necessary matrices for the beam elements can be set up using Tables 4.3 and 5.2. However, only matrix elements are to be adopted which are linked to the deforma-tions mentioned previously. According to Table 4.3, these elements of the stiffness matrix are the ones depending on the stiffnesses EIy, EI and GIT. The values to be taken from the geometric element stiffness matrix (see Table 5.2) depend on the par-ticular task. As a general rule, only axial forces N are considered in the stiffeners for plate buckling and therefore the components depending on N, which are linked to the deformation values stated above, need to be considered. In doing so, it has to be noted that Mrr is influenced by the axial force N – see Table 5.1. An additional look at Ta-ble 4.3 shows that distributed springs cw and c can also be considered for plate buckling.

For beam elements in the y-direction, which run from a to c, the allocation of the degrees of freedom can be also carried out with Figures 4.2 and 10.12. The selection of the matrix elements is done similarly as for beam elements in the x-direction.


FEM for stiffened buckling plates

Figure 10.13 shows a buckling plate which is stiffened by a longitudinal stiffener and stressed by constantly distributed stresses x only.

Figure 10.13 Buckling panel with a longitudinal stiffener and cross section properties of the stiffener

According to Figure 10.13a, the stress x is set to be constant over the total height since the buckling plates are usually components of total cross sections. For FE modelling, the plate is laterally and longitudinally divided into finite plate elements, providing a bending stiffness and being stressed by “in-plane stresses”.

Figure 10.13b shows six equal elements in the transverse direction. Stiffeners are usually arranged on one side regarding the plate centre face, which is also the case for the T-stiffener in Figure 10.13. It is idealised using beam elements, for which the stiffnesses EIy, GIT and EI as well as, with respect to the stability risk, the compression force N = x A and N 2

pi are required. The values are to be calculated as follows:

10.9 Plates with Longitudinal and Transverse Stiffeners 437

Moment of inertia Iy (bending) The stiffened plate acts as the upper flange of the stiffener. Therefore, Iy has to be calculated considering the effective flange width b1 + b2 – also see Figure 10.18 and Table 10.3. Since the bending of the plate is already included in the stiffness matrix (see Table 10.1), the upper flange may only be considered with the Steiner part for the Iy of the stiffener.

Torsion constant IT (primary torsion) For this value, only the stiffener itself may be considered since the correspond-ing stiffness of the plate is already included in the stiffness matrix (see Table 10.1). This becomes obvious with the virtual work of Eq. (10.28) and the term depending on the shear modulus G. For thin-walled stiffeners (open cross sec-tion!) we have

3 3T w w f f

1 1I b t b t3 3

(10.38)

Warping constant I (secondary torsion) For the calculation of this value, the pivot point is assumed to be in the centre face of the plate since the plate cannot displace laterally. With this assumption, the warping ordinate for the edges of the lower flange of the stiffener is

f w f ft 2 b t 2 b 2 and the warping constant is (10.39)

2 2f f f

A

1I dA b t3 (10.40)

Compression force N for flexural buckling For the flexural buckling of the stiffener the axial force N = x A is of impor-tance. A is the cross section area of the stiffener, i.e. here in this case it is A = bw tw + bf tf. Parts of the plate are not considered since the compression stresses x are already included in the geometric stiffness matrix (see Table 10.2).

N 2pi for torsional buckling

For the torsional buckling of the stiffener, the origin of the rotation is assumed in the centre face of the plate as done for the warping constant I . For Mrr of Tables 5.1 and 5.2 with x = const., we therefore have

2 2 2 2rr x x

A A2

x z y x p x p

M y z dA y z dA

I I I A i (10.41)For the stiffener in Figure 10.13, we obtain

3z f fI t b 12 (10.42)

2 23y w w w w w f f w fI t b 12 t b t 2 b 2 t b t 2 b b 2 (10.43)


For hand calculations, the torsional stiffnesses GIT and EI of stiffeners are usually neglected and a “minimum torsional buckling stiffness” is demanded for compressed stiffeners with an open cross section shape. Specifications on this are given in DIN 18800 Part 3, for example element 1004. Often, FE calculations are also conducted for EI = GIT = 0. If these stiffnesses are considered, which is possible without any difficulty, the danger of torsional buckling of the stiffener has to be captured with

2pN i in the geometric stiffness matrices of the beam elements.

With the eigenvalue analysis of the plate of Figure 10.13a using FEM, a buckling of a single panel (in between the stiffeners), of the entire panel (whole plate) or a buckling of the stiffener can occur. For the single panel buckling in Figure 10.13c the right partial plate is decisive because it is wider than the left one. While for hand calculations the single panels do not influence each other, for a calculation using FEM the left panel stabilises the right one and in addition the eigenvalue is increased due to the torsional stiffness of the stiffener, provided that the torsional buckling of the stiffener is not decisive. For a buckling of the entire panel, which is sketched in Figure 10.13d, the stiffener will translate downwards (or upwards). Here, the flexural buckling of the stiffener is of primary importance. If the support of the longitudinal edges has no or only a slight influence, we speak of a buckling panel behaviour “similar to buckling members”.

10.10 Verifications for Plate Buckling

Designation plate buckling

The stability problem of plate buckling occurs when plates are stressed in their planes by compression stresses or shear stresses. These stresses are x, y, xy and

yx of an in-plane loading as shown in Figure 10.3. However, displacements perpen-dicular to the plane w(x,y) emerge for the buckling, characterising the stability problem combarable to the stability modes of beams (flexural buckling and lateral torsional buckling).

Verification methods

For the verification against flexural- and lateral torsional buckling of beam members, the following verification methods can be distinguished:

procedure (see Section 9.4 and 9.6)

procedure with equivalent geometric imperfections (see Section 9.8)

plastic zones theory (see Section 5.12)

Verifications for plate buckling have been conducted with a corresponding modified method so far, since this is regulated in the relevant engineering standards. As an

10.10 Verifications for Plate Buckling 439

alternative, the method of effective widths becomes more important. A procedure with equivalent geometric imperfections for plates is perfectly possible, but it has not been developed to the point that it could be put into use. There is a lack of appropriate equivalent geometric imperfections as well as of methods for the computation of the stresses and for the verification and load-bearing capacity. Calculations according to plastic zones theory are also possible for plates – also see [5]. However, they are not adequately regulated for practical issues, especially regulations for the geometric imperfections, and also the residual stresses to be applied are missing.

For plate buckling, the verifications are usually conducted using the method. For this method, the critical buckling stresses (eigenvalues) are needed, which are often determined with the help of formulas or diagrams. Since only a limited number of practical cases is covered with these, the determination of eigenvalues for plate buckling is an important task for FEM. Moreover, the corresponding eigenmodesare often also needed.

Verification against buckling according to DIN 18800 Part 3

FE computations have to be executed so that the results can be used for buckling verifications going along with the regulations of the standards. Therefore, important principles according to DIN 18800 Part 3 are compiled below.

Figure 10.14 Differentiation of different buckling panels, [8]

Rectangular plates of structural components susceptible to buckling are referred to as buckling panels. Their longitudinal edges are oriented in the direction of the longitu-dinal axis of the component. Buckling panels can be strengthened by stiffeners. Stiffeners in the direction of the longitudinal edges are longitudinal stiffeners, those in direction of the transverse edges transverse stiffeners.


Figure 10.15 Examples for plate edges of webs and parts of the flanges, [8]

As shown in Figure 10.14, buckling panels are differentiated as single, partial and entire panels. Entire panels are stiffened or unstiffened plates, which are generally supported nondisplaceably at their logitudinal and transverse edges (see Figure 10.15). The edges can also be supported elastically; longitudinal edges may also be unsupported. Partial panels are logitudinally stiffened or unstiffended plates located in between adjacent transverse stiffeners or between a transverse edge and an adjacent transverse stiffener and the logitudinal edges of the entire panel. The decisive widths of the buckling panels bG for entire and partial panels and bik for single panels are defined in Figure 10.16.

Figure 10.16 Decisive width of the buckling panels, [8]

Figure 10.17 Stresses x, y and in a buckling panel, [8]

DIN 18800 Part 3 contains the following specifications for the assumption of the boundary conditions:


For plate edges supported perpendicularly to the plate, a hinged support (sim-ple support) is assumed.

For the edges of single panels, which are composed by stiffeners, nondisplace-able, hinged bearings may be assumed for the verification of the single panels. For the transverse edges of partial panels, which are composed of transverse stiffeners, nondisplaceable bearings may be assumed for the verification of the partial panels.

For boundary stiffeners, which elastically support a longitudinal edge, a nondisplaceable bearing may be assumed, if a stability verification according to DIN 18800 Part 2 is executed for the stiffeners.

Supporting and fixing effects of the adjacent structural components may be considered if the total stability of the cooperating parts is considered.

Table 10.3 Reduction factors for an individual acting of x, y or , [8]

Bucklingpanel Supports Stress

Non-dimensionalslenderness

Reduction factor

Singlepanel

At all edges Axial stresses with an edge stress ratio T 1*)

y,kp

cr

f2

p p

1 0.22c 1

with c = 1.25 – 0.12 T 1.25 At all edges Shear stresses

y,kp

cr

f

3p

0.84 1

Partialandentirepanel

At all edges Axial stresses with an edge stress ratio 1

y,kp

cr

f2

p p

1 0.22c 1

with c = 1.25 – 0.25 1.25 At three edges

Axial stresses y,k

pcr

f **)2p

1 10.51

At three edges

Constant edge displacement u y,k

pcr

f **)

p

0.7 1

At all edges, withoutlongitudinal stiffeners

Shear stresses y,k

pcr

f

3p

0.84 1

At all edges, withlongitudinal stiffeners

Shear stresses y,k

pcr

f

3 p

0.84 1 if p 1.38

2p

1.16 if p 1.38

*) For single panels, T is the edge stress ratio of the partial panel in which the single panel is located.**) For the determination of cr the buckling value min k ( ) for = 1 has to be considered.


Table 10.4 Reduction factos and for panels supported at all edges ( 1), [29]

Comp., = 1*) Comp., 0*) Shear Comp., = 1*) Comp., 0*) Shear

p

b/t b/t p

b/t b/t b/t for � = 0 for� b/t for � = 0 for�

S235 S355 S235 S355 S235 S355 S235 S355 S235 S355 S235 S355

0.60 33.7 27.6 1.000 47.2 38.5 1.000 51.3 41.9 1.000 1.85 104 85.0 0.476 145 119 0.595 158 129 0.454

0.65 36.6 29.8 1.000 51.1 41.7 1.000 55.6 45.4 1.000 1.90 107 87.3 0.465 149 122 0.582 162 133 0.442

0.70 39.4 32.1 0.980 55.0 44.9 1.000 59.9 48.9 1.000 1.95 110 89.5 0.455 153 125 0.569 167 136 0.431

0.75 42.2 34.4 0.942 58.9 48.1 1.000 64.1 52.4 1.000 2.00 112 91.8 0.445 157 128 0.556 171 140 0.420

0.80 45.0 36.7 0.906 62.9 51.3 1.000 68.4 55.9 1.000 2.05 115 94.1 0.435 161 132 0.544 175 143 0.410

0.85 47.8 39.0 0.872 66.8 54.5 1.000 72.7 59.4 0.988 2.10 118 96.4 0.426 165 135 0.533 180 147 0.400

0.90 50.6 41.3 0.840 70.7 57.7 1.000 77.0 62.8 0.933 2.15 121 98.7 0.418 169 138 0.522 184 150 0.391

0.95 53.4 43.6 0.809 74.7 61.0 1.000 81.2 66.3 0.884 2.20 124 101 0.409 173 141 0.511 188 154 0.382

1.00 56.2 45.9 0.780 78.6 64.2 0.975 85.5 69.8 0.840 2.25 127 103 0.401 177 144 0.501 192 157 0.373

1.05 59.1 48.2 0.753 82.5 67.4 0.941 89.8 73.3 0.800 2.30 129 106 0.393 181 148 0.491 197 161 0.365

1.10 61.9 50.5 0.727 86.4 70.6 0.909 94.1 76.8 0.764 2.35 132 108 0.386 185 151 0.482 201 164 0.357

1.15 64.7 52.8 0.703 90.4 73.8 0.879 98.4 80.3 0.730 2.40 135 110 0.378 189 154 0.473 205 168 0.350

1.20 67.5 55.1 0.681 94.3 77.0 0.851 103 83.8 0.700 2.45 138 113 0.372 193 157 0.464 210 171 0.343

1.25 70.3 57.4 0.659 98.2 80.2 0.824 107 87.3 0.672 2.50 141 115 0.365 196 160 0.456 214 175 0.336

1.30 73.1 59.7 0.639 102 83.4 0.799 111 90.8 0.646 2.55 143 117 0.358 200 164 0.448 218 178 0.329

1.35 75.9 62.0 0.620 106 86.6 0.775 115 94.3 0.622 2.60 146 119 0.352 204 167 0.440 222 182 0.323

1.40 78.7 64.3 0.602 110 89.8 0.753 120 97.8 0.600 2.65 149 122 0.346 208 170 0.433 227 185 0.317

1.45 81.6 66.6 0.585 114 93.0 0.731 124 101 0.579 2.70 152 124 0.340 212 173 0.425 231 189 0.311

1.50 84.4 68.9 0.569 118 96.2 0.711 128 105 0.560 2.75 155 126 0.335 216 176 0.418 235 192 0.305

1.55 87.2 71.2 0.554 122 99.5 0.692 133 108 0.542 2.80 157 129 0.329 220 180 0.411 239 196 0.300

1.60 90.0 73.5 0.539 126 103 0.674 137 112 0.525 2.85 160 131 0.324 224 183 0.405 244 199 0.295

1.65 92.8 75.8 0.525 130 106 0.657 141 115 0.509 2.90 163 133 0.319 228 186 0.398 248 203 0.290

1.70 95.6 78.1 0.512 134 109 0.640 145 119 0.494 2.95 166 135 0.314 232 189 0.392 252 206 0.285

1.75 98.4 80.4 0.500 138 112 0.624 150 122 0.480 3.00 169 138 0.309 236 192 0.386 257 209 0.280

1.80 101 82.7 0.488 141 115 0.610 154 126 0.467 3.05 172 140 0.304 240 196 0.380 261 213 0.275

*) for partial and entire panels

For the verification against buckling according to DIN 18800 Part 3, reduction factors are required, which can be determined according to Table 10.3. An aid for the de-

termination of and for panels supported at all edges is given in Table 10.4. The factors are determined using a nondimensional plate slenderness ratio p :

y,k y,kp p

cr cr

f f or

3(10.44)

The slenderness ratios depend on the critical buckling stresses cr,x, cr,y and cr,which are to be calculated according to the linear buckling theory. For their calculation, the following assumptions are valid:

unlimited validity of Hooke’s Law ideal isotropic material ideal planarity of the plate ideal centric load application


no residual stresses in equilibrium conditions, only linear terms of displacements are considered

The critical buckling stresses are always calculated with x, y or acting individually. This can be accomplished with the finite element method as shown in Section 10.11 and exemplified in Section 10.12.

Table 10.5 Buckling values k and k for unstiffened panels with simple, nondisplaceable supports at all edges compiled in [29]

Stress = a/b 1 = a/b < 1 = 1 k = 4 (const. compression)

1 > 08.2k1.05

21 2.1k1.1

0 -1 k = 7.81 – 6.29 + 9.78 2 -

= -1 k = 23.9 (pure bending) -

-1 -2 k = 5.97 (1 - )2 -

24k 5.34

: k 5.342

5.34k 4

Besides computer-oriented solutions, critical buckling stresses are usually determined with the product of “buckling value times reference stress”, that is with buckling val-ues k x, k y or k (see Table 10.5) and the reference stress e, as shown in the following compilation. In addition, the compilation contains the most important specifications of DIN 18800 Part 3. The subscript “P” designates the plate buckling.

a, b length or width of the analysed buckling panel = a/b aspect ratio

t plate thickness 22

e 2E t

b12 1

the reference stress e, equal to Euler’s buckling stress of a plate strip with the length b and the width t, which is supported at both ends free of fixing stresses; its bending stiffness is replaced by the plate stiffness, and with the numeric values E = 210 000 N/mm2 and = 0.3 we get

2 2

e 2 2t N 100 t kN189800 1.898b bmm cm

.

k x, k y, k buckling values of the analysed buckling panel with an exclusive acting of the boundary stresses x, y or

cr,x = k x e critical buckling stress with an exclusive acting of boundary stresses x


cr,y = k y e critical buckling stress with an exclusive acting of boundary stresses y

cr = k e critical buckling stress with an exclusive acting of boundary stresses

ay,k

Ef

the reference slenderness ratio a, calculated with the characteristic material parameters:

a = 92.9 for St 37 (S 235) with fy,k = 240 N/mm2

a = 75.9 for St 52 (S 355) with fy,k = 360 N/mm2

pPi

E or pPi

E3

plate slenderness ratio

p p a nondimensional plate slenderness ratio

x, y, reduction ratios for plate buckling

For stiffened buckling panels, the following cross section and system values are used for the stiffeners:

I second degree area moment (formerly moment of inertia), calculated with the effective flange width b

A area of cross section without effective plate parts 2

3G

I12 1b t

referenced second degree area moment (stiffness);

for = 0.3 it is 3G

I10.92b t

G

Ab t referenced cross-sectional area

The effective flange width of stiffeners can be calculated with Figure 10.18 and Table 10.6.

Figure 10.18 Effective flange width of compressed longitudinal and boundary stiffeners, [8]


Table 10.6 Effective flange width of longitudinal stiffeners, [8]

Flange width of …

compressed longitudinal stiffeners compressed boundary stiffeners

non-compressedlongitudinal and boundary

stiffeners

i,k 1ik bbb2 2

with

aik a

ik

tb 0.605 t 1 0.133b

but ik ikb b and iik

ab3

i1i0

bb b2

with

i0 ab 0.138 t

or

i0 i0p

0.7b b

but

i0 i0b b and ii0

ab6

ik ikb b but ia3

i0 i0b b but ia6

For single, partial and entire panels it is to be verified that the stresses due to the loading do not exceed the limit of the buckling stresses. For an exclusive acting of x,

y or the verifications are to be performed as follows:

p,Rd1 with y,k

p,RdM

f (for x and y) (10.45)

p,Rd1 with y,k

p,RdM

f3 (10.46)

If stresses x, y and occur simultaneously, the verification has to be performed ac-cording to element 504 of DIN 18800 Part 3 (also see [61] and [75]). However, this case is not shown here, since it does not have great relevance in practice. In this con-text the combination of x and is more interesting, for which the following verification condition is valid:

1 3e ex

xp,Rd p,Rd1

with: 41 xe 1 and 2

3 xe 1

(10.47)

With Eq. (11.34), the critical buckling stresses cr and cr have to be determined for an individual acting of x and as mentioned previously. Further regulations of DIN 18800 Part 3 will not be responded to at this point, but it should be mentioned that the following points are of special importance:


behaviour of the buckling panel “similar to buckling members” structural design of stiffened plates and compressed stiffener requirements interaction of plate buckling with flexural buckling

For a quick estimation of the resistance of unstiffened panels, Figure 10.19 contains an evaluation of the verification conditions for different stress states. Using the panel ratio b/t, the figure allows a direct determination of the maximum possible stress for the following cases:

max x with = 0 max x with = 5 kN/cm2

max x with = 10 kN/cm2

max with x = 0

The evaluation for max has been performed with k = 5.34 (a/b ), k = 6.34 (a/b = 2) and k = 9.34 (a/b = 1) according to Table 10.5. For max x with = 5 and 10 kN/cm2, the value k = 5.34 is used, which is min k .

Figure 10.19 Maximum stresses for unstiffened partial and entire panels supported on all sides using according to Table 10.3


Verification against buckling according to DIN Technical Report 103/Eurocode 3

The DIN Technical Report 103, Steel Bridges [11], is valid for motorway bridges and railway bridges and replaces the present national regulations for these bridges. The basis of the technical report is Eurocode 3 [10] or its status when the technical report was published. The guidelines of DIN Technical Report 103, Steel Bridges [81], contain comments and calculation examples which should make the application of the technical report easier.

For plate buckling, two methods are regulated in [11] – see [81] as well:

1. A procedure where the stress of a beam is divided into longitudinal stresses, shear stresses and transverse stresses due to the loads at the transverse edges. For each stress component, an individual verification against buckling is con-ducted. For the consideration of the mixed stress state, the verifications are then combined using in interaction relationships. This procedure works with effective cross sections regarding the longitudinal stresses.

2. A procedure with a limitation of stresses, where the limit loading is deter-mined for each buckling panel of the cross section regarding the mixed stress state, for which the contribution of the total cross section is applied.

According to [11], the procedure with a limitation of stresses is usually to be used for the design of steel bridges. This procedure and its methodology widely comply with DIN 18800 Part 3, as long as the individual acting of the stresses x, y and is ana-lysed. The corresponding buckling values can be determined with appropriate tables or with computer calculations (FEM).

For the combination of stresses x, y and acting, the von Mises criterion (equiva-lent stress) is used as boundary condition in [11]. In the notation of [81], it is:

22 2y,Ed y,Edx,Ed x,Ed Ed

x,Rd y,Rd x,Rd y,Rd Rd1.0 (10.48)

The boundary stresses x,Rd, y,Rd and Rd are determined with buckling curves and the coaction is considered in the system slenderness

ult,kp

crit(10.49)

The amplification factor crit is the critical load factor cr for the combined acting of the stresses x, y and . For the determination of crit, an FEM analysis or an ap-propriate solution from the corresponding literature can be used. The execution of the verifications is dealt with using the example of Section 10.12.2.


10.11 Determination of Buckling Values and Eigenmodes with FEM

In this chapter, the determination of buckling values and eigenmodes is covered with the help of finite elements. Using the matrices in Sections 10.7 10.9, the stiffness matrix K and geometric stiffness matrix G can be set up for the buckling panels. With these, a homogeneous matrix equation yields for the eigenvalue problem “plate buck-ling”:

(K + cr,r G) vr = 0 (10.50)

In Eq. (10.50), cr,r is the critical load factor for plate buckling, which is required for the calculation of buckling values or critical buckling stresses. vr is the eigenvector, with which the eigenmode (buckling shape) is described. The subscript “r” designates the number of the eigenvalue.

In Chapter 6 the determination of eigenvalues and eigenmodes is dealt with in detail regarding the buckling of beam structures. Since Eq. (10.50) is formally in accor-dance with Eq. (6.30), the method suggested in Section 6.2 can also be used for plate buckling, i.e. the combination of a matrix decomposition method with the inverse vector iteration is recommended. The iterative calculation can be structured accord-ing to Section 6.2.5 as follows:

1. search for interval (matrix decomposition)

2. reduction of interval (matrix decomposition)

3. determination of critcal load factor and eigenmode (inverse vector iteration)

4. check

In comparison to the beam structures, three fundamental differences can be recog-nised for plate buckling:

The band width of the matrix equation (10.50) is often significantly larger, so that the complexity (calculating time) for the matrix decomposition increases considerably in comparison to the vector iteration.

In many cases, not only the first eigenvalue is needed for plate buckling but higher eigenvalues as well. In Section 10.12.2, the buckling of the entire panel corresponds to the 11th eigenvalue, for example.

The eigenvalues are located very close to each other for plate buckling. For that reason, the solution method must be modified accordingly.

10.11 Determination of Buckling Values and Eigenmodes with FEM 449

For plate buckling it is necessary to determine the searched eigenvalue r with relatively high accuracy using the matrix decomposition method. Since the eigenvalues can be very close together, the decomposition should be executed until an exactness of approximately 0.001 is reached. Afterwards, the inverse vector iteration can be started, as described in Section 6.2.4. It serves the determination of the eigenvectors and the exact eigenvalues. As initial value 0, the approximated eigenvalue of the matrix decomposition is used and with that the vector iteration is displaced to the area of the searched eigenvalue (spectral displacement). As a general principle, the following is valid: the closer the initial value is located to the searched eigenvalue and the more the start vector and the eigenvector are alike, the faster the vector iteration converges to the right solution. As discussed in Section 6.2.4, the use of a start vector generated using random numbers is recommended for plate buckling as well. In [2], it suggests setting all elements of the start vectors to the value 1.

In order to determine the eigenmode correctly, the inverse vector iteration should generally be conducted until the exactness of the eigenvalue is of at least 10 5. After-wards, the check described in Section 6.2.5 has to be carried out. If more than one eigenvalue is located in the considered interval, it can be reduced. However, as shown in Section 10.12.1, this is useless if several equal eigenvalues occur. The procedure for the calculations is explained with the following example. Here, the case is also covered when the vector iteration does not converge to the searched eigenvalue and eigenvector.

Example:

For the stiffened buckling plate in Figure 10.23, the 11th eigenvalue and the corre-sponding eigenmode are to be determined. According to Section 10.12.2, this is buckling of the entire panel. This case results for the chosen partition with 20 16 fi-nite elements as 11th eigenvalue. With a different partition into elements, the buckling of the entire panel could, for example, also be the 10th eigenvalue. With the FE calculation, we obtain

10th eigenvalue: cr,10 = 2.8353



As it can be seen, the 10th and the 12th eigenvalue are close to the 11th eigenvalue (96.0% or 101.3%), so that an exactness of 0.001 is reasonable for the determination of the eigenvalues with the matrix decomposition method. With this, the decomposi-tion is carried out 13 times in the search for the 11th eigenvalues, that is for the following values of : 1; 5; 3; 2; 2.5; 2.75; 2.875; 2.9375; 2.9688; 2.9531; 2.9453; 2.9492; 2.9512.


As a result of the interval reduction, we find that the 11th eigenvalue is located between 2.9512 and 2.9531. If the following vector iteration is started with the mean value of the interval range, i.e. a spectral displacement of 0 = 2.9521 is conducted, we obtain cr,11 = 2.95311 after some iterations with an exactness allowance of 10 5.

The number of iterations required depends on the choice of the start vector. For this example, even using different random start vectors, it always converges after six iterations. The concluding check with the matrix decomposition method for u = 0.9999 2.95311 and o = 1.0001 2.95311 shows that the 11th eigenvalue is the only one located in the interval. For this reason, it is clearly verified that the eigenvector, which is determined with the vector iteration, belongs to the 11th eigenvalue. This is not self-evident, since the vector iteration can also converge to the next higher or lower eigenvalue and eigenvector. However, if the check shows that the searched eigenvalue has not been found, the exactness of the matrix decomposition method can be increased to 0.0001. Another possibility is a vector iteration with a correspond-dingly changed initial value o. In the case that previously the next higher eigenvalue has been determined, a smaller initial value has to be chosen, where at minimum the lower boundary determined with the reduction of the interval has to be used. Since the eigenvector is needed anyway, the last method stated is to be preferred.

FE modelling of buckling panels

As a general rule, it is sufficient to divide buckling panels into plate elements of equal size. However, the partition has to consider the location of longitudinal and transverse stiffeners, i.e. the element borders have to be arranged at these positions. The number of plate elements to be chosen in longitudinal and transverse direction depends on the specific problem. In general, the degrees of freedom, which arise due to the discretisation, have to be able to describe the modal shape in a proper way. For that reason, the calculated eigenmode should always be inspected in order to ascertain whether it shows many waves which can be covered with the chosen partition into elements or not. 10 to 20 plate elements in longitudinal direction and 5 to 20 in trans-verse direction are a reasonable choice for many structural problems.



In the following examples, compression stresses are assumed to be positive as in DIN 18800 Part 3. When using computer programs, it is important to pay attention to whether tension or compression stresses are defined as positive. The following FE calculations are conducted with the RUBSTAHL-program FE-Plate Buckling – also see Section 1.10.

10.12.1 Single Panel with Constant x and 1.5

The buckling panel shown in Figure 10.20 is analysed and the lowest eigenvalue and the respective eigenmode are determined. The buckling panel is part of the track system girders for the magnetic levitation train Transrapid, which was built some years ago in large quantities for the Transrapid testing facility in Emsland [37]. Ac-cording to [80], it was later developed further in terms of its structural aspects. The middle part of the cover plate between main girder webs and transverse compart-ments is analysed with the assumption of simple supports at the edges of the buckling panel.

Figure 10.20 Buckling panel of the cover plate of a Transrapid track system girder

Since the aspect ratio of the buckling panel is larger than 1, the critical buckling stress is usually calculated with the buckling value k = 4.0. With the reference stress

2

e 2100 1.6 kN1.898 1.239

198 cm(10.51)

it follows that 2

cr 4.0 1.239 4.956 kN cm . (10.52)


Due to the large number of the girders, the buckling value was calculated more pre-cisely. It is generally known that with an aspect ratio of 2 1.41 the single-waved eigenmode turns into a two-waved. The buckling value k = 4.50 corresponds to 2 according to [42] and with = 1.564 for the buckling panel considered the buckling value is

22 1.564k 4.2471.564 2

(10.53)

The ideal buckling stress is then cr = 5.262 kN/cm2, which is 6.2% larger than the one with k = 4.0.

For comparison, this value is calculated with an FE program. As shown in Figure 10.21, the buckling panel is divided into 10 10 finite elements, i.e. a total of 100 elements. The figure also shows the numbering of the elements and nodes and fur-thermore the boundary conditions for the simple supports (Navier’s support). As a result of the FEM analysis, we obtain the critical load factor cr = 5.264 and the buckling value k = 4.248 with, as expected, the two-waved eigenmode shown in Figure 10.22. The second eigenvalue is, by the way, cr = 6.016 or k = 4.854 and has a single-waved buckling shape.

Figure 10.21 FE modelling of the buckling panel in Figure 10.20


Figure 10.22 Two-waved modal shape at the first eigenvalue

With cr = 5.262 kN/cm2 and fy,k = 24.0 kN/cm2, the non dimensional slenderness is

y,kp

cr

f 24.0 2.1365.262

(10.54)

and the reduction factor according to DIN 18800 part 3 (see Table 10.3) is

21 0.221.25 0.12 0.475

2.1369 2.136(10.55)

For the limit compression stress, we have

x 0.475 24.0/1.1 = 10.36 kN/cm2. (10.56)

Due to the postcritical reserves, a compression stress is permitted which is clearly larger than the critical buckling stress.

Variant with = 2

As a variation to the buckling panel of Figure 10.20, the case of an aspect ratio equal to 2 is now considered. For a better understanding of eigenvalue calculations, the corresponding buckling panel is shown in Figure 10.23.

As already mentioned, the transition from a single- to a two-waved buckling shape is at = 2 and the first and second eigenvalue are equal:

cr,x,1 = cr,x,2 = 4.50 e = 8.541 kN/cm2. (10.57)

An FEM analysis with a 10 10 mesh leads to cr = 8.54182, a value that complies with the theoretical solution within the scope of the computation exactness. The


Figure 10.23 Buckling panel with x = const. and = 2

check described in Section 10.11 shows that the intervals 0.9999 cr to 1.0001 crinclude two eigenvalues, namely the first and second one. Even if the interval is arbi-trarily reduced, this result does not change. Therefore, the question is whether the FE calculation leads to an eigenmode belonging to the first or the second eigenvalue. This basically depends on the programming. Since the initial value 0 for the vector iteration is of significant impact, with 0 < cr, we usually obtain an eigenmode belonging to the first eigenvalue. In contrast to that, it is most likely that for 0 > cr,the eigenmode will correspond to the second eigenvalue. The vector iteration will only lead to the other eigenvalue if the start vector strongly tends to the buckling shape of this eigenvalue. Even if it is often clear which eigenvalue the eigenmode corresponds to (as in this example), with mathematics mean that this can only be de-termined with big efforts. Computer programs require a corresponding programming.

10.12.2 Beam Web with Longitudinal Stiffeners

The buckling panel shown in Figure 10.24 is analysed in [62] in detail, calculating the buckling values with the help of formulas. Here, they are determined with an FE pro-gram and the longitudinal stiffeners are regarded as in [62]: A = 8.73 cm2,I = 410 cm4 and IT = I = 0.

Figure 10.24 Beam web with longitudinal stiffeners


For the FE calculation, the buckling panel is divided into 20 16 = 320 equal plate elements. A look into the design aid of Figure 10.19 shows that the buckling panel without longitudinal stiffeners is not able to carry the existing stresses. This is also by trend confirmed by very small critical buckling stresses, which result to 1cr = 4.1656 kN/cm2 (eigenmode with two longitudinal waves) and to cr = 2.1471 kN/cm2

(eigenmode with a large diagonal wave) using the finite element method. The longitudinal stiffeners are therefore necessary.

The FE calculation of the buckling panel with longitudinal stiffeners leads to cr = 1.7993 as first eigenvalue for the axial stress and 1cr = 27.1693 kN/cm2. The buck-ling of single panel 1 is decisive since the buckling shape shows seven half waves there and the others only have small deflections. A comparative analysis for an iso-lated single panel being simply supported at all edges leads to

1cr = k e = 8.2/(0.625 + 1.05) 4.86 = 23.79 kN/cm2. This value is smaller than 1cr = 27.25 kN/cm2 because in the FE calculation the stiffening (supporting) effect of

panels 2 and 3 is taken into consideration.

For the calculation of the second to tenth eigenvalue, critical load factors between 1.7995 and 2.8323 occur. The buckling shapes with 6 to 13 distinct waves within panel 1 show that this panel is decisive. Partially, also large amplitudes result in panel 2. The buckling of the entire panel does not occur until the 11th eigenvalue with cr= 2.9531 and 1cr = 44.59 kN/cm2 is reached, for which the stiffeners show a lateral deflection. The eigenmode is shown in Figure 10.25. For the buckling of the entire panel, the example in Section 10.11 contains additional calculation results and explanations. In [62], 1cr = k e = 77.9 0.304 = 23.7 kN/cm2 and after a tightening of the calculation, 1cr = 98.5 0.304 = 29.9 kN/cm2 is determined with formulas or using the tables for buckling values of [47], respectively. The difference from the FE computation of 33% is significant and the formulas of [62] are apparently well on the safe side.

Figure 10.25 Buckling of the entire panel in Figure 10.24 for x (11th eigenvalue)


For the exclusive acting of the shear stresses we obtain cr = 1.8941 and cr = 7.1599 kN/cm2 (first eigenvalue) using an FE calculation. The buckling shape is located in panel 3 and shows two diagonal waves. Panel 3 is decisive for shear buckling, as also mentioned in [62], and there the corresponding critical buckling stress is determined with cr = 23.2 0.304 = 7.05 kN/cm2, a value, that complies with the FE result. The verifications for structural safety are not shown here since they are shown in [62] in detail.

DIN Technical Report 103

The buckling panel of Figure 10.23 is also treated in [81] and in 29(!) pages different approaches for the verification are analysed. Here, it is compared with the calculations in Section II-3.4 of [81] and the factor crit is calculated with the help of FEM.

If the first eigenvalue is calculated with the program FE-Plate Buckling regarding the acting of and , it leads to cr = crit = 1.705 and a buckling shape that shows six distinct waves (longitudinal) in panel 1 and small amplitudes in panels 2 and 3. The first eigenvalue therefore corresponds to the buckling of single panel 1, for which an amplification factor crit = 1.52 is calculated in [81]. The corresponding hand calcu-lation is relatively time-consuming, so that the use of an FE program has advantages. Moreover, the value 1.705 is larger (12%) and more exact than crit = 1.52, since in comparison to the hand calculation, the FE analysis takes into account the stiffening (supporting) effect of the adjacent panels. This effect has already been pointed out at the beginning of the chapter.

For the investigation of the buckling of the entire panel, the following values are taken as a basis in [81]:

1cr = k e = 77.90 0.304 = 23.68 kN/cm2 (10.58)

cr = k e = 23.2 0.304 = 7.05 kN/cm2 (10.59)

With these critical buckling stresses, an amplification factor crit = 1.17 for the com-bined acting of and is determined. While the value for cr is relatively exact, 1cris comparatively far on the safe side, so that crit must be larger than 1.17.

A determination of crit with an FE program is relatively difficult since the buckling of the entire panel has to be identified visually on the basis of the eigenmode. This is made even more difficult due to the acting of and . Because eigenvalues 1 to 5 show five to nine distinct waves in panel 1, they are to be associated with a single panel buckling. The sixth eigenvalue leads to a long waved eigenmode, which is indeed irregular, as shown in Figure 10.26, but it totally covers the upper stiffener.

cr,6 = 1.984 = crit is determined as the sixth eigenvalue, a value significantly larger than 1.17 according to [81] (see above).


Figure 10.26 Eigenmode for the acting of and (sixth eigenvalue)

The result should therefore be checked using the critical buckling stresses considering the individual effect. With the values of the FE computation specified above, we ob-tain

crit,44.59 2.9515.1

(10.60)

crit,7.16 1.893.78

(10.61)

For the combined acting of both of the stresses a crit usually results which is smaller than the lower single value. If the two single values are far apart (which is the case here), crit is slightly smaller than the lower single value. Therefore, a value of about

crit = 1.8 would be expected here, so that the FE result seems to be inconsistent. However, an examination shows that the crit, = 1.89 belongs to the shear buckling in panel 3 and that tension stresses act there which lead to a decrease of the buckling danger.

For the analysis of the buckling of the entire panel it is often reasonable to roughly subdivide the buckling panel into finite elements and to arrange element borders only in the range of the stiffeners. Therefore, the buckling panel of Figure 10.23 is now divided transversly into four and longitudinally into 20 elements. With this, the first eigenvalue represents a buckling of the entire panel for which the upper stiffener shows a lateral deflection, and with cr = 1.936, a value which tends to confirm cr,6 = 1.984 (see above).

10.12.3 Web Plate of a Composite Bridge with Shear Stresses

In Figure 10.27, the web plate of a composite bridge at the bridge end is shown. Axial stresses are small and therefore only the plate buckling due to shear stresses is


examined. For the calculation with FEM, the web plate is divided into 10 identical plate elements in longitudinal and transverse direction.

Figure 10.27 Web plate of a composite bridge

If a simple support is assumed for each edge of the buckling panel, as is common practice, the calculation with FEM leads to cr = 4.357 and the modal shape mainly shows a large, diagonal wave. A computation leads to

24k 5.34 7.063

320 210 (10.62)

22

e100 1.21.898 0.6197 kN cm

210(10.63)

2cr 7.063 0.6197 4.377 kN cm (10.64)

The verification for the structural safety according to DIN 18800 Part 3 with this critical buckling stress reveals that the existing shear stress can not be carried. How-ever, the assumed support at the upper edge is on the safe side since the concrete slab is adjacent there. Due to the structural design between the concrete slab and the web plate, a restraint can be applied at the upper edge of the buckling panel. The calcula-tion with FEM then leads to cr = 5.528 kN/cm2, a value which is 27% larger than for the simple support. An adequate structural safety can now be verified according to DIN 18800 Part 3:

p36 1.939

5.528 3(10.65)

0.84 0.4331.939

(10.66)

8.05 8.05 0.984 18.1820.433 36 3 1.1 (10.67)


10.12.4 Web Plate with High Bending Stresses

The example of Section 10.12.3 is continued here and the bending stresses within the field range of the composite bridge are examined. At the upper edge of the web, axial compression stresses occur, and at the lower edge, due to the bending moment and the asymmetric cross section of the bridge, high tensile stresses occur – see Figure 10.28.

Figure 10.28 Web plate with high bending stress

An FE calculation with a restrained upper edge as described in Section 10.12.3 leads to cr = 4.1614 and 1P,i = 41.614 kN/cm2. The corresponding eigenmode shows four waves in longitudinal direction, which are concentrated in the upper part of the buckling panel due to the compression stress. Whether the existing loadings can be carried, will not be discussed here. With this example priority is given to the evalua-tion of the calculation results.

A computer program could possibly determine cr = 0.6352 as a result of the calculation. The minus sign of the critical load factor can easily be missed. If it is detected, an inexperienced user could be irritated and might conclude that the program had calculated incorrectly. This view could be strengthened with the consideration of the corresponding buckling shape. Where the program provides and displays the eigenmode, it could have two waves in the longitudinal direction being strongly developed in the lower area of the buckling panel. This described case can occur if the computer program uses the inverse vector iteration as solution method, since it determines the lowest absolute eigenvalue. In the example considered here, the first negative eigenvalue of cr = 0.6352 is located closer to the origin of the iteration with = 1 than the first positive eigenvalue with cr = (+)4.1614. Using the inverse vector iteration, the first positive eigenvalue can only be determined, if, as described in the Section 10.11 and 6.2.4, a spectral displacement close to the first positive eigenvalue is carried out. This should of course be done by the program and it should be ensured by corresponding checks (point 4 in Section 6.2.5). However, we can also input higher stresses and cause a spectral displacement. If stresses three


times as high as those in Figure 10.28 are entered, we are closer to the first positiveeigenvalue than to the first negative one.

11 FEM for Cross Sections

11.1 Tasks

For the calculation of beams and frames, the finite element method has become generally accepted. Even for the examination of simple, statically determinate systems FEM is used since it provides the state variables for the verification of a static calculation with little effort quickly and safely. The method has developed into the standard method in this field, so that it seems likely to be used for other problems such as the analysis of cross sections as well, which is addressed in this chapter.

To define an appropriate FEM application area for cross sections, it is in the first in-stance reasonable to classify the cross section shapes occurring in steel structures. As shown in Figure 11.1, the following classification can be made:

a) thin-walled, open cross sections b) thin-walled, closed cross sections c) arbitrary thick-walled cross sections

Figure 11.1 Classification of cross sections

Cross sections are called thin-walled if through a reduction to the profile centre line and the application of simplified theories sufficiently exact calculation results are obtained – see Figures 11.2 and 3.17. The cross sections predominantly consist of rectangular sections, whereas a distinction is made between open and closed cross


11 FEM for Cross Sections 462

sections. From a proportion of approx. / t 10 plates are classified as thin-walled, so that steel cross sections with the usual plate dimensions are generally considered as thin-walled.

Figure 11.2 Example of the reduction of a cross section to the profile centre line (cross section of a hanging railway)

In exceptional cases it is necessary to analyse cross sections with a more precise and more complex theory. This is the case for the category of arbitrary thick-walled cross sections, such as solid sections or rolled sections, if precise solutions are needed.

The analysis of cross sections is connected to a variety of tasks (see Chapters 2, 7 and 8). Table 11.1 gives an overview with reference to the application of FEM. Here, it must be pointed out that in structural engineering and industrial construction the plastic bearing capacity of the cross sections is of particular interest and also, in bridge building the focus is on the stress analysis for large multi-part cross sections.

Table 11.1 clearly shows that for thin-walled, open cross sections (Figure 11.1.a), the use of FEM may not be needed. For thin-walled hollow sections (Figure 11.1b) a statically indeterminate problem emerges for calculation of the standardised warping ordinate and for the shear stresses. The solution involves a relatively large effort (see Sections 2.5 2.7), so that here the use of FEM is advisable. For thick-walled cross sections (Figure 11.1c), analytical solutions for the determination of , xy and

xz only exist for a few basic shapes such as rectangles, equilateral triangles and ellipses. Therefore, numerical methods like FEM are inevitable for such cross sections.

Incidentally, with the method of the finite elements, thin-walled cross sections especially can be analysed as fast and safely as can beams and frames. For this reason, it can be assumed that in this application field, FEM will also prevail as the standard method even for simple cross sections.

11.1 Tasks 463

Table 11.1 Tasks for the analysis of cross sections

Task Calculation with or without FEM?

Standardised reference systems

a) Position of the centre of gravity S, principal axes y and z

FEM not necessary, calculation with FE modelling is advisable

b) Position of the shear centre M, standardised warping ordinate

FEM advisable / necessary for closed and thick-walled cross sections

Cross section properties

a) Area A, Principal moment of inertia Iy and Iz

FEM not necessary, calculation with FE modelling advisable

b) Warping constant I ,Torsion constant IT

FEM advisable / necessary for the calcula-tion of the required warping ordinate

Stresses

a) x due to N, My and Mz FEM not necessary, calculation with FE modelling is advisable

b) x due to M FEM advisable / necessary for the calcula-tion of the required warping ordinate

c) due to Vy, Vz, Mxp and Mxs FEM advisable / necessary for closed and thick-walled cross sections

Plastic bearing capacity of the cross section see Chapter 8 and [25]

For the calculation of cross section properties according to point in Table 11.1, standardised reference systems relating to point are required, as explained in detail in Chapter 2. Unless they do not result from the symmetry conditions, they have to be determined arithmetically. For reasons of clarity it is divided into two parts, as shown in Chapter 2:

Part I: Determination of the y-z principal system

position of the centre of gravity S (yS, zS)direction of the principal axes (angle )ordinates in the y-z principal system cross section properties A, Iy, Iz

Part II: Determination of the main system

position of the shear centre M (yM, zM)standardised warping ordinate cross section properties I , IT


If the standardisation is regarded with reference to the application of the method of finite elements, it can be shown that an analysis based on FEM is not required for the first part. For all cross section shapes, the position of the centre of gravity, the direc-tion of the principal axes as well as the principal moments of inertia can be unproblematically determined without FEM as explained in Section 2.3. For this reason, in the following sections it is always assumed that part I of the standardisa-tion has already been completed.

Note: The standardisation parts I and II is the content of Sections 2.3 and 2.5. InTables 2.3 and 2.15, the corresponding calculations are summarised. In principle, it certainly suggests itself to use the division of a cross section into elements con-nected to FEM not only for standardisation part II, but also for the determination of the y-z principal system. To do so, the integrations in Table 2.3 have to be carried out, which can be done for thin-walled cross sections on the basis of Table 11.2. For the calculation of the principal moments of inertia Iy and Iz using division into elements with FEM, the formulation for the bending resistance I according to Eq. (11.30) can be adapted accordingly. This can be done without any difficulty, so that going more into detail is not necessary. The same goes for arbitrary, thick-walled cross sections. Carrying out standardisation part I as well as for the following calculation of the cross section properties, is explained in Table 11.6 and Eq. (11.61). The adaptation to the integration of the first part of the standardisation is possible without any problems.

11.2 Principle of Virtual Work

According to Section 3.4.2, the condition

W = Wext + Wint = 0 (11.1)

is the general requirement for an equilibrium. This condition was used in Chapters 3 and 4 for the derivation of beam elements and it is also to be used for the cross sec-tion elements. The aim is the formulation of stiffness relationships for cross section elements which comply with Eq. (4.1) for the linear beam theory:

e e e es K v p (11.2)

At first, only the internal virtual work is considered but later the external virtual work is added in connection with the cross section elements.

According to [25], the internal virtual work under consideration of axial stress x and shear stresses xy as well as xz for beams is

int x x xy xy xz xzx A

W dA dx (11.3)


Warping ordinate With the warping ordinate, also called the warping function, the displacements u in the longitudinal direction of a beam as a result of torsional stress are described. According to Eq. (2.57),

u(x, y,z) (y,z) (x) , (11.4)

that is to say for a derivative of the angle of twist 1, then u = , so that the warping is equal to the longitudinal displacements of the cross sections. This warping belongs to the primary torsion, because all other shear stresses have been neglected for the beam theory. The corresponding virtual work is compiled in Table 3.2 and the primary torsional moment Mxp = GIT is given in Table 3.3. Figure 11.3 shows the warping for a rectangular solid cross section and of a rolled I-profile.

u1 u

Figure 11.3 Warping as a result of primary torsion

By defintion, no axial stresses occur for the primary torsion, but exclusively shear stresses. Thus, in Eq. (11.3), x can be set to zero. For the shear equations (1.4d) and (1.4e)

xy Mu v z zy x y

(11.5a)

xz Mu w y yz x z

(11.5b)

are used. Equations (11.5) already contain translations with Eqs (1.1) (1.3), so that the terms on the right side (with the square brackets) are valid for the beam theory. Since the virtual work is to be formulated with reference to the warping ordinate, we set 1 and the virtual shearing strains result as follows:

xy( )

y(11.6a)


xz( )

z(11.6b)

With the constitutive equations, (1.6) and (1.7), the shear stresses are replaced by the shearing strains and we obtain:

xy xyG (11.7a)

xz xzG (11.7b)

With x = 0, Eqs (11.5) (11.7) as well as 1, Eq. (11.3) can be transformed and the following formulation for the internal virtual work results:

e

int M M eA

( ) ( )W G z z y y dAy y z z (11.8)

Compared to Eq. (11.3), Eq. (11.8) was reduced to the cross section plane, because the warping of a cross section is to be calculated. The index “e” refers to a cross section element. In Sections 11.3 and 11.4 Eq. (11.8) is used as a starting point for the derivation of finite cross section elements.

Shear deformations u as a result of shear force and secondary torsion The internal virtual work in Eq. (11.3) is to be transformed so that the connection with the internal forces Vy, Vz and Mxs and the corresponding deformations in the cross section plane is established. With that, the finite cross section elements are to be derived, which enable the calculation of deformations due to Vy, Vz and Mxs with the help of FEM. However, the ultimate objective is the following determination of the shear stresses.

Figure 11.4 Connection between the stresses x and and the corresponding internal forces


Figure 11.5 shows the displacements u for a rectangular solid cross section and a rolled I-profile determined with FEM. With the courses shown, the shear stresses can be calculated; this is shown in detail in Sections 11.3 and 11.4.

The first term in Eq. (11.3) is transformed with the help of an integration by parts.With Eq. (1.4a), x u and this gives:

xx x xx 0

x A A xu dA dx u u dx dA (11.9)

The transformation is necessary because here it is not the axial stresses x that are to be calculated but the shear stresses . Figure 11.4 shows the familiar connection between the stresses and the internal forces. The derivation of the axial force x is therefore equal to – / s.

Figure 11.5 Shear deformations u due to shear forces and secondary torsion

In Eq. (11.3), the terms are now treated which contain shear stresses xy and xz. Still, the constitutive equations xy = G xy and xz = G xz are valid – see Eq. (11.7). However, the shearing strains in the formulation of Eq. (11.5) cannot be used here


because they are based on constitutive equations of the beam theory. Since the shear deformations due to Vy, Vz and Mxs given in Figure 11.5 are not considered, a corresponding addition is required. With the help of Figure 11.6, it is obvious what significance the individual terms in the basis equations have.

xyu vy x (11.10a)

xzu wz x

(11.10b)

The example of shearing strains xz shows that the shear stresses xz in the z-direction belong to w/ x. Since the two stress components are equal ( zx = xz), it is sufficient to consider only one of them. Thus, in Eqs (11.10) only the first term is considered each time, which links the shear stresses in the x-direction with the corresponding displacements u:

yx xyuGy

(11.11a)

zx xzuGz

(11.11b)

Figure 11.6 Shear stresses zx and xz and corresponding shearing strains

With x according to Figure 11.4 and Eqs (11.10) and (11.11), the following formulation for the internal virtual work emerges from Eq. (11.3):

e

y xszint e

z yA

V u uMV u uW u y z G dAI I I y y z z

(11.12)

Compared to Eq. (11.3), Eq. (11.12) has been reduced to the cross section plane, be-cause the displacements due to Vy, Vz and Mxs are to be calculated. For this reason the boundary condition in Eq. (11.9) has not been considered and x is set to zero. The subscript “e” refers to a cross section element. Equation (11.12) is used in

11.3 One-Dimensional Elements for Thin-Walled Cross Sections 469

Sections 11.3 and 11.4 as the starting point for the derivation of finite cross section elements and after that the calculation of the shear stresses is covered.

Note: The virtual work for beam cross sections is also derived in [54] and [55], but with a slightly different approach.

11.3 One-Dimensional Elements for Thin-Walled Cross Sections

11.3.1 Virtual work

As shown by Figures 2.12, 2.23, 11.1 and 11.2, thin-walled cross sections can be reduced to their profile centre line and described with the help of the profile ordinates. Usually, cross sections consist of plane plates with a constant plate thickness t in sections.

Figure 11.7 Shear stresses xs in a thin-walled cross section

According to Figure 11.7, shear stresses xs, which are constant across t, result in the individual plates due to an acting of Vy, Vz and Mxs. Due to Mxp, linearly varying shear stresses xs emerge in the through-thickness direction and also shear stresses perpendicular to xs. Additional information on this is given in Figure 11.13 and Chapter 7.

Moreover, constant shear stresses xs due to Mxp occur in the hollow cells of cross sections over t. Figure 11.8 shows the shear stresses in two cross sections and the corresponding warping u = – of the profile centre line.

For the formulation of the virtual work, the cross section element depicted in Figure 11.9 is considered. Compared to the y-direction, its profile ordinate s has the angle and, according Figure 11.6, in the nodes a and b the shear stresses cut free ( sxa and

sxb) act in the x-direction.


Figure 11.8 Shear stresses xs due to Mxp in thin-walled cross sections and warping of the profile centre line

They are combined in the shear flows Tsxa and Tsxb (T = t). With that, the external virtual work for the cross section element in Figure 11.9 results to

Wext = ub Tsxb – ua Tsxa (11.13)

Figure 11.9 Cross section element with two nodes and the boundary shear flows Tsxa and Tsxb

Warping ordinate To formulate the virtual work for thin-walled cross section elements, we act on the assumption of Eq. (11.8). In doing so, shear stresses xs and corresponding shearing strains xs in the direction of the profile ordinate s are considered. With a similar ap-proach as in Section 11.2, the following equations result:

u(x,s) = – (s) (x) (11.14)


xs tv cos w sinu r

s x s(11.15)

with: rt = (y – yM) sin – (z –zM) cos (11.16)

xs xsG (11.17)

For 1, we have

xs trs(11.18)

and

xs s(11.19)

With these formulas, the virtual work for thin-walled cross section elements can be formulated and the following condition for the equilibrium at the element (in the x-direction) results:

* *b sxb a sxa t

sW T T G t r ds 0

s s(11.20)

The shear flows of the external virtual work are marked with an asterisk (*). This highlighs the fact that these are shear flows corresponding to 1 acting at .

Shear force and secondary torsion

For the shear stresses due to shear force and secondary torsion, the following relation for thin-walled cross sections is valid in accordance with Eq. (11.11):

xsuGs

(11.21)

Here, the shear stresses xs as well as the corresponding shear deformations u are as-sumed to be constant across the plate thickness. By an adjustment of the virtual work according to Eq. (11.12) and the consideration of the external work according to Eq. (11.13), and with Condition (11.1), the virtual work results to:

b sxb a sxa

y xsz

z ys

W u T u T

V MV ( u) uu y z G t ds 0I I I s s

(11.22)

This allows determining the shear deformations u due to shear force and secondary torsion.


11.3.2 Element Stiffness Relationships

Section 3.5.5 showed which displacement approaches precisely describe the deformations in the longitudinal direction of the beam for primary torsion as well as shear force and secondary torsion. In this chapter, these displacement approaches are introduced into the virtual work formulations of Section 11.3.1 and the stiffness relationships for a finite thin-walled cross section element are the result. In Figure 11.9, the element is shown for which the local element coordinate is introduced in Figure 11.10. Furthermore, the nodal displacements and the shear flows at the element ends for the warping ordinate as well as the shear deformations u are de-fined. A constant plate thickness t is assumed in the element.

Figure 11.10 Cross section element with degrees of freedom and boundary shear flows

First of all, for the determination of the standardised warping ordinate the devel-opment of the element relationships is addressed below. Afterwards, the corres-ponding explanations for the determination of the shear deformations u due to shear force and secondary torsion follow. Note that the displacement approaches are formulated using the local ordinate of the element, which is why the derivations of the displacements according s have to be differentiated with the chain rule. For the integration, the differential of the profile ordinate ds therefore has to be transformed into d .

Warping ordinate

For the determination of the warping ordinate , the virtual work according to Eq. (11.20) is considered. Using the local coordinate , the following relationship results:


b sxb a sxa1 1

t1 1

W T T

2 G t ( ) ( )d G t r d 0(11.23)

From Section 3.5.5, the deformations can be described precisely with the linear approach using Eq. (3.63). If it is introduced into Eq. (11.23) and the integrations are performed, the following stiffness relationship results:

a asxat

b bsxb

: 1 1 1T G t G t r: 1 1 1T

e ee e e: t K f

(11.24)

Equation (11.24) is a stiffness relationship for cross section elements comparable to (11.2). The values have the following meaning:

et vector of the shear flows in the nodes for 1

eK element stiffness matrix

e vector of the warping ordinates in the nodes

ef “load vector” for 1

The warping ordinate and the distance rt in Eq. (11.24) refer to the shear centre M. As already explained in Sections 2.5 and 2.6, their position is often unknown at first. Therefore, warping ordinates and distances tr are considered initially in Eq. (11.24), and these refer to an arbitrary centre of rotation D – see Sections 2.6 and 7.3.4.

Shear deformations u due to shear force and secondary torsion

For the determination of the stiffness relationship, the virtual work is used according to Eq. (11.22). In terms of the local coordinate , it turns out to be:

1

b sxb a sxa1

1y xsz

z y1

2 G t ( u) uW u T u T d

V MVt u y z d 02 I I I

(11.25)

From Section 3.5.5 it is clear that the shear deformations u can be represented accurately with the help of a cubic displacement approach. If the course of the ordi-nates y, z and is described with the local element coordinate and the interpolation polynomial is applied according to Table 3.4 on the right, the stiffness relationship of an element with four nodes results. Through a static liquefaction, the element with


four nodes can be transformed into an element with two nodes. According to [54], the following stiffness relationship is gained for the element with two nodes:

1 1a ba sxa a 3 6

1 1a bb sxb b 6 3

F Fu : T u1 1G t tF Fu : T u1 1

e ee e eu : t K u f

with: y xsza a a a

y z

V MVF z yI I I

y xszb b b b

y z

V MVF z yI I I

(11.26)

If required, the matrices of the equivalent element with four nodes, which is not shown explicitly here, can be taken from [54]. In Eq. (11.26), the shear flows at the element ends are linked to the corresponding shear deformations through the stiffness matrix Ke. The load vector fe arises from the derivation of the axial stresses x (s) ,which have been converted into equivalent “nodal loads” with the help of the shape functions f.

11.3.3 Equation Systems

For a thin-walled cross section, which commonly consists of several plates, each of the rectangular parts of the cross section can be represented by a single element. At every point where parts are connected to each other, as well as at all end points of the parts, there has to be a node – see Figure 11.12. As shown in Section 4.3, for beams, the equilibrium at the nodes is formulated, whereas for the cross section elements given here, the equilibrium of the shear flows Tsx is considered, see Figure 11.11. For example, with the stiffness relationship (11.26), the virtual work at node k is

i i i 1 i 1k k k 1 k k 1

i i i 1 i 1

t t t tW u G u u u (11.27)

It becomes obvious that the stiffness ratios of the elements i and i+1 which corre-spond to node k superpose with each other through addition. If the corresponding equilibrium relationships are established for every node, the equation system for the total cross section results. With the element stiffness relationship (11.24) for the determination of the warping ordinate and relationship (11.26) for the shear deformation due to shear force and secondary torsion, two equation systems arise:

K f and K u f (11.28a, b)


Since both problems show identical element stiffness matrices Ke, the total stiffness matrix K of a cross section only needs to be set up once and can then be used for both equation systems. The formulation of the equation systems from the element relations is conducted as for beams, which is described in detail in Section 4.5.

Figure 11.11 Nodal equilibrium at node k

The equation systems evolved (11.28) cannot be solved for the time being since they are singular due to a missing support of the cross section. For this reason, an arbitrary node i has to be held nondisplaceably in the u-direction:

i 0 or iu 0 (11.29a, b)

These boundary conditions are included in the equation systems – see Section 4.5.4. Although the solution vector of the warping ordinates or the shear deformations, re-spectively, do not contain the exact node displacements then, the values being determined only differ from the actual deformations by a rigid-body ratio.

Figure 11.12 Example of the discretisation of thin-walled cross sections


11.3.4 Calculation of Cross Section Properties and Stresses

After the equation systems (11.28) have been solved, the deformations are available in the elements’ nodes. With them, the cross section properties (standardisation part II) and the shear stress distributions can be determined.

Cross section properties and standardisation part II

In consideration of tr in the element load vectors ef , the warping ordinate results as the solution of the equation system (11.28a). For the solution, an arbitrary node is fixed, as a result of which the ordinates with reference to the centre of rotation D contain a rigid-body ratio complying with the integration constant k, cf. Section 2.5. According to Table 2.15, the warping ordinate must be standardised. For that purpose, the corresponding calculation formulas for the cross section element of Fig-ure 11.9 are compiled in Table 11.2.

Table 11.2 Determination of A , yA and zA according to [25], [54]

en

a b ej 1 j

1A A2

en

y a b a a b b ej 1 j

1A 2 y y y 2 y A6

en

z a b a a b b ej 1 j

1A 2 z z z 2 z A6

ne: Number of cross section elements

After having calculated the position of the shear centre with Table 2.15 and when the standardised warping ordinate is available in the element nodes, the warping con-stant of the cross section can be calculated as follows:

en2 2 2

a a b bj 1 jA

tI dA3 (11.30)

Note that, for the torsion constant IT the finite element does not capture the shear stresses as a result of the impact of the individual plates which are linearly distributed across the plate thickness. Only the torsion constant of closed cross sections, which is here identified by IT,closed, is obtained with the warpings being assumed to be constant across the plate thickness. According to [54], the following conditional equation results for cross sections consisting of linear elements:

en2

T,closed t t t t a b jj 1s

I r r t ds r t rs (11.31)

The ratio of rectangular cross section parts at the torsion constant IT has to be deter-mined via a separate calculation – also see Figure 11.7 on the right. To do so, the


approximate solution according to [41] shown in Figure 11.13 can be used. Here, it is often sufficiently accurate to assume the factor 1 with 1.0 for plate dimensions t / commonly occurring in steel structures. If the ratio of individual plates is identi-fied by IT,open, the following is valid for ne elements:

en31

T,open 13 jj 1I t (11.32)

The total torsional constant is obtained by the superposition of the two portions:

T T,open T,closedI I I (11.33)

Figure 11.13 Shear stresses due to primary torsion and IT for rectangular cross sections according to [41]

Shear stresses due to primary torsion

As already shown for the torsion constant, only the shear stresses that are evenly distributed across the plate thickness, which emerge for closed cross section shapes, can be calculated with the help of warping ordinates of the profile centre line. With Eq. (11.15), the following is valid for shear stresses:

txs rs

G (11.34)

With the help of the basis equation of the primary torsion according to Table 3.3, the derivative of the angle of twist is linked to the torsional moment by the following relationship:


xp

T

MG I

(11.35)

With the help of Eqs (11.34) and (11.35), a conditional equation for the shear stresses in the plates of hollow cells results, which are designated by xs,closed:

xpa bxs,closed t

T

M r

I (11.36)

Equation (11.36) can be used for all plates of a cross section, independent of whether they are part of a hollow cell. When using it for “open” cross section parts with Eq. (11.36), xs,closed = 0 results.

The maximum stresses due to the action of the individual rectangular parts can be determined with Figure 11.13. The stresses xs in the direction of the profile centre line of a plate comply with the xy of the figure. Consequently, the following is valid for the “open” ratio, which is identified by xs,open:

xpxs,open 2

T

Mt

I (11.37)

For plate dimensions occurring in steel structures, 2 = 1.0 is frequently valid (cf. Figure 11.13). The total shear stresses result from the superposition of the ratios

xs,open and xs,closed.

Shear stresses due to shear force and secondary torsion

The shear stresses due to shear force and secondary torsion are calculated from the corresponding shear deformations which can be determined with the help of equation system (11.28b). As a result, the displacements in the element nodes are available, so that through their insertion into the element stiffness relationship (11.26), the shear flow in the nodes can be determined. Here, the rigid-body ratio in the deformations due to the boundary condition (11.29b) does affect the calculation. The following re-lationships result for the shear stresses in the element nodes:

xsa b a a bG u u 2 F F

6(11.38a)

xsb b a a bG u u F 2 F

6(7.38b)

In order to determine the shear stresses inside the element as well, Eq. (11.21) is used. If the deformations in Eq. (11.21) are described by the cubic displacement ap-proach, a relationship can be derived according to [54] with which the shear stresses at an arbitrary point of the finite element with two nodes can be determined:


2 2xs b a a b

G ( ) u u F 1 6 3 F 1 6 324

(11.39)

The nodal displacements ua and ub, determined according to Section 11.3.3 are in-serted into equations (11.38) and (11.39). The values Fa and Fb are defined in Eq. (11.26).

11.3.5 Compilation

To make the determination of the deformations with FEM and the following identification of cross section properties and shear stresses easier, the required calculation steps are compiled in Table 11.3. The approach is based on the general calculation procedure described in Section 11.5. A detailed calculation example can be found in Section 11.6.2.

Table 11.3 Overview of the calculation for thin-walled cross sections

Calculation of the warping ordinate :Equation system (boundary condition: i 0 ): K f Eq. (11.28a)

Element matrices: e1 1G tK1 1

, e t1

f G t r1

Eq. (11.24)

with: t a b b a1r y z y z for D = S

Position of the shear centre and standardisation of the warping ordinate:

Tables 2.15 and 11.2: yzM D M D

y z

AAy y ; z z

I I

k A A

k M D M Dz z y y y z

Calculation of the cross section properties I and IT:en

a a b bj 1 j

2 2tI3 Eq. (11.30)

T T,open T,closedI I I Eq. (11.33)

en

T,openj 1 j

31

1I t3

,en

T,closed a bj 1

t t jI r t r Eqs (11.31),

and (11.32)

1 see Figure 11.13, yzt,j a M a M

j

r y y z z


Table 11.3 continuation

Calculation of shear deformations due to shear forces and secondary torsion:Equation system (boundary condition: iu 0 ): K u f Eq. (11.28b)

Element load vector: 1 13 6a b

e 1 16 3a b

F Ff t

F F with: y xsz

i i i iy z

V MVF z y

I I IEq. (11.26)

Note: The stiffness matrix is identical to the one for the warping ordinate (see point )and has to be calculated only once for a cross section. Calculation of shear stresses xs:Shear stresses due to primary torsion:

linearly via t with: xpxs,open 2

T

M t

I ( 2 see Figure 11.13) Eq. (11.37)

constantly via t: xpa bxs,closed t

T

M r

I (with: rt see above) Eq. (11.36)

Shear stresses due to shear forces and secondary torsion: 2

axs b a 2

b

F 1 6 3G ( ) u u24 F 1 6 3

Eq. (11.39)

11.4 Two-Dimensional Elements for Thick-Walled Cross Sections


In the following sections the FE formulation for two-dimensional elements is treated. With these, thick-walled cross section shapes can be handeled. Since several theoretical basics have already been presented in connection with thin-walled cross sections in Section 11.3 and these are likewise valid for the arbitrary cross sections regarded here, this section is based on the explanations of the previous one.

Figure 11.14 shows different oblique elements where the boundary is curvilinear, to be used for the analysis of cross sections. For the development of corresponding stiff-ness relationships, the displacements u are approximated by means of appropriate approaches. However, the exact course of the warpings or shear deformations of thick-walled cross sections is generally not known, so that the accuracy of the dis-placement approaches in Section 3.5.6 cannot be ascertained as being universally valid. It is obvious that the FE calculation can only provide an approximate solution for the use of two-dimensional elements, which converges towards the accurate so-

11.4 Two-Dimensional Elements for Thick-Walled Cross Sections 481

lution when refining the FE model. If corresponding experience is missing, several calculations are always required for a cross section in order to be able to estimate the quality of the attained solution and the convergence behaviour. Besides the displace-ment approach, further influences arise which affect the results of the FE calculation. This is covered in the following sections.

Figure 11.14 Examples of oblique elements and elements with curvilinear boundaries

Moreover, for the elements where the boundaries are curvilinear, there is the problem that as well as the displacements the element geometry also has to be represented or described properly. To do so, the shape functions in Section 3.5.6 can be used. If the same functions are used for the description of the displacements as well as for the representation of the element geometry, one speaks about an isoparametric element formulation. This concept is realised for the development of the elements in this section. In Section 3.5.6, the steadiness of Langrange’s interpolation polynomials with reference to the displacement approaches has already been discussed and it was shown that steadiness is not violated. Besides the steadiness, a finite element must also meet presentability demands to ensure that the numeric results converge towards the exact solution for an increasingly fine elementing of the cross section. However, the element must be capable of representing rigid-body displacements, i.e. it must be displaceable without generating improper strains. It also has to be ensured that the element is capable of accurately capturing constant states of strain. With the isoparametric element formulation, these demands are always met – see, for example, [48].

Note: Besides the isoparametric representation we speak of a subparametric rep-resentation if the description of the geometry is supported by fewer nodes than the one for the displacement approach. A convenient field of application could here be rectangular or oblique elements whose geometry can be accurately represented with the help of bilinearly varying basis functions and for which a quartic or bicubic approach is more suitable for the description of deformations. For this pro-ceedure, however, steadiness and presentability demands have to be checked in


detail. It is generally not thought good to use a superparametric representation where the description of the geometry is supported by more nodes than the one for the displacements, because, as shown in [48], presentability demands are usually violated.

The formulation of two-dimensional oblique finite cross section elements or elements with curvilinear boundaries is carried out in a general way, so that the shown rela-tionships are valid for all of the elements. Via the consideration of the bilinear or quartic shape functions of Table 3.5, the element in Figure 11.14 with four or nine nodes, respectively, arises. However, higher approaches can also be chosen, so that, for example, the element with 16 nodes results in a quartic approach. Theoretically, boundary node elements like the element with 8 or 12 nodes could be generated with corresponding shape functions (cf. Figure 11.15), but this will not be addressed in this book. As a sort of special case, quadratic or rectangular elements are also certainly captured with the element formulations of this section, and this will be responded to in Section 11.4.9.

Figure 11.15 Boundary node elements

11.4.2 Virtual Work for Thick-Walled Cross Section Elements

By considering a two-dimensional cross section element, shear stresses yx and zx or x and x, respectively, emerge on the element boundaries, which are combined in

the element nodes as equivalent shear flows Tx. These are shown for an element with four nodes in Figure 11.16. Compared to thin-walled cross sections, they are, to simplify matters, marked as positive in the x-direction, without the consideration of the actual direction of action, which is comparable to the sign definition II for beams. The external virtual work performed by the shear flows for an element with n nodes is

n

ext i xii 1

W u T (11.40)


Figure 11.16 Element with four nodes and node shear flows T

The internal virtual work has been formulated in Section 7.2. For the warping ordi-nate the following relationship arises with Eqs (11.8) and (11.40) and Condition (11.1):

e

e

n

i xi ei 1 A

M M eA

( ) ( )W T G dAz z y y

( ) ( )G y y z z dA 0z y

(11.41)

Here, all the ratios depending on the warping ordinate have been summed up. As for the thin-walled cross sections, the shear flows T are marked by a * to clarify that they correspond to the derivative of the angle of twist 1.

For the load due to shear force and secondary torsion, the following virtual work comes from Eqs (11.12), (11.40) and (11.1):

e

e

n

i xi ei 1 A

y xsze

z yA

( u) u ( u) uW u T G dAz z y y

V MVu y z dA 0I I I

(11.42)


11.4.3 Element Geometry

The element with nine nodes shown in Figure 11.17 is regarded, the number of the nodes is chosen as an example and the element could be replaced by another one, cf. Figure 11.14. Figure 11.17 shows an element with curvilinear boundaries and in the picture on the right, the element is depicted in the form of a uniform square with the local - -system for which shape functions for the description of the displacements in Section 3.5.6. are given.

Figure 11.17 Two-dimensional element with curvilinear boundaries

The local - -system of the uniform square becomes an element-specific coordinatesystem in the element with curvilinear boundaries for which the element boundaries coincide with the coordinate lines. With the shape functions and the node coordinates yi and zi, the curvilinear geometry of an element can be described with n nodes in the y-z principal system:

ey( , ) f y and ez( , ) f z (11.43a, b)

with: 1 2 nf f f f

T1 2 ne

y y y y , Te 1 2 nz z z z

(11.44a, b, c)

To put it differently, Equations (11.43) are transformation relationships describing the course of y and z of the element with curvilinear boundaries in the - system of the uniform square. The position of the nodes is captured accurately and the courses be-tween the nodes are captured approximately by the shape functions used – see Figure 11.18.


Figure 11.18 Example of the transformation of the courses of y and z of an element with curvilinear boundaries to the uniform square

For the transformation, a uniqueness of the representation in the uniform square must of course be provided, i.e. every point of the element with curvilinear bounda-ries must exactly comply to one point of the uniform square.

As an example of a nonunique representation, Figure 7.19a shows an element with four nodes which strongly deviates from its rectangular shape due to the position of the fourth node. On closer inspection, it becomes clear that two points in the uniform square correspond to the point P of the oblique element, so that this is a nonunique representation. Such an element is therefore unacceptable. For elements with four nodes, a clear criterion can be specified in this context: A representation is always unique if no interior angle is larger than 180°. For elements of higher grade this is not that easily possible, so that in case of doubt, an optical checking is required. For that purpose, Figure 7.19b shows an example of an element that shows an overlap due to the arrangement of the nodes and for which therefore a unique representation is not possible.


Figure 11.19 Elements without a unique mapping according to [48]

11.4.4 Transformation Relationships

Since the displacement approaches are, from Eq. 3.67, formulated in the - system, the partial derivations with respect to y or z, respectively, occurring in the virtual works (see Section 11.4.2) must be transformed. For the general case of the element with curvilinear boundaries, the following can be formulated according to [4]:

y zy z

y zy z

y zy y

Jy z

z z

(11.45)

The inversion of the relationship of the matrices from Eq. (11.45) gives a transformation relationship for the differentiations / y and / z:

1

z zy 1J

det J y yz

with: y z y zdet J

(11.46)


Here, J is referred to as Jacobian matrix and det(J) as Jacobian determinant. If Eq. (11.43) is considered for the description of the element geometry, we have

e e1 f fz z

y det J(11.47a)

e e1 f fy y

z det J(11.47b)

e ee ef f f fdet J y z y z (11.48)

Also the area differential dA = dx · dy has to be transformed into d · d The direc-tions of the coordinates at a point P of an element with curvilinear boundaries or an oblique element result from the tangents to the coordinate lines, thus, from the deri-vations of the coordinate functions of Eq. (11.43). With the help of these the following tangent vectors can be stated according to Figure 11.20:

y zy zg d e d e (11.49a)

y zy zg d e d e (11.49b)

Figure 11.20 On the determination of the tangent vectors

The surface area and thus the area differential is determined by the cross product of the vectors:

y z y zg g d d d d det J d d dA (11.50)


Figure 11.21 Example of elements with nonuniform and constant Jacobian determinant

It becomes obvious from Eq. (11.50) that the Jacobian determinant det(J) · d · d is a factor describing the size of the surface elements dA and that it is thus a value for the finite surface elements being enclosed by the coordinate lines. This connection is illustrated in Figure 11.21 showing elements with four nodes. In the oblique element, the area elements show different sizes, therefore, the Jacobian determinant is nonuniform. In contrast to that, the coordinate lines span an equidistant grid for the rectangular element. All area elements are of equal size, and the Jacobiandeterminant is constant in the total element.

11.4.5 Stiffness Relationships

In Figure 11.22, a two-dimensional element is shown with the definition of the de-formations of the nodes as well as the nodal shear flows. As for the thin-walled beam cross sections, the element stiffness relationship for the determination of the warping ordinate is developed as a first step. Afterwards, the relationship for the determi-nation of the shear deformations due to shear force and secondary torsion follows.

Figure 11.22 Element with nodal shear flows and nodal degrees of freedom


Warping ordinate

By introducing the displacement approach of Eq. (3.67) into the virtual work according to Eq. (11.41) and describing the element in the - -system with the help of the Equations (11.43), the following stiffness relationship results for a finite element:

e ee e e: t K f (11.51)

T T1 1

e1 1

f f f fK G det J d dz z y y

(11.52)

T T1 1

e eM Me1 1

f ff G f y y f z z det J d dz y

(11.53)

In Eqs (11.52) and (11.53), the transformations for the partial derivative according to Eqs (11.47a) and (11.47b) are to be considered. For the Jacobian determinant, Eq. (11.48) or Eq. (11.59) is valid.

Since the position of the shear centre (yM, zM) is not known, first of all an arbitrary centre of rotation D (yD, zD) is chosen and considered in Eq. (11.51), as was done for the thin-walled cross sections. With that, the warping ordinate, which is for dis-tinction designated by , refers to this centre of rotation.

The integration for the determination of the element stiffness matrix (11.52) and the element load vector (11.53) can generally not be solved analytically and an explicit depiction of the element matrices is not possible. A numeric integration is required; this is treated in Section 11.4.6.

Shear deformations u due to shear force and secondary torsion

The shear deformations due to shear force and secondary torsion are determined with the help of the virtual work according to Eq. (11.42). Comparable to the explanations on the warping ordinate , the following element stiffness relationship results through the introduction of the displacement approach of Eq. (3.67) and the description of the element in the - -system with the help of Eqs (11.43):

e ee e eu : t K u f (11.54)

T T1 1

e1 1

f f f fK G det J d dz z y y

(11.55)

1 1yT xsz

e e eey z1 1

V MVf f f z y det J d dI I I

(11.56)


Equations (11.47) and (11.48) have to be considered here for the transformations and the Jacobian determinant (also see Eq. (11.59)), and, as for the warping ordinate ,the integration has to be carried out numerically – see Section 11.4.6.

Total stiffness relationships and boundary conditions

In principle, the explanations in Section 11.3.3 are also valid for thick-walled cross sections. A cross section is divided into finite elements with the help of an element mesh. Through the formulation of the equilibrium at the element nodes, the equation systems shown in Eq. (11.28) result with the element relationships (11.51) and (11.54). Here, identical total stiffness matrices emerge as well, which is directly shown by the comparison of Eqs (11.52) and (11.55). Through the consideration of the boundary conditions (11.29), the equation systems can be solved, but the corresponding solution deviates from the actual deformations by a rigid-body ratio cf. Section 11.3.3.

11.4.6 Numerical Integration

As already mentioned in the previous section, the integrations occurring in the ele-ment stiffness relationships (11.51) and (11.54) must generally be solved numerically. Here, the integration of a polynomial function g( ) is transformed into a weighted summation. Formally, this turns out to be as follows for the one-dimen-sional case:

n1

P kk 11g( ) d w g( ) (11.57)

At selected points – the so-called integration points P with the position P – the function g is evaluated and weighted by the factor w . Through the summation via the number of integration points n , the weighted function values are afterwards added up – this corresponds to the solution of the integration.

For the realisation of this, different integration methods exist, as for example the Newton Cotes quadrature where the function values are evaluated at equidistantly arranged integration points and provided with corresponding weighting factors. An-other method, the so-called Gauss quadrature, uses optimised integration point positions with corresponding weights. Thus, the equidistant arrangement is no longer considered here, but this is irrelevant anyway for software implementations. Through optimisation, a limited number of integration points are required for the exact solution of an integral as a result of which the effort is reduced. For this reason, Gauss quadrature has become accepted for isoparametric elements and it is used below for the solution of the integrations occurring in the element stiffness relationships (11.51) and (11.54).


For the exact position of the integration points see Table 11.4

Figure 11.23 Position of the integration points for Gauss quadrature

For the element relationships, Eqs (11.52), (11.53), (11.55) and (11.56), it is a matter of the integration of two-dimensional functions depending on and They can be depicted in the following form and transformed into the weighted summation with the help of Eq. (11.57):

n n1 1

,k ,l P,k P,l P,k P,lk 1 l 11 1

g( , ) det J d d w w g( , ) det J( , )

(11.58)

eP,k P,l P,k P,l P,k P,le

eP,k P,l P,k P,le

f fwith: det J( , ) ( , ) y ( , ) z

f f( , ) y ( , ) z(11.59)

Here, n and n indicate the number of integration points for the corresponding coor-dinate direction. For each integration point with the position ( P, P), the function value g is to be calculated and to be weighted by the factors w and w . Furthermore, the Jacobian determinant has to be determined for each integration point, as shown in Eq. (11.59) – also cf. Eq. (11.48). Figure 11.23 shows the position for a different number of integration points, and Table 11.4 indicates the corresponding weighting factors.

Table 11.4 Position of integration points and weighting factors for Gauss quadratureaccording to [2]

Number of points n , n Position of points P, P Weighting factor w , w1 0 2. (15 zeros) 2 ±0.577 350 269 189 626 1. (15 zeros)

3 ±0.774 596 669 241 483 0

0.555 555 555 555 556 0.888 888 888 888 889

4 ±0.861 136 311 594 053 ±0.339 981 043 584 856

0.347 854 845 137 454 0.652 145 154 862 546


The question is how many integration points are required for the different elements. The number necessary for an exact integration depends on the degree of the polynomial that is to be integrated. Gauss quadrature is, with n integration points, capable of exactly integrating a (2n – 1) degree polynomial. A four-node rectangular element with bilinearly nonuniform shape function f and the calculation of the stiff-ness matrix Ke is taken as an example. Equation (11.52) provides:

1 1

e1 1

K G g( , ) det J d d with:T Tf f f fg( , )z z y y

(11.60)

The shape function f of the bilinearly nonuniform approach of Table 3.5 shows the polynomial degree n = a + b = 2 with a · b = 1 · 1. With that and with the trans-formation relationships (11.47), also a second degree polynomial results for the function g in Eq. (11.60). Since the Jacobian determinant is constant for rectangular elements (cf. Section 11.4.4), the degree of the polynomial that is to be integrated for the calculation of Ke is exclusively determined by the function g. For solving the in-ner integral d , two integration points are required since with that, because 2n – 1 = 3, first and second degree polynomials can be integrated exactly. With the performance of the inner integration d , cubic terms emerge for the outer integral, so that the solution of a third degree polynomial again requires two integration points. It becomes obvious, that for an exact integration, 2 × 2 integration points are necessary for the rectangular four-node element.

If now, instead of the rectangular, an oblique four-node element is considered, the integration with 2 × 2 integration points is no longer exact. Since, as shown in Section 11.4.4, the Jacobian determinant is not constant in such cases, it has an influence on the function that is to be integrated, so that higher polynomials occur. Table 11.5 gives an overview of the number of integration points necessary for a reli-able integration. Reliable integration means the minimum number of integration points being required for the exact integration of an rectangular element (det(J) = const.). Although the integration for oblique and for curvilinear elements is not exact with the equal number of integration points, it is usually sufficiently accurate. Here, the natural position of the inner nodes has to be always observed. That means that they have to be arranged midline for the 9-node and in the tripart points for the 16-node element. If the nodes deviate from these positions, they have a disadvantageous influence on the Jacobian determinant and thus on the solution of the integration (see Section 11.4.8).

People talk about a reduced integration if the number of integration points is reduced by one for each coordinate direction. As a result, calculation time decreases corre-spondingly. However, a reduced integration must be used with caution since it can lead to so-called zero-energy eigenmodes. That means that the stiffness matrix has eigenvalues (which can be interpreted as strain energies) which have the value zero, but whose eigenvectors do not represent a rigid-body state of displacement – see [48].


Figure 11.38 shows this with the example of the standardised warping ordinate. In principle, this effect can be prevented for a system of multiple elements only by cor-responding boundary conditions or through connecting the elements with each other, respectively. Nevertheless, generally, only one node is held for the cross sections used for the solution of the equation systems, as a result of which the zero-energy eigenmodes cannot be totally prevented. As a consequence, the solutions are useless.

Table 11.5 Number of integration points for a reliable integration

Element type Number of integration points

4-node element 2 x 2



11.4.7 Cross Section Properties and Stresses

First of all, the warping ordinate resulting from the FE calculation must be standardised with the help of Table 2.15. The integrations to be solved are compiled in Table 11.6 for two-dimensional elements, for which the cross section geometry, as well as the warping ordinate is described using the shape function f. The integration is conducted element by element and afterwards it is totalled via the element number ne of a cross section.

After the standardisation of the warping ordinate, so that is given in the element nodes, the warping constant can be determined. Here, again, the course of the warp-ing ordinate within the element is described with the help of the shape functions:

e 1 1n2

e ej 1A 1 1 j

I dA f f det(J) d d (11.61)


Table 11.6 Determination of A , yA and zA according to [54] or [70]

e 1 1n

ej 1 1 1 j

A f det(J) d d

e 1 1n

ez ej 1 1 1 j

A f z f det(J) d d

e 1 1n

y eej 1 1 1 j

A f y f det(J) d d

The same applies to the torsional constant, which can, according to [54] or [55], be determined by the following equation:

e

T M M M MA

1 1n

M Me e ej 1 1 1

e eM Mej

I y y y y z z z z dAz y

f f y y f y yz

f f z z f z z det(J) d dy

(11.62)

The integrations occurring in Table 11.6 as well as in Eqs (11.61) and (11.62) have to be solved numerically – see Section 11.4.6. Here, the Jacobian determinant det(J) is described with the help of Eqs (11.48) or (11.59), and partial derivations / z and

/ y in Eq. (11.62) have to be transformed using Eqs (11.47).

The shear stresses resulting from primary torsional moments can be determined with the help of Eqs (11.7) and (11.5) or (1.6) and (1.7). With the description of the element geometry and the warping ordinate using the shape functions, the following shear stresses result at the position ( , ) of an element:

xpexy Me

T

Mf( , ) ( , ) f ( , ) z zy I

(11.63a)

xpxz Me e

T

Mf( , ) ( , ) f ( , ) y yz I

(7.63b)

For the calculation of the shear stresses due to shear force and secondary torsion,the shear deformations u of the FE calculation are used. With the help of Eq. (11.11), the following relationships for the calculation of the shear stresses result:


xy ef( , ) G ( , ) uy

(11.64a)

xz ef( , ) G ( , ) uz

(11.64b)

11.4.8 Performance of the Approximate Solutions

As already explained at the beginning of this chapter, using the elements with curvi-linear boundaries can generally only lead to an approximate solution whose result approaches the accurate solution through a refinement of the FE model. The resulting error has several causes for the cross sections considered here. These causes are compiled below.

The first error source is the inaccuracy of the displacement approach. Since the actual deformation behaviour in generally not known beforehand, the chosen poly-nomial degree does not exactly reflect this. For example, for 9-node elements, a quartic course of the deformations is assumed, which usually deviates from the actual course.

Figure 11.24 Error when depicting a rolling fillet with a 9-node element

A further error source results from the inaccuracy of the cross section illustration. The geometry of the elements and thus the geometry of the total cross section is described with the help of the shape functions. Here, it is important that a clear representation of the finite elements is provided, which has already been covered in detail in Section 11.4.3. Through the description with the shape functions, the boundary of the cross section is described by the corresponding polynomial degree. If, for example, the fillet radius of a rolled profile is to be depicted using a 9-node element (circular course), this leads to inaccuracies – see Figure 11.24a. A further aspect in this context is the position of the element nodes. As is customary, we have a natural position with nodes on the centre lines (9-node elements) or nodes in the tripart points (16-node


element). If the nodes deviate from their natural position, the course of the element boundary changes, which is shown for the 9-node element in Figure 11.24b. This often leads to a worse depiction.

As third error source is the inaccuracy of the numeric integration, which has already been treated in Section 11.4.6. From the explanations it can be seen that, for a nonuniform Jacobian determinant and the choice of a number of integration points, as it is used for a reliable integration, the numeric integration is not exact. For this reason, elements should be avoided which show a distinct change of det(J). This is clarified in Figure 11.25. In chart a), a distorted element is shown where the areas between the coordinate lines and with this det(J) change distinctly (cf. hatched areas) in contrast to chart b), which must be avoided. Besides the geometry, also the position of the element nodes has a considerable influence. If they deviate from their natural position, as shown in Figure 11.25c, this has an unfavourable effect on the course of the coordinate lines and thus on the Jacobian determinant. This makes it clear that for the numeric integration, the maximum accuracy is achieved if the element nodes are in their natural position.

Figure 11.25 Influence of the element geometry on the numeric integration

A final aspect is the singularities in the shear stresses. If cross section shapes show re-entrant corners without fillets, singularities are to be expected at these corners.


11.4.9 Special Case: Rectangular Elements

For the isoparametric elements, it is not possible to explicitly determine the element matrices. One exception are rectangular finite elements. In order to avoid the numeric integration for such elements, the element matrices for 4-node and 9-node elements, which were developed in [54], are compiled in the following table. The precondition for the specified relationships is that the element edges coincide with the y-z principal coordinate system – see Figure 11.26.

Figure 11.26 Rectangular element

To make the calculation of the cross section properties easier, i.e. in order to avoid numeric integration, conditional equations are formulated in [54]; these are given in Table 11.9.

Table 11.7 Bilinear 4-node element according to [54]

Stiffness matrix:

2 2y ze

e

GK6 A

2 1 1 2 2 2 1 11 2 2 1 2 2 1 11 2 2 1 1 1 2 22 1 1 2 1 1 2 2

Load vector for the determination of the standardised warping ordinate:

D D

1 1

y z2 2e

3 3

4 4

y zy z

Gf y z12

y zy z

6 2 1 1 2 6 2 2 1 16 1 2 2 1 6 2 2 1 16 1 2 2 1 6 1 1 2 26 2 1 1 2 6 1 1 2 2


Table 11.7 Continuation

Load vector for the displacements due to the shear of shear force and secondary torsion:

1

2ee

3

4

FFA

f36 F

F

4 2 1 22 4 2 11 2 4 22 1 2 4

with: y xszi i i i

y z

V MVF z y

I I I

Table 11.8 Quartic 9-node element according to [54]

Stiffness matrix:

ee

GK90 A

28 7 1 4 14 8 2 32 167 28 4 1 14 32 2 8 161 4 28 7 2 32 14 8 164 1 7 28 2 8 14 32 16

14 14 2 2 112 16 16 16 1288 32 32 8 16 64 16 16 322 2 14 14 16 16 112 16 128

32 8 8 32 16 16 16 64 3216 16 16 16 128 32 128 32 256

2y

32

2 2

28 4 1 7 32 2 8 14 164 28 7 1 32 14 8 2 161 7 28 4 8 14 32 2 167 1 4 28 8 2 32 14 16

32 32 8 8 64 16 16 16 322 14 14 2 16 112 16 16 1288 8 32 32 16 16 64 16

14 14 16 16 16 112 12816 16 16 16 32 1

2z

28 32 128 256

Load vector for the displacements due to the shear of shear force and secondary torsion:

1

2

3

4e

5e

6

7

8

9

FFFF

Af F

900FFFF

16 4 1 4 8 2 2 8 44 16 4 1 8 8 2 2 41 4 16 4 2 8 8 2 44 1 4 16 2 2 8 8 48 8 2 2 64 4 16 4 322 8 8 2 4 64 4 16 322 2 8 8 16 4 64 4 328 2 2 8 4 16 4 64 324 4 4 4 32 32 32 32 256

Fi: see Table 11.7



Load vector for the determination of the standardised warping ordinate:

eGf

180

30 12 3 1 4 6 4 2 16 830 3 12 4 1 6 16 2 4 830 1 4 12 3 2 16 6 4 830 4 1 3 12 2 4 6 16 8

120 6 6 2 2 48 8 16 8 640 4 16 16 4 8 0 8 0 0

120 2 2 6 6 16 8 48 8 640 16 4 4 16 8 0 8 0 00 8 8 8 8 64 0 64 0 0

D

1

2

3

4y

5

6

7

8

9

yyyyyyyyyy

30 12 4 1 3 16 2 4 6 830 4 12 3 1 16 6 4 2 830 1 3 12 4 4 6 16 2 830 3 1 4 12 4 2 16 6 8

0 16 16 4 4 0 8 0 8 0120 2 6 6 2 8 48 8 16 64

0 4 4 16 16 0 8 0 8 0120 6 2 2 6 8 1

D

1

2

3

4z

5

6

7

8

9

zzzzzzzzzz

6 8 48 640 8 8 8 8 0 64 0 64 0

Table 11.9 Calculation of cross section properties with rectangular elements, [54]

Conditional equations

Standardisation of the warping ordinate: (see Table 2.15 )

• Position of the shear centre:

enTe ff e jj 1

M Dy

z Ay y

I,

enT

ff ee jj 1M D

z

y Az z

I

• Transformation constant:

en

e jj 1k

A

A

Warping constant and torsional constant:en

Tffe e jj 1

I A

en e M M M MT T Tj 1 M Mff ff e ff ff e f fe e ee e j

A y y 2 y z z 2 zI

y A A y z A A z y A z A



Auxiliary values for bilinear 4-node elements

1 2 3 4 4 , 1 2 3 4 4y y y y y , 1 2 3 4z z z z z 4

yf 2A 1 1 1 1 , zf 2A 1 1 1 1 ,

eff

4 2 1 22 4 2 1AA

36 1 2 4 22 1 2 4

, yff

2 1 1 21 2 2 1

A12 1 2 2 1

2 1 1 2

, zff

2 2 1 12 2 1 1

A12 1 1 2 2

1 1 2 2

Auxiliary values for quartic 9-node elements

1 2 3 4 5 6 7 8 9 364 16

1 2 3 4 5 6 7 8 9 36y y y y y 4 y y y y 16 y

1 2 3 4 5 6 7 8 9 36z z z z z 4 z z z z 16 z

yf 6A 1 1 1 1 4 0 4 0 0 , zf 6A 1 1 1 1 0 4 0 4 0

eff

16 4 1 4 8 2 2 8 44 16 4 1 8 8 2 2 41 4 16 4 2 8 8 2 44 1 4 16 2 2 8 8 4

AA 8 8 2 2 64 4 16 4 32900

2 8 8 2 4 64 4 16 322 2 8 8 16 4 64 4 328 2 2 8 4 16 4 64 324 4 4 4 32 32 32 32 256

yff

12 3 1 4 6 4 2 16 83 12 4 1 6 16 2 4 81 4 12 3 2 16 6 4 84 1 3 12 2 4 6 16 8

A 6 6 2 2 48 8 16 8 64180

4 16 16 4 8 0 8 0 02 2 6 6 16 8 48 8 64

16 4 4 16 8 0 8 0 08 8 8 8 64 0 64 0 0

11.5 Calculation Procedure 501


zff

12 4 1 3 16 2 4 6 84 12 3 1 16 6 4 2 81 3 12 4 4 6 16 2 83 1 4 12 4 2 16 6 8

A 16 16 4 4 0 8 0 8 0180

2 6 6 2 8 48 8 16 644 4 16 16 0 8 0 8 06 2 2 6 8 16 8 48 648 8 8 8 0 64 0 64 0

11.5 Calculation Procedure

This section gives an overview on the calculation process which is valid for the thin-walled cross sections in Section 11.3 and the thick-walled cross sections of Section 11.4. For the thin-walled cross sections, an overview has already been given with Table 11.3 being based on the following approach:

Preconditions: Cross section geometry, internal forces, y, z, Iy, Iz

The y-z principal coordinate system of the cross section and the associated area moments Iy and Iz are required – see Section 2.3. In addition, it is assumed that the internal forces Vz, Vy, Mxp and Mxs for the determination of the shear stress distribution are known.

Meshing of the cross section In a first step the cross section must be discretised via a suitable division into finite elements.

Calculation of the warping ordinate with FEMUsing the element stiffness matrices and the element load vectors (thin-walled: Eq. (11.24), thick-walled: Eqs (11.51) (11.53)), the total stiffness relationship (11.28a) of a cross section can be set up. After the introduction of the boundary condition (11.29a) by the fixing of an arbitrary node, the equation system K f can be solved.

Position of the shear centre and standardisation of the warping ordinateWith the warping ordinates according to point , the position of the shear centre M and the transformation constant k is determined using Table 2.15 and Table 11.2 (thin-walled) or Table 11.6 (thick-walled).


Calculation of the cross section properties I and IT

After determining the warping ordinate , it can be used for the calculation of the warping constant I and the torsion constant IT. The associated conditional equations are given with the Relationships (11.30) (11.33) for thin-walled and Eq. (11.61) and (11.62) for thick-walled cross sections.

Calculation of shear deformations u due to shear force (Vy, Vz) and secondary torsion (Mxs) with FEM

The total load vector can be set up with the help of the element load vectors (thin-walled: Eq. (11.26), thick-walled: Eqs (11.54) (11.56)). The total stiffness matrix has already been calculated under point , so the equation system (11.28b) results. Through the introduction of the boundary condition (11.29b) (fixing of an arbitrary node), the equation system K u f can be solved and u determined.

Calculation of the stresses Using the standardised warping ordinate according to point , the shear

stress distribution due to a primary torsional moment Mxp can be calculated with Eqs (11.36) and (11.37) for thin-walled and Eq. (11.63) for thick-walled cross sections. Also, the warping ordinate can be used for the calculation of axial stress x due to a warping moment M .

With the deformations u according to point , the shear stresses due to shearforce and secondary torsion (Vy, Vz and Mxs) result from Eq. (11.39) for thin-walled cross sections. For thick-walled cross sections, the shear stresses result with Eq. (11.64).




In these examples, the following values or shear distributions, respectively, are cal-culated:

position of the shear centre, i.e. yM and zMstandardised warping ordinate torsion constant IT and warping constant Ishear stresses due to Vy, Vz, Mxp and Mxs

Since the shear centre is required as starting point for the calculations of beam structures, the determination of its position is a central task of structural engineering. As explained in detail in Section 2.7, there are different methods for doing this. Com-puter-oriented calculations using the warping ordinate as the basis for the determination of the cross section properties is the most suitable method – also see Table 2.15. Furthermore, the standardised warping ordinate is the basis for the determination of the cross section values IT and I as well as for the calculation of shear stresses due to Mxp. Moreover, it is needed for the determination of the shear stresses due to Mxs and for the determination of the corresponding axial stresses

x = M / I · It becomes obvious that the warping ordinate is required for the calculation of several values (see Table 2.17 as well). For this reason, this is repeat-edly covered in the examples. The following calculations have been carried out with the FE programs CSP-FE (two-dimensional, isoparametric cross section elements) and CSP-FE ML (thin-walled cross section elements) – see Section 1.10.

11.6.2 Single-Celled Box Girder Cross Section

In order to clarify the approach for the examination of cross sections with the help of the finite element method, the calculation of the example cross section given in Fig-ure 11.27 is shown in detail in this section, also see [54]. In the following, all the numeric values are expressed in kN and cm. The box girder cross section is exposed to torsional moments Mxp and Mxs as well as to the shear forces Vy and Vz. Due to the given plate dimensions, it can be regarded as thin-walled and discretised using the one-dimensional finite elements. The numbering of the nodes and elements which results from this is presented in Figure 11.27. It is assumed that the position of the centre of gravity, the cross section area as well as the principal moments of inertia given in the picture are known due to a previously conducted calculation.

In the first step, the standardised warping ordinate is determined. To do so, ele-ment relationships are formulated according to point 1 in Table 11.3. They result


Figure 11.27 Single-celled box girder cross section and discretisation

from Eq. (11.24), although the position of the shear centre is not known and the centre of gravity S is chosen as centre of rotation D. For the first element, we get

1sx1

2sx2

1

2

1 1 1T 8100 1,5 8100 1,5 19 30 1 1 1T

405 405 230850

405 405 230850

1 11 1t K f

With the numbering of nodes in Figure 11.27 on the right, the direction of the profile ordinate s is defined since it is always assumed to be from the node with the smaller number to the node with the larger number. The s-direction, i.e. the node numbering, defines the sign of the element load vectors. With 19rt , a negative value results for element 1 since, due to the s-direction from node 1 to node 2, the direction of rotation is in the opposite direction to the definition for (like !). If the formula for the determination of tr given in Table 11.3 is used, which is valid for D = S, this is considered automatically. The stiffness matrices and load vectors of the remaining elements become

163 KKK ,8

12158

12158

12158

1215

2K ,13

243013

243013

243013

2430

54 KK

132 fff ,498150498150

f 6

134677750

134677750

54 11

65191541405,18100ff


Using the element matrices the stiffness relationship according Eq. (11.28a) can be set up for the whole cross section. At first, the emerging equation system is singular.For this reason, node 1 is supported ( 1 = 0), which is acknowledged by crossing out the corresponding row and column in the equation system.:

131798200

131798200

134677750

134677750

6

5

4

3

2

1

137695

132430

137695

132430

132430

10477355

81215

132430

81215

10477355

230850

230850

4050004050000040540500

04050004050000405405

fKAuxiliary calculation:

132430

81215

10477355 405 , 13

243013

7695 405 , 498150134677750

131798200

The solution of the equation system leads to the following warping ordinates: T 0 570 110 460 432.5 27.5 in cm2

The warping ordinate can be calculated with Table 2.15, point 6. In addition, the transformation constant k and the position of the shear centre are determined using the formulas of Table 11.2:

k

2

1,5 30 570 110 460 432.5 27.52 45080 570 110 65 570 432.5 110 27.5

230 cm

M Dy y 0 , 187M D 42z z 4.4524 cm

With that, the standardised warping ordinate results as follows:

230 4.4524 y

2

541.67161.90

161.90 cm

541.67135.71

135.71Figure 11.28 Standardised warping ordinate


With the help of , the warping constant can be determined with Eq. (11.30): 6

2 2 6a a e e

jj 1

tI 12612000 cm3

The torsion constant consists of two parts according to Eq. (11.33). The part of the individual rectangular sections results with Eq. (11.32), where the factor 1 is assumed to be 1.0. The second part is determined with the standardised warping ordinate according to Eq. (11.31):

63 4

T,openjj 1

1I t 337.5 cm3

64

T,closed t t a e jj 1

I r t r 272250 cm

4T T,open T,closedI I I 272587.5 cm

The standardised warping ordinate also serves the calculation of primary shear stresses, which also consist of two parts. Figure 11.29 shows the stress distributions regarding the torsional moment Mxp = 25000 kNcm and shows the application of the associated conditional equations (11.36) and (11.37) for 2 = 1.0.

Figure 11.29 Shear stresses of the primary torsion due to Mxp = 25000 kNcm

For the determination of shear deformations as a result of shear due to shear force and secondary torsion, the element stiffness relationships (11.26) have to be set up. Since the stiffness matrices have already been formulated for the warping ordinate, only the element load vectors arising from Eq. (11.26), which for the loads Vz, Vy and Mxs consist of three vectors each, are quoted below:


a e a e a e

a e a e a e

z z y yy3 6 3 6 3 6xsz

e z z y yy z6 3 6 3 6 3

V MVf tI I I

48375y 7xsz

1 22875y z 14

427.5 1350V MVfI I I427.5 1125

2168000

2168000

xs

z

y

y

z2 I

M800

800IV

11401140

IVf

2287514y xsz

3 48375y z 7

427.5 1125V MVfI I I427.5 1350

31362542y xsz

4 21125y z 3

48.75 1543.75V MVfI I I1023.75 1137.5

31362542y xsz

5 21125y z 3

48.75 1543.75V MVfI I I1023.75 1137.5

71257y xsz

6 7125y z 7

922.5 112.5V MVfI I I922.5 112.5

The total load vector becomes

483757

635007

63500y 7xsz

48375y z 7

16925021

16925021

427.5 13501518.75 3468.751518.75 3468.75V MVf

I I I427.5 13501946.25 12501946.25 1250

with:233550

500IV

y

z ,5040001000

IV

z

y ,12612000

15000I

Mxs

The total stiffness matrix has already been set up for the calculation of the standard-ised warping ordinate and can be used for the calculation of the shear deformations. Analogous to the calculation of the standardised warping ordinate, node 1 is held as boundary condition (u1 = 0), so that the equation system K · u = f can be solved. The displacements u for the individual portions of the load vector are determined sepa-rately. That way, it can be subsequently shown which shear stresses result from the individual internal forces. The equation system provides the following displacements:


3 3 2

3 2 3

z y xs 2 2

2 2 2

2 2

0 0 02.260 10 6.614 10 2.029 102.260 10 5.062 10 1.125 10

u u V u V u M0 5.724 10 2.142 10

2.455 10 2.201 10 2.212 102.455 10 3.523 10 47.028 10

Due to the boundary conditions, these differ by a rigid-body portion from the actual displacements, but this has no influence on the calculation of shear stresses. The stress distributions shown in Figure 11.30 result from the shear deformations with the help of Eq. (11.39). If only the shear stresses in the nodes are of interest, also Eq. (11.38) or the element stiffness relationship (11.26) can be used for their determination.

Figure 11.30 Shear stresses in kN/cm2 due to Vz = 500 kN, Vy = 1000 kN and Mxs = 15000 kNcm

11.6.3 Bridge Cross Section with Trapezium Stiffeners

Figure 11.31 shows an open bridge cross section with trapezium stiffeners. In order to determine the torsional stress for cross sections of this kind resulting from lateral load effects, knowledge of the shear centre is required. This will be determined here taking account of the trapezium stiffeners, which leads to corresponding hollow cells and


with conventional calculation methods to a considerable, error-prone calculating effort. In contrast to that, the calculation can be carried out safely and without problems with FEM. For this purpose, the cross section is discretised with one-dimensional elements according Eq. (11.24), so that the elementing shown in Figure 11.32 results.

Figure 11.31 Bridge cross section

Figure 11.32 FE model of the bridge cross section


With that, the equation system (11.28a) can be set up and the warping ordinate for the cross section can be determined; this has already been shown in detail in the previous example. Using the warping ordinate, the position of the shear centre can be determined according to Table 2.15 and Table 11.2. The corresponding calculation with the program CSP-FE ML leads to the following position of the shear centre in the reference system:

Mz 53.82 cm , My 225 cm

a)

b)

Figure 11.33 Standardised warping ordinate with and without consideration of the stiffeners


With the standardisation the warping ordinate given in Figure 11.33a is obtained. To clarify the influence of the trapezium stiffeners, the cross section is now discre-tised and calculated disregarding the stiffeners. The resulting standardised warping ordinate is given in Figure 11.33b and the shear centre has the following position:

Mz 61.98 cm

It becomes clear that, due to the consideration of the trapezium stiffeners, the shear centre is approx. 8 cm closer to the cover plate. This has a favourable effect on the torsional load due to loads in the y-direction. The difference depends on where the lateral load is acting. For the assumption of the transverse load in the middle of the cross section, as for example the resultant of a wind load, an increase of approx. 6.3% results, neglecting the trapezium stiffeners.

The influence of the trapezium stiffeners on the torsion constant of the bridge cross section is also interesting. While when neglecting the stiffeners, a torsion constant of

4T cm1910I

results, this increases to 4

T cm66265I

when considering the stiffeners. In principle, this value also results if for the IT of the open cross section the neglected portions of the trapezium stiffeners are superim-posed with the help of the Bredtschian formula:

2 24m

T4 A 4 625.5I 1910 6 =1910 6 =66230 cmds 146

t(s)

11.6.4 Rectangular Solid Cross Section

In Section 11.4.8, it was explained that the calculation with two-dimensional finite elements is an approximate solution. In this Section, the convergence behaviour of the solution is to be examined taking the example of a rectangular solid cross section with the lateral dimensions 5 × 10 cm. In doing so, the bilinear 4-node elements as well as the quartic 9-node elements are used. Quadratic or rectangular finite elements are used, whose edges coincide with the principal axes y and z of the cross section. It is discretised in Figure 11.34a with 128 9-node elements (561 degrees of freedom). The standardised warping ordinate resulting from this elementing is shown in Figure 11.34b.


Figure 11.34 Discretisation of a rectangular cross section and standardised warping ordinate

For rectangular solid cross sections, an analytic solution exists for the calculation of the standardised warping ordinate, which results with the help of a so-called stress function, see, for example, [25]. Table 11.10 contrasts the solutions at selected points of the cross section resulting with the help of the finite element method and different discretisations with the analytic solutions and gives the percentaged deviations.

For the rectangular elements used, the numeric integration is exact and the shape of the cross section is also represented exactly. For this reason, the error in the approxi-mation exclusively results from the displacement approach, which captures the cross section warpings imprecisely. Due to the refinement of the elementing, the warping approximates the analytic course, which is shown for the 9-node as well as for the 4-node element in Table 11.10. The predominance of the quartic element in contrast to the bilinear approach is distinguishable and becomes even more obvious with the example of the cross section properties.


Table 11.10 Standardised warping ordinate using FEM and analytic solution

Analytic solution

1 = 4.96728 cm2

2 = 5.73961 cm2

3 = 2.22105 cm2

4 = 1.89578 cm2

Solutions with the FE method

Position 45 degrees of freedom

153 degrees of freedom



y z Error Error Error Error cm cm cm2 % cm2 % cm2 % cm2 %

Bilinear 4-node elements

32 elements 128 elements 512 elements 2048 elements (4 x 8) (8 x 16) (16 x 32) (32 x 64)

1 -2.50 2.50 5.06748 2.02 4.99199 0.497 4.97343 0.124 4.96882 0.0312 -2.50 5.00 5.81555 1.32 5.75845 0.328 5.74431 0.082 5.74079 0.0203 -1.25 2.50 2.28953 3.08 2.23701 0.719 2.22497 0.177 2.22202 0.0444 -1.25 5.00 1.88030 -0.82 1.89450 -0.068 1.89543 -0.018 1.89569 -0.005

Quartic 9-node elements

8 elements 32 elements 128 elements 512 elements (2 x 4) (4 x 8) (8 x 16) (16 x 32)

1 -2.50 2.50 4.95065 -0.33 4.96695 -0.007 4.96728 0.000 4.96728 0.0002 -2.50 5.00 5.73707 -0.04 5.73960 0.000 5.73961 0.000 5.73961 0.0003 -1.25 2.50 2.22901 0.36 2.22106 0.001 2.22105 0.000 2.22105 0.0004 -1.25 5.00 1.86357 -1.70 1.89996 0.221 1.89586 0.004 1.89578 0.000

For this purpose, the convergence behaviour is graphically shown in Figure 11.35 for the torsion constant when using the 4-node and 9-node elements. The analytic solu-tion, against which the FE calculations converge with an increasingly fine discretisation, results to:

IT = 285.852 cm4


Figure 11.35 Torsion constant and convergence behaviour of the FE calculations

This solution is shown in Figure 11.35 with the percentaged deviations of the FE so-lutions. Using 128 (8 × 16) 4-node or 8 (2 × 4) 9-node elements results in solutions with deviations in the ‰-range from the analytic solution. Figure 11.35 shows that when using the 9-node elements with 153 degrees of freedom (32 elements) the de-viation is already significantly smaller than 1‰. For this accuracy, 2048 4-node elements with 2145 degrees of freedom are required.

A similar tendency results for the warping constant I . When using quadratic ele-ments, the elementings shown in Figure 11.36a result. While for the minor discretisation with 15 degrees of freedom the 4-node elements have a better agree-ment with the limit of I = 317.54 cm6, here the deviations of 12.6% and 6.9%, respectively, are unacceptable. For the finer elementing a clear predominance of the quartic approach can be observed. In this context it is also interesting to notice the convergence behaviour with a different elementing, so that rectangular elements re-sult – see Figure 11.36b. Due to the rougher elementing, the standardised warping ordinate is not calculated as accurately. However, with the integration of this warping ordinate, the warping constant is approximated better in the integral middle. Especially with the 4-node element, a comparably good agreement with the exact solution is already achieved with 25 degrees of freedom. Nevertheless, as expected, with finer elementings, the 9-node element converges against the limit in a better manner.


Figure 11.36 Warping constant and convergence behaviour of the FE calculations

If the cross section is subjected to a shear force Vz, shear stresses xz in the z-direction result. It is common that, with this load, the course in the y-direction is constant for the rectangular solid profile, while in the z-direction a parabolic course results – see Figure 11.37. For the calculation with bilinear approaches, the distribution is approxi-mated by element-wise constant shear stresses. These result with xz = G · u/ z from the derivations of the shear deformations being linearly approximated in the z-direction. With the quartic approach, the shear stresses in the z-direction are represented with a linearly varying course across the element, which leads to a better overall approximation to the actual quartic distribution for the rectangular solid cross section cf. Figure 11.37 on the left.


Figure 11.37 Distributions of the shear stresses with finite elements due to Vz = 10 kN

Generally, the distributions given in Table 11.11 can be calculated with the rectangular finite elements for the shear stresses xy and xz as a result of shear forces and secondary torsion. The coherencies can be stated since the element-specific -and -directions coincide with the principal axes of the cross section and thus, without the complex transformations according to Eq. (11.47), a direct connection to the principal directions is given. For example, the shear stress xy results from the derivation of the shear deformations u according to y. With the bilinear approach, the direct representation of a constant shear stress distribution in the y-direction and a linearly varying distribution in the z-direction within the element is possible. This can immediately be seen from the associated shape functions in Table 3.5.


Table 11.11 Shear stress distribution for rectangular elements

Distribution in the element for

Shear stress Biliniarly varying 4-node elements

Quartic9-node elements

uGyxy

y-direction: constant y-direction: linearly varying

z-direction: linearly varying z-direction: quartic uGzxz

y-direction: linearly varying y-direction: quartic z-direction: constant z-direction: linearly varying

Finally, with the standardised warping ordinate we will show how a reduced inte-gration of the element stiffness relationships can affect the rectangular cross section cf. Section 11.4.6. The cross section is discretised by eight 9-node elements (see Fig-ure 11.38a) and the equation system is set up for the determination of according to Eq. (11.28a). In the process of solving, an arbitrary node is held with the boundary condition Eq. (11.29a). With a reliable integration of the element stiffness relation-ships this leads to the deformations shown in Figure 11.38b. When using a reduced integration, a bearing of a single node cannot prevent zero-energy eigenvalues cf. Section 11.4.6. A standardised warping ordinate results as shown in Figure 11.38c. The reduced integration here leads to a useless solution. Therefore, we would gen-erally advise against using it. An exception may be if cross section symmetries are used, where additional boundary conditions (bearings of nodes) can prevent the zero-energy eigenvalues.

Figure 11.38 Standardised warping ordinate for a reliable and a reduced integration


11.6.5 Doubly Symmetric I-Profile

A rolled I-profile HEM 600 is considered. Figure 11.39 shows the cross section as well as the internal forces it is subjected to. The stress calculation for this example with conventional methods is shown in [25]. Here, priority is given to the calculation with FEM. For this reason, the cross section is first of all discretised with five one-dimensional finite elements – see Figure 11.39.

Figure 11.39 Cross section and discretisation with one-dimensional elements

Due to the symmetry properties, the position of the shear centre can be specified di-rectly. The standardised warping ordinate and the torsional properties given in Figure 11.40 result from the calculation with the program CSP-FE ML.

Figure 11.40 Standardised warping ordinate and cross section properties (one-dimensional elements)


The results can be checked without any problem with known solutions using centre-line models via hand calculations and an exact agreement will be recognised. For the calculation of the torsion constant 1 = 1.0 has been considered (cf. Figure 11.13). A degradation of IT due to coefficients for the individual plates is generally not advisable for rolled I-cross sections since, due to the rolling fillets, the torsion constant is considerably larger. This matter is further clarified below.

Figure 11.41 Shear stresses due to torsion with the one-dimensional elements

Figure 11.42 Principal moments of inertia and stresses due to shear forces with one-dimensional elements


For the calculation of the shear stresses due to the torsional moments, the cross sec-tion properties according to Figure 11.40 are used, resulting in the shear stress distributions given in Figure 11.41. For the calculation of shear stresses due to shear force, the principal moments of inertia Iy and Iz are used, which result with the cross section idealisation in Figure 11.42a. Figure 11.42b shows the shear stress distributions for the shear forces.

To evaluate the solutions obtained with the one-dimensional elements on the basis of the centre-line model, calculations with the two-dimensional 9-node elements with curvilinear boundaries are now conducted. As it has already been shown for the rec-tangular solid cross section, several calculations with an increasing refinement of the elementing have to be carried out. Compared to the rectangular solid cross section, the inaccuracy of the calculations no longer exclusively depends on the inaccuracy of the displacement approach. The error in the approximation has several causes here, which have already been explained in detail in Section 11.4.8.

Figure 11.43 Discretisation of the HEM 600 with two-dimensional 9-node elements

Figure 11.43 shows the divisions of the cross section into finite elements. To minimise the effort, also analyses utilising the cross section symmetry are conducted, so that only a quarter of the cross section has to be discretised. For carrying out the calculation with FEM, additional boundary conditions then have to be formulated. The bearing of an arbitrary node connected with Eq. (11.29) is often no longer correct


and sufficient. Table 11.12 gives an appropriate overview regarding the equation systems to be solved.

Table 11.12 Utilisation of the cross section symmetry

Warpingordinate Shear deformations u

Symmetry of the cross section

Bearing of the nodes i on the axes of symmetry (and bearing condition) for the equation system

K f zK u f(V ) yK u f(V ) xsK u f(M )

Single symmetric to the y-axis (horizontal symmetry axis) Discretisation of the half cross section

all nodes (condition:

zi = 0)


zi = 0)

one arbitrary node

(condition: zi = 0)


zi = 0)

Single symmetric to the z-axis (vertical symmetry axis) Discretisation of the half cross section


yi = 0)

one arbitrary node

(condition: yi = 0)


yi = 0)


yi = 0)

Doubly symmetric (horizontal and vertical) Discretisation of a quarter of the cross section


yi = 0 or zi = 0)

all nodes on the horizontal symmetry axis(condition: zi = 0)

all nodes on the vertical

symmetry axis (condition: yi = 0)


yi = 0 or zi = 0)

After the determination of the warping ordinate, it should be noted that for the cal-culation of the cross section properties, i.e. for all integrations relating to the total cross section area A, with the utilisation of the symmetries only a part of the cross section is available as FE model. If this is used, the solutions of the integrations must thus be increased by the factor 2 for the modelling of half or by 4 for the quarter of the cross section.

Figure 11.44 shows the exact solution for the standardised warping ordinate resulting for the HEM 600 with the program CSP-FE. For the purpose of illustration, it is shown for the total cross section, without utilising the symmetries. The maximum ordinate results on the insides of the flange ends to:

max = 466.0 cm2

The maximum value of = 442.3 cm2 calculated with the centre-line model deviates by 5.1% from this ordinate. Here, it is to be observed that the value resulting from the centre-line model is a medial ordinate with regard to the plate thickness. The dis-tribution shown in Figure 11.44 shows a warping ordinate of = 439.8 cm2 at the flange end in the middle of the plate. The deviation of the centre-line model is now


+0.56 %. An interesting aspect is that the exact solution is a little bit smaller than the one of the centre-line model. This is caused by the rolling fillets, which lead to a mi-nor deformation of the cross section. This effect has already been examined in [54].

Figure 11.44 Exact standardised warping ordinate

With the exact standardised warping ordinate, the torsional properties are:

IT = 1574 cm4, I = 15 700 000 cm6

If, first of all, the warping constant is compared to the one of the centre-line model, it becomes obvious that the value calculated with the one-dimensional elements shows a good agreement and is only 1.3% larger. With the corresponding integration the smaller warpings lead to a smaller I when considering the rolling fillets than it would if the fillets are neglected. This phenomenon can be observed for all common I-profiles (rolled sections). For further details refer to [54], where rolled U-profiles are also analysed, for example, which show a different behaviour due to an additional effect.


When comparing the torsion constant, a deviation of 5.9% results (centre-line model compared to the exact solution). For this profile, it is comparably small, but it can be up to 50% for other profiles. The rolling fillets in the area of the transition from the web to the flanges lead to an increase of the IT. This can be clarified with the shear stress course due to a primary torsional moment in Figure 11.45. In the transition area mentioned, a stress concentration emerges which suggests an increased torsion constant. This is naturally not captured with the centre-line model. With the one-dimensional elements good solutions can be achieved concerning the IT for welded profile shapes, but for rolled I-cross sections, they are generally insufficient. For this reason, the use of two-dimensional elements is advisable here. Alternatively, the relevant literature offers approximate solutions with which the IT can be calculated considerably better – see, for example, [25], [67], [41] and [84]. Moreover, exact cross section properties on the basis of the FE method are tabulated for a multitude of common rolled profiles in [29] and [87].

Figure 11.45 Shear stresses due to torsion calculated with two-dimensional elements


Figure 11.45 shows the shear stress distribution in one quarter of the cross section resulting with the two-dimensional elements for the torsional stresses. The results of the centre-line model for the primary torsion agree relatively well with the stresses resulting in large areas of the web and the flanges. The deviations here result almost exclusively from the inaccurate torsion constant being used for the calculation of the shear stresses with the one-dimensional elements. Naturally, the increased stresses in the area of the rolling fillets cannot be captured with the centre-line model.

The shear stresses calculated with the centre-line model for the secondary torsion agree well with the accurate solution except for the rolling fillets. In the transition area from the flange to the web, stress concentrations result for the accurate solution which cannot be captured by the one-dimensional elements.

Figure 11.46 Shear stresses due to shear force calculated with two-dimensional elements


The shear stress distributions determined with the help of the isoparametric elements due to shear force are shown in Figure 11.46. The maximum shear stress due to Vzshows a very good compliance to the centre-line model and the accurate solution. The notes on the stress distribution of Mxs are also valid for the shear stresses due to Vy,since the distribution is qualitatively comparable.

11.6.6 Crane Rail

Figure 11.47 shows an A 100 crane rail – depicted according to DIN 536-1 – which can be used for crane runway girders. For the sake of clarity, only the external measurements of the rail as well as the position of the centre of gravity are shown in the figure. Needless to say, for the determination of the geometry of the rail, further measurements are required. These can be taken from DIN or for example from [29]. For the determination of the cross section properties, the rail is discretised by quartic 9-node elements. Figure 11.47 shows an elementing of half of the cross section utilising the symmetry, reducing calculating efforts correspondingly.

Figure 11.47 Crane rail A 100 and example discretisation

From the calculation with the program CSP-FE

zM = 0.996 cm

results for the position of the shear centre and for the standardised warping ordinate shown in Figure 11.48. With the help of the warping ordinate, the cross section values IT and I are determined:

IT = 670.7 cm4 , I = 3994 cm6

The standard defines zM = 0.98 cm for the position of the shear centre and IT = 666 cm4 for the torsion constant, which represents a deviation of 1.6% or 0.7%, respectively. The warping torsion and the warping constant determined with the FE


calculation only have a subordinate significance for the cross section shape given here.

Figure 11.48 Standardised warping ordinate of rail A 100

Also, it should be mentioned that for crane rails in static calculations a rail head abrasion of 15% must be considered. If this is considered in the calculation, the fol-lowing cross section properties for rail A 100 result with the FE method:

zM = 0.805 cm, IT = 499.3 cm4, I = 2437 cm6

The properties of further “complete” and “worn down” crane rails of the shapes A and F can be taken from [29] or [55].

Besides the calculation of the cross section properties, the standardised warping ordi-nate is also used for the calculation of shear stresses due to primary torsion. For rail A 100, these are shown in Figure 11.49a as a result of a torsional moment Mxp = 100 kNcm utilising the cross section symmetry. In the area of the fillets below the rail head, stress concentrations are recognisable.

The FE calculation of the stress distribution due to a shear force Vz is conducted us-ing the corresponding shear deformations u. Figure 11.49b shows the distribution resulting for Vz = 10 kN. Here, stress concentrations occur in the fillet areas as well. As expected, the largest stresses occur in the area between rail head and rail foot.


Figure 11.49 Shear stresses in rail A 100

References

[1] Argyris, J. H., Kelsey, S.: Energy theorems of structural analysis. Butterworth Publ., London 1960

[2] Bathe, K.-J.: Finite-Elemente-Methoden. Springer-Verlag, Berlin/Heidelberg 1990

[3] Beier-Tertel, J.: Geometrische Ersatzimperfektionen für Tragsicherheits-nachweise zum Biegedrillknicken von Trägern aus Walzprofilen. Shaker Verlag, Aachen 2009

[4] Bronstein, I. N., Semendjajew, K. A.: Taschenbuch der Mathematik, 25. Auf-lage. Verlag Harri Deutsch, Thun/Frankfurt am Main 1991

[5] Brune, B.: Stahlbaunormen – Erläuterungen und Beispiele zu DIN 18800 Teil 3. Stahlbau-Kalender 2000, Verlag Ernst & Sohn, Berlin

[6] Bürgermeister, G., Steup, H., Kretschmar, H.: Stabilitätstheorie Teil I. Akademie-Verlag, Berlin 1966

[7] DIN 1055: Einwirkungen auf Tragwerke Teil 100: Grundlagen der Tragwerksplanung, Sicherheitskonzept und Bemes-sungsregeln

[8] DIN 18800 (11/90 und 11/2008) Teil 1: Stahlbauten, Bemessung und Konstruktion Teil 2: Stahlbauten, Stabilitätsfälle, Knicken von Stäben und Stabwerken Teil 3: Stahlbauten, Stabilitätsfälle, Plattenbeulen

[9] DIN EN 1991: Eurocode 1 – Einwirkungen auf Tragwerke [10] DIN EN 1993: Eurocode 3 – Bemessung und Konstruktion von Stahlbauten

Teil 1-1 (07/05): Allgemeine Bemessungsregeln und Regeln für den HochbauTeil 1-5 (02/07): Plattenbeulen Teil 1-8 (07/05): Bemessung von Anschlüssen For calculations and design the National Annexes have to be considered.

[11] DIN Fachbericht 103 “Stahlbrücken”. Beuth Verlag, Berlin 2003 [12] ECCS-CECM-EKS, Publication No. 33: Ultimate Limit State Calculation of

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Index

area element: see Element approximate solutions, performance 495f assumptions 8ff, 13 – for buckling panels 440f –, realistic calculation 210ff

back substitution 219, 221 band width 148, 217 band structure 148, 150, 154, 217 basis functions 87ff – for beam elements 87ff – –, loading with axial forces 88f, 111, 181 – –, lateral torsional buckling and further combined loadings 95 – –, bending with axial compression, second order theory, flexural buckling 91ff – –, bending with tension force, second order theory 93f – –, bending about the y-axis 90, 113, 181 – –, bending about the z-axis 89f, 181 – –, St Venants torsion 89 – –, trigonometric and hyperbolic functions 91ff, 117 – –, warping torsion 94, 117, 162, 181 – for cross section elements – –, one-dimensional functions 99ff – –, two-dimensional functions 103ff – for plate buckling 95ff – for plate elements 98, 429 beam elements 5f – for linear calculations 108ff – –, arbitrary loading 120ff – –, arbitrary reference system 146 – –, bending stress 113ff – –, axial stress 110ff – –, torsional stress 116ff – for non-linear calculations 168ff – –, arbitrary loading 180ff – –, bending with compression or tension stress 185ff bending about the y-axis 90, 113ff bending about the z-axis 89f, 114ff bending stiffness 144, 177 – of a plate 426f bending stress 113ff bending with compression, second order theory 91ff, 185ff

bending with tension force, second order theory 93f, 188ff Bernoulli‘s–hypothesis 13 bifurcation loads: see critical loads boundary conditions –, for cross sections 475 –, geometrische 73f, 87, 151ff –, physikalische 87 boundary node elements 482 boundary shear flows 470, 472, 483, 488 Bredtschian formula 511 bridge cross section with trapezium stiffeners 319ff buckling curves 330ff, 341f, 365, 370ff buckling panel 98, 439ff

– eigenmodes 448ff –, single panel with constant compression stresses 451 –, stiffened 436ff – width 440 buckling shapes: see eigenmodes buckling values, determimation 448ff

C0- continuity: see continuity C1- continuity: see continuity calculations – according to second order theory 193ff –, linear 6f –, – of beams and frames 108ff –, –, –, example 149ff, 155f –, nonlinear 6f –, –, of beams and frames 168ff –, –, –, Beispiel 198ff, 103ff centre line model 48, 462 centre of gravity 9, 29f, 31ff Cholesky method 220 coefficient of thermal expansion 159 component method 414 compression members: see columns computer programs 24 concentrated loads 157 conditions for bearings in the local coordinate system 152 conservative forces 83 constitutive relationships 13ff – for plates 423ff continuity

0

–, C1- 107 convergence behaviour 225, 240, 513ff

–, beam web with longitudinal stiffeners 454


–, C - 104


Index 535

coordinate system 8f, 127, 131, 135, 484 coordinates and ordinates 9 – for beam elements – –, global 8, 127, 152 – –, local 9, 127 – –, nondimensional 88, 108 – for plate elements 95f, 99 – for cross section elements – –, one-dimensional 100f, 470, 472 – –, two-dimensional 104f, 484, 487 crane rail 525ff critical loads 171, 201ff, 214, 342ff critical load factor 80, 171, 205, 344ff – for plate buckling 448 critical moments 375ff cross section 25ff, 462ff cross section deformations 144 cross section elements –, one-dimensional 469ff –, rectangular 497ff –, two-dimensional 480ff cross section properties 11, 25ff –, with FE-calculation 476f, 493f, 499ff cross section symmetry 29ff, 36, 62, 521

decomposition 219, 221, 223, 224 decrease of interval 242 degrees of freedom 4, 95 –, introduction of additional 164 denominator determinant 171 design values 2, 16ff designations 7ff – for beams and members – –, further symbols and assumptions 12ff – –, parameters in the local x-y-z coordinate system 9ff – –, values in the global X-Y-Z coordinate system 8 – for plate buckling 443ff determinante 232 diagonal elements 233f diagonal matrix 204, 221, 231 differential equations 80, 84ff –, flexural buckling about the strong axis 85 – for beam elements – –, bending about the z-axis 89 – –, bending with compression, second order theory and flexural buckling 91 – –, bending with tension force, second order theory 93

– –, loading with axial force 88 – –, St Venants torsion 89 – –, warping torsion 94 –, linear beam theory 85 –, plate buckling 86 –, second order beam theory 85 direct solution methods 224 Dirichlet‘s variational principle 83 displacement approach: see basis functions displacement method 3f, 72ff –, proceedure 78f distributed springs: see springs

eigenmodes 203ff, 227ff –, beam 238, 363, 377 –, beam with cantilever 395 –, beam with distributed torsional spring 230 –, cantilever 322 –, determination 235f –, frame beam 411 –, frame regarding joint stiffness 417 –, plate, single panel 453 –, plate, entire panel 455, 457 –, single haunched frame 227, 346 –, stiffened plate 436 –, two-hinged frame 348, 350 –, two-span beam 231, 363 –, two-span beam with intermediate hinge 229, 346 –, zero-energy 493f, 517 eigenvalue 201, 225ff –, explanations for understanding 225ff eigenvalue problem 80, 201, 225, 448 eigenvector 203, 225ff element –, area 4f –, beam: see beam elements –, cross section: see cross section elements –, line 4f –, plate: see plate elements – typs 4ff –, volume 4 element geometry 484ff element length 98, 119, 186f, 189, 397 element load vector – for bar elements – –, due to initial imperfections 192 – –, linear bar theory – –, –, arbitrary loading 122 – –, –, axial force loading 112 – –, –, bending about the y-axis 114, 115

536 Index

– –, –, bending about the z-axis 114 – –, –, reduced, with hinge 166 – –, –, St Venant‘s torsion 120 – –, –, warping torsion 117 – for cross section elements – –, thick walled 489 – –, thin walled 473, 474 – –, rectangular 497ff – for plate elements 431 element matrix of shear diaphragms 164 element spring matrix 163 element stiffness matrix – for beam elements – –, geometric 180f – –, linear beam theory – –, –, arbitrary loading 120ff – –, –, axial force loading 112 – –, –, bending about the y-axis 114, 115 – –, –, bending about the z-axis 114 – –, –, reduced, with hinge 166 – –, –, St Venant‘s torsion 120 – –, –, warping torsion 117, 118 – for cross section elements – –, thick walled 489 – –, thin walled 473, 474 – –, rectangular 497ff – for plate elements – –, linear theory 431 – –, geometric 433 element stiffness relationship –, complete 112 –, incomplete 112 – for beam elements – –, linear theory 76, 110, 134 – –, –, arbitrary loading 120ff – –, –, axial force loading 112 – –, –, bending about the y-axis 114 – –, –, bending about the z-axis 114 – –, –, reduced, with hinge 166 – –, –, St Venant‘s torsion 120 – –, –, warping torsion 117 – –, second order theory 167, 180 – for cross section elements – –, thick walled 489 – –, thin walled 473, 474 – for plate elements – –, linear theory 431 – –, plate buckling 433 entire panel 439 equation system 217ff – for cross sections 474f – for beams and frames 147ff

– –, plane frame 77 – – according to second order theory 193f – additional notse 224 – solution methods 218f equilibrium 80ff, 320 – at the deformed system 168ff, 178 equilibrium internal forces: see internal forces equivalent geometric imperfections 189ff, 381ff – function 191 equivalent imperfections method 326 equivalent stress 1f, 270 examples – for beams and frames 393ff – for cross sections 503ff – for plate buckling 451ff extensional stiffness of a plate 426

flexural buckling 85f, 91ff, 168, 185f, 228, 322ff –, frame column 404ff –, verification with reduction factors 329ff force method 2f frame corner 145, 401 fundamental relationships 13ff – for plates 423ff

GAUCHO method 154, 220ff, 230, 240ff –, calculation example 222ff Gauss quadrature 490ff Gaussian algorithm 154, 219 geometric imperfections 210f geometric nonlinearity: see nonlinearity geometric stiffness matrix: see element stiffness matrix global coordinate system: see coordinate system

Hermitian interpolation polynomials: see interpolation polynomials hinge springs: see springs hinges 164ff Hooke‘s law 14, 424

inaccuracy – of the numeric integration 496 – of the cross section illustration 495f – of the displacement approach 496

Index 537

initial bow imperfection 191f, 382 initial sway imperfection 189f, 382 influence of imperfections 213f integration–, by parts 467 –, numerical 53ff, 490ff –, reduced 492f, 517 –, reliable 492, 517 integration constant 66, 91, 117, 255, 476 integration points 491 interaction conditions 1, 297 internal forces and moments 10f – as resultants of stresses 14f – of plates 421ff –, definition 11 –, definition I/II: s. sign definition I/II –, determination 2ff, 154ff –, equilibrium –, –, local 179, 194 –, –, global 179 –, verification 179, 193ff interpolation polynomials –, Hermitian 90, 107 –, Lagrangian 88f, 99ff, 104ff inverse 154 inverse vector iteration 202, 226, 236ff, 242ff, 448ff –, example 240 –, recommendations 240 –, modified 236ff isoparametric formulation 481 iterative elastic calculation 207ff iterative solution methods 224

Jacobian determinant 486f, 491f, 494, 496Jacobischian matrix 486f joint stiffness 413ff

kinematic chain 207 Kirchhoff‘s plate theory 429

lateral torsional buckling 95, 177, 228, 322ff, –, beam with scheduled torsion 396ff –, frame beam 406ff –, frame column 405 –, single span girder with cantilever 394f –, verification with reduction factors 360ff loads 10, 138f, 157ff

–, equivalent nodal 112, 114, 116, 117, 157ff, 192f

Lagrangian interpolation polynomials: see interpolation polynomials left triangular matrix 219 limitation of rotation 198 line elements: see element line model: see centre line model linear beam theory 15 linear calculations: see calculations linearisation 170, 190, 195 liquefaction: see static liquefaction load vector: see element load vector, total load vector loadings 157ff local coordinate system: see coordinate system

material behaviour 12f, 14, 212 material law 14 material properties 12 matrix elements 121, 182 matrices 12 matrix decomposition 224 matrix decomposition method 202, 218ff, 230ff, 241ff, 448ff –, example 228ff member characteristics 86, 91, 93, 94, 117, 185, 188 modal shapes: see eigenmodes modelling of buckling panles 450 moment rotation characteristic 414f

natural position of nodes 492 Newton Cotes quadrature 490 nodal equilibrium 75, 123ff – with consideration of the deformations 140ff nodal shear flows: see boundary shear flows nonlinear calculation: see calculations nonlinearity –, geometrical 6f, 130, 170 –, physical 6f, 130, 212 numerical integration: see integration

partial internal forces method 1, 285ff, 290ff, 296ff, 393f, 396, 412, 416

538 Index

partial panel 439 partial safety factors 12, 17f Pascal‘s triangle of polynomials 96 physical nonlinearity: see nonlinearity performance of the approximate solutions 495f plane load bearing structures 420ff plastic hinge theory 1, 206ff plastic theory 1, 206ff, 210ff, 273ff plastic zone theory 1, 210ff –, application areas 210 –, calculation example 214 plate 420, 428f plate buckling 31, 40ff –, beam web with longitudinal stiffeners 454ff –, designation 438 –, single panel with compression 451ff –, verifications 438ff –, web of a composite bridge with shear stresses 457f –, web with high bending stresses 459ff plate elements 420ff – for linear calculations 429ff – for plate buckling 432ff point springs: see springs position stability 16 potential energy 83f primary torsion: see torsion principal axes 9, 30, 31ff principal coordinate system 25 principal moments of inertia 11, 31ff principle of minimum of potential energy 80, 83f principles of FEM 72ff progress of the calculations 78ff Pythagoras´s theorem 176f

Rayleigh quotient 241 realistic calculation assumptions 210ff reference points 9 rectangular solid cross section 511ff reduced integration: see integrations reduction of the element stiffness matrix 165ff reference systems 126ff –, finite elements for arbitrary 146 reliable integration: see integration residual stresses 211f, 213 resistance Rd 1, 16ff resistance of cross sections 245ff, 274ff

right triangular matrix 219 rotation 198 rotational stiffness: see rotational springs

search of interval 242 secondary torsion: see torsion series expansions 92, 94 servicability limit state 16 settlements 158f shape functions 88f, 90, 91, 100f, 105f, 484 shear centre 67ff, 479 shear deformations 466ff shear diaphragms 105ff shear stress 14, 65, 69, 245, 250ff, 262, 264ff, 469ff, 477f, 494f – distributions – –, for rectangular cross sections 477, 516f – –, for a gutter cross section 268 – –, doubly symmetric I-profils 271, 519, 523, 524 – –, single celled cross section 506, 508 – –, crane rail 527 sign defintion I/II 10f single panel 439 singular 151, 475, 505 singularities 496 solution of equation systems 154, 217ff spectral displacement 226, 236ff, 449, 459 spring constant 160 spring forces 164 spring law 164, 190 springs 160ff –, distributed 162f –, hinge 164ff –, point 160ff –, rotational 349ff, 352ff, 410, 221, 415 –, warping 142ff St Venant‘s torsion: see torsion stability analyses 201ff, 319ff standardisation part I 31ff standardisation part II 58ff, 476, 493f static liquefaction 473 stiffeners 142 –, longitudinal and transverse of plates 434ff stiffness relationship: see element- , total stiffness relationship stiffness matrix: see element- , total stiffness matrix strain iteration 307ff strains 14

Index 539

stress Sd 1, 16ff stresses 11 – for plates 420ff – for beams 11, 14 –, calculation of 245ff, 476ff, 493ff subparametric representation 481 superparametric representation 482 support conditions 73, 78 –, determination 156 Sylvester test 242

temperature coefficient of expansion 159 temperature influences 159 torsion–, primary/St. Venant’s 63, 89, 102, 116, 119, 261ff, 264ff, 465, 469f, 477 –, secondary 102f, 261ff, 266ff, 466ff, 471, 473f, 489, 494 –, warping 94, 116ff torsion constant 63, 265f, 476f, 494 torsional buckling 227, 323, 362ff torsional rigidity 116 total load vector – for cross sections 474 – for beams and frames 78, 124f, 149ff, 156 total stiffness matrix 77f – for cross section 474f – for beams and frames 147ff, 150 –, geometric 80, 184 total stiffness relationship 77, 490 transformation 126ff –, beam elements in arbitrary reference systems 146 –, beam elements in the X–Y–Z COS 134ff –, beam elements in the X–Z plane 76, 131ff –, cross section elements 486ff –, derivation of twist, warping bimoment 139ff –, element stiffness relationship 134 –, element load vector of initial imperfections 192 –, equilibrium-/verification internal forces 195ff –, loads 138f –, nodal internal forces 125 –, of a curvilinear boundary element to the uniform square 484f triangular matrix 219f, 224

ultimate limit state 16

uniform square 484

variational principle 83, 174f vectors 12 vector iteration according to von Mises 236 verification(s)– against plate buckling 438ff – –, according to DIN 18800 439ff – –, according to DIN Technical Report/EC 3 447f – –, methods 438f – of beams 319ff – – against flexural buckling – –, –, with reduction factors 1, 7, 329ff – –, –, with equivalent imperfections 1, 7, 381ff – – against lateral torsional buckling – –, –, with reduction factors 1, 7, 360ff – –, –, with equivalent imperfections 1, 7, 389ff – –, procedures 1f – –, –, Elastic–Elastic 7, 245ff – –, –, Elastic–Plastic 7, 273ff – –, –, Plastic–Plastic 7, 206ff, 210ff – –, required 1f – –, sufficient cross section capacity 245ff, 273ff verification internal forces: see internal forces virtual work: see work virtual work principle 74f, 81ff, 425ff, 464ff volume elements: see elements

Wagner’s–hypothesis 13 warping bimoment 140ff warping constant 63, 71, 476, 493 warping ordinate 58ff, 63ff, 71, 472f, 489f –, bridge section 510 –, C–section 64 –, crane rail 526 –, doubly symmetric I-section 518, 522 –, rectangular solid section 512, 517 –, single celled section 505 warping springs: see springs warping stiffness 116 warping torsion: see torsion weighting factors 491 work 81, 83 –, virtual 81

540 Index

–, virtual, for beams –, –, according to first order theory 82 –, –, –, axial force 110f –, –, –, at nodes 123f –, –, –, bending stress 113ff –, –, –, torsional stress 116f –, –, according to second order theory 172ff –, –, –, internal forces 172, 175ff, 181 –, –, –, load values 172ff –, virtual, for cross sections 464ff –, –, one-dimensional elements 469f –, –, two-dimensional elements 482f –, virtual, for plates/shells 425ff –, virtual, for shear diaphragm 161 –, virtual, for springs 160ff work components 82, 161, 172, 177

zero-energy eigenmodes: see eigenmodes

Documents

Rolf Kindmann, Matthias Kraus Steel Structures Design