Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapsecradpdf.drdc-rddc.gc.ca/PDFS/unc104/p534276_A1b.pdf · 2012-08-03 · Prediction of the Effects of Cold

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

Prediction of the Effects of Cold Bending on

Submarine Pressure Hull Collapse

Liam Gannon

Technical Memorandum

DRDC Atlantic TM 2010-065

April 2010

Copy No. _____

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

This page intentionally left blank.

Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapse

Liam Gannon

Defence R&D Canada – Atlantic



April 2010

Principal Author

Liam Gannon

Defence Scientist

Approved by

Kevin McTaggart

Acting Head/Warship Performance

Approved for release by

Calvin Hyatt

DRP Chair

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2010

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale,

2010

Original signed by Liam Gannon

Original signed by Kevin McTaggart

Original signed by Ron Kuwahara for

DRDC Atlantic TM 2010-065 i

Abstract

Submarine pressure hull frames and shell plating are shaped by cold bending during fabrication.

Cold bending introduces significant residual stress in these components which can be detrimental

to the strength of the structure. This study evaluates different methods of incorporating cold

bending residual stresses in the analysis of pressure hulls considering different out of circularity

mode shapes. Several methods of pressure hull collapse analysis are compared considering

interframe and overall collapse modes. These include an empirical method, a finite difference

method and the finite element method.

Collapse pressures predicted using the methods prescribed in the UK MoD submarine structure

design standard, SSP 74, are found to be conservative when compared with results from finite

element analysis. Collapse pressures predicted using effective stress-strain curves to incorporate

the effects of cold bending residual stress in finite element models agree well with those predicted

by explicitly modelling the cold bending process. This indicates that the use of effective stress-

strain curves is an acceptable means of accounting for the influence of cold bending residual

stress on the collapse pressure of a submarine pressure hull.

Résumé

Le bordé extérieur et les couples de coque épaisse des sous-marins sont formés par pliage à froid

pendant la fabrication. Le pliage à froid introduit une contrainte résiduelle importante dans ces

pièces, ce qui peut diminuer la résistance de la structure. La présente étude évalue différentes

méthodes permettant d’incorporer des contraintes résiduelles de pliage à froid dans l’analyse des

coques épaisses en considérant différentes formes rondes déformées. Plusieurs méthodes

d’analyse d’écrasement de coque épaisse sont comparées en considérant l’écrasement global et

l’écrasement entre couples. Ces méthodes sont la méthode empirique, la méthode par différences

finies et la méthode par éléments finis.

Les pressions d’écrasement prédites à l’aide des méthodes prescrites dans la norme de conception

de structure de sous-marin du ministère de la défense du Royaume-Uni, SSP 74, sont considérées

conservatrices lorsqu’on les compare aux résultats de l’analyse par éléments finis. Les pressions

d’écrasement prédites à l’aide de diagrammes effort-déformation efficaces pour incorporer les

effets des contraintes résiduelles de pliage à froid dans des modèles par éléments finis

correspondent bien à celles prédites en modélisant explicitement le procédé de pliage à froid.

Cela indique que l’utilisation de diagrammes effort-déformation efficaces est un moyen

acceptable de tenir compte de l’influence de la contrainte résiduelle de pliage à froid sur la

pression d’écrasement d’une coque épaisse de sous-marin.

ii DRDC Atlantic TM 2010-065


DRDC Atlantic TM 2010-065 iii

Executive summary


Liam Gannon; DRDC Atlantic TM 2010-065; Defence R&D Canada – Atlantic;April 2010.

Background: An accurate assessment of the structural performance of submarine hulls must

consider fabrication-induced residual stress and distortion. These arise from forming and joining

operations such as cold bending and welding. There are several ways of incorporating these

imperfections in a pressure hull collapse analysis. One such method is the effective stress-strain

curve method, which is a relatively simple means of characterizing these imperfections in a

structural analysis.

Results: This memorandum evaluates several methods of pressure hull collapse analysis

considering different ways of accounting for cold bending residual stress. Collapse pressures

predicted using the various methods are compared and it is found that the effective stress-strain

curve method of accounting for residual stress is both accurate and efficient.

Significance: An accurate, simple way of accounting for fabrication-related imperfections in

submarine structural analysis will make future investigations of complicated structural behaviour

easier to conduct and save a significant amount of time in future structural analyses.

Future plans: The effective stress-strain curve method could potentially be used to characterize

residual stresses due to welding. A future study will examine the use of this method to account for

residual stresses from both cold bending and welding.

iv DRDC Atlantic TM 2010-065

Sommaire


Liam Gannon; DRDC Atlantic TM 2010-065; R & D pour la défense Canada –Atlantique; avril 2010.

Introduction : Une évaluation précise du rendement de la structure de la coque des sous-marins

doit tenir compte des distorsions et des contraintes résiduelles produites par la fabrication. Ces

distorsions et ces contraintes sont produites par les opérations de formage et de liaison comme

le soudage et le pliage à froid. Il y existe plusieurs façons d’incorporer ces imperfections à

une analyse d’écrasement de coque épaisse. Une de ces méthodes est le diagramme

effort-déformation efficace, qui est un moyen relativement simple de caractériser ces

imperfections dans une analyse de structure.

Résultats : Le présent document évalue plusieurs méthodes d’analyse d’écrasement de coque

épaisse considérant différentes façons de tenir compte de la contrainte résiduelle de pliage à

froid. Les pressions d’écrasement prédites à l’aide des diverses méthodes sont comparées, et il

s’avère que le diagramme effort-déformation efficace qui permet de tenir compte de la contrainte

résiduelle est précis et efficace.

Portée : Une façon précise et simple de tenir compte des imperfections liées à la fabrication lors

d’une analyse de structure de sous-marin devrait rendre d’ultérieures études sur le comportement

complexe des structures plus faciles à réaliser et permettra d’économiser beaucoup de temps dans

la réalisation des analyses de structure à venir.

Recherches futures : Le diagramme effort-déformation efficace pourrait être utilisé pour

caractériser les contraintes résiduelles dues au soudage. Une étude à venir examinera l’utilisation

de cette méthode pour tenir compte des contraintes résiduelles découlant du soudage et du pliage

à froid.

DRDC Atlantic TM 2010-065 v

Table of contents

Abstract ... .......................................................................................................................................i

Résumé.... .......................................................................................................................................i

Executive summary ..................................................................................................................... iii

Sommaire .....................................................................................................................................iv

Table of contents ...........................................................................................................................v

List of figures ...............................................................................................................................vi

List of tables ................................................................................................................................vii

Acknowledgements ....................................................................................................................viii

1 Introduction .............................................................................................................................1

2 Geometry .................................................................................................................................3

3 Cold Bending...........................................................................................................................5

3.1 SSP 74 ..........................................................................................................................5

3.2 Finite Element Cold Bending Simulation ....................................................................6

3.3 Effective Stress-Strain Curves ......................................................................................7

4 Methods of Analysis ................................................................................................................8

4.1 SSP 74 – Design of Submarine Structures ....................................................................8

4.2 Finite Element Analysis..............................................................................................10

5 Results and Discussion ..........................................................................................................12

5.1 Cold Bending Residual Stress.....................................................................................12

5.2 Effective Stress-Strain Curves ....................................................................................13

5.3 Model A Collapse Analysis ........................................................................................14

5.4 Model B Collapse Analysis ........................................................................................16

6 Conclusions ...........................................................................................................................18

References....................................................................................................................................19

Annex A .. Displacement Contours...............................................................................................21

A.1 Contours .....................................................................................................................21

Annex B .. Model B Stress History...............................................................................................25

List of symbols/abbreviations/acronyms/initialisms ....................................................................29

Distribution list ............................................................................................................................31

vi DRDC Atlantic TM 2010-065

List of figures

Figure 1: Overall and interframe collapse modes (n=4) ..................................................................1

Figure 2: Model A scantlings .........................................................................................................3

Figure 3: Model B scantlings ..........................................................................................................4

Figure 4: Frame and plate after cold bending .................................................................................6

Figure 5: Ring stiffened cylinder finite element model ................................................................10

Figure 6: Ring stiffened cylinder with out of circularity (n =4 OOC mode) .................................11

Figure 7: Comparison of cold bending residual stress predicted by K79 and 3D FEA .................12

Figure 8: Effective stress-strain curves for model A (interframe model) ......................................13

Figure 9: Effective stress-strain curves for Model B (overall model) ...........................................13

Figure 10: Displacement vector sum contour plot for Model A at failure for n = 3 OOC .............15

DRDC Atlantic TM 2010-065 vii

List of tables

Table 1: Methods used to predict cold bending residual stresses for the methods of analysis

used in this study ..........................................................................................................8

Table 2: Collapse pressure predictions for Model A (interframe collapse) ...................................14

Table 3: Collapse pressure predictions for Model B (overall collapse except where noted) .........16

viii DRDC Atlantic TM 2010-065

Acknowledgements

The author would like to thank Mr. John MacKay for preparing and plotting the effective stress-

strain curve data.

DRDC Atlantic TM 2010-065 1

1 Introduction

Submarine pressure hulls typically consist of internally stiffened cylindrical and conical

compartments separated by watertight bulkheads. The primary load for which pressure hulls are

designed is due to external water pressure acting on the hull when the submarine is submerged.

Pressure hulls are designed considering two primary failure modes: interframe collapse and

overall collapse. Interframe buckling occurs with little deformation of the stiffening frames. It is

characterized by large deformations of the shell plating between frames for a small increase in

external pressure. Overall buckling involves significant deformation of the frames and occurs

over the entire length of a compartment between bulkheads. Interframe and overall collapse

modes of a ring stiffened cylinder are illustrated in Figure 1.

During fabrication, the shell and frames of a pressure hull are shaped by cold bending. This

process induces significant residual stress in the material that may cause a reduction in the load

carrying capacity of the pressure hull. Consequently, an accurate pressure hull collapse analysis

must account for the cold bending residual stresses. Several different methods are available both

for determining the cold bending residual stresses and for incorporating them in a collapse

analysis. Collapse pressures determined from empirical data [1] implicitly account for cold

bending residual stress, while analytical approaches [2] often use an iterative elasto-plastic

approach based on the assumptions of beam theory to explicitly calculate the residual stresses due

to cold bending before the collapse analysis is carried out. The latter typically considers residual

stresses only in the circumferential direction. Accuracy in prediction of cold bending residual

stresses may be improved by application of the finite element method (FEM) to simulate the cold

bending process, allowing the full three dimensional residual stress field induced by cold bending

to be determined. This method has been used by Graham [3] to account for the effects of cold

bending residual stress in the collapse analysis of a ring stiffened cylinder.

Figure 1: Overall and interframe collapse modes (n=4)

The UK MoD submarine design standard, SSP 74 [1] may be used to determine the design

collapse pressure for both interframe and overall failure modes. The standard specifies that the

interframe collapse pressure for a cylindrical stiffened shell be determined by first calculating the

Von Mises interframe buckling pressure [4] normalized with respect to the pressure at which the

2 DRDC Atlantic TM 2010-065

mean circumferential stress at mid-bay reaches yield. This value is then compared with empirical

data to determine the interframe collapse pressure.

Kendrick [5], [6], [7] developed the first acceptable theory for overall elastic buckling of a ring

stiffened cylinder using energy methods. Three versions of the theory were developed

considering different buckled shapes and stress distributions. Kendrick's Part I theory for overall

buckling of a finite length cylinder assumes a buckled shape with a half sine wave between

bulkheads with no undulations due to the presence of the frames and uses an approximate stress

distribution prior to buckling. The Part II theory considered a uniformly framed cylinder of

infinite length, but with a more refined buckling shape where interframe undulations were

allowed and the correct axisymmetric stress distribution. Kendrick's Part III theory reverts to the

approximate stress distribution, but applies to finite length cylinders. Using his Part III theory,

Kendrick [8] found that compartment length is much more important for small out-of-circularity

mode numbers, n than for large ones. For n > 3, a compartment length of 11 frame spaces is

effectively infinite; however for n = 2, 28 frame spaces still resulted in an overall buckling

pressure significantly higher than that of an infinitely long cylinder.

In SSP 74 [1], the overall collapse pressure is taken as either the pressure to cause yielding in the

frame flange or in the plating in way of the frame. This is determined using either an elastic

method of analysis or an elasto-plastic method. In the elastic method, the stress is calculated by

adding the hoop stress and the bending stresses induced by the magnification of the initial out of

circularity. The bending stresses are calculated using a magnification factor which relies on a

simplification of Kendrick's Part I theory, derived by Bryant [9]. The Bryant pressure is a

simplification of the Von Mises buckling pressure where the Von Mises buckling pressure for a

cylinder without ring frames and the Bresse pressure for one frame plus a frame space of plating

are added together. Collapse is assumed to occur when the sum of the hoop and bending stresses

reaches yield. For the elasto-plastic analysis, the overall collapse pressure is determined by

analysis of a single frame and one bay of attached plating using the method of Kendrick [10]. A

number of approximate correction factors are applied to the collapse pressure of the single frame

and plating model to determine the collapse pressure for a complete compartment. Among the

limitations of this method is that it only applies to axisymmetric geometry. The elasto-plastic

method allows the limit state to be determined and may be more conservative than the elastic

method since the latter does not account for residual stresses. The application of the finite element

method allows many of the limitations inherent in analytical procedures to be overcome. Finite

element analysis (FEA) allows the actual geometry of the pressure hull to be modelled including

penetrations, geometric imperfections and the three-dimensional residual stress field resulting

from fabrication processes such as cold bending. The disadvantage of a detailed finite element

analysis is that significant time and effort are required on the part of the engineer to generate the

model and to specify initial conditions.

The objective of this study is to evaluate various methods of calculating cold bending residual

stresses and of incorporating them in a pressure hull collapse analysis. The effective stress-strain

curve method of incorporating cold bending residual stresses in a collapse analysis is of particular

interest due to its relative simplicity. This approach involves adjusting the material properties of

the model to implicitly account for the effects of cold bending residual stress in a collapse

analysis. This method may be significantly faster to implement than explicitly defining cold

bending residual stresses at each integration point in a three-dimensional finite element model

and may result in significant time savings for future projects.


2 Geometry

Two different geometries are considered, designated Model A and Model B. The cylinder

scantlings were chosen such that one will fail by interframe buckling (Model A) and the other

will fail by overall buckling (Model B). The two geometries are similar; however the frame web

depth and flange width is reduced for Model B. This reduces the frame stiffness in order to

precipitate overall collapse. The scantlings for Models A and B are shown in Figure 2 and Figure

3 respectively. The material used is high strength steel used in submarine construction such as

Q1N or HY 80 with a yield strength of 550 MPa and elastic modulus of 207 GPa. For the finite

element analyses wherein cold bending is explicitly simulated and where residual stresses

determined by the method of Mitchell [2] are used, the hardening modulus is taken as 3.2 GPa as

measured by Bayley [11]. Although strain hardening should have little influence on the results, it

is included to enhance numerical stability of finite element solutions. For the collapse pressure

calculation using the methods of SSP 74 and for determination of effective stress-strain curves,

strain hardening is neglected.

Figure 2: Model A scantlings


Figure 3: Model B scantlings


3 Cold Bending

3.1 SSP 74

The elasto-plastic collapse analysis described in SSP 74 [1] requires that the cold bending

residual stresses be estimated using an iterative elasto-plastic procedure developed by Mitchell

[2]. This procedure is used to determine the overbend moment and springback curvature

necessary to achieve the required final curvature. The procedure begins by dividing the cross-

section of a single ring frame and one half bay of plating on either side of the web into a number

of horizontal strips. The material of each fibre in the cross-section is assigned an elastic,

perfectly plastic stress-strain behaviour.

The initial estimation of the required overbend moment is related to the maximum elastic moment

and the plastic moment by an approximate function, f(s), representing the variation in moment

from the limiting elastic moment to the fully plastic moment. The overbend moment is given by

[1]:

(1)

where is the initial estimate of overbend moment; Mel is the limiting elastic bending moment

and Mpl is the plastic moment capacity of the section. Assuming that the springback curvature is

linearly related to the elastic curvature, the initial overbend moment is used to calculate the initial

overbend curvature given by:

(2)

where Cob is the overbend curvature, Cr is the final required curvature and Cel is the curvature at

the maximum elastic bending moment. Recognizing that the final required curvature is equal to

the overbend curvature minus the springback curvature, (2) can be arranged to express the

springback curvature, Cs as:

(3)

After springback occurs, the section should be left with zero bending moment; however due to the

approximation in (1), the net internal bending moment will not be in balance with the externally

applied moment after springback, which is zero. This inconsistency is resolved by iteratively

adjusting the springback curvature until the net internal bending moment is zero. A new estimate

of the overbend curvature is then obtained and the process is repeated until Cs and Cob converge.


3.2 Finite Element Cold Bending Simulation

Cold bending residual stresses are also predicted by finite element analysis. Three dimensional

FEA enables calculation of both the longitudinal and circumferential cold bending stress whereas

the iterative method prescribed by [1] only allows the circumferential residual stress to be

calculated. In the finite element simulations of cold bending, the shell plating and stiffener are

bent separately. Separate models are created for the frame and the plating using the ANSYS

SHELL181 element type with reduced integration option. In order to obtain a sufficient through

thickness resolution of residual stress, 21 integration points are used through the thickness of

shell plating and frame flange elements and 5 integrations points through the thickness of the

frame web elements.

Three bay widths of plating were used in the shell plating model in order to reduce end effects

and to reduce anticlastic curvature which is quite pronounced if only one bay width of plate is

used. Reducing the anticlastic curvature is thought to better represent the constraint of a typical

cold bending processes where three rollers are used to progressively shape the part. Both the

plating and frame models have lengths equal to one quarter of the total circumference and are

initially straight in order to reduce analysis times. The overbend curvature is determined using an

iterative procedure where rotations are applied to the nodes of each cross-section along the

circumference of the part to generate the overbend curvature as shown in Figure 4. Rotations

were applied at each cross-section in order to avoid excessive local deformations at the end of the

plate when rotations are only applied at the end cross-section. A small extension of the

frame/plate is added to both ends of both models in the circumferential direction. Rotations are

also applied to these end sections, so that a nearly axisymmetric stress field is generated and end

effects in the circumferential direction are eliminated. Once the overbend curvature is attained,

the rotations are removed and the frame or plate springs back elastically. Due to numerical

instability associated with springback to a new equilibrium position, the cold bending simulation

is run as a “slow dynamic” analysis which uses the Newmark algorithm for solution, allowing the

numerical instability to be overcome.

Figure 4: Frame and plate after cold bending


3.3 Effective Stress-Strain Curves

The effective stress-strain curve technique is used to implicitly account for the effects of cold

bending residual stress by modifying the material stress-strain relationship. Once the residual

stresses are estimated, a frame or plate cross-section is divided into a number of horizontal strips

and incremental strains are applied to the cross-section. For each strain increment, the

corresponding stress is calculated and added to the residual stress in each fibre of the cross

section such that:

� �iyiiiri E )(,)(max �� for �i < 0 (4)

where �i is the total stress in the ith fibre; (�r)i is the residual stress; Ei is the elastic modulus, (�y)i

is the yield stress and �i is the strain in the ith fibre. The average stress, , acting on the cross-

section is then given by:

�i

ii

A

A�� (5)

where Ai is the area of the ith fibre. A stress-strain curve for each structural component (shell

plating, frame web or flange or complete frame) is constructed by applying compressive axial

strains until the entire cross-section yields. Effective stress-strain curves "soften" the material

response to account for the early onset of yielding due to residual stress. Instead of using the

entire cross-section to produce a single effective stress-strain curve, several subsets of fibres can

be selected to produce effective stress-strain curves representing the nonlinear behaviour of the

web and flange separately. Any number of subsets of adjacent fibres can be used to incorporate

the effects of residual stresses in this way. Although effective stress-strain curves should differ

for tensile and compressive loading, most material models for numerical analysis assume the

same material behaviour in both tension and compression. In the case of pressure hulls, the

external pressure results in primarily compressive stress until buckling occurs; therefore the use

of a compressive effective stress-strain curve should adequately represent the material behaviour

accounting for residual stress.


4 Methods of Analysis

The following describes the methods of analysis used to determine the collapse pressure of a

cylindrical stiffened shell. A summary of the methods of used in this study is provided in Table 1.

Table 1: Methods used to predict cold bending residual stresses for the methods of analysis used

in this study

Cold Bending

Residual Stress

Prediction

Method

Method of Analysis

SSP 74

Interframe

SSP 74

Overall

FEA

InterframeFEA Overall

Implicit/Empirical x - - -

Mitchell [2] - x x x

3D FEA - - x x

Effective Stress-

Strain Curves- - x x

None - - x x

4.1 SSP 74 – Design of Submarine Structures

Interframe Collapse

Evaluation of the interframe collapse pressure of a stiffened cylinder begins with calculation of

the Von Mises buckling pressure, Pm1, normalized with respect to the mean circumferential stress

mid-way between frames, Pc5. The interframe collapse pressure, Pc, is then determined from an

empirical curve of the normalized interframe collapse pressure, Pc/Pc5 versus the normalized Von

Mises pressure, Pm1/Pc5. The pressures Pm1 and Pc5, are given by:


(6)

(7)

where E is the modulus of elasticity; h is the of thickness of the shell plating; �yp is the yield

strength of the shell plating; n is the out of circularity mode; a is the radius of the cylinder to the

mid-plane of the shell plating, L is the length of the cylinder; � is Poisson’s ratio and G and � are

parameters defined in Chapter 6 of reference [1]. It is important to note that the empirical data on

which the interframe collapse pressure curve is based is only valid for an out of circularity of

0.5% of the radius a, or less.

Overall Collapse

The overall collapse pressure is taken as the lower of the pressure to cause yielding in the frame

flange or the plating in way of the frame. SSP 74 [1] requires that the overall collapse pressure be

calculated using both an elastic method and an elasto-plastic method. This is because the elastic

approximation is not always conservative if residual stresses are present or if elastic buckling

occurs. In the elastic method, the stress is calculated by combining the direct stress from linear

elastic axisymmetric analysis of an infinite ring-stiffened cylinder and the bending stresses arising

from the initial out of circularity. The calculations are carried out for a single frame including an

effective breadth of attached plating, recognizing that shear lag reduces the effectiveness of the

plating in bending. A range of out of circularity modes is investigated in order to find the

minimum collapse pressure.

The elasto-plastic collapse analysis is carried out using an algorithm developed at DRDC Atlantic

called K79 [12]. The collapse analysis begins with estimation of the cold bending residual

stresses using the method of Mitchell [2]. Once the cold bending residual stresses have been

determined, the collapse pressure for a single frame with one half bay of plating on either side of

the web is determined using the method of Kendrick [13], where the finite difference method is

used to solve the differential equation describing the response of the ring frame to increasing

pressure until a valid solution no longer exists. Bending moments and stresses are corrected to

account for the actual stress distribution in a stiffened cylinder and for the finite length of the

cylinder as specified in reference [1]. As in the elastic analysis, the elasto-plastic analysis is only


valid for a single mode shape and so must be repeated for several out of circularity modes until

the minimum collapse pressure is found.

4.2 Finite Element Analysis

Three dimensional finite element models, created using ANSYS finite element software, are used

to determine both the overall and interframe collapse pressures of a ring stiffened cylinder.

Details on the geometry of the models were given in Section 2. Each model contains

approximately 54,000 elements and 54,000 nodes. All elements are 4-node shells of type

SHELL181 with six degrees of freedom at each node and employ reduced integration to prevent

shear locking and reduce analysis time. Two hundred elements are used around the circumference

with 10 elements between each frame. There are 10 elements through the depth of the frame web

and 6 across the width of the stiffener. For consistency between the cold bending simulation and

the collapse analysis, 21 integration points are used through the thickness of the shell plating and

frame flange and 5 integration points through the thickness of the frame web in all collapse

analyses. Residual stresses due to cold bending are included by mapping the residual stresses

from the cold bending simulation described in Section 2.2 onto the complete model. Cold bending

stresses determined using K79 [12] are also applied to this model in order to compare collapse

pressures predicted using the two methods of residual stress calculation.

Figure 5: Ring stiffened cylinder finite element model


A preliminary elastic analysis is used to apply geometric imperfections in the form of out of

circularity mode shapes where the maximum amplitude of out of circularity is 0.5% of the

cylinder radius. This is done by starting with an initially flat stiffened plate and applying

displacements to each node to produce the required cylinder shape including out of circularity in

the desired mode n. A n = 13 interframe OOC mode with an amplitude of 0.01 times the shell

plating thickness was used in addition to the overall OOC mode for all analyses to enable

interframe collapse. This resulted in a combined maximum OOC slightly larger than 0.5% of the

cylinder radius at mid-bay. The initial elastic analysis allows the nodes to be moved to the correct

locations by specifying displacements and eliminates the need for time consuming equilibrium

iterations where convergence difficulty might be encountered. Following the elastic analysis, the

model geometry is updated so that its new undeformed shape corresponds to the deformed shape

from the initial elastic analysis. Figure 6 shows a model with an exaggerated n = 4 out-of-

circularity created in this manner. For the nonlinear collapse analyses, displacements of all nodes

at one end of the cylinder are constrained. The cylinder is loaded by a uniform external pressure

in addition to nodal loads applied in the longitudinal direction at the axially unrestrained end to

represent the hydrostatic pressure acting on the end closures. With the exception of the models

where effective stress-strain curves are used, the material model incorporates bilinear kinematic

hardening with Von Mises plasticity and associated flow rule. All geometry properties, material

properties, and solution steps are implemented through the ANSYS parametric design language,

allowing the entire analysis to be run from scripted macros.

Figure 6: Ring stiffened cylinder with out of circularity (n =4 OOC mode)


5 Results and Discussion

5.1 Cold Bending Residual Stress

Cold bending residual stresses are predicted using the methods described in Section 3. The flange

and web of the frame are bent together with the shell plating bent separately. Circumferential

residual stresses in Model B, predicted using the method of reference [2] are compared with the

residual stress distribution predicted by the FEM in Figure 7. Results from the two methods of

predicting cold bending residual stress agree well with each other. The small difference between

peak values is attributed to different integration point locations between the two models.

Figure 7: Comparison of cold bending residual stress predicted by K79 and 3D FEA

465

470

475

480

485

490

495

500

505

-600 -400 -200 0 200 400 600

Rad

ial L

ocati

on

(m

m)

Circumferential Residual Stress (MPa)

3D FEA

K79

Flange

Web

Plate


5.2 Effective Stress-Strain Curves

Figures 8 and 9 show the effective stress-strain curves calculated using the method described in

Section 3.3.

Figure 8: Effective stress-strain curves for model A (interframe model)

Figure 9: Effective stress-strain curves for Model B (overall model)

The 'softening' effect of the cold bending residual stresses is observed for all components once the

applied strain reaches approximately one half of the yield strain. The modified material model

allows the web to reach stresses higher than yield due to the net tensile residual stress state that

can be seen in Figure 7.

0

100

200

300

400

500

600

700

800

0.000 0.001 0.002 0.003 0.004 0.005 0.006

Str

ess

(MP

a)

Strain (-)

Shell Plating

Frame

Flange

Web

Stress-Relieved

0

100

200

300

400

500

600

700

800

0.000 0.001 0.002 0.003 0.004 0.005 0.006

Str

ess

(M

Pa)

Strain (-)

Shell Plating

Frame

Flange

Web

Stress-Relieved


5.3 Model A Collapse Analysis

Table 2 provides a summary of collapse pressures predicted using the various methods to analyse

Model A. Separate effective stress-strain curves were used for the frame flange and web for all

effective stress-strain curve analyses.

Table 2: Collapse pressure predictions for Model A (interframe collapse)

OOC

ModeAnalysis Method - Cold Bending RS Method Collapse Pressure (MPa)

2

SSP 74 - Implicit 4.33

FEM - Mitchell 5.41

FEM - FEM 5.35

FEM - ��-� 5.18

FEM - None 5.54

3


FEM - Mitchell 4.95

FEM - FEM 4.97

FEM - ��-� 4.81

FEM - None 5.21

4


FEM - Mitchell 4.95

FEM - FEM 4.98

FEM - ��-� 4.81

FEM - None 5.20

Collapse pressures predicted by the method of SSP 74 [1] are somewhat conservative when

compared with those predicted by finite element analysis. SSP 74 predicts interframe failure at

approximately 80% to 90% of the collapse pressure determined by FEA. This discrepancy is

greater than the ±10% scatter of the empirical data about the mean curve, although it should be


noted that the empirical data are derived from a variety of different sources and consider

internally, externally stiffened and unstiffened cylinders. Comparison of collapse pressures from

the cases with no residual stress and those with cold bending residual stress shows that cold

bending residual stresses reduce the interframe collapse pressure by 3-5%.

Effective stress-strain curves are an effective means of representing the effects of cold bending

residual stress in a finite element model without the need for time-consuming mapping of the

stress field from one model to the other. Interframe collapse pressures predicted using effective

stress-strain curves are within 1-3% of those predicted by explicitly specifying the cold bending

residual stress as an initial stress. Figure 10 shows contours of the displacement vector sum at the

interframe collapse pressure for the n = 3 out of circularity mode shape predicted by the finite

element model with residual stresses predicted by FEA. A comparison of displacement contours

from the effective stress-strain model and the explicit cold bending model considering the n = 3

OOC mode over regular increments of external pressure is shown in Annex A.

Figure 10: Displacement vector sum contour plot for Model A at failure for n = 3 OOC


5.4 Model B Collapse Analysis

Table 3 provides a summary of collapse pressures predicted using the different methods of

analysis for Model B.

Table 3: Collapse pressure predictions for Model B (overall collapse except where noted)

OOC Mode Analysis Method - Cold Bending RS Method Collapse Pressure (MPa)

2

SSP 74 - Elasto-plastic 4.44*

FEM - Mitchell 5.45*

FEM - FEM 5.33*

FEM - ��-� 5.14*/5.14*†

FEM - None 5.53*

3

SSP 74 - Elasto-plastic 3.84

FEM - Mitchell 4.19

FEM - FEM 4.33

FEM - ��-� 4.16/4.29†

FEM - None 4.5

4

SSP 74 - Elasto-plastic 3.18

FEM - Mitchell 3.57

FEM - FEM 3.86

FEM - ��-� 3.65/3.85†

FEM - None 3.99

* Failure occurred in interframe collapse mode

† Separate curves for web and flange / combined curve for web and flange


For the n = 2 OOC mode, all analysis methods predict failure by interframe buckling and follow

similar trends to those discussed in Section 5.2. The increase in overall collapse pressure above

the interframe pressure for the n = 2 OOC mode is consistent with Kendrick's [8] observation that

a finite compartment length is more important for small n than for large n. The SSP 74 method of

analysis provides conservative predictions of overall collapse pressure, predicting collapse

pressures 80-90% of those determined by finite element analysis using finite element predictions

for cold bending residual stress. Cold bending residual stresses are found to decrease the overall

collapse pressure by an amount comparable to the reduction in interframe collapse pressure. Cold

bending residual stresses reduce overall collapse pressures by 3-4%.

The effective stress-strain curve method of accounting for cold bending residual stresses predicts

collapse pressures in good agreement with those predicted using residual stresses from cold

bending simulations. Displacement contours from Model B with a n = 3 OOC mode and using

one stress-strain curve for the complete frame are compared with those from the explicit cold

bending model in Annex A. The use of a single effective stress-strain curve for the complete

frame provides collapse pressures in better agreement with the explicit cold bending simulation

model than the case where separate stress-strain curves are used for the frame flange and web.

The overall collapse pressures predicted when using separate effective stress-strain curves for the

frame flange and web are 4-6% lower than those predicted by the explicit cold bending

simulation. When one effective stress-strain curve is used for the entire frame, the effective

stress-strain curve analysis predicts collapse pressures within 1% of those predicted when cold

bending is explicitly simulated.

Circumferential stresses from Model B are compared at several critical locations for the n = 3

OOC mode in Annex B. When separate effective stress-strain curves are used for the frame flange

and web, yielding of the flange at approximately 400 MPa leads to plastic redistribution of stress

to the web causing a rapid rise in web stress. This is accompanied by a reduction in the effective

moment of inertia of the frame leading to premature failure in the overall buckling mode. This

behaviour is not observed in the case where a single effective stress-strain curve is used for the

entire frame. In that case, the flange is able to maintain a higher circumferential stress (525 MPa)

and yielding does not occur until a higher external pressure is applied. Otherwise, differences

between the stresses at these locations for cold-bent and effective stress-strain models can be

attributed to the differences in the initial residual stress values that are non-zero for the explicitly

cold-bent models and zero for the effective stress-strain models. Based on these observations, it is

recommended that a single effective stress-strain curve is used for the entire frame as opposed to

separate curves for the flange and web.

The reduction in overall collapse pressure due to cold bending residual stress is significantly less

in the present study than reductions predicted by Creswell and Dow [14] who found that cold

bending residual stresses have the potential to lower the overall collapse pressure by as much as

20-30%. They used separate effective stress-strain curves in the frame flange and web to

incorporate cold bending residual stress effects. Using the same approach, this study found that

there is an 8% decrease in overall collapse pressure compared to the stress-free structure.

Creswell and Dow [14] demonstrated that the reduction in collapse pressure due to cold bending

residual stress also depends on cross-section dimensions and both overall and interframe out of

circularity. This may be the reason for the large discrepancy between their results and those of

this study.


6 Conclusions

Both interframe and overall collapse pressures predicted using the methods of SSP 74 were found

to be fairly conservative. SSP 74 under-predicts the interframe collapse pressure by 13-20%

compared with finite element solutions where cold bending is explicitly simulated. Overall

collapse pressures predicted by K79 to implement the elasto-plastic collapse analysis method

prescribed in SSP 74 are conservative and fall 11-18% below finite element predictions where

cold bending is explicitly simulated.

Circumferential residual stresses predicted by K79 using Mitchell's method agree well with those

predicted by simulating the cold bending process in a 3D finite element analysis. Cold bending

residual stresses are found to reduce interframe collapse pressure by as much as 3% and overall

collapse pressure by as much as 4%. Interframe collapse pressures predicted using cold bending

residual stresses predicted by Mitchell's method are similar to those predicted by the finite

element simulation where cold bending is explicitly simulated. Overall collapse pressures

predicted using cold bending residual stresses calculated using Mitchell's method were 3-8%

lower than those predicted by explicitly simulating cold bending. Since Mitchell's method

predicts only circumferential residual stress and the finite element cold bending simulation

provides the complete biaxial residual stress distribution, this result indicates that neglecting the

longitudinal component of residual stress is a conservative assumption.

The use of effective stress-strain curves to incorporate the effects of cold bending in a pressure

hull collapse analysis is efficient and allows collapse pressures to be predicted accurately when

compared with analyses wherein cold bending is explicitly simulated. Models using a single

effective stress-strain curve for the complete frame predict overall collapse pressures in better

agreement with models where cold bending is simulated explicitly than models using separate

stress-strain curves for the frame flange and web. When separate curves are used, the early onset

of yielding in the frame flange causes plastic redistribution of stress and a reduction in the

effective moment of inertia of the cross-section leading to premature failure. Although using

separate curves for the frame flange and web is less accurate, collapse pressure predictions are

conservative and fall below those predicted by explicit cold bending simulation by 1-3% for

interframe collapse and 3-4% for overall collapse. It is possible that further refinement of the

effective stress-strain curve method by using several effective stress-strain curves for multiple

fibres through the thickness of the flange may result in collapse pressures in better agreement

with the predictions where cold bending is explicitly simulated than using a single effective-stress

strain curve for the flange.


References

[1] Defence Procurement Agency (2001). SSP 74 Design of Submarine Structures. Sea

Systems Publication No. 74. Defence Procurement Agency, Sea Technology Group, United

Kingdom.

[2] Mitchell, G.C. (1986). Overbend Prediction for Cold-Bent Beams. Computers and

Structures, 24(2), 187-196.

[3] Graham, D. (2007). Predicting the Collapse of Externally Pressurised Ring-Stiffened

Cylinders using Finite Element Analysis. Marine Structures, 20, 202-217.

[4] Von Mises, R. (1929). Stodola Festschrift.

[5] Kendrick, SB. (1953). The Buckling Under External Pressure of Circular Cylindrical Shells

with Evenly Spaced, Equal Strength, Circular Ring Frames - Part I. Naval Construction

Research Establishment Report R211.


with Evenly Spaced, Equal Strength, Circular Ring Frames - Part II. Naval Construction



with Evenly Spaced, Equal Strength, Circular Ring Frames - Part III. Naval Construction


[8] Kendrick, SB. (1964). Structural Design of Submarine Pressure Vessels. Naval

Construction Research Establishment Report NCRE R483.

[9] Bryant, A.R. (1954). Hydrostatic Buckling Pressure of a Ring-Stiffened Tube. Naval

Construction Research Establishment Report R306.

[10] Kendrick, S.B. (1977). The Elasto-Plastic Collapse of Ring Stiffened Cylinders. Naval

Construction Research Establishment Report R653.

[11] Bayley, C. (2007). Stress-Strain Characterization of NQ1 Pressure Hull Material. DRDC

Atlantic TN 2007-328. Defence Research and Development Canada.

[12] Smith, M.J. and MacKay, J.R. (2004). Overall Elasto-Plastic Collapse of Submarine

Pressure Hull Compartments. (DRDC Atlantic TM 2004-243). Defence Research and

Development Canada.

[13] Kendrick, S. (1979). The Influence of Shape Imperfections and Residual Stresses on the

Collapse of Stiffened Cylinders. Conference on Significance of Deviations from Design

Shape, 25-35. the Institute of Mechanical Engineers.


[14] Creswell, D.J. and Dow, R.S. (1986). The Application of Nonlinear Analysis to Ship and

Submarine Structures. Advances in Marine Structures, Eds. C.S. Smith and J.D. Clarke,

175-200.


Annex A Displacement Contours

A.1 Contours

Figure A-1 through A-6 show contours of the displacement vector sum at regular pressure

increments for Model A with an n = 3 OOC mode. Results of the effective stress-strain curve

method and results from the explicit simulation of cold bending by FEA are compared.

Displacements differ at failure because the failure loads differ slightly and near failure,

displacements increase significantly for a small increase in applied load.

Figure A-1: Model A displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 1 MPa (mm)



Figure A-3: Model A displacement vector sum contours from cold bending simulation (left) and

effective stress-strain curves (right) at 3 MPa (mm)



effective stress-strain curves (right) at 4.5 MPa (mm)



effective stress-strain curves (right) at failure (mm)

Figures A-7 through A-11 show contours of the displacement vector sum at regular pressure

increments for Model B comparing the results of the effective stress-strain curve method and

results from the explicit simulation of cold bending by FEA.

Figure A-7: Model B displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 1 MPa (mm)





Figure A-11: Model B displacement vector sum contours from cold bending simulation (left) and

effective stress-strain curves (right) at failure (mm)


Annex B Model B Stress History

Figures B-1 through B-4 show the stress at critical locations for increasing external pressure for

Model B considering an n = 3 OOC mode. The cases where separate effective stress-strain curves

for the flange and web and a single stress-strain curve for the entire frame are compared with the

results of the analysis where cold bending is explicitly simulated.

Figure B-1: Stress history at the flange mid-width for a frame at mid-length, top of cylinder

-600

-500

-400

-300

-200

-100

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Cir

cum

fere

nti

al

Str

ess

(MP

a)

External Pressure (MPa)

Cold Bending

Effective Stress-Strain

Frame Effective Stress-Strain


Figure B-2: Stress history at the web mid-height for a frame at mid-length, top of cylinder

Figure B-3: Stress history for shell plating directly above a frame at mid-length, top of cylinder

-800

-700

-600

-500

-400

-300

-200

-100

0

100

200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Cir

cum

fere

nti

al

Str

ess

(MP

a)


Cold Bending



-400

-300

-200

-100

0

100

200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Cir

cum

fere

nti

al

Str

ess

(MP

a)


Cold Bending




Figure B-4: Stress history for shell plating at mid-length of cylinder (mid-bay)

-700

-600

-500

-400

-300

-200

-100

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Cir

cum

fere

nti

al

Str

ess

(MP

a)


Cold Bending






List of symbols/abbreviations/acronyms/initialisms

Symbols

a Radius of cylinder to mid-plane of shell plating

Ai Area of ith fibre

Cel Limiting elastic curvature

Cob Overbend curvature

Cr Required curvature

Cs Springback curvature

E Modulus of elasticity

Ei Elastic modulus of ith fibre

f(s) Function representing variation in moment from limiting elastic to plastic

moment

h Thickness of shell plating

L Length of cylinder

Mel Limiting elastic bending moment

Mob Overbend moment

Mpl Plastic bending moment

n Out of circularity mode shape

Pc Interframe collapse pressure

Pc5 Mean circumferential stress between frames

Pm1 Von Mises buckling pressure

� Strain

�i Strain in ith fibre

µ Poisson's ratio

� Stress

Average stress

�i Total stress in ith fibre

��r)i Residual stress in ith fibre

��y)i Yield stress of ith fibre

�yp Yield stress of shell plating


Abbreviations, acronyms and initialisms

DND Department of National Defence

DRDC Defence Research and Development Canada

DRDKIM Director Research and Development Knowledge and Information

Management

FEA Finite element analysis

FEM Finite element method

MoD Ministry of Defence

OOC Out of circularity

R&D Research & Development

RS Residual stress

SSP Sea Systems Publication


Distribution list

Document No.: DRDC Atlantic TM 2010-065

LIST PART 1 – Internal Distribution by Centre:

4 Author (2 hardcopies, 2 CDs)

3 DRDC Atlantic Library (1 hardcopies, 2 CDs)

1 Scientific Authority, Submarine Scientific Support SLA (Mr. John Porter)

1 Project Manager, Submarine Scientific Support SLA (LCdr Wade Temple)

9 TOTAL LIST PART I

LIST PART II: External Distribution within Canada by DRDKIM

1 NDHQ/DRDKIM 3

2 NDHQ/DMEPM(SM) 4

2 NDHQ/DMEPM(SM) 4-2

1 Library & Archives Canada (Attn: Military Archivist, Government Records Branch)

6 TOTAL LIST PART II

15 TOTAL COPIES (3 hardcopies, 12 CDs)



DOCUMENT CONTROL DATA(Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified)

1. ORIGINATOR (The name and address of the organization preparing the document.

Organizations for whom the document was prepared, e.g. Centre sponsoring a

contractor's report, or tasking agency, are entered in section 8.)

Defence R&D Canada – Atlantic9 Grove StreetP.O. Box 1012Dartmouth, Nova Scotia B2Y 3Z7

2. SECURITY CLASSIFICATION (Overall security classification of the document

including special warning terms if applicable.)

UNCLASSIFIED

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Gannon, L

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April 2010

6a. NO. OF PAGES(Total containing information,

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13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable

that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification

of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include

here abstracts in both official languages unless the text is bilingual.)

Submarine pressure hull frames and shell plating are shaped by cold bending during fabrication.

Cold bending introduces significant residual stress in these components which can be

detrimental to the strength of the structure. This study evaluates different methods of

incorporating cold bending residual stresses in the analysis of pressure hulls considering

different out of circularity mode shapes. Several methods of pressure hull collapse analysis are

compared considering interframe and overall collapse modes. These include an empirical

method, a finite difference method and the finite element method.

Collapse pressures predicted using the methods prescribed in the UK MoD submarine structure

design standard, SSP 74, are found to be conservative when compared with results from finite

element analysis. Collapse pressures predicted using effective stress-strain curves to incorporate

the effects of cold bending residual stress in finite element models agree well with those

predicted by explicitly modelling the cold bending process. This indicates that the use of

effective stress-strain curves is an acceptable means of accounting for the influence of cold

bending residual stress on the collapse pressure of a submarine pressure hull.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be

helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model

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published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select

indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

submarine structure; cold bending; residual stress; pressure vessel; nonlinear finite element

analysis; ring stiffened cylinder


Documents

Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapsecradpdf.drdc-rddc.gc.ca/PDFS/unc104/p534276_A1b.pdf · 2012-08-03 · Prediction of the Effects of Cold