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UDC534.832.08:624.

Michael Willford and Peter Young have over 30 years combinedexperience in the area of structural dynamics. The methodspresented in this guide have been developed by them over the last10 years and have been used extensively within Arup for the past 5 years.

A Design Guide for Footfall Induced Vibration of Structures

A Design GuInduced VibA tool for designers to engineer the footfall vibratcharacteristics of buildings or bridges

M R Willford CEng MIMechEP Young CEng MIMechEWhilst footfall induced vibrations on buildings or bridges

is not normally signicant in terms of structural integrity,footfall vibration can be a critical serviceability condition.This publication presents a new method for evaluatingthe vibration due to a single pedestrian walking on a atsurface, such as a oor slab or bridge deck. The method wasdeveloped by Arup, and has been calibrated and rened withverication measurements taken on completed structuresover a period of ten years.

This publication provides an informative description of the factorseffecting footfall induced vibration and guides the engineer throughthe process of designing for vibration. It includes owcharts forcalculation procedures and a useful glossary. It also includes workedexamples on a concrete footbridge, a low frequency multispan posttensioned concrete oor and a high frequency ribbed slab.

CCIP-016Published December 2006ISBN 1-904482-29-5Price Group Q© The Concrete Centre

Riverside House, 4 Meadows Business Park,Station Approach, Blackwater, Camberley, Surrey, GU17 9ABTel: +44 (0)700 4 500 500www.concretecentre .com

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C C I P - 0 1 6

A D e s i gn G ui d ef or F o o t f a l l I n d u c e d Vi b r a t i on of S t r u c t ur e s

Footfall Induced Vibration cover1 1Footfall Induced Vibration cover1 1

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Published for The Concrete Centre by The Concrete Society

CCIP-016Published November 2006ISBN 1-904482-29-5 Price Group Q© The Concrete Centre

The Concrete CentreRiverside House, 4 Meadows Business Park, Station Approach, Blackwater, Camberley, Surrey GU17 9ABTel: +44 (0)1276 606800 Fax: +44 (0)1276 606801 www.concretecentre.com

CCIP publications are produced byThe Concrete Society (www.concrete.org.uk) on behalf ofthe Cement and Concrete Industry Publications Forum – an industry initiative to publish technicalguidance in support of concrete design and construction.

CCIP publications are available from the Concrete Bookshop atwww.concretebookshop.com Tel: +44 (0)7004 607777

All advice or information from The Concrete Centre is intended for those who will evaluate the signicance and limitations of itscontents and take responsibility for its use and application. No liability (including that for negligence) for any loss resulting fromsuch advice or information is accepted by The Concrete Centre or its subcontractors, suppliers or advisors. Readers should notethat publications are subject to revision from time to time and should therefore ensure that they are in possession of the latest

version.

Printed by Cromwell Press, Trowbridge, UK.

AcknowledgementsThe authors would like to acknowledge the contribution of their colleagues at Arup to the validationof the methods described in this guide, in particular Caroline Field, Kathy Gubbins, Kubilay Hicyilmazand Mark Arkinstall. Professor Aleksander Pavic of Shefeld University also provided additionalvalidation data from his own independent research projects. The authors would like to thank DrStuart Kerr for providing the data which is the basis of the footfall forces used here.

Finally, special thanks are due to Professor Tom Wyatt of Imperial College London, who was theindependent peer reviewer for this document.

Footfall Induced Vibration cover2 2Footfall Induced Vibration cover2 2

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A Design Guide for Footfall InducedVibration of Structures

Contents

Nomenclature 2

1. Introduction 5

2. Understanding footfall induced vibration 6

3. Quantifying vibration 13

4. Predicting footfall induced vibration of structures 22

5. Worked examples 41

References 61

Glossary 62

Appendix A. Evaluating modal properties of a structure 66

Appendix B. Procedures for measuring oor response 75

Appendix C. Validation 79

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Nomenclaturea momentary accelerationat total acceleration response in all modesah acceleration due tohth harmonicareal,h aimag,h real and imaginary parts of the acceleration due to thehth harmonicah,m acceleration due tohth harmonic inmth modeareal,h,m aimag,h,m real and imaginary parts of the acceleration in themth mode due to

the hth harmonicb shorter dimension of ribC, C x , C y torsional rigidity of ribs, torsional rigidity of ribs in shorter spanning

direction, torsional rigidity of ribs in longer transverse direction

d longer dimension of ribD x exural rigidity in the shorter spanning directionD y exural rigidity in the longer transverse directionD xy slab torsional rigidityDLF dynamic load factorE Young’s modulusF h harmonic forcef , f m frequency, frequency of modem (Hz)f n natural frequency (Hz)f w walking frequency (paces per second) (Hz)f 0 natural frequency of bay assumed to be one-way spanning onlyh harmonic numberG shear modulusH torsional stiffness parameterIeff effective impulse j number of half sines in the longer, transverse directionk number of half sines in the shorter spanning directionK f rst mode frequency correction factor for a two-way spanning slabK fm frequency correction factor for slab with back spanK im multiplier on rst mode impulsive response to account for other

modesK mm modal mass correction factor for a slab with back spanK rm multiplier on rst mode resonant response to account for other modes

l stride lengthL spanm mass per unit length or mass per unit aream, mm modal mass, modal mass of themth modeN number of footstepsP static weight of walkerr ratio of the natural frequency of a structure,f n, to walking speed,f w R, Rh response factor, response factor due tohth harmonicRMS root mean square s beam spacingt time

2


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Nomenclature

t slab thickness

T period of vibrationT a RMS averaging periodv momentary velocityv peak velocityv m velocity response in themth modev m peak velocity in themth modeVDV vibration dose valueW width x momentary position or displacement x when referring to a direction or property of a oor plate, the sufx x

denotes the transverse (or wider) direction of the oor (see Figures4.8 and A2)

x RMS , v RMS , aRMS RMS displacement, velocity or acceleration x R = 1 v R = 1 aR = 1 displacement, velocity or accelerations corresponding toR = 1 X displacement amplitude y when referring to a direction or property of a oor plate, the sufx y

denotes the spanning (or shorter) direction of the oor (see Figures4.8 and A2)

ζ , ζ m modal damping, modal damping of modem (fraction of criticaldamping or percentage of critical damping)

μ mode shape valueμe , μr mode shape value at excitation point, mode shape value at response

pointμe,m μr,m mode shape values at excitation and response points inmth modev Poisson’s ratioρ , ρh,m resonant correction factor, resonant correction factor for thehth

harmonic and themth modeω angular frequency, radians per second


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5

Introduction

1. IntroductionThis guide describes a reliable methodology for predicting the vertical vibration inducedby pedestrians crossing structures such as oors and bridges.

Many methods already exist1, 2, 3,4, 5 but each of these is signicantly limited in some respect.A full discussion of these limitations can be found elsewhere6, 7.

The method presented here addresses all of the issues described below in a consistentmanner:n It is applicable to any type of structure on which people walk, including floors and bridges.n It is applicable to structures of any form or construction material, e.g. steel, composite,

reinforced or pre-stressed concrete or timber structures, and enables reliablecomparisons to be made between designs of different forms and materials.

n Complex irregular structures can be assessed as reliably as simple regular ones.n The footfall forcing functions recommended are based on a very extensive set of

measured data, and these loads and the likelihood of their occurrence have beenstatistically quantified.

n The methodology has been extensively validated and independently peer reviewed.n The method has been used routinely and regularly on design projects around the world

for the past five years, and its accuracy is validated by many measurements ofcompleted structures.

n It interfaces well with modern engineering design methods and software packages.

The method described in this guide is based on the well-established principles of modalanalysis. This enables rst principles calculations to be made, and unlike most other methods,it does not require the introduction of arbitrary or empirical factors. This makes it a robustapproach for the assessment of any type of structure of any construction material.

The method is appropriate for calculating the vibration caused by a single pedestrianwalking on a at surface on any structure which is signicantly heavier (by at least a factorof 10) than the individual. Whilst very simple regular structures can be assessed entirely byhand or spreadsheet calculation, it is envisaged that the method will be used principallyin conjunction with nite element analysis as a means of estimating the modal propertiesof oor and bridge structures. Whilst this might be seen as an added complexity, in practicethe additional accuracy that nite element analysis of less regular structures brings to the

assessment more than outweighs the modest additional effort associated with buildingand analysing a model.

The method was developed within Arup and rened by reference to the measured per-formance of completed structures over a period of ten years. It has been independentlyreviewed by Professor T A Wyatt of Imperial College London. It is published here as aCement and Concrete Industry publication but it is applicable to structures of any con-struction material.


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6

-800

-600

-400

-200

0

200

400

600

800

D y n a m

i c f o r c e

( N )

0 0.5 1.0 1.5 2.0

Time (secs)

Force time history

b10105 (757 N)

-800

-600

-400

-200

0

200

400

600

800

D y n a m

i c f o r c e

( N )

0 0.5 1.0 1.5 2.

Time (secs)

Force time history

a50503 (1023 N)

-800

-600

-400

-200

0

200

400

600

800

D y n a m

i c f o r c e

( N )

0 0.5 1.0 1.5 2.

Time (secs)

Force time history

820n05 (701 N)

-800

-600

-400

-200

0

200

400

600

800

D y n a m

i c f o r c e

( N )

0 0.5 1.0 1.5 2.

Time (secs)

Force time history

126n15 (925 N)

2. Understanding footfall induced vibrationWhen people walk on oors, staircases, bridges or other structures they apply dynamicforces that cause the structures to vibrate. The vibration displacement amplitude isnormally very small (micrometres rather than millimetres) and is not perceptible visuallyor signicant in terms of structural integrity. However it can often be felt by other peopleand, if excessive, can be distracting and detrimental to the perceived quality of the structure.Vibration can also impair the function of sensitive equipment in buildings such aslaboratories and precision manufacturing facilities.

In order to assess how a structure responds to forces induced by walking persons it isnecessary to understand:n The forces applied to the structure.n The dynamic properties of the structure (mode shapes, natural frequencies, damping etc.).n The way in which the structure responds to the applied forces.

Each of these is discussed in the following sections.

The dynamic forces applied to the ground by a walking person can be measured directlyby walking experiments using instrumented force plates or platforms. Four examplemeasured footfall time histories11 are shown in Figure 2.1. These show the total dynamic

2.1 Footfall forces

Figure 2.1Examples of measured footfall

force time histories.


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Understanding footfall induced vibration

component of the force applied to a structure by both feet during the period of several

footfalls. In normal walking there are typically between 1.5 and 2.5 steps per second(average about 1.8) and so the period of one footfall is typically just over 0.5 seconds. Thestatic weight of the individual has been subtracted from the measurements as it does notvary with time and so does not induce any dynamic response.

The four time histories are very different in both shape and magnitude. Extensive research10 has shown that factors which contribute to this variation include height, gender, footwear,weight, walking surface and walking speed. Given the extent of this variability, there is nosingle ‘correct’ force that can be used to calculate structural response, but it is insteadappropriate to adopt values that are statistically representative and have an arbitrary butknown probability of exceedence.

In this guide, mean dynamic forces and their coefcients of variation are presented whichhave been derived from 882 measured footfall time histories10. Design forces are proposedwhich have a 25% probability of exceedence.

All structures have what are known as natural modes of vibration, and each of these modeshas a unique spatial distribution of displacement (known as the mode shape), naturalfrequency of vibration, modal mass and modal damping associated with it. The naturalmodes are a structure’s ‘preferred’ patterns of vibration – the modes in which it will vibrateif excited by a sudden impact. These concepts are dened formally in the glossary andillustrated by example below.

Consider the short bridge shown in Figure 2.2a, idealised as a uniform simply supportedbeam. Some of its lowest modes of vibration could be excited by someone performing asingle jump at mid-span. This would cause the bridge to vibrate, and the displacementhistory at mid-span in the rst mode would be as shown in Figures 2.2b and 2.2c. Thevibration displacement, velocity and acceleration all vary sinusoidally with time at thenatural frequency of the mode, which depends on the stiffness, mass and span of thebridge. The amplitude of vibration decays with time at a rate dependent on the damping,but during this decay the frequency and spatial distribution of the vibration do notchange. Neither the frequency nor mode shape of the vibration depends on how high theperson jumps (assuming linear elastic response) – but the initial amplitude of vibration is

affected by the jump height.

The motion in one vibration mode following a single impact is known as damped simpleharmonic motion, and a rigorous mathematical description of this can be found indynamics textbooks11.

In fact, the bridge has many modes of vertical vibration, the rst three of which are shownin Figure 2.3. A mode can’t be excited by forces applied at a position of zero displacementin the mode. In this case, therefore, whilst the rst and third modes would be excited by a jump at mid-span, the second would only respond if the person jumped elsewhere on the

2.2 Dynamic properties of astructure


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bridge. Jumping at the quarter point, for instance, would excite all three modes, but would

excite the rst mode (also known as the fundamental mode) less effectively than would a jump at mid-span.

Figure 2.2A simply spanning bridge excited impulsively

at mid-span.(a) A bridge as a simply supported beam

(b) The rst mode of v ibration

(c) The displacement of mid-span after

someone jumps once at mid-span

Figure 2.3The rst three bending modes of a simplysupported beam.

The methodology described in this guide requires an assessment of the modal properties(that is, frequency, mode shape, modal mass and damping) of all the relevant modes ofthe structure. Details of how this can be done are described in Appendix A. Whilst for somesimple and regular structures (e.g. uniform beams, regular simply supported oor bays)analytical solutions can be used, the easiest general method is nite element analysis,which is now widely accessible in design ofces.

Mode 1

Mode 2

Mode 3


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Rather than jumping on the bridge, the person described above is more likely to want to

walk across it. In so doing, he or she applies a periodically varying dynamic force (such asshown in Figure 2.1) to the bridge, and the bridge responds dynamically to this force timehistory. Initially, let us assume that the bridge is long and that the point of application ofthe load can be considered as stationary at mid-span (i.e. the pedestrian is effectivelywalking on the spot at the middle of the bridge).

The response of the bridge to the repeated application of this force depends on severalfactors, the most important being:n the stiffness and mass of the bridge.n the damping of the bridge.n the ratio (r ) of the bridge’s natural frequency (f n) to the pacing frequency (f w ), as given

in Equation 2.1:

(2.1)

The inuence of the frequency ratior can be illustrated using one of the force timehistories described in section 2.1 above. Consider one in which the person is walking at2.0 footfalls/second (Hz) and assume that the bridge has a modal mass of 10 tonnes,damping of 1% of critical and that the natural frequency of the bridge is either 3Hz or4Hz. As natural frequency is proportional to √(stiffness/mass) and the masses of the twobridges are identical, the 4Hz bridge must be about 77% stiffer than the 3Hz bridge. Thevelocity time histories calculated at the centre of the bridge are as shown in Figure 2.4.

2.3 Structural response tofootfall forces

r = f n

f w

0 1 2 3 4 5 6 7 8 9 10−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10−3

Time [secs]

V e l o c i t y

[ m / s ]

Bridge response at midspan, 1% damping

f n = 3Hz

f n = 4Hz

Figure 2.4Velocity time histories calculated at mid-spanfor a person walking at 2.0 steps per second

on a 3Hz and a 4Hz bridge.


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It is seen that the vibration response of the two structures is signicantly different. For the

4Hz bridge, the vibration builds up over time such that the response after several footfallsis signicantly greater than after the rst. This is a ‘resonance’ phenomenon, in which thevibrations induced by each footfall reinforce the vibration generated by previous footfalls.For the 3Hz bridge this is not the case, and the response is similar at each footfall. Themaximum response of the 4Hz (stiffer) bridge is signicantly greater than that of the lessstiff, but otherwise identical, structure.

A natural extension of this analysis is to calculate such velocity time histories for manymore values ofr and to graph the variation of maximum response velocity withr . This issimilar to the development of a velocity ‘response spectrum’ under seismic excitation.Figures 2.5a and 2.5b below show the results of such analysis, averaged across all of the 882measured force time histories (rather than just the one mentioned above) for a 10 tonnestructure with 3% of critical damping.

0 2 4 6 8 10 120

1

2

3

4

5

6x 10

−3 Peak velocities, 3% damping

r = f n / f

w

P e a

k v e

l , [ m / s ]

0 2 4 6 8 10 120

1

2

3

4

5

6x 10

−4 Peak velocities, 3% damping

r = f n / f w

P e a

k v e

l , [ m / s ]

Figure 2.5aAveraged peak velocity response spectrum for

882 footfall force time histories for r > 0.6.

Figure 2.5bAs Figure 2.5a, but showing more detail for

r > 1.5.


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In these graphs it can be seen that, in general, the vibration response reduces asr (the

ratio of a structure’s modal frequency to footfall rate) increases. However, there are ‘peaks’in the response spectra at particular ‘whole number’ values ofr . Ifr is close to 1.0 then thefootfall rate equals the natural frequency of the mode, and the mode responds stronglyin resonance to the ‘rst harmonic’ of the footfall forces. Ifr is close to 2.0, the moderesponds resonantly to the ‘second harmonic’ of the footfall forces, and so on. Theseresonances lead to signicantly greater response whenr is close to 1.0, 2.0, 3.0 or 4.0than arises for intermediate values ofr .

Forr > 4 there is much less sensitivity to the exact value ofr . Inspection of the timehistories shows that the response does not build up over time, and the response to eachindividual footfall is comparable. At high values ofr the velocity response is characterisedby an initial peak response (such as that produced by a single impulse) followed by a de-caying sinusoid similar to that illustrated in Figure 2.2c (and 3.2). The non-resonant responsecan therefore be treated as repeating impulsive responses to individual foot impacts.

Figure 2.6 shows the effect of varying the damping of the structure. It can be seen that themagnitude of resonant response for whole number values ofr is strongly dependent onboth the damping of the structure and the harmonic number. Impulsive response (that is,away from ‘whole number’ values ofr and forr > 4) is much less sensitive to damping.

Figure 2.6aAverage peak velocity response spectrum for

different values of damping.


0 2 4 6 8 10 120

0.002

0.004

0.006

0.008

0.01

0.012

0.014Peak velocities

r = f n / f w

P e a

k v e

l [ m / s ]

1% damping3% damping5% damping


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The above generic analyses show that if a person can walk at a footfall rate that is a factorof 1, 2, 3 or 4 lower than the natural frequency of a mode, then resonance is possible, andthis will lead to vibration levels greater than walking at slightly different footfall rates. Forvalues ofr greater than approximately 4.2, there is a general trend of decreasing responsewith higher structural frequency, but there is little evidence of resonant response and farless sensitivity to the exact value ofr .

As footfall rates typically vary between 1.5Hz and 2.5Hz, then any structure with modalfrequencies between 1.5Hz and 10.5Hz (~4.2 × 2.5Hz) is potentially susceptible to the higherresponses due to resonance. For structural modes with natural frequencies sufciently highthat they cannot be excited by the fourth harmonic (i.e. natural frequencies greater thanapproximately 10.5Hz), then vibrations will not signicantly exceed the impulsive responsedue to a single footfall. If the damping is very low, then in theory resonance to the fthand sixth harmonics could produce a higher response, but this is not usually of practicalconcern because successive footfalls are not completely identical, and this reduces themagnitude of the higher harmonics.

Many structures have several modes that may simultaneously experience signicantresponses to footfalls, and practical prediction methods must allow for this. Whilst thedetermination of modal properties of the structure is the same, it is convenient to havedifferent approaches for the calculation of resonant and impulsive vibration responses.Therefore structures with vertical natural frequencies less than 4.2 times the fastest walkingfrequency should be designed for resonant response using the methods described insection 4.2 below, and structures with all vertical natural frequencies higher than 4.2times the fastest walking frequency should be designed for impulsive response using themethods described in section 4.3.

Figure 2.6b

Peak velocities.

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

x 10−3 Peak velocities

r = f n / f

w

P e a

k v e

l [ m

/ s ]

1% damping3% damping5% damping


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Quantifying vibration

3. Quantifying vibrationBefore attempting to predict vibration levels, it is necessary to understand how vibration isdescribed quantitatively. Generic terms, and those that refer specically to footfall inducedvibration, are described below. Typical performance targets are also given for structures ofdiffering usage.

At least two parameters are required to dene vibration quantitatively. Formal denitionsof some of these are included in the Glossary but the metrics generally refer either to themagnitude of vibration (e.g. the amplitude of motion) or thecyclic rate at which it occurs(e.g. the frequency).

Consider rst a system which vibrates such that its displacement is a sinusoidal functionof time, as shown in Figure 3.1. This is known as simple harmonic motion. The relationshipbetween displacement ( x ) and time (t ) is given in Equation 3.1:

x = X sinωt (3.1)

3.1 Generic measures ofvibration

x

Time

T

Figure 3.1Displacement time history for vibration with a

period T and an amplitude X.

This vibration has a period ofT and a frequency ( f ) of 1/T cycles per unit time. IfT is mea-sured in seconds then the unit of frequency is Hertz (Hz) andf is the number of vibrationcycles per second. Frequency is sometimes expressed asangular frequency (ω) which isthe number of radians per second; there are 2π radians in a cycle. Period, frequency andangular frequency are therefore related by the following equations:

ω = 2πf (3.2)f = 1

T


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Theamplitude of the vibration is the maximum displacement from rest. The time history

shown in Figure 3.1 has an amplitude of X .

It is more common in the context of oor vibration to talk about velocity (v ) and accele-ration (a) than displacement. Velocity is the rst derivative of displacement with respectto time, and acceleration is the derivative of velocity with respect to time. For simpleharmonic motion, as shown in Figure 3.1, the derivatives of the displacement function are:

v = Xω cosωt a = – Xω2 sinωt (3.3)

and so the amplitudes of the velocity and acceleration are Xω and Xω2 respectively. Thefrequency of the velocity and the acceleration are the same as those of the displacement.

A single value of amplitude and frequency completely dene simple harmonic motion.However, the measured velocity of a oor excited by walking may be like that shown inFigure 3.2, and clearly neither the amplitude nor the frequency are constant.

Figure 3.2A measured velocity time history for a

pedestrian walking at 100 paces per minute.


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The peak velocity amplitude (0.12mm/s in this case) can be a useful measure in certain

circumstances, but may not be very representative of the vibration as a whole if, for instance,one isolated peak is signicantly higher than the rest of the signal. A measure of the averageamplitude is theroot mean square, or RMS, evaluated over a certain averaging period,T a.The RMS response is calculated using Equation 3.4 and involves rst squaring the velocityat every instant (making all values positive), nding the average of the squared valuesover the specied time interval, and taking the square root of this average. For simpleharmonic motion the RMS value is equal to 1/√2, or 70.7% of the peak value. For themeasured trace in Figure 3.2, the RMS averaged over the 4 second period is 0.036mm/s,which is approximately 30% of the peak value:

(3.4)

3.2 Human response tovibration

Where human perception of vibration is the primary concern (as opposed to the operationof sensitive equipment) vibration performance is sometimes expressed in terms of peak orRMS acceleration or velocity. Sometimes a ‘response factor’ is specied, and sometimesthe ‘Vibration Dose Value’, or VDV, is used. The background to these parameters is out-lined in the following section.

People are more sensitive to vibration at some frequencies than at others. BS 6472 8 species base curves for human perception of continuous vibration which depend on thefrequency and the orientation of the vibration relative to the axes of the human body. Forthe head-to-toe axis ( z ), the RMS acceleration level corresponding to the averagethreshold of perception is plotted in Figure 3.3. In the most sensitive frequency range,between 4Hz and 8Hz, the criterion is one of constant acceleration. Above 8Hz thecriterion is one of constant velocity, which appears on Figure 3.3 as an accelerationincreasing linearly with frequency.

3.2.1 Perception of vibration

= 1 T a ∫ 2dttt+T a


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Figure 3.3

Baseline RMS acceleration in the head-to-toe( z) axis.

The response factor (R) is simply a multiplier on the level of vibration at the average thres-hold of human perception. Therefore a response factor of 1 represents the magnitude ofvibration that is just perceptible by a typical human, a response factor of 2 is twice that,and a response factor of 8 is eight times that.

If the vibration is predominantly at one frequency, then the baseline RMS acceleration for

R = 1 can be found from Figure 3.3, and the response factor is calculated simply by dividingthe calculated (or measured) RMS acceleration by the baseline RMS acceleration at thatfrequency. If the vibration contains a range of frequencies, then a rigorous calculation ofthe response factor involves passing the predicted or measured acceleration time historythrough a band pass weighting lter before evaluating the RMS value. This process isdescribed in Appendix B and is particularly useful for determining the response factor frommeasured data.

3.2.2 Response factor

Building vibration Z-axis Base curve for acceleration (RMS)

Frequency/Hz

A c c e l e r a t

i o n

( R M S

) / m / s 2


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The response factor corresponding to footfall induced vibration predicted by analysis may

be estimated in the following ways.

Resonant responseEvaluate the acceleration response to each of the rst four harmonic forces and calculateresponse factors for each using the base value forR = 1 from Figure 3.3 for each of thefour frequencies. Then calculate the total response factor as the ‘square root sum of thesquares’ of the four (Equation 4.8). This procedure is explained in section 4.2.2.

Impulsive responseCalculate the total RMSvelocity over a 1-second averaging period (which will be approxi-mately equal to that evaluated over the period of one footstep) and convert this to aresponse factor using the RMS base value forR = 1 for the frequency of the dominant modein the response. This dominant mode will often be the fundamental mode. This procedureis explained in section 4.3.2. Usually the impulsive response calculation will be performedfor oors having natural frequencies above 8Hz where the constant velocity criterion applies(v R=1 = 1 x 10–4 m/s RMS).

The response factorR is a measure of perceptibility relative to the threshold of perceptionof humans tocontinuous vibration of constant amplitude. However, footfall induced vibrationis never of constant amplitude and seldom continuous and these factors (i.e. distributionof vibration amplitude and how often it occurs) affect people’s acceptance of vibration.The ‘Vibration Dose Value’, or VDV, is a measure of vibration over a longer period of timethat endeavours to account for these factors. It is dened in Appendix B of BS 6472 8 (Equation 3.5 below), and can be readily measured using standard equipment and simplesignal processing. The VDV is based on the fourth power of acceleration, which means thatthe magnitude of vibration is weighted much more than the duration. For example,doubling the vibration amplitude will double the VDV, but doubling the duration willresult in an increase of just 19%.

(3.5)

In order to predict a VDV at the design stage, the engineer needs to assess how oftenvibration of particular levels will occur and for how long the ‘recipient’ is likely to be exposed

to it.

Human acceptance of vibration is very subjective. A level of vibration that causes oneindividual to complain might be unnoticed by another. Similarly, vibration that causesconcern or distraction for an individual sitting in a quiet ofce could be quite acceptable tothe same person walking around a shopping centre. Typical current vibration acceptancecriteria attempt to take this subjectivity into account by specifying different targets fordifferent environments.

3.2.3 Duration effects

3.3 Vibration performancetargets

3.3.1 Background toperformance targets for

human acceptance

VDV =( ∫ a4(t) dt)0.25T 0


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When assessing the dynamic performance of a oor or similar structure at the design stage,

it is common practice to calculate the maximum representative vibration level that islikely to occur and to compare this to acceptance criteria. Measurements on completedoors are conventionally assessed in a similar manner, e.g. oor response is recorded for asignicant period and the worst 1 second of response is used as a basis for compliance.

BS 6472 8sets out criteria in terms of levels ofcontinuous vibration at which the probabilityof adverse comment is low . These are summarised in Table 3.1 as multiples (R values) ofthe threshold of perception curve:

Environment Frequency (Hz)Critical working areas 1

Residence - day 2–4 - night 1.4

Ofce 4

Workshop 8

BS 6472 8 advises that if these levels are doubled, then adverse comment may result, andthis may increase signicantly if values are quadrupled. This places into context thefuzziness of these criteria; achievement of these levels should result in alow probability ofadverse comment (but not zero probability), but at twice these levelsadverse commentmay result. Signicant changes to a oor design may be required to halve the expectedvibration level, and hence there is a cost associated with achieving a rather vague degreeof improvement in subjective performance. The selection of criteria is therefore a matter ofrisk/cost balance, and this cannot be laid down rigidly for every circumstance.

Criteria for non-continuous vibration would be expected to be somewhat less onerous thanthose for continuous. The vibration dose value (VDV) method described in Appendix B ofBS 6472 8 (and section 3.2 above) can be used to assess intermittent vibrationexplicitly ,but conventional criteria for footfall induced vibrationimplicitly consider theintermittency of vibration that is normal for particular types of building usage. Theacceptability of a given dynamic performance is based on a limited base of unsolicitedadverse feedback from users.

Finally, it should be emphasised that since these criteria refer to human acceptance, smallvariations in performance (10% to 20%) are unlikely to be noticeable; a signicant changein terms of perceptibility would generally require a change in vibration level by a factor oftwo. Similarly a response factor of 3.9 is little different from 4.1, both would be perceivedequally by an individual and so binary pass/fail decisions based on the second signicantgure should be avoided.

Table 3.1Levels of critical vibration at which probability

of adverse comment is low.


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3.2.2.1 Commercial buildingsIf footfall vibration is intermittent and the worst vibration levels experienced fall belowR = 8,this is almost always satisfactory for commercial buildings such as ofces, retail, restaurants,airports and the like where some people are seated. This recommendation is consistent withthose of the SCI 076 2, CEB, AISC 1 and Canadian Codes. This is twice the level proposed byBS 6472 8 for continuous vibration. However, there are instances in which some peoplehave registered complaints at this vibration level, particularly when it occurs regularly.Whilst a performance ofR < 8 can be achieved with most current forms of oorconstruction without special changes, checks should be made on a case-by-case basis.

At the design stage, the vibration prediction methods proposed here (and in other recom-mendations) are based on a single person walking at the most critical footfall rate. However,somewhat higher levels of vibration can sometimes be generated by many people walking,and in busy places the vibration may be close to ‘continuous’ rather than intermittent. Inaddition, for special, high-quality spaces, a lower target is advisable to further reduce the(low) risk of complaint.

The target ofR < 8 for single person excitation should therefore be reduced by a factor ofup to 2 in circumstances such as the following:n Premium quality open-plan offices and when precision tasks are to be undertaken.n Open-plan offices with busy corridor zones near mid-span.n Heavily trafficked public areas with seating.

This target might be relaxed by a factor of up to 1.5 for oors with many full height partitions(when partitions have not been explicitly included in the prediction analysis, as describedin section 3.3.2.4 below).

3.3.2.2 Residential buildingsThe BS 6472 8 criteria for residential buildings are largely based on experience withexternal sources of vibration such as railways and roads. The criteria are more onerousthan for ofces, especially during night-time, and it is not clear that such stringent levelsare really necessary for vibration caused by occupants of the residence themselves walking.In general it is recommended that values of twice those proposed by BS 6472 forcontinuous vibration in residences be adopted for footfall induced vibration.

3.3.2.3 HospitalsRequirements for NHS hospitals are specied in HTM 2045 9. These take the form of theBS 6472 8 criteria for continuous vibration and VDV values for intermittent vibration. Forward areas the criteria are the same as those for residences in BS 6472. If the VDV approachis to be used then assumptions about how often predicted levels of vibration will occur arerequired; at the design stage these assumptions should be agreed with the client and anyother interested parties, based on the volume, continuity and speed of pedestrian move-ment expected in each part of the building. In the absence of such agreed assumptions, itis conservative to consider the predicted footfall induced vibration to be continuous.


3.3.2 Performance targets foroors


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HTM 2045 9 states that for operating theatres the requirement forR < 1 should be applied

to intermittent as well as continuous vibration, and any relaxation of this should beagreed with the client.

3.3.2.4 Effect of partitionsThe presence of full height partitions of masonry, studwork or glass signicantly alters thedynamic properties of oors, leading to lower vibration levels. In many cases (particularlyresidential and hospital usage) the presence of partitions is integral to the function of theoor spaces. In these circumstances it is appropriate to include the partitions in theirproposed layout in an analysis model of the oor plate (if this is necessary) to demonstratethat the required criteria are met.

Part-height and moveable partitions cannot be relied upon to alter the footfall response.

BS 5400 3 provides guidance on the acceptable levels of vertical vibration for footbridges.This is well founded for long span, low frequency (

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Criterion curve Max. velocitylevel*µm/sec (RMS)

Detail size**microns

Description of use

Workshop(ISO2631 and BS6472)R = 8, ASHRAE J

800 N/A Distinctly perciptible vibration. Appropriateto workshops and non-sensitive areas.

Ofce(ISO2631 and BS6472)R = 4, ASHRAE I

400 N/A Perceptible vibration. Appropriate to ofcesand non-sensitive areas.

Residential day(ISO2631 and BS6472)R = 2, ASHRAE H

200 75 Barely perceptible vibration. Appropriate tosleep areas in most instances. Probably ade-quate for computer equipment, probe testequipment and low-power (to 20X) micro-scopes.

Operating theatre(ISO2631 and BS6472)R = 1, ASHRAE F

100 25 Threshold of perception. Suitable for sensi-tive sleep areas. Suitable in most instancesfor microscopes to 100X and for otherequipment of low sensitivity.

VC-A(BBN-A or ASHRAE E)R = 0.5

50 8 Adequate in most instances for opticalmicroscopes to 100X, microbalances,optical balances, proximity and projectionaligners, etc.

VC-B(BBN-B or ASHRAE D)R = 0.25

25 3 An appropriate standard for optical micro-scopes to 1000X, inspection and lithographyequipment (including steppers) to 3 micronline widths.

VC-C(BBN-C or ASHRAE C)R = 0.125

12.5 1 A good standard for most lithography andinspection equipment to 1-micron detailsize.

VC-D(BBN-D or ASHRAE B)R = 0.0625

6 0.3 Suitable in most instances for the mostdemanding equipment including electron

microscopes (TEMs and SEMs) and E-Beamsystems, operating to the limits of theircapability.

VC-E(BBN-E or ASHRAE A)R = 0.03125

3 0.1 A difcult criterion to achieve in mostinstances. Assumed to be adequate for themost demanding of sensitive systemsincluding long path, laser-based, smalltarget systems and other systems requiringextraordinary dynamic stability.

Notes* As measured in one-third octave bands of frequency over the frequency range 8 to 100 Hz.** The detail size refers to the line widths for microelectronics fabrication, the particle(cell) size for medical and pharmaceutical research, etc. the

values given take into account the observation that the vibration requirements of many items depend upon the detail size of the process.

Table 3.2

Generic vibration criteria for sensitiveequipment.



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4. Predicting footfall induced vibration ofstructuresThis chapter presents methods for calculating the footfall induced vibration of structures.A detailed and generally applicable method is outlined, utilising the modal properties ofthe structure which may be estimated by a number of methods, as described in AppendixA. In addition, a simplied method is given, suitable for certain simple regular structuresand for checking more detailed assessments.

As discussed in Chapter 2, there are two generic types of response, requiring different cal-culation types. One (section 4.2) is for resonant response and should be used for structures

with vertical natural frequencies less than approximately 10Hz (‘low-frequency structures’).The other (section 4.3) is for impulsive response and should be used for structures whoselowest vertical natural frequency is above 10.5Hz (‘high frequency structures’). If a structureis potentially susceptible to both resonant and impulsive response, i.e. if the rst verticalnatural frequency is close to 4 times the maximum walking frequency, then both calcula-tions should be used and the results compared, to assess which excitation method islikely to generate the highest response.

The detailed methods require the calculation of response in several modes of vibration.As this effectively means repeating the same calculation for all the relevant modes, theassessment is best done using a spreadsheet. Flowcharts outlining the spreadsheet proce-dure are included as are all the required equations. The simplied methods described in

section 4.4 comprise an approximate assessment of the response of the rst mode, and ameans for accounting for the additional response in other modes.

Worked examples for both low- and high-frequency structures are included in Chapter 5.

All calculation methods require the use of dynamically consistent units. The followingtwo sets of units are dynamically consistent.

Set 1 Set 2Mass kg tonnesForce and weight N kNLength m m

Footfall induced vibration is sensitive to walking speed. In general, people walking fasterhave the potential to induce greater levels of vibration. For design purposes the followingmaximum footfall rates are proposed:

Footbridges 2.5 footfalls/secCorridor and circulation zones in any building 2.5 footfalls/secWithin ofce bays and residential rooms (i.e. not corridor zones) 2.0 footfalls/secWithin laboratories, operating theatres, and the like 1.8 footfalls/sec

4.1 Overview

Table 4.1Sets of dynamically consistent units.

Table 4.2Proposed maximum footfall rates for different

environments.


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Predicting footfall induced vibration of structures

Figure 4.1Procedure for evaluating the dynamic

performance of a structure under footfallloading.

Clearly, faster footfall rates are possible, but are associated with a low probability. The

possibility of occasional exceedences of target acceptance criteria is implicit in the levelsat which the criteria are set.

The overall assessment and design procedure is summarised in a owchart given in Figure 4.1.

f 1

f 1

Find all modes up to2 x f 1

Find all modesup to 15Hz

(Fig. 4.6) (Fig. 4.3)


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Structures with natural frequencies less than about 10Hz (depending on maximum expec-

ted footfall rate) are potentially susceptible to resonant response induced by the rst fourharmonics of footfall forces as described in chapter 2. These harmonics are sinusoidal forcesapplied to the structure at frequencies of 1, 2, 3 and 4 times the walking frequency. That is,someone walking at 2 steps per second applies sinusoidal forces that can generate resonantresponse at the harmonic forcing frequencies of 2Hz, 4Hz, 6Hz and 8Hz. To assess theresponse of the structure to walking at 2 steps per second, it is necessary to calculate theresponse in all modes to each of these harmonics and combine them. Further, as theresponse will be very sensitive to the exact walking speed (faster speeds not necessarilybeing worse), it is necessary to assess the response for all walking speeds that might inducesignicant resonant response.

Harmonic force data and a calculation procedure are presented below for assessing theresonant response of a structure at a particular walking frequency. Section 5 includesworked examples which illustrate how the method should be used in practice.

As discussed in chapter 2 there is considerable variation in both the shape and magnitudeof footfall forces due to different individuals. Any measured footfall time history can bebroken down into harmonic components and these can be plotted to illustrate the trendsand variability. Kerr11 measured 882 such traces and Figure 4.2 plots the rst four harmonicsof each of these against excitation frequency (footfall rate × harmonic number) as dynamicload factors. (Dynamic load factor, or DLF, is the dynamic force expressed as a fraction ofa person’s static weight. See Glossary for full denitions of DLF and Fourier analysis.)

DLFs calculated by others are also included for comparison. It can be seen that there is aclear trend of increasing force with faster walking speed in the rst harmonic, but that thisis less clear in the higher harmonics. That there is no single ‘correct force’ is also apparent,but it is possible to calculate a ‘mean’ force, and to propose a ‘design force’ that has aquantiable likelihood of exceedence. A simple expression for the mean force associatedwith each harmonic is given in Table 4.3 as a function of excitation frequency. Also givenare the coefcients of variation for each harmonic and a ‘design’ force, suitable forserviceability assessment, calculated to have a 25% chance of exceedence.

4.2 Assessment of footfallinduced vibration for

structures with naturalfrequencies less than 10Hz

4.2.1 Harmonic footfall forceloads


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1st Harmonic

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Frequency (Hz)

D L F

4th Harmonic

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Frequency (Hz)

D L F

3rd Harmonic

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.0 2.0 4.0 6.0 8.0 10.0

Frequency (Hz)

D L F

2nd Harmonic

0.00

0.05

0.10

0.15

0.20

0.25

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Frequency (Hz)

D L F

Kerr Galbraith & Barton Wheeler Ohlssen Rainer Ellis ~Mean Value Design Value

Figure 4.2

The rst four harmonics of footfall forces.

Harmonicnumber, h

Harmonic forcingfrequency (Hz)

Mean value(DLF)

Coefcient ofvariation

Design value(DLF)

1 1–2.8 0.37(f - 0.95), | 0.5 0.17 0.41(f - 0.95), | 0.56

2 2–5.6 0.054+0.0044f 0.40 0.069+0.0056f

3 3-8.4 0.026+0.0050f 0.40 0.033+0.0064f

4 4–11.2 0.010+0.0051f 0.40 0.013+0.0065f

h>4 >11.2 0 0

>

>

Table 4.3Average and design values and coefcients of

variation of the footfall harmonics.


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ρh,m = 1 – e–2πζ mN

This calculation is valid only for structures with vertical natural frequencies less than 4.2

times the maximum footfall rate. All modes with frequencies up to 15Hz can potentiallycontribute signicant response and should be identied and included in the calculation.Appendix A describes how to calculate the natural modes of a structure. The frequency,modal mass and damping in each modem needs to be known, denoted asf m, mm andζ m respectively in the equations below. The mode shape values at the excitation and responsepoints in each mode,μe,m andμr,m, are also needed.

As the total response is sensitive to footfall rate it is recommended that this calculation bedone for all rates that are likely to occur. The output should therefore be a graph such asthat shown in Figure 4.4, which plots total response factorR as a function of footfall rate.

The calculation procedure is set out below and illustrated in Figure 4.3. The calculationpredicts the extent of resonant build up that is possible under a limited number of cyclesof harmonic loading; full details of how this is done for a structure with many modes canbe found in dynamics textbooks11.

The maximum steady-state acceleration at resonance that could be attained with anunlimited number of forcing cycles (force amplitudeF ) at the most critical point on astructure of modal massmm is given by:

(4.1)

Particularly for lightly damped structures excited by lower harmonics, many footfalls(perhaps greater than 30) would be required to generate this full resonant response. Inpractice, by the time this many steps have been taken the pedestrian may have movedoff the structure completely, or at least he or she is likely to have moved away from themaximum point on the mode shape. To account for this, a correction factorρ, for crossinga span of typical mode shape with a limited number of footsteps,N, is introduced:

where N = 0.55h L (4.2)

l

Hereh is the harmonic number,m is the mode number,L is the span andl is the stridelength. The equation assumes that an individual walks in the spanning direction from oneline of support to the next. This factor effectively reduces the extent of full resonant build-up by a factor related to the damping and the number of footsteps to cross the span. The

reduction factor is likely to be more signicant for bridges than for oors, and can be(conservatively) omitted by assumingρ = 1 (orN >> 10).

The full procedure is set out step-by-step below. The concepts of real and imaginaryaccelerations are used when combining the responses in different modes. Although thismight appear unnecessarily complex it is merely a mathematically convenient way ofexpressing the magnitude and phase relationship between the responses in each mode.The same calculation could be effected using a number of sine and cosine functions, thoughthe equations become rather unwieldy.

4.2.2 Resonant responsecalculation procedure

a = μe,m μr,mF h

2ζ mm


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Figure 4.3

Procedure for evaluating resonant response ofa structure under footfall loading.

damping and mode shape

h


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For a particular footfall rate,f w , it is required to calculate the response in each mode to

each of the rst four harmonics:

Firstly, for each harmonich, fromh = 1 toh = 4.

1. Calculate the harmonic forcing frequency,f h:

f h = hf w (4.3)

2. Calculate the harmonic force,F h, at this harmonic frequency from Table 4.3:

For each mode,m

3. Calculate the real and imaginary acceleration (areal,h,m , aimag, h, m) in each mode:

F h = DLF.P

whereDLF is calculated from Table 4.3 (Design value) andP equals the staticweight of the walker.

(4.4)

where Am = 1 – ( f hf m )2 , Bm = 2ζ m

f hf m ,

ρ h,m is from equation 4.2

ζ m can be estimated from Table A2 andf m, mm andμ are found using methodsdescribed in Appendix A.

4. Sum the real and imaginary responses in all modes to yield the total real andimaginary acceleration to this harmonic force,ah:

areal,h = ∑ areal,h,m ; aimag,h = ∑ aimag,h,m (4.5)

m

m

5. Find the magnitude of this acceleration|ah| which is the total response in allmodes to this harmonic (at this frequency):

(4.6)|ah| =√ areal,h2 + aimag,h2

aimag,h,m = ( f h )2 F h µr,mµe,mρh,m Bm

f m mm^ Am2+ Bm2

areal,h,m = (f h )2 F h µr,mµe,mρh,m Am

f m mm^ Am2+ Bm2

^


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6. Convert this acceleration to a response factor,Rh. First calculate the baseline peak

acceleration for a response factor of 1 at this harmonic frequency,aR=1,h. Dividethis into the total acceleration response for this harmonic:

Iff h < 4Hz,aR=1,h =0.0141 m/s2

√ f h If 4Hz 8Hz,aR=1,h = 2.82πf h x 10–4 m/s2

Rh =|ah|

aR=1,h (4.7)

7. Find the total response factor,R, which is the ‘square root sum of the squares’combination of the response factor for each of the four harmonics.

R =√ R12 + R22 + R32 + R42 (4.8)

8. Repeat this calculation for other footfall rates and find the critical rate thatgenerates the maximum response.

The result of this calculation may be presented on a graph such as Figure 4.4 below. Theresponse factors due to each harmonic, and the total response factor, are plotted againstfootfall rate. The maximum value of the ‘Total Response Factor’ curve (i.e.R = 8.5 forwalking at 2.3Hz) is that which should be compared against the criteria given in section 4.3.

Figure 4.4Graph of response factors against walking

frequency.

The impulsive response of structures to footfall forces is characterised by an initial peakvelocity followed by a decaying vibration at the natural frequency of the structure beforethe next footfall (e.g. Figure 3.2). Subsequent footfalls generate comparable responsesand there is no tendency for the response to build up over time, as is the case forresonant response.

4.3 Assessment of theimpulsive response ofstructures to footfall

Walking frequency (Hz)

R e s p o n s e

f a c t o r First harmonic

Second harmonic

Third harmonic

Fourth harmonic

Total


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The same footfall time histories for which the harmonic loads of Table 4.3 were derived

have been used to calculate effective impulses for footfall on structures. The basis of theseis a number of curves such as those in Figure 2.5. The effective impulse can be thought ofas the equivalent ‘perfect impulse’ of innitesimally short duration that induces the samepeak structural response as the direct application of that footfall time history. It is foundthat this effective impulse is empirically related to the walking speed and the naturalfrequency of the structure. The mean effective impulse is given in Equation 4.9 below; thecoefcient of variation is 0.4, and the serviceability ‘design’ impulse with a 25% chanceof exceedence is given in Equation 4.10.

Mean Ieff = 42 f w 1.43 / f n1.30 Ns (4.9)

Design Ieff = 54 f w 1.43 / f n1.30 Ns (4.10)

Note that these formulae provide the effective impulse in Ns units and that the walkingfrequency,f w , must be in Hz. If tonne/kN units are being used, then they are to be dividedby 1,000.

This calculation is valid for structures with vertical natural frequencies above 4Hz andshould use all modes with frequencies less than twice the fundamental vertical frequencyand with signicant vertical participation at the excitation and response locations. Forexample, if the rst active vertical mode is at 12Hz, then all vertical modes between 12Hzand 24Hz should be found and used in the response calculation.

This procedure calculates the velocity time history due to a single footfall at a particularlocation. Faster walking speeds will induce greater responses, so it is only necessary toperform this calculation for the fastest anticipated walking speed. The direct output fromthe calculation is a velocity time history such as that shown in Figure 4.5a. From this,either peak or RMS velocities can be calculated. The RMS velocity may then be used tocalculate a response factor and a VDV can be derived from the time history if required.

The simulated time history for a series of footfalls is obtained simply by combining a seriesof these time histories, as shown in Figure 4.5c and this can be compared directly againstthe measured time history of Figure 3.2, included again here as Figure 4.5b. Although thetwo traces appear slightly different visually, the measured and predicted RMS velocities(by which the dynamic performance of the oor would be judged) correlate very well.

A full calculation is included in example 3 of Chapter 5.

4.3.2 Impulsive responsecalculation procedure

4.3.1 Impulsive footfall loads


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Figures 4.5b and 4.5cMeasured and predicted velocities at

walking rate of 100 paces per minute.

Figure 4.5a

Simulated velocity time history due to a singlefootfall.

Measured response

Predicted response


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Figure 4.6

Procedure for evaluating impulsive dynamicresponse to footfall loading.

shape for each of these modes.

might occur.


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The procedure is set out step-by-step below:

1. All modes with frequencies up to twice the fundamental frequency are calculatedand an effective footfall impulse (Ieff ) for each mode is calculated from Equation4.10.

2. The peak velocity in each mode,v m, is given by:

(4.11)

and from this the velocity response in each mode over the period of one footfall(v m(t) fromt = 0 tot =T ) is calculated using Equation 4.12:

v m(t) = v me–2πζf mt sin 2πf mt (4.12)

3. The total response to each footfall is found by summing the velocity responses ineach mode in the time domain using Equation 4.13:

(4.13)

wherev(t) is the total velocity at timet andN is the number of modes.

From the resulting velocity time history, a RMS response can be evaluated overthe period of one footfall:

(4.14)

4. A response factor can be calculated by dividing this by the baseline RMS velocityforR = 1 (v R=1) at the fundamental frequency,f 1:

Iff 1 < 8Hz v R=1 = 5 x 10–3

m/s

Iff 1 > 8Hzv R=1 = 1.0 x 10–4 m/s

andR = v RMS (4.15) v R=1

The variation of response with footfall rate can be determined as shown in Figure 4.7. Theeffective impulse, and thereforev(t) is proportional tof w 1.43, but for faster footfall rates thenext footfall occurs sooner (i.e.T is smaller), which increases the overall RMS response.The relationship betweenv RMS and footfall rate is therefore not quite linear, having anexponent slightly greater than 1. To calculate the maximum possible response, it is onlynecessary to calculate the response at the fastest walking speed.

2πf 1

v m = µe,mµr ,ml eff,mmm

v(t) = ∑ v m(t)N

m=1

= 1 T ∫ (t)2dtT o


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Figure 4.7

Graph of impulsive response versus footfallrate.

Whilst the detailed methods of the previous sections enable responses to be predicted forany structure, simplied methods are included below for certain regular rectangular oorbays of uniform properties which can be approximated reasonably well as simply supportedplates having equal or different stiffnesses in the two orthogonal directions. This mightapply to bays composed of at slabs, ribbed slabs, coffer slabs and steel beam/compositeslabs. Floor layouts that may be assessed using this method are shown in Figure 4.8.

4.4 Simplied andapproximate methods for

oors

Figure 4.8Floor layouts suitable for assessment using

the approximate method.


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In addition to use as quantitative estimators of oor response, the simplied methods also

give some insight into the effects of changing key parameters on the vibration performance,indicating how a design might be developed to improve its dynamic performance.

The most important limitation to the use of these methods is the degree to which thetheoretical idealisation is representative of the actual structure.

The methods work well when:n The edges of the floor bay are supported along their length by walls, a facade or rela-

tively closely spaced columns (cf. Appendix A3, point 8).n The bay has uniform structural properties (e.g. constant slab thickness, identical ribs or

beams at the same centres).

It is further assumed that the response of the oor is characterised by the vibration at themiddle of the oor due to walking near there.

This method differs from those already presented in that it is based on the modal proper-ties of a single mode and response multipliers to account for the response of higher modesand the participation of adjacent bays.

These multipliers are presented in simple design charts as a function of:n The aspect ratioW/L of the floor plate.n The flexural stiffnesses of the floor in the two orthogonal directions.n Relationships related to the properties of adjacent spans.

The methodology is otherwise very similar to that already presented. The rst stage is tond the modal properties of the rst mode of the oor bay, and then either a resonant oran impulsive response is calculated, depending on the frequency of the oor. The effect ofhigher modes and adjacent spans is then incorporated. The calculation procedures aredescribed below.

The modal properties of oors such as those shown in Figure 4.8 can be calculated from theexural rigidities of the slab in the two orthogonal directions (D x andD y ). The calculationof these rigidities is described in Appendix A for different oor constructions.

For oors with adjacent parallel bays such as type (ii) of Figure 4.8, the modal propertiesof the long bay under consideration is rst calculated ignoring the parallel bay, and thenthese properties are modied to reect the presence of the adjacent bay, as described insection 4.4.1.2


4.4.1 Calculation of rst modeproperties


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4.4.1.1 Calculation of the rst mode properties of single long bayThe natural frequency and modal mass of the rst mode of the oor bay is given by:

; f 1 = K f f 0 (4.16)

(4.17)

K f is a multiplier on the natural frequency of the elements spanning in the ‘primary span’direction to account for ‘two-way’ spanning, and is given in Figures 4.9a and 4.9b below asa function of the width/span ratio of the oor and the ratio of stiffnesses of the oor inthe two orthogonal directions,D x /DY . Figure 4.9a should be used for solid or ribbedconcrete oors which can have signicant torsional stiffness. Figure 4.9b should be usedfor steel composite oors in which the torsional stiffness of the steel members is likely tobe negligible.

m = mLW

4

Figure 4.9aFrequency factors for solid and ribbed

concrete.

Figure 4.9bFrequency factors for composite oors.

f 0 = π DY2 mL4


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4.4.1.2 Incorporating the stiffness and mass contribution of adjacent baysFor oors with an adjacent row of parallel bays, the modal mass and frequency should befurther multiplied by factorsK mm andK fm respectively from Figures 4.10a and 4.10b.

Here the main bay under consideration has spanLa and the adjacent bay has spanLb . It isassumed that the mass per unit area,m, of the two bays is the same. The degree of inter-action between the bays is related to the ratio of their natural frequencies,f b /f a, in the spandirection. In general, this may be expressed as:

f b = D y, bLa4 (4.18)

f a D y, aLb4

where the sufxes a and b refer to the two spans.

And so, for oor arrangements such as type ii) in Figure 4.8:

f 1 = Kfm K f f 0 (4.19)

andm = mLWK mm (4.20) 4

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20.5

1

1. 5

2

2.5

3

3.5

4

4.5

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

Lb /L

a=2.0

Lb /L

a=0.5

Lb /L

a=1.0

Lb /L

a=1.5

K m m

f b /f a

L b /L a =2.0

L b /L a =1.5

L b /L a =1.0L b /L a =0.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 21

1.05

1. 1

1.15

1. 2

1.25

1. 3

1.35

1. 4

1.45

1. 5

Lb /L a =0.5

L b /L a =1

Lb /L a =1.5Lb /L a =2

K f m

f b

/f a

Figure 4.10aModal mass factors for a bay with an adjacent

parallel bay.

Figure 4.10bNatural frequency factors for a bay with an

adjacent parallel bay.


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The resonant response for oors withf 1 less than 10Hz is calculated as follows:

1. Calculate the peak resonant acceleration in the first modea1 for the most criticalwalking speed using Equation 4.21:

a1 =

ρF h 2ζm (4.21)

whereF h is the harmonic force amplitude appropriate to the oor bay frequencyf 1 (Table 4.3) andρ is the resonant correction factor calculated from Equation 4.2.The critical walking speed is that which gives the highest harmonic force amplitude.

Determine resonant response multiplierK rm from Figure 4.11a or 4.11b. The resonantresponse multiplier is the factor by which the resonant response in the rst mode mustbe increased to account for the contribution of all the other modes. Again this factoris different for oors with signicant torsional stiffness and so two plots are provided,one for solid and ribbed concrete oors and a second for steel composite oors.

2. Calculate total peak acceleration response:

at =K rm a1 (4.22)

3. Calculate response factor,R

R =

at (4.23) 0.0071

4.4.2 Simplied calculation ofresonant response

Figure 4.11aResonant response multipliers for solid and

ribbed concrete oors.

R e s o n a n t m u

l t i p l i e r , K

r m


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Figure 4.11b

Resonant response multipliers for compositeoors.


The impulsive response of oors to footfall can be calculated as follows:

1. Calculate impulsive peak velocity response in the first mode using Equation 4.24

v 1 =

Ieff (4.24) m

whereIeff is the effective impulse appropriate to the floor bay frequencyf 1

(Equation 4.10)

2. Determine impulsive response multiplierK im from Figure 4.12a or 4.12b. 3. Calculate total impulsive peak velocity response as:

v = K imv I (4.25)

4. Calculate the RMS velocity from the peak velocity as follows:

v RMS = 0.3v forK im > 2.0 (4.26) v RMS = 0.3 + 0.12(2 –Kim) for 1.0

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Figure 4.12a

Impulsive response multipliers for solid andribbed concrete oors.

Figure 4.12bImpulsive response multipliers for composite

oors.

I m p u

l s i v e m u

l t i p l i e r

, K i m

I m p u

l s i v e m u

l t i p l i e r

, K i m


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5. Worked examplesThis chapter uses three worked examples to show how modal properties may be used topredict the response of structures to footfall loading. Each example is an existing structure,and each has been physically tested to check the modal properties and footfall inducedvibrations. Predictions are compared against measurements in Appendix C.

The three examples are summarised below.

Example 1 is a concrete footbridge over a motorway, having constant cross-section andtwo continuous equal spans of 20m. Its modal properties are estimated by hand calculation,a simple nite element analysis and have been veried by physical testing. The lowestmeasured natural frequency is 4.65Hz, and it is assessed as a low-frequency structureusing the methods described in section 4.2.

Example 2 is a multi-span 225mm thick post-tensioned concrete oor slab spanningbetween pre-cast concrete beams. The fundamental natural frequency of the longestspan (10m) is 7.0Hz and so it is assessed as a low-frequency oor using the methoddescribed in section 4.2. Its layout is slightly unusual and so its modal properties are bestassessed using an FE model. The simplied method of section 4.4 can also be used, andgives a reasonable estimate of the likely dynamic performance.

Example 3 is a ribbed concrete laboratory oor comprising a 130mm thick slab supportedon concrete ribs 300mm wide and 350mm deep at 1100mm centres. The span is 8.6m andthe oor is 38m wide. Its fundamental frequency is 12.5Hz and so it is assessed as a high-frequency oor. Its layout is regular and it may be assessed using either the simpliedmethod described in section 4.4 or the detailed method described in section 4.3. As thisoor can be well approximated as a simply supported rectangular plate the two methodsgive very similar predictions.

Worked examples


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Figure 5.1Worked example 1: footbridge over a

motorway.

Figure 5.2Approximate dimensions of bridge.

Example 1 is a concrete bridge over a motorway shown in Figure 5.1, and its main dimen-

sions are shown in Figure 5.2.

5.1 Example 1:A footbridge over a

motorway


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Worked examples

Figure 5.3Mode shapes of the rst three modes of the

footbridge.

The natural frequencies and modal masses may be calculated by hand. Mode 1 is equiva-lent to the rst simply supported mode of both spans, mode 2 is equivalent to the xed-

free mode of each span and mode 3 the second simply supported mode of both spans.Assuming un-cracked reinforced concrete withE = 38GPa, and negligible non-structuralmass, the mass per unit length is 1848kg/m and the natural frequencies obtained fromthe FE analysis and Equation A1 are identical. These are compared with measured valuesin Table 5.1. The modal mass of each mode is approximately 20 × 2 × 1848/2 = 36,960kg.

Hand Calculation Finite Element Solution Measured Frequency f 1 (Hz) 4.22 4.22 4.65

f 2 (Hz) 6.59 6.59 6.56

f 3 (Hz) 16.90 16.88 14.66

The bridge has two modes that might be excited to resonance by walking. Based on themeasured natural frequencies, the rst mode may be excited in either the second or thirdharmonic of walking at 2.32 or 1.55 footfalls per second respectively, and the second modeby the third or fourth harmonics of walking at 2.18 or 1.64 footfalls per second.

Table 5.1Calculated and measured natural

frequencies

5.1.2 Footfall inducedvibration calculation

The cross-sectional area of the beam section is approximately 0.77m2 and the second

moment of area 0.056m4. From these the modal properties can be calculated usingeither standard equations for the natural frequencies of beams or a simple nite elementmodel. The shapes of the rst three modes obtained from the nite element analysis areshown in Figure 5.3.

5.1.1 Calculation of modalproperties

Mode 1

Mode 2

Mode 3


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Harmonic number 1 2 3 4Harmonic frequency (Hz) (Eq.4.3) 1.55 3.10 4.65 6.20Real and Imag. Responses (m/s 2)(Eqs. 4.2 & 4.4)

Real Imaginary Real Imaginary Real Imaginary Real Imaginary

Mode 1 5.79E-04 6.51E-06 1.31E-03 4.70E-05 0.00E+00 3.96E-02 -2.30E-031.18E-04 Mode 2 2.52E-04 1.90E-06 4.33E-04 7.91E-06 1.10E-03 4.72E-05 7.27E-031.93E-03 Mode 3 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+000.00E+00 Total (Eq. 4.4) 8.32E-04 8.41E-06 1.74E-03 5.50E-05 1.10E-03 3.97E-02 4.97E-032.05E-03Magnitude of response (m/s 2)(Eq. 4.6)

8.32E-04 1.74E-03 3.97E-02 5.38E-03

Base curve acceleration (m/s 2) 1.13E-02 8.01E-03 7.10E-03 7.10E-03

Response factor (Eq. 4.7) 0.07 0.22 5.59 0.76

TOTAL RF AT 1.55 Hz (Eq. 4.8) 5.65

Table 5.2Response factor calculations for walking at

1.55Hz

The damping was calculated as 1.5% of critical from measurements. This value is used for

the response calculation, though at the design stage a lower value (say 0.9%, See tableA2) would be more conservative. The detailed calculation of section 4.2 is summarised inTable 5.2 and Table 5.3 below for walking speeds of 1.55Hz and 2.18Hz. In the calculation,it is conservatively assumed thatρ =1.0. Figure 5.4 shows the maximum response for allfootfall rates between 1Hz and 2.8Hz. In practice it is difcult to predict which frequency islikely to generate the maximum response and the calculation should be performed foreach of those likely to produce resonance, in this case 1.55Hz, 1.64Hz, 2.18Hz and 2.32Hz.

Here it was found that the maximum response will be due to a pedestrian walking at2.32Hz and will be approximatelyR = 8.5.

A simple check can be made using Equation 4.21. Assuming conservatively thatρ = 1.0,the check for walking at 2.32Hz (second harmonic) gives:

F h = 700 × (0.069 + 0.0056 × 2 × 2.32) = 66.5N (Refer table 4.3)

a = 1.0 × 66.5/(2 × 0.015 × 36960) = 0.06m/s2 at 4.65Hz (see eq. 4.21)

R = 0.06/0.0071 = 8.5 (see eq. 4.23)


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Worked examples

Harmonic number 1 2 3 4Harmonic frequency (Hz) (Eq.4.3) 2.18 4.36 6.54 8.52Real and Imag. Responses (m/s 2)(Eqs. 4.2 & 4.4)


Mode 1 2.78E-03 5.09E-05 1.41E-02 3.83E-03 -2.82E-03 1.19E-04 -1.84E-03 4.05E-05 Mode 2 1.13E-03 1.28E-05 1.33E-03 4.88E-05 -1.53E-02 3.78E-02 -2.75E-03 1.38E-04

Mode 3 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Total (Eq. 4.4) 3.91E-03 6.37E-05 1.55E-02 3.88E-03 -1.82E-02 3.79E-02 -4.59E-03 1.79E-04

Magnitude of response (m/s 2)(Eq. 4.6)

3.91E-03 1.59E-02 4.21E-02 4.60E-03

Base curve acceleration (m/s2

) 9.51E-03 7.10E-03 7.10E-03 7.80E-03Response factor (Eq. 4.7) 0.41 2.25 5.93 0.59

TOTAL RF AT 2.2 Hz (Eq. 4.8) 6.38

Table 5.3

Response factor calculations for walking at2.18Hz

Figure 5.4Response at the centre of the span for all

walking frequencies.

Walking frequency (Hz)

R e s p o n s e

f a c t o

r


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7.5 m7.5 m10.0 m 10.0 m

3 0 0 0

3 0 0 0

9 0 0 0

2 6 0 0

2 6

0 0

Figure 5.5Structural drawing of the

concrete oor.

Figure 5.6Photo showing the 10m span between the

nearest two columns at the time of testing(bare condition).

The oor is a post-tensioned 225mm at slab spanning between 7.5m and 10m onto pre-

cast concrete beams as shown in Figure 5.5. The response of the 10m span section –marked in Figure 5.5 – will be considered here. It was tested when bare and without tout as shown in Figure 5.6. Modal properties and responses are calculated for both theas-tested condition and the nished state.

Finite element analysis is used to nd the modal properties and the method described insection 4.2 is used to calculate the response in both cases. The simplied method ofsection 4.4 is also used to predict the response of the oor in its nished condition.

5.2 Example 2:A post-tensioned at slab

assessed as a low-frequency oor


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Worked examples

Figure 5.7Finite element model of the oor.

The FE analysis predicts the rst mode of the 10m bay for completed oor to have anatural frequency of 7.0Hz and modal mass of 24,100kg; its shape is shown in Figure 5.8.

There are no other modes with signicant participation of the 10m bay below 15Hz.

Figure 5.8Mode shape of the rst mode of the 10m span

section.

The nite element model shown in Figure 5.7 was used to calculate the modal properties

of the oor. For the nished condition, 1kPa non-structural dead load is included and theedges of the slab are restrained vertically to represent the support that would be provided bythe facade. For the bare (i.e. test) condition, no additional loads or restraints are included.

5.2.1 Calculation of modalproperties


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Figure 5.9Mode 1 of the as-tested oor.

Figure 5.10Mode 2 of the as-tested oor.

For the unnished state at the time of testing, the absence of the facade leads to there

being three modes below 15Hz with signicant participation of the 10m span. These areshown in Figure 5.9 to Figure 5.11 and their modal properties summarised in Table 5.4. Tofacilitate comparison of the measured and predicted data, the measured frequencies willbe used in the prediction calculation for the as-tested case. This will have little effect onthe magnitudes of the response but will ensure that the peaks of the frequency responsegraphs coincide.


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Worked examples

Figure 5.11

Mode 3 of the as-tested oor.

Mode j CalculatedFrequency,f , (Hz)

MeasuredFrequency,f , (Hz)

ModalMass (kg)

Measured modaldamping(% crit)

1 6.43 6.55 35300 1.5

2 7.66 7.85 14800 1.5

3 9.64 9.90 15300 1.5

For the oor in its nished condition, only one mode, at 7.0Hz, is potentially susceptible toresonant response. It could be excited either by the third harmonic of walking at 2.35Hzor the fourth harmonic of walking at 1.76Hz.

5.2.2.1 Detailed methodAssuming 3% of damping, the predicted resonant response for walking at 2.35Hz is calcu-lated in Table 5.5. The response at all walking frequencies between 1Hz and 2.8Hz is plottedin Figure 5.12.

Table 5.4Calculated and measured modal properties of

oor in bare condition.

5.2.2 Response calculations –nished condition

Harmonic number 1 2 3 4Harmonic frequency (Hz) (Eq.4.3) 2.35 4.70 7.05 9.40

Real and Imag. Responses (m/s 2)(Eqs. 4.2 & 4.4)


Mode 1 1.56E-03 3.51E-05 2.09E-03 1.50E-04 -3.53E-04 3.73E-02 -4.85E-03 4.98E-04 Mode 2 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

Mode 3 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Total (Eq. 4.4) 1.56E-03 3.51E-05 2.09E-03 1.50E-04 -3.53E-04 3.73E-02 -4.85E-03 4.98E-04

Magnitude of response (m/s 2)(Eq. 4.6)

1.56E-03 2.09E-03 3.73E-02 4.88E-03

Base curve acceleration (m/s 2) 9.20E-03 7.10E-03 7.10E-03 8.33E-03

Response factor (Eq. 4

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