guideline on the design of floor for vibration due to human ed

Structural Engineering Branch, ArchSD of 39 File code : WalkingVibration.doc

Guidelines on Walking Vibration MKL/KYL/RL//CYW Issue No./Revision No. : 1/- Issue/Revision Date : June 2009

GUIDELINES

ON

THE DESIGN FOR

FLOOR VIBRATION DUE TO HUMAN ACTIONS

PART II: FLOOR VIBRATION DUE TO WALKING LOADS

STRUCTURAL ENGINEERING BRANCH

ARCHITECTURAL SERVICES DEPARTMENT

June 2009



CONTENTS

Content Page

1. Introduction ............................................................................................................3

2. Minimum Required Fundamental Frequency and Acceptable Peak Acceleration

for Offices or Similar Building Structures ...........................................................4

3. Floor Vibration .......................................................................................................5

4. Footbridge Vibration............................................................................................20

5. Design Examples ...................................................................................................26

6. Design References .................................................................................................37

Copyright and Disclaimer of Liability

This Guideline or any part of it shall not be reproduced, copied or transmitted in any

form or by any means, electronic or mechanical, including photocopying, recording, or

any information storage and retrieval system, without the written permission from the

Architectural Services Department. Moreover, this Guideline is intended for the internal

use of the staff in the Architectural Services Department only, and should not be relied by

any third party. No liability is therefore undertaken to any third party. While every effort

has been made to ensure the accuracy and completeness of the information contained in

this Guideline at the time of publication, no guarantee is given nor responsibility taken by

the Architectural Services Department for errors or omissions in it. The information is

provided solely on the basis that readers will be responsible for making their own

assessment or interpretation of the information. Readers are advised to verify all relevant

representation, statements and information with their own professional knowledge. The

Architectural Services Department accepts no liability for any use of the said information

and data or reliance placed on it (including the formulae and data). Compliance with

this Guideline does not itself confer immunity from legal obligations.



1. Introduction

1.1 Part I of this set of Guidelines has already mentioned that human-induced vibration

is a problem for lightweight floors with large span, and has identified the sources of

vibration to include rhythmic activities, walking, operating machines, etc. Part I of

this set of Guidelines focuses on the human-induced vibration caused by rhythmic

activities, and is applicable to buildings which house gymnasium, indoor games

halls, etc. Part II of this set of Guidelines will focus the human-induced vibration

due to walking.

1.2 Human-induced vibration due to walking usually occurs in shopping malls with

lightweight, long span and open areas without partitions. For the types of structures

in ArchSD, PSE shall note that human-induced vibration may also be a problem for

open-plan offices, libraries, museums, exhibition centres, and air or cruises

terminals with long-span and few partitions.

1.3 Structurally, it is not acceptable for the vibration of floor to be very large and

occurring frequently, as this may lead to the fatigue failure of the floor if the stress

level is high. Excessive vibration, however, causes nuisance and discomfort to the

occupants. Human perceptibility of vibration depends on a number of interrelated

factors, i.e. serviceability problem. Among these factors, ISO 2631-2:1989 and

Murray et al (1997) consider that the type of activity is one of the dominant factors

in setting the acceptable level. People in offices, libraries, museums and exhibition

centres are especially most sensitive to vibration; whilst those taking part in

rhythmic activities can tolerate much more vibration without discomfort. This is a

particular problem for quiet locations (such as meeting rooms, reading rooms), or

offices where occupants are required to perform tasks requiring prolonged special

concentration without disturbance by floor vibration.

1.4 Human-induced vibration due to walking will be a particular problem for hospitals,

as they will inevitably house delicate and expensive medical equipment whose

accuracy will be sensitive to vibration. For medical and other sensitive equipment,

Part III of this set of Guidelines will provide further guidance.

1.5 The Code of Practice for the Structural Use of Steel 2005 for steel structures states

that for lightweight and long-span structures where excessive vibration is

anticipated, floor vibration assessment may be necessary; but does not suggest any

limits on the vibration. It recommends designers to consult specialist literature.

The Code of Practice for the Structural Use of Concrete 2004 for concrete

structures states that excessive vibration due to fluctuating loads that may cause

discomfort or alarm to occupants from people should be avoided. Again, it

recommends designers to consult specialist literature. However, it states that for a

building floor structure with the natural frequency less than 6 Hz, or a footbridge

superstructure with the natural frequency less than 5 Hz, a dynamic analysis may be

desirable. The purposes of this Guideline are therefore to provide:

(a) guidance to designers on the acceptable limits for structures (including

building floors and footbridges) subjected to human-induced vibration by

walking; and



(b) methods to calculate the dynamic responses of structures under human-

induced vibration by walking.

1.6 The analysis and design of human-induced vibration on structures due to walking

are similar to those due to rhythmic activities in Part I of this set of Guidelines.

That is first to compare the natural frequency of structure with the required

minimum frequency of the structure, and if the natural frequency of the structure is

less than the required minimum frequency, then it is necessary to compute the peak

acceleration due to walking, which is then to compare with the maximum

acceptable acceleration.

1.7 The following sections will first provide a summary of the minimum fundamental

frequencies and the maximum acceptable peak accelerations required for the

structure, and will then provide methods to calculate the peak acceleration due to

walking. The discussion will mainly focus on open-plan buildings, including

offices, libraries, museums, exhibition centers, and air or cruises terminals. Again,

as in Part I of this set of Guidelines, a simplified method and a computer analytical

method will be presented. The analysis and design footbridges for human-induced

vibration will be included as a separate section. It will then be followed by design

examples to illustrate the procedures in using such methods. This Guideline

provides basic knowledge on the subject, and indeed, no one single guideline or

reference can solve all problems, and designers should therefore carry out their own

research to suit their own problems. A list of design references is included at the

end of this Guideline.

2. Minimum Required Fundamental Frequency and Acceptable Peak

Acceleration for Offices or Similar Building Structures

2.1 The Code of Practice for the Structural Use of Concrete 2004 for concrete

structures states that for a building floor system with the natural frequency less than

6 Hz, a dynamic analysis is required; but it does not specify the minimum required

fundamental frequency. Ellis (2000) considers that the fundamental frequency of

the floor system is only of reference value and the peak acceleration should be the

dominant factor affecting vertical vibration of floors. Nevertheless, Ellis (2003)

suggests that floors having fundamental frequencies above 9Hz would not be

expected to suffer from human-induced vibration problem due to walking.

Similarly, Murray (1975, 1981) suggests that if the fundamental frequency is greater

than 10Hz, the structure is generally not susceptible to walking excitation. Murray

et al (1997) recommend that floor with fundamental frequencies of less than 3Hz

should be avoided; otherwise the floor should be designed for rhythmic excitation.

For fundamental frequencies between 3Hz and 10Hz, Murray et al (1997)

recommend that the peak acceleration should be checked.



2.2 The first step in checking walking excitation is to compare the fundamental

frequency of the structure against the minimum required fundamental frequency. If

the fundamental frequency of the structure is less than the minimum required

fundamental frequency, then the peak acceleration of the structure under walking

excitation should be compiled. Murray et al (1997) suggest that the acceleration

limit as percentage of gravity shall be 0.5% g.

2.3 Table 1 summarises the minimum required fundamental frequency and acceptable

peak acceleration of structures for walking excitation as discussed above.

Table 1 Minimum Required Fundamental Frequency fn(req’d)

and Acceptable Acceleration for Offices or Similar Building Structures

Minimum Required Fundamental

Natural Frequency fn(req’d) (Hz) Acceptable Peak Acceleration

10 0.5% g

The above criterion for the minimum required fundamental frequency is higher than

that recommended in the Code of Practice for the Structural Use of Concrete 2004,

which states that for a building floor structure with the natural frequency less than 6

Hz, a dynamic analysis may be desirable. For the minimum required fundamental

natural frequency and acceptable peak acceleration for footbridges, designers should

refer to Section 4.2.

3. Floor Vibration

3.1 The following paragraphs will describe how to calculate the fundamental natural

frequencies and peak accelerations of flooring systems subjected to walking

excitation. Two methods of analysis will be presented: the first on a simplified

method which can be used for preliminary design or simple structures, and the

second by analysing the dynamic behaviour of the floor system by modelling the

walking load, which should be used for structures with critical responses to

vibration or with complex structural forms.

3.2 Simplified method

3.2.1 Fundamental Frequency of Structure

3.2.1.1 For building structures, Part I of this set of Guidelines has already provided the

following formula adopted from Murray et al (1997) to calculate the fundamental

natural frequency fn of a building structure:

cgj

n

gf

∆+∆+∆= 18.0

where

∆j is the elastic deflection of the floor joist or beam at mid-span due to

bending and shear

∆g is the elastic deflection of the girder supporting the beams due to

bending and shear

∆c is the elastic shortening of column or wall due to axial strain



Designers should note that the formulae given in Murray et al (1997) are

applicable to steel and/or composite structures, although Murray et al (1997) do

not expressly state its inapplicability to rc structures.

3.2.1.2 In calculating ∆j for concrete slab on structural steel joists, Murray et al (1997:12)

notes that full composite action of the slab can be included even though shear

connectors are not used, because the shear forces at the slab/joist interface can be

resisted by the friction between the concrete and steel surfaces. Designer should

further note that the above formula is only applicable for a simply-supported floor

on rigid supports (usually found with structural steel floor with pinned joints). For

r.c. structures, the monolithic casting would cause rotational restraints to the

beams, and designer should estimate the natural frequency by computer analysis.

3.2.1.3 As in Part I of this set of Guidelines, in using the above formula the loading on the

floor is the dead and actual live loads. Design live loads from the Hong Kong

Building (Construction) Regulations (for offices, the design live load being 3.0

kPa plus a minimum partition load of 1.0 kPa) should not be used for vibration

analysis. Murray et al (1997) suggest that for office floors the actual live load may

be taken as 0.5 kPa (although ASCE 7-05 specifies a design live load of 2.4kPa in

addition to the partition load), and claimed that this value had already included the

live load due to desks, file cabinets, bookcases, etc. Murray et al (1997) further

suggest that the actual live load for residential floors may be taken as 0.25 kPa,

and that for footbridges, gymnasiums and shopping centres may be taken as zero.

Designers should note that these specified actual live loads (e.g. 0.5 kPa or 10

lb/ft2 for office floors) are very light as compared with the design live loads (4.0

kPa for office floors), and should therefore make your own assessment of the

actual live load in your individual case.

3.2.1.4 If the calculated fundamental natural frequency fn of the structure is greater than

the minimum natural frequency fn(req’d) stated in previous section, then the structure

is not susceptible to vibration. However, if the natural fundamental frequency fn of

the structure is less than minimum natural frequency fn(req’d) stated in previous

section, then the peak acceleration of the structure must be calculated and checked

against the acceptable peak acceleration in Section 2.

3.2.1.5 Sometimes, designers may use commercial computer software to calculate the

fundamental natural frequency fn of the structure. In such a case, for floor system

of narrow width (e.g. corridor or footbridge) or multi-level framed system (e.g.

mega-trussed floors), the “fundamental” mode shape obtained from computer

analysis may not be a purely vertical translation. If such fundamental frequency is

then to be used for checking vertical acceleration by hand calculation, the designer

should exercise judgment in choosing the lowest frequency that corresponds to

dominantly vertical translational mode.

3.2.2 Peak Acceleration of Structure

3.2.2.1 Murray et al (1997) propose the following formula to calculate the peak

acceleration ap due to walking:

W

eP

g

a nfp

β

35.0

0

−

= --------- Eqt. (1)



where

=g

ap estimated peak acceleration (in units of g)

=nf fundamental natural frequency of the structure

=0P 0.29kN

β = damping ratio of the floor system (Table 2(a) or Table 2(b))

=W effective weight of the structure = loading on the floor system

The term P0nfe

35.0− represents R×α×P, where R is the reduction factor taking into

account the factors that full steady resonant motion is not achieved for walking

and that the walking person and the person annoyed are not simultaneously at the

location of maximum nodal displacement (may be taken as 0.5), α is the ratio of

peak sinusoidal force to the weight of a person (= 0.83 nfe35.0−

), and P is the weight

of a person (usually taken as 0.7kN).

3.2.2.2 As an alternative, Ellis (2000) proposes the following method to calculate the peak

acceleration:

Step 1: Obtain the characteristic dimension D, which may be taken as the

span of the floor.

Step 2: Obtain the walking velocity of a single pedestrian V.

Step 3: Calculate the circular frequency of the structure ω = 2πfn.

Step 4: Calculate the steady-state acceleration Pa(ms-2

) using the equation:

kN

FP

eff

a β

ω

2

2

= ------ Eqt. (2)

whereβ is damping ratio of the floor system, and k is the stiffness

of the floor system.

The term F represents the product of the Fourier coefficient rn and

the weight of a person (Ellis (2000) assumes it to be 76kgf or

746N). For the choice of rn, 95% of people walk at a pacing

frequency between 1.65Hz and 2.35Hz (Pretlove et al (1995)), the

acceleration will be maximum when the fundamental natural

frequency of the floor system lies within this range. Ellis (2000)

therefore advises the use of different Fourier coefficient rn for

different range of the fundamental natural frequency nf of the floor

system (see Section 3.3.2.4) as follows:

fn rn F

1.5 ~ 2.5Hz n =1, r1 = 0.45 336

3 ~ 5Hz n =2, r2 = 0 .14 104

> 5Hz n ≥ 3, rn = 0 .1 74.6

The stiffness of the floor system k = 48EI/l3, where I = moment of

inertia of the supporting beam and l is the span of the beam. The

effect of the floor slab and adjacent beams is taken into account by

Neff, and Murray et al (1997: 19) gives the following formula to

calculate Neff:



24

9 00059.0)100.9(2.3449.0

−×++= −

S

L

I

L

S

dN

j

t

jeeff

where:

beam.-T theof inertia ofmoment

spab, beam

spacing, beam

depth, slab effective

=

=

=

=

t

j

e

I

L

S

d

Step 5: Calculate the parameters of resonance build-up factor R1 and the

window effect factor R2 by the following equations:

V

D

eR

βω−

−= 11

8

1

220

=V

DR

βω

Ellis (2000) notes that a steady state response to the excitation

takes time to develop, and R1 and R2 (both less than 1) are

included to compensate this delay.

Step 6: The peak acceleration ap = Pa×R1×R2(ms-2

).

For the walking velocity V of a person, Pachi and Ji (2005), who have taken

measurements of over 800 pedestrians on two footbridges and two shopping

centres, found that people walk faster on the shopping centres with a velocity of

1.4ms-1

than on the footbridges with a velocity of 1.3ms-1

. Rather using different

velocities for different structures, Ellis (2000) recommends a typical value of

1.57ms-1

for V.

3.2.2.3 In both methods, the value of the damping ratio β of the floor system is required.

Part I of this set of Guidelines has already stated that the damping ratio β has an

important effect on the peak acceleration, especially when resonance occurs,

Naeim (1991) notes that a damping ratio of up to 20% can be achieved with

partitions. Hewitt and Murray (2004) give typical values for different layout of

partitions for offices in Table 2(a). They also illustrate the arrangement of the

layout with photos, and designer may refer to their paper (available:

http://www.modernsteel.com/Uploads/Issues/April_2004/30728_vibration.pdf, accessed:

4 September 2008).

Table 2(a) Damping ratio β with different partition layout

Layout Damping Ratio

Full-height partitioned office with suspended ceiling

and ductwork attached below the slab

5%

Electronic office with no full-height partitions 2%-2.5%

Open office plan with no full-height partitions 2%-3%

Office library with full-height bookshelves 3%-4%

(Source: Adapted from Hewitt and Murray (2004))



Designers should note that these values are applicable to steel and/or composite

building structures. However, adopting such values for rc structures is generally on

the conservative side. Designers should further note that these values are suggested

for offices with partitions (although in some cases not full-height partitions). For

offices with only few partitions, Ellis (2000) used a β value of 1.65%. For

footbridges, Pretlove et al (1995) found that for rc and steel footbridges the mean β

values are 1.3% and 0.4% respectively. Alternatively, designers may use the

damping ratio β in Table 2(b) recommended by Hicks (2006), which gives

damping values for different types of floors, furniture and finishes.

Table 2(b) Damping ratio β with different types

of floors, furniture and finishes

Components of damping Damping Ratio

1. Damping of bare floors

Reinforced concrete

Steel

Steel-concrete composite

1.5%

1.3%

1.8%

2. Damping due to furniture

Traditional floors for 1 to 3 persons with separation walls

Open plan office

Library

Schools

Gymnastic

2%

0.5%

1%

0%

0%

3. Damping due to finishes

Ceiling under the floor Free floating floor

0.5% 0%

(Source: Adapted from Hicks (2006: 7))

3.2.3 Point Stiffness Criterion

Murray et al (1997) consider that if the natural frequency of the floor is between

9Hz and 10Hz, the deflection due to a person walking across the floor will

superimpose on the floor vibration, and recommend that in addition to checking the

peak acceleration, a point load stiffness criterion should be followed. The point

stiffness criterion requires the point stiffness of the floor to be at least of 1kN per

mm. The point stiffness is the inverse of a displacement in the load direction on a

node where a unit load is applied. Hence, the stiffness is a function of position and

direction. Again, a conservative but quick way to calculate the point stiffness of the

floor system is to ignore the contribution of the floor slab and the adjacent beams,

and the point stiffness of the floor system is estimated by 48EI/l3, where I = moment

of inertia of the supporting beam and l is the span of the beam. The details of the

procedures in calculating the point stiffness of the floor are to be referred to Murray

et al (1997: 19).

3.3 Computer Analytical Method

3.3.1 When the boundary condition or the floor structure is more complicated, or the

loading is not uniformly distributed, or the peak accelerations by the simplified

method of analysis are critical, computer analysis can be used to investigate the

dynamic behaviour due to walking excitation. With the use of computer model, the



value of the fundamental natural frequency and the peak acceleration can then be

calculated.

3.3.2 Loading

3.3.2.1 In order to carry out dynamic analysis to assess the structural behaviour of the

floor system, the characteristic of walking load should first be evaluated. The

simplest scenario is the vibration produced by one person walking along the

footbridge (the ‘single pedestrian loading’), with the walking assumed to be at

constant pace and in a straight line. BS 5400: 1978 recommends a maximum

acceleration of footbridge deck when one pedestrian walked over the main span in

step.

3.3.2.2 As the computer analytical method aims at simulating a person walking from one

end to the other end of the span of the structure, the dynamic loading applied by a

walking person has to be entered as a time-history function for detailed analysis.

For the single-pedestrian loading, the force-time graph of a single footfall is shown

in Figure 1. The two peaks occur characteristically under “heel strike” and “toe

off” for single footfall.

Figure 1 Force-Time Function for a Single Footfall

3.3.2.3 Clark (1981) combined the combined effect of consecutive single-foot in Figure

2, and noted that the combined effect can be represented by a sine wave with an

amplitude of about 25% of the static single walking person load of 0.7kN. This

relationship then formed the assumed loading specified in BS 5400-2:2006, which

states that the maximum vertical acceleration should be calculated assuming that

the dynamic loading applied by a walking person can be represented by F(t),

moving across the main span of the superstructure at a constant speed v as

follows:

F(t) = 700+180 sin 2πfo t (in N), where t is the time (in s)

and v = 0.9 fo (in m/s)

where fo is the pacing frequency of the walking person. The term 180N represents

the walking load, which is the product of the weight of a walking person (assumed

to be 700N) by a dynamic load factor of 0.257.



Figure 2 Moving Pulsating Force

(Source: Adapted from Clark (1981: 153))

3.3.2.4 Rather than a single dynamic load factor, the following generalized Fourier series

can be used to represent the successive footfall of a walking person:

))2

cos(1()(1

n

n

n tT

nrGtF φ

π++= ∑

∞

=

where G is the weight of the walking person,

T = 1/pacing frequency fo,

and rn is the Fourier coefficient.

Ellis (2000) found the values of the Fourier coefficients by actual measurements.

He found that the values vary with the pacing frequency, and their found first and

second Fourier coefficients are shown in Figures 3 and 4. For the third to eight

terms, Ellis (2000) suggested that an value of 0.1 can be adopted.



Figure 3 Values of First Fourier Coefficient against Pacing Frequency

(Source: Adapted from Ellis (2000: 20))

Figure 4 Values of Second Fourier Coefficient against Pacing Frequency

(Source: Adapted from Ellis (2000: 21))

By comparing the values of the coefficients, Ellis (2000) noted that the first

Fourier coefficient is much more significant, and that the contribution of the other

coefficients can be neglected. Hence, the above generalised Fourier series can be

re-written as:

)2cos1(700)( 1 tfrtF oπ+= (in N)

where r1 is the first Fourier coefficient.



In the above equation, there are two parameters r1 and fo. For the value of fo,

Pretlove et al (1995) found that the average pacing frequency is 2Hz with a

standard deviation of 0.175Hz, and that 95% of people walk at a pacing rate

between 1.65Hz and 2.35Hz. Pachi and Ji (2005), who have recently taken

measurements of over 800 pedestrian on two footbridges and two shopping centres,

found that people walk at an average pacing frequency of 2.0Hz on shopping

centres, and at an average pacing frequency of 1.8Hz on footbridges.

Bachmann and Ammann (1987) found the different values of fo in Table 3 for

different walking mode. For the value of r1, Ellis (2000) and Pretlove et al (1995)

gave numerical values of r1 for different pacing frequency fo in Table 3.

Table 3 Values fo and r1 for Different Mode of Walking

r1 Mode of Walking fo

Ellis (2000) Pretlove at al (2008)

Slow Walk 1.7Hz 0.25 0.25

Normal Walk 2.0Hz 0.45 0.40

Fast Walk 2.4Hz 0.55 0.50

(Source: Modified from Pan et al (2008), Ellis (2000)

and Bachmann and Ammann (1987))

3.3.2.5 With these values of r1 and fo, the force-time functions F(t) in Table 4 for different

pacing frequencies are recommended:

Table 4 Force-Time Function for Different Pacing Frequency

Force-Time Function Pacing

Frequency fo Ellis (2000) Pretlove at al (2008)

1.7Hz ttF 68.10cos175700)( += ttF 68.10cos175700)( +=

2.0Hz ttF 56.12cos315700)( += ttF 56.12cos280700)( +=

2.4Hz ttF 07.15cos385700)( += ttF 07.15cos350700)( +=

3.3.2.6 Unlike BS 5400-2:2006, Zivanovic et al (2005) suggest that the force time

histories acting on supporting objects by people walking are complicated functions

which consist of a series of footfall force time histories separated in time and

space (Figure 5(a) and (b)). So modelling the successive single-pedestrian

loading as a simple smooth sine function with time and without due consideration

on the spatial dimension may be over-simplified.



(a) Time Dimension (b) Spatial Dimension

Figure 5 Typical Pattern of Walking Footfall

(Source: Adapted from Zivanovic et al (2005:7))

Figures 6(a), (b) and (c) show the force-time history functions for a single footfall

with different modes of walking and durations of contact provided by Wheeler

(1982). Hence, as an alternative to model the force-time histories by sinusoidal

functions in BS 5400-2:2006, the impulse force due to a single football can be

summated and averaged to form a point load acting at a particular point at a

particular time.

(a) Slow Walk



(b) Normal Walk

(c) Brisk Walk

Figure 6 Force-Time history for a Single Footfall

at Different Pacing Frequencies

(Source: Adapted from Zivanovic et al (2005:7))

3.3.2.7 Table 5 gives the average point load, and with such point load due to a single

footfall, and a computer model will be built in next paragraph with this average

point load.

Table 5 Average Force due to a Single Footfall

Mode of Walk Average Force F

Slow Walk 952N

Normal Walk 616N

Brisk Walk 700N

3.3.3 Computer Model

3.3.3.1 The above paragraph explains how to model the loading from a single footfall in

term of time, and concludes that the simplest way to model the force due to a



single footfall is to represent it by the point load in Table 5. This paragraph will

describe a method to input the average force onto the span of the floor with time.

3.3.3.2 Pan et al (2008) advise that to model the span of the following system into discrete

elements using a finite element model in order to input such loading into the

computer program. This Guideline recommends that the length of each element

may be chosen as the distance between successive strides of a single footfall.

Wheeler (1982) found the stride length for different values of fo in Table 6. For

example, if the single pedestrian is walking at a pacing frequency of 2Hz (i.e. two

strides in a second), from Table 6 the stride length is 0.75m and designers may

choose an element of length 0.75m, which corresponding to the distance between

successive strides as shown in Figure 7(a). With a pacing frequency of 2Hz, each

footfall will take a time of interval of 0.5s. At t=0.5s, the averaged point load in

Table 5 due to the right footfall can then be applied on the node 1 (Figure 7(b)).

Similarly, at t=1s, the average point load in Table 5 due to the left footfall can

then be applied on the node 2 as a point load (Figure 7(c)).

Table 6 Stride Length and Duration of Footfall

fo Stride Length

1.7Hz 0.6m

2.0Hz 0.75m

2.3Hz 1m

(Source: Adapted from Wheeler (1982))

Figure 7(a) Elements along Span of Floor

Figure 7(b) Load on Node 1 at t=0.5s

Figure 7(c) Load on Node 2 at t=1s



3.3.3.3 If the length of each element (say L) is chosen at a value larger than the distance

between successive strides, the single footfall in Table 5 is required to be

distributed linearly to nodes (i on the left and j on the right) at the two ends of an

element according to the following equations when the foot falls at a distance x

from the node I:

L

xFF

L

xLFF ji ×=

−×= ;

3.3.3.4 The commercial package SAP 2000 can be employed for such computer analysis.

It is also necessary to input the geometry of the structure into the program to build

up the model. Multiple cross sections can also be entered to give a more realistic

analysis. An example will be presented in the Section 5 to show the results of

analysis of a footbridge.

3.4 Crowd Effect

3.4.1 A more complex scenario happens when there is more than one person walking.

The load is obviously larger, and so the static deflection of the floor would be

increased. An early-recorded incident occurred in 1831. When 60 soldiers were

marching across the Broughton Suspension Bridge near Manchester, UK, the

footbridge collapsed. As a result, the ‘break step’ command was created when

soldiers were marching across bridges. Earlier studies, however, concluded that

when a group of people walked across a floor system or footbridge, the resulting

acceleration is not necessarily larger. Wheeler (1982) pointed out that the

likelihood of numbers of people both moving in step and not in normal walking

mode is remote. At the normal walking frequency, the probability of occurrence of

walking in step is also not common. The response acceleration due to numbers of

people is similar to that caused by a walking person. Wheeler (1982) concluded

that it is appropriate to analyse the peak acceleration of the footbridge by

considering the loading exerted by a single walking person.

3.4.2 In order to investigate the crowd effect, Ellis (2000) measured the peak vertical

accelerations generated by a crowd on two floors at BRE’s Cardington laboratory.

In the first test, a group of 32 persons were asked to walk on a steel framed building.

In the second test, a group of 300 persons were asked to walk on the first floor of a

seven-storey rc building. In both cases, Ellis (2000) found that the peak

accelerations produced by crowd are of an order similar to that by a single person.

Ellis (2000: 24) further advised that although it is possible that a crowd may

intentionally walk in step at the critical walking frequency, the ‘chances of this

occurring naturally are negligible and …. should not considered to be normal crowd

loading.’

3.4.3 However, the possibility of synchronization of people walking in groups and crowds

should not be neglected, especially in a limited space (e.g. at the close of play in

theatres), where people would likely to be forced to synchronise their steps with the

crowd. This issue has recently attracted a lot of research, especially after the

London Millennium Footbridge problem in 2000 (which will be discussed in

Section 4). BS EN 1995-2:2004 is one of the first international codes that requires



designers to check the crowd effect on timber footbridge. The Structures Design

Manual for Highways and Railways (2006) also gives the following formula to

calculate the peak vertical acceleration due a crowd moving continuously at a speed

of Vms-1

over a simply-supported footbridge:

EI

lVWa sl

p768

5222π

=

where ap = peak vertical acceleration (in ms-2

)

V= speed of the moving crowd (Structures Design Manual for Highways and

Railways (2006) advises to be 3ms-1

)

l = span of the footbridge (in m)

Wsl = unit live load on the footbridge (in kN/m)

E = modulus of elasticity (in kN/m2)

I = the second moment of inertia of the cross-section

However, there is no detailed information in the Structures Design Manual for

Highways and Railways (2006) on the derivation of this formula, or the degree of

synchronization of people included.

3.4.4 Subsequent to the study in 2000, Ellis (2003) conducted further measurements on

the effect of walking crowds in an rc flat-slab building at BRE’s Cardington

laboratory. Ellis (2003: 20) found that the peak accelerations increase with group

size, and the maximum peak acceleration produced by a crowd is ‘no more than

double those generated by an individual’ (emphasis not in the original). He

advised that doubling the peak acceleration calculated or determined for an

individual could reflect crowd loading.

3.4.5 Indeed, the crowd effect depends on the size of the crowd (which in turn depending

on the floor area) and the possibility of synchronisation between people in the

crowd. Matsumoto et al (1978), using a Poisson distribution for the arrivals of

pedestrians who walk in an unsynchronised manner, found that the peak

acceleration due to a single-pedestrian loading is required to be multiplied by a

magnification factor m order to model the dynamic behaviour of the group effect.

They derived the magnification factor m to be N , where N is the number of

persons on the footbridge at any one time. Grundmann et al (1993), using a

probability of synchronisation of 0.225, found the magnification factor m to be

0.135N; but the following table shows that such factor would mean that for a value

of N up to 55 people, their magnification factor is smaller than that (= N )

suggested by Matsumoto et al (1978), although in the latter case no synchronization

has been taken into account.



Magnification factor m Number of

People N Matsumoto et al (1978)

m= N

Grundmann et al (1993)

m=0.135N

1 1 0.135

10 3.2 1.4

20 4.5 2.7

30 5.5 4.1

40 6.3 5.4

50 7.1 6.8

100 10.0 13.5

200 14.1 27.0

Grundmann et al (1993) therefore recommend that for a group up to 10 people, the

value of m should be taken as 3, when the fundamental natural frequency lies

between 1.5Hz and 2.5Hz (see Figure 8).

0

1

2

3

4

0 1 2 3 4 5

Fundamental Natural Frequency fn

Magnification Factor m

Figure 8 Relationship between Magnification Factor m

and the Fundamental Natural Frequency fn (Source: Adapted from Zivanovic et al (2005:46))

3.4.6 Zivanovic et al (2005), after a review of 2000 references on the subject published

before 2003, concluded that where the area of the floor structure is less than 37m2,

the crowd effect can be obtained by multiplying the peak acceleration due to a

single-pedestrian loading by a magnification factor m of 3, if the fundamental

natural frequency lies between 1.5 to 2.5Hz. Zivanovic et al (2005) further

cautioned that if the area of the floor structure is greater than 37m2, the

magnification factor m should be increased further (which should be dependent on

the area), because there is a higher probability of synchronization among the

crowd.

3.4.7 Therefore, although a ‘single-pedestrian loading’ is specified in the international

codes and/or recommended in various literatures, designers should bear the crowd



effect in mind in modelling the walking loading on the structure, especially where

there is a possibility of crowd walking in an unsynchronised or synchronised

manner. Based on the above recent research on the crowd effect, this Guideline

recommends that the magnification factor m should be based on the floor area, the

fundamental natural frequency fn of the floor structure, and the use of the structure.

The floor area determines the crowd size, and Bachmann and Ammann (1987) noted

that the maximum physically possible crowd density for footbridge deck can be 1.6-

1.8 person/m2, and recommended that a value of 1 person/m

2 for footbridge. The

use of the structure determines the probability of synchronisation of the crowd, and

a higher probability is expected for pedestrians in footbridges. The fundamental

natural frequency of the floor structure determines whether the pacing frequency of

the crowd will cause the structure to be in resonance. With these three parameters,

the recommendations in Table 7 are suggested.

Table 7 Magnification Factor m for Crowd Effect

Building Structures* Footbridges

Fundamental

natural frequency

Magnification

factor

Floor area#

< 37m2

Floor area#

> 37m2

fn > 3Hz m = 2

fn = 1~3Hz m = 3

Same as building

structures

m = N

or 0.135N

where N may be

taken as 1×Area

of the footbridge

deck (in m2)

Note: * Designers should note that there may be cases in building structures, where there are

a significant probability of crowd effect in either synchronized (e.g. at the close of

play in theatres) or unsynchronized (e.g. in the gallery of a museum) manner. In

such a case, rather than designing it as a building structure, the crowd effect shall be

included as that for a footbridge. Moreover, designers should also note that if an

office, library, meeting room, etc. (where occupants are very sensitive to vibration)

is situated near such a structure, the acceptable criteria should follow those for

offices or similar buildings structures in Table 1.

#

Floor area is the area of the floor panel between column grids.

3.4.8 Designers should also note that the design of human-induced vibration caused by

crowd effect is a rather new area of research, and should therefore always check

with the latest development when using the recommendations.

4. Footbridge Vibration

4.1 As stated in earlier section, human–induced vibration due to walking excitation may

also be a serviceability problem in footbridges. Incidences of human-vibration

problems in footbridges have also appeared as headlines in both Hong Kong and

overseas. In 1982, a footbridge in the Shatin Racecourse was found to be vibrating

excessively transversely at their supporting columns (Tung and Wong (1983)).

Recently, when the London Millennium Footbridge opened on 10 June 2000, it was

also reported that there was excessive lateral vibration of the footbridge. The design

approach for walking excitation of footbridges is the same as that for building

structures, i.e. first to compare the natural frequency of the footbridge with the



required minimum frequency, and if the natural frequency of the footbridge is less

than the required minimum frequency, then it is necessary to compute the peak

acceleration due to walking, which is then to compare with the maximum

acceptable acceleration. Again, two methods can be adopted: a simplified method

for preliminary design or simple structures, and a computer analytical method for a

more accurate analysis.

4.2 Minimum Required Frequency and Acceptable Peak Acceleration

4.2.1 Fundamental Frequency

BS 5400-2:2006 and the Structures Design Manual for Highways and Railways

(2006) issued by the Highways Department state that the vibration serviceability

requirement for footbridges is deemed to be satisfied if the fundamental frequency fn

is greater than 5Hz for an unloaded bridge in the vertical direction (Table 8). For

vibration in the horizontal direction, BS 5400-2:2006 specifies that the fundamental

frequency fn shall be greater than 1.5Hz (Table 8); but does not specify the

vibration limits for supporting columns. The Structures Design Manual for

Highways and Railways (2006), on the other hand, specifies minimum fundamental

natural frequencies for the supporting columns, which shall not be less than 2Hz

transversely and 1Hz horizontally.

Table 8 Minimum Required Fundamental Frequency fn(req’d)

Usage Minimum Required Fundamental

Natural Frequency fn(req’d) (Hz)

Footbridge (vertical vibration) 5

Footbridge (horizontal vibration) 1.5

4.2.2 Vertical Acceleration

4.2.2.1 BS 5400-2:2006 and the Structures Design Manual for Highways and Railways

(2006) state that if the fundamental natural frequency of vibration fo of the

unloaded bridge exceeds 5Hz in the vertical direction, the vibration serviceability

requirement is deemed to be satisfied. BS 5400-2:2006 and the Structures Design

Manual for Highways and Railways (2006) further states that if fn is equal to or

less than 5Hz, the maximum vertical acceleration of any part of the superstructure

shall be limited as2ms5.0 −

nf , where fn (Hz) is the fundamental natural frequency

of the unloaded structure in the vertical direction.

4.2.2.2 Murray et al (1997) proposes numerical values for the acceptable peak

acceleration rather than values depending on the fundamental frequency. His

proposed values are 1.5% g and 5% g for indoor and outdoor footbridges

respectively. For a footbridge with a fundamental frequency less than 5Hz (say

the average pacing frequency 2Hz), the acceptable peak acceleration shall be, in

accordance with BS 5400-2:2006 and Structures Design Manual for Highways and

Railways (2006), in the order of 0.7ms-2

(or 7% g), which far exceeds those limits

(i.e. 1.5 or 5% g) proposed by Murray et al (1997). Part I of this set of Guidelines



has stated that the values proposed by Murray et al (1997) are based on ISO 2631-

2, which is an adaptation using a multiplying factor of a curve obtained for a

standing subject, and Pimentel and Waldron (1997: 340) commented that since

footbridges are designed for movement of pedestrians, evaluating the vibration

serviceability using such ‘standing subjects would result in a conservative

solution’ (emphasis not in the original). Hence, this Guideline will adopt the

criteria set in BS 5400-2:2006 and the Structures Design Manual for Highways

and Railways (2006) as the governing criteria for footbridges.

4.2.3 Lateral Vibration

4.2.3.1 Apart from vertical vibration, lateral vibration of footbridges can also lead to

serviceability problems. Dallard et al (2001) note that vertical forces generated by

pedestrians are generally random whilst the lateral forces are strongly correlated

with the lateral movement of the bridge. This is because pedestrians are less stable

laterally than vertically. The pedestrians are therefore more sensitive to lateral

vibration than vertical vibration and they will modify their walking patterns when

they experience such vibration.

4.2.3.2 For the footbridge columns, the Structures Design Manual for Highways and

Railways (2006) states that the fundamental natural frequencies shall not be less

than 2Hz transversely and 1Hz horizontally, and contains a simplified formula to

calculate the natural frequency of a free-standing cantilever column supporting a

footbridge.

4.2.3.3 For footbridges generally, BS 5400-2:2006 states that if the fundamental natural

frequency fn of vibration exceeds 1.5 Hz for the loaded bridge in the horizontal

direction, the vibration serviceability requirement is deemed to be satisfied.

Where the fundamental frequency of horizontal vibration is less than 1.5 Hz, BS

5400-2:2006 states that special consideration shall be given to the possibility of

excitation by pedestrians of lateral movements of unacceptable magnitude.

Neither BS 5400-2:2006 nor the Structures Design Manual for Highways and

Railways (2006) contains formula(e) to find the natural frequency of the structure

of the footbridge in the horizontal direction.

4.2.3.4 In the past, there is little research on lateral vibration due to walking loads.

Following the London Millennium Bridge wobble in year 2000, larger amount of

research (e.g. Dallard et al (2001), Roberts (2005)) on this topic is now available.

Dallard et al (2001) had done tests and research on the Millennium Bridge wobble,

and concluded that the excessive sway of the Millennium Bridge was because the

pedestrians walked in synchronization with the natural swaying of the footbridge

(i.e. at the resonant frequency), amplifying the motion of the footbridge. Both

Dallard et al (2001) and BS 5400-2:2006 state that lateral vibration problems are

especially critical on footbridges having low mass and damping and expected to be

used by crowds of people.

4.2.3.5 Dallard et al (2001) further found footbridge is dynamically stable in horizontal

direction until a critical number of people are walking, and then it increases very

rapidly. The implication of this is that unless a footbridge has experienced its



critical number of pedestrians lateral vibration may not be a problem. The limiting

number of people to avoid instability can be estimated from:

k

McfN n

L

π8=

where

LN = limiting number of people

fn = natural frequency in the horizontal direction

c = critical damping ratio

M = mass of the bridge

k = lateral walking force coefficient (in Ns/m)

4.3 Simplified Method

4.3.1 Fundamental Frequency

4.3.1.1 Pretlove et al (1995), by fitting the fundamental natural frequencies of 67

footbridges in vertical direction around the world, found the following empirical

relationship between the fundamental natural frequency fn (in Hz) and the span

length l (in m):

Concrete: fn = 39l -0.77

Steel: fn = 35l -0.73

Composite: fn = 42l -0.84

Pretlove et al (1995), however, state that these formulae cannot replace a proper

design prediction, and hence this Guideline considers that these formulae should

only be used in preliminary design when only the span length is available and the

other structural dimensions of the footbridge are not available.

4.3.1.2 Once the structural dimensions of the footbridge, BS 5400-2:2006 gives the

following formula to calculate the fundamental natural frequency fn for single span,

or two-or-three-span continuous, symmetric footbridge of constant cross-section:

M

EIg

l

Cf n 2

2

2π=

where g is the acceleration due to gravity (in m/s2)

l is the length of the main span (in m)

C is the configuration factor (Table 9)

E is the modulus of elasticity (in kN/m2)

I is the second moment of inertia of the cross-section

M is the weight per unit length (including superimposed dead load but

excluding pedestrian live load) (in kN/m)



Table 9 Value of C

4.3.2 Vertical Acceleration

4.3.2.1 BS 5400-2:2006 gives the following formula to calculate the peak vertical

acceleration ap from the passage of a single-pedestrian loading with a pacing

frequency equal to the fundamental frequency fn for single span, or two-or-three-

span continuous, symmetric footbridge of constant cross-section:

ψπ kyfa snp

224= ------ Eqt. (3)

where fn is the fundamental frequency (in Hz)

ys is the static deflection at the mid-span (in m) with a vertical point load of

0.7kN applied at this point

k is the configuration factor (Table 10)

ψ is the dynamic response factor (Figure 9)

Table 10 Value of k



Figure 9 Value of ψ

Note: δ (i.e. β in Section 3) is the damping ratio.

4.3.2.2 BS 5400-2:2006 states that if the fundamental natural frequency fn of the structure

is greater than 4Hz, the calculated peak vertical acceleration ap from Eqt. (3) may

be reduced by an amount varying linearly from zero reduction at 4Hz to 70%

reduction at 5Hz. Presumably, this reduction takes into account the fact that the

pacing frequency of a normal person is in the range of 1.5-2.5Hz, and hence the

footbridge will be less susceptible to walking excitation above this range. This is

also consistent with Eqt. (1) by Murray et al (1997) and Eqt. (2) by Ellis (2000) in

Section 3.2.2, where both equations consider that the calculated peak acceleration

decreases with the fundamental natural frequency fn of the structure. Eqt. (1) by

Murray et al (1997) further shows that the calculated peak acceleration decreases

exponentially with the fundamental natural frequency fn of the structure, and hence

the linear reduction in BS 5400-2:2006 should be in the conservative side. No

reduction formula is suggested in BS 5400-2:2006 if the fundamental natural

frequency fn of the structure exceeds 5Hz, as BS 5400-2:2006 states that if the

fundamental natural frequency of vibration fo of the structure exceeds 5Hz in the

vertical direction, the vibration serviceability requirement is deemed to be

satisfied.



4.4 Computer Analytical Method

The computer analytical method for the analysis of footbridge is the same as that for

floor system, i.e. by simulating the loading generated by the footfall of a person

walking along the main span of the footbridge as a function of time and space, and

then using a structural program to obtain the fundamental frequency and the peak

acceleration due to the walking excitation.

5. Design Examples

5.1 The following four examples illustrate the structural analysis of different types of

structures subjected to walking excitation. Examples 1, 2 and 3 are respectively a

footbridge, rc floors and structural steel floors, and the analysis is carried out by the

simplified method of analysis. Example 4 uses the analytical method to calculate

the peak acceleration of a footbridge of two spans 8m and 4m.

5.2 Example 1: Footbridge

Consider a simply-supported footbridge of span 20m with cross-section as shown in

Figure 10.

Figure 10 Cross-section of Footbridge in Example 1

Concrete slab properties:

mc = 2400 kg/m3

Ac = 600000 mm2

For a 28-day compressive strength fcu = 35 N/mm2,

modulus of elasticity = 23.7 kN/mm2 (according to the Code of Practice for

the Structural Use of Concrete 2004)

The dynamic concrete modulus of elasticity Ec = 23.7 × 1.35 = 32 kN/mm2

Ic = 1125 × 106 mm

4

838x292x194kg/m UB properties:

ms = 193.8 kg/m

As = 24700 mm2

Es = 205 kN/mm2

Is = 2792 × 106 mm

4



n = Es/Ec = 205/32 = 6.41

y =

2470041.6

600000

)2

7.840150(24700

2

150

41.6

600000

+

+×+×

= 178 mm

I = 41.6/])2

150178(600000101125[ 26 −×+×

+ 26 )178

2

7.840150(24700102792 −+×+×

= 7763 × 106 mm

4

M = 2400 × 4 × 0.15 + 193.8

= 1634 kg/m

C = π for single span

Using the formulae given in BS 5400-2:2006 (see Section 4),

nf = 1000/81.91634

81.910763.710205

)20(2

36

2

2

×

×××× −

ππ

= 3.88 Hz which is less than the minimum required fundamental frequency

(5Hz) in Table 8, and hence it is necessary to compute the peak

acceleration.

ys = EI

Fl

48

3

= 6

3

10776320548

200007.0

×××

×

= 0.073 mm

k = 1.0 for single span

δ = 0.03

ψ = 8.6

pa = ψπ kyf sn

224 = 4 × π2 × 3.88

2 × 0.073/1000 × 1.0 × 8.6

= 0.373 ms-2

= 3.80 % g for a single-pedestrian loading

The acceptable peak acceleration limit is 0.5× 88.3 =0.985ms-2

(i.e. 10.04% g), and

hence the footbridge is satisfactory for a single-pedestrian loading.

The footbridge deck area is 4×20=80m2, which is greater than 37m

2, and the

fundamental natural frequency is less than 3Hz. From Table 7, an appropriate

magnification factor m may be N (=8.94) or 0.135N (=10.8). Therefore, with

crowd effect taken into account and take m=10.8, the peak acceleration 4.03ms-2

(i.e. 41% g), which far exceeds the acceptable peak acceleration limit of 10.04% g.

Using the formula given in the Structures Design Manual for Highways and

Railways (2006) and with Wsl=0.5×4=2.0kN/m=2.0×103N/m and V=3ms

-1, the peak

vertical acceleration due a crowd moving continuously at a speed of 3ms-1

is

0.68ms-2

(i.e. 6.93% g).



5.3 Example 2: RC Floors

Consider a typical r.c. floor system as shown in Figure 11. For simplicity, assume

that all beams are simply supported. Assume actual live load = 0.5 KPa.

Figure 11 Interior Floor Framing for the Example 2

Murray et al (1997)’s method:

Beam 600×400

mc = 2400 kg/m3

For a 28-day compressive strength fcu = 35 N/mm2,

modulus of elasticity = 23.7 kN/mm2 (according to the Code of Practice for

the Structural Use of Concrete 2004)

The dynamic concrete modulus of elasticity,

Ec = 23.7 × 1.35 = 32 kN/mm2

Ib = 12740 × 106 mm

4

Self-weight of concrete (slab + beam) = [(0.15×4)+(0.6-0.15) ×0.4] ×24

= 18.72 kN/m

Supported weight wb = 0.5×4+18.72

= 20.72 kN/m

∆b = bc

bb

IE

Lw

384

54

= 63

4

10127401032384

800072.205

××××××

= 2.711 mm

Beam 800×400

mc = 2400 kg/m3

fcu = 35 N/mm2



Ec = 23.7 × 1.35 = 32 kN/mm2

Ig = 29570 × 106 mm

4

Self-weight of concrete (beam), wg = (0.8-0.15) ×0.4 ×24

= 6.24 kN/m

∆g = gc

gbb

gc

gg

IE

LLw

IE

Lw

48384

534

+

= 63

3

63

4

1029570103248

8000800072.20

10295701032384

800024.65

××××

××+

××××

××

= 2.220 mm

With supports of square columns size 600x600 and floor height of 3500mm and

assume the stress in column is 7.5MPa, the elastic shortening,

∆c =

cc

c

AE

PL =

31032

35005.7

××

= 0.820 mm

Combined Mode Properties

fn = cgb

g

∆+∆+∆18.0 =

820.0220.2711.2

981018.0

++= 7.43 Hz

As stated in Section 3.2.1.2, the above formula has not taken into account the

monolithic casting at the supports of the beams, and designer may estimate the

natural frequency with the effect of monolithic casting included by computer

analysis.

For office occupancy without full height partitions, β = 0.03 and Po = 0.29 kN

W = 20.72 × 8+6.24×8 = 216kN

g

a p =

W

fP n

β)35.0exp(0 −

= 21603.0

29.0 )43.735.0(

×× ×−e

= 0.00332 = 0.33 %g

which is less than the 0.5%g limit in Table 1, and the floor is therefore judged

satisfactory for walking vibration.

Ellis (2000)’s method:

fn = 7.43 Hz

Since fn>5Hz, use rn=0.1, and hence F=74.6.

ω = 2π fn = 46.68 Hz

β = 0.03

k = 3

48

L

EI = 1000

8000

10127401032483

63

×××××

= 38220000

Neff =

24

9 00059.0)100.9(2.3449.0

−×++ −

S

L

I

L

S

d j

t

je = 1.77



Pa =k

w

eff

2

N 2

6.74

β×

= 0.04003 m/s2

R1 = V

wD

e

β−

−1 = 57.1

868.4603.0

1

××−

− e = 0.999

R2 = 8

1

20

V

wDβ=

8

1

57.120

868.4603.0

×

××= 0.879

pa = Pa × R1 × R2 =0.04003 × 0.999 × 0.879

= 0.0352 m/s2 = 0.358 % g (which is less than 0.5% g in Table 1)

Similar calculations can be carried out with the same sizes of beams for span

lengths of the secondary beam from 5m to 10m using the methods of Murray et al

(1997) and Ellis (2000), and the results are summarised in the following table.

Designers should also note that the crowd effect is not included in these methods.

Since fn>3Hz, Table 7 recommends that the magnification factor m is taken as 2,

and the results are also given in the following table.

Span (m) 5 6 7 8 9 10

fn (Hz) 10.74 9.62 8.50 7.43 6.46 5.60

pa (%g) (Murray et al (1997)) 0.15 0.19 0.25 0.33 0.43 0.53

pa (%g) (Ellis (2000)) 0.18 0.25 0.31 0.36 0.38 0.39

pa (%g) for crowd effect 0.36 0.50 0.62 0.72 0.76 0.78

For the beam size of the example, it is noted that taking into account of the crowd

effect, human-induced vibration due to walking excitation for the rc floor with few

partition of over 7m requires further attention. Of course, deeper rc beam can

improve the performance.

5.4 Example 3: Structural Steel Floors

Consider a typical r.c. floor supported by steel beams as shown in Figure 12. For

simplicity, assume that all beams are simply supported. Assume that actual live

load = 0.5 kPa.

Figure 12 Interior Floor Framing Details for the Example 3



Murray et al (1997)’s method:

UB533×210×92kg/m

Ms = 92.1 kg/m

Es = 205 kN/mm2

For a 28-day compressive strength fcu= 35 N/mm2, the dynamic concrete modulus of

elasticity

Ec = 23.7 × 1.35 = 32 kN/mm2

n = Es/Ec = 205/32 = 6.41

y =

1170041.6

1600150

)2

1.533150(11700

2

150

41.6

1600150

+×

+×+×× = 156 mm

Ib = 552.3 × 106 + 11700× (150-156+533.1/2)

2 + (1600/6.41)×150

3/12

+ (1600/6.41) ×150× (156-75)2

= 1662 × 106 mm

4

Self-weight of concrete slab+steel beam = 0.15×4×24+92.1×0.00981

= 15.3 kN/m

Supported weight wb = 0.5×4+15.3

= 17.3 kN/m

∆b = bs

bb

IE

Lw

384

54

= 63

4

10166210205384

80003.175

××××

××

= 2.708 mm

UB762×267×147kg/m

Ms = 146.9 kg/m

y =

1870041.6

1600150

)2

754150(18700

2

150

41.6

1600150

+×

+×+×× = 226 mm

Ig = 1685 × 106 + 18700× (150-226+754/2)

2 + (1600/6.41)×150

3/12

+ (1600/6.41) ×150× (226-75)2

= 4303 × 106 mm

4

∆g = gs

gbb

gs

gg

IE

LLw

IE

Lw

48384

534

+

= 63

3

63

4

1043031020548

800080003.17

10430310205384

800047.15

××××

××+

××××

××

= 1.762 mm



With supports of square columns size 600x600 and floor height of 3500mm and

assume the stress in column is 7.5MPa, the elastic shortening,

∆c =

cc

c

AE

PL =

31032

35005.7

×

× = 0.820 mm

Combined Mode Properties

fn = cgb

g

∆+∆+∆18.0 =

820.0762.1708.2

981018.0

++= 7.75 Hz

W = (17.3 × 8+1.47×8) = 150kN

For office occupancy without full height partitions, β = 0.03 and Po = 0.29 kN

g

a p =

W

fP n

β)35.0exp(0 −

= 15003.0

29.0 )75.735.0(

×× ×−e

= 0.00428 = 0.43 %g

which is less than the 0.5%g limit in Table 1, and the floor is therefore judged

satisfactory for walking vibration.

Ellis(2000)’s method:

fn = 7.75 Hz

Since fn>5Hz, use rn=0.1, and hence F=74.6.

ω = 2π fn = 48.69 Hz

β = 0.03

k = 3

48

L

EI = 1000

8000

10166310205483

63

×××××

= 31960781

Neff =

24

9 00059.0)100.9(2.3449.0

−×++ −

S

L

I

L

S

d j

t

je = 1.79

Pa =k

w

eff

2

N 2

6.74

β×

= 0.0515 m/s2

R1 = V

wD

e

β−

−1 = 57.1

869.4803.0

1

××−

− e = 0.999

R2 = 8

1

20

V

wDβ=

8

1

57.120

869.4803.0

×

××= 0.884

pa = Pa × R1 × R2 =0.0515 × 0.999 × 0.884

= 0.0455 m/s2 = 0.46% g (which is less than 0.5% g in Table 1)

Similar calculations can be carried out with the same sizes of beams for span

lengths of the secondary beam from 5m to 10m using the methods of Murray et al

(1997) and Ellis (2000), and the results are summarised in the following table.

Designers should also note that the crowd effect is not included in these methods.

Since fn>3Hz, Table 7 recommends that the magnification factor m is taken as 2,

and the results are also given in the following table.



Span (m) 5 6 7 8 9 10

fn (Hz) 11.59 10.26 8.96 7.75 6.68 5.75

pa (%g) (Murray et al (1997)) 0.17 0.23 0.32 0.43 0.56 0.70

pa (%g) (Ellis (2000)) 0.25 0.34 0.42 0.46 0.48 0.49

pa (%g) for crowd effect 0.50 0.68 0.84 0.92 0.96 0.98

For the beam size of the example, it is noted that taking into account of the crowd

effect, human-induced vibration due to walking excitation for the steel/composite

floor with few partition of over 6m requires further attention. Of course, deeper

beam will improve the performance.

5.5 Example 4: Computer Analytical Method

5.5.1 Another rc footbridge of with two spans 8m and 4m as shown in Figure 13 and with

the cross-section as shown in Figure 14 is input into SAP2000:

Figure 13 Footbridge for Example 4

Figure 14 Cross-Section for the Example 4

The computed sectional properties are as follows:

2

43

kN/mm205

kN/m3.15

m10243.2

=

=

= −

sE

w

xI

5.5.2 Using the formula given in BS 5400-2:2006 (see Section 4),

fn = 17.02Hz

As the fundamental frequency fn is greater than 5Hz, there is no need to compute the

peak acceleration. The following computation is aimed at illustrating the computer

model used to compute the peak vertical acceleration.



Using the formulae given in BS 5400-2:2006 (see Section 4),

fn = 17.02Hz, ys = 1.62×10-5

m, and pa = 0.584ms-2

In Section 4, it has been mentioned that BS 5400-2:2006 states that if the

fundamental natural frequency fn of the structure is greater than 4Hz, the calculated

peak vertical acceleration ap may be reduced by an amount varying linearly from

zero reduction at 4Hz to 70% reduction at 5Hz. In the present case, the fundamental

natural frequency is 17.02Hz, the reduction shall be over 70%, although BS 5400-

2:2006 does not provide the reduction factor. For a conservative design, take the

reduction factor to be 70%, the calculated peak acceleration ap is 0.584×0.3=

0.175ms-2

(or 1.79% g).

5.5.3 A computer model is now used to compute the peak acceleration. The model is

built to suit a pacing frequency of 2Hz. The nodal distance is chosen as 0.75m in

the model as shown in Figure 15. With a pacing frequency of 2Hz, the single

footfall is modelled by an equivalent point load of 616N (see Table 5), which is

then applied to successive nodes of the span of the footbridge at a time interval of

0.5s as discussed in Section 3.

Figure 15 Computer Model for Example 4

Figures 16 and 17 show the vertical acceleration along the span of the footbridge at

t=0.5s (i.e. after the first footfall) and t=1.0s (after the second footfall) respectively.



-0.15

-0.1

-0.05

0

0.05

0.1

0 2 4 6 8 10 12

span (m)

Acceleration (ms-2)

Figure 16 Vertical Acceleration at t=0.5s along the Span

-0.15

-0.10

-0.05

0.00

0.05

0.10

0 2 4 6 8 10 12

span (m)

Acceleration (ms-2)

Figure 17 Vertical Acceleration at t=1s along the Span

The accelerations at the mid-span of the footbridge due to the successive footfalls

are then summated. The envelope of the peak acceleration at the mid-span is also

superimposed in Figure 18. The peak acceleration at mid-span is 0.077ms-2

(i.e.

0.785% g) at t=2.5s as against 1.79% g calculated using the simplified formula in

BS 5400-2:2006.



-0.0769

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0 2 4 6 8 10 12 14

Time (sec)

Peak Acceleration (ms-2)

Figure 18 Envelope of Acceleration at Mid-Span of the Footbridge

5.5.4 Designers should, however, note the discussion in Section 3.4, which concludes that

the crowd effect is not included in the above calculation, and the calculated peak

acceleration represents that produced by a single-pedestrian loading. Using Table 7,

as the footbridge deck area is 12×3=36m2, which is less than 37m

2 and the

fundamental natural frequency is greater than 3Hz, an appropriate magnification

factor m is 2. Therefore, with crowd effect taken into account, the peak acceleration

at mid-span is 0.154ms-2

(i.e. 1.55% g). Using the formula given in the Structures

Design Manual for Highways and Railways (2006) and with Wsl=0.5×3=1.5kN/m

and V=3ms-1

, the peak vertical acceleration due a crowd moving continuously at a

speed of 3ms-1

is 0.121ms-2

(i.e. 1.23% g). The following table summarises the

computed results for Example 4:

Methods

Peak acceleration ap at

mid-span

BS 5400-2:2006 0.175ms-2

Sin

gle

-

ped

est

ria

n

Lo

ad

ing

Computer model using

SAP2000 0.077ms

-2

Table 7 0.154ms-2

Cro

wd

Lo

ad

ing

Structures Design

Manual for Highways

and Railways (2006)

0.121ms-2



6. Design References

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Canadian Journal of Civil Engineering, 3(2), pp. 165-73.

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guideline on the design of floor for vibration due to human ed

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