Floor Vibration Guide

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

Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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File code : WalkingVibration.doc MKL/KYL/RL//CYW Issue/Revision Date : June 2009

CONTENTS Content Page 1.

Introdu Introduction ction ............................. ............................................. ................................ ................................ ............................... ...............................3 ................3

2.

Minimum Required Fundamental Fundamental Frequency and Acceptable Peak Acceleration for Offices or Similar Building Structures .................... .......... ................... ................... .................... ................... ..........4 .4

3.

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

4.

Footbrid Footbridge ge Vibration................... Vibration................................... .............................. .............................. ................................ ...........................20 ...........20

5.

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

6.

Design References........................ References........................................ .............................. .............................. ................................ ...........................37 ...........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. party. No liability is therefore undertaken undertaken to any any third party. 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). data). Compliance with this Guideline does not itself confer immunity from legal obligations. Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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1.

Introduction

1.1

Part I of this set of Guidelines has has already mentioned that human-induced human-induced vibration is a problem for lightweight floors with large span, and has identified the sources of vibration to include rhythmic 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 human-induced vibration due to walking.

1.2

Human-induced vibration due to walking usually occurs in shopping shopping malls with lightweight, long span and open 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 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, however, causes nuisance and discomfort to the occupants. Human perceptibility of of vibration depends on a number of of interrelated factors, i.e. i.e. serviceability problem. Among these these factors, factors, ISO 2631-2:1989 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, offices, libraries, museums and exhibition centres are especially most sensitive to vibration; whilst those taking part in rhythmic activities can tolerate much 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, hospitals, as they will inevitably house delicate and expensive medical equipment whose accuracy will be sensitive to vibration. For medical and other other sensitive equipment, 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 the vibration. It recommends recommends designers to 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 occupants from people should be avoided. 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 this Guideline are therefore to provide: (a)

guidance to designers designers on the acceptable limits for structures (including (including building floors and footbridges) subjected to human-induced vibration by walking; and


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(b)

methods to calculate the dynamic responses of structures under humaninduced 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 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.


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Acceptable

Peak


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 f n(req’d) and Acceptable Acceleration for Offices or Similar Building Structures Minimum Required Fundamental Acceptable Peak Acceleration Natural Frequency f n(req’d) (Hz) 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 f or 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 f n of a building structure: f n where ∆ j ∆g ∆c

= 0.18

g

∆ j + ∆ g + ∆ c

is the elastic deflection of the floor joist or beam at mid-span due to bending and shear is the elastic deflection of the girder supporting the beams due to bending and shear is the elastic shortening of column or wall due to axial strain


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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 2 lb/ft 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 f n of the structure is greater than the minimum natural frequency f n(req’d) stated in previous section, then the structure is not susceptible to vibration. However, if the natural fundamental frequency f n of the structure is less than minimum natural frequency f n(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 f n 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 a p due to walking: a p P 0 e −0.35 f n = --------- Eqt. (1) g β W Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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where a p g

= estimated peak acceleration (in units of g )

= fundamental natural frequency of the structure P 0 = 0.29kN f n

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 P 0 e − f n 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 e −0.35 f n ), and P is the weight of a person (usually taken as 0.7kN). 0.35

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 = 2π f n. -2 Step 4: Calculate the steady-state acceleration P a(ms ) using the equation: P a

=

F ω 2 2 β N eff k

------ Eqt. (2)

where β 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 r n and the weight of a person (Ellis (2000) assumes it to be 76kgf or 746N). For the choice of r n, 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 r n for different range of the fundamental natural frequency f n of the floor system (see Section 3.3.2.4) as follows: f n 1.5 ~ 2.5Hz 3 ~ 5Hz > 5Hz

r n n =1, r 1 = 0.45 n =2, r 2 = 0 .14 n ≥ 3, r n = 0 .1

F 336 104 74.6 3

The stiffness of the floor system k = 48EI/l , 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 N eff , and Murray et al (1997: 19) gives the following formula to calculate N eff : Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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N eff

L j4 = 0.49 + 34.2 + (9.0 × 10 ) S I t d e

−9

 L  − 0.00059 j   S 

2

where: d e = effective slab depth,

= beam spacing, L j = beam spab, I t = moment of inertia of the T - beam. S

Step 5:

Calculate the parameters of resonance build-up factor R1 and the window effect factor R2 by the following equations: βω D

R1

=1− e

−

V 1

R2

Step 6:

βω D  8 =     20V 

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. -2 The peak acceleration a p = P a× R1× R2(ms ).

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 -1 1.4ms than on the footbridges with a velocity of 1.3ms . Rather using different velocities for different structures, Ellis (2000) recommends a typical value of -1 1.57ms 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 5% and ductwork attached below the slab 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))


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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 1. Damping of bare floors Reinforced concrete Steel Steel-concrete composite 2. Damping due to furniture Traditional floors for 1 to 3 persons with separation walls Open plan office Library Schools Gymnastic 3. Damping due to finishes Ceiling under the floor Free floating floor

Damping Ratio

1.5% 1.3% 1.8% 2% 0.5% 1% 0% 0% 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, 3 and the point stiffness of the floor system is estimated by 48EI/ l , 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


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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:

and

F(t) = 700+180 sin 2π f o t (in N), where t is the time (in s) v = 0.9 f o (in m/s)

where f o 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. Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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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: ∞ 2nπ F (t ) = G (1 + r n cos( t + φ n )) T n =1 where G is the weight of the walking person, T = 1/pacing frequency f o, and r n 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.


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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: F (t ) = 700(1 + r 1 cos 2π f o t ) (in N) where r 1 is the first Fourier coefficient.


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In the above equation, there are two parameters r 1 and f o. For the value of f o, 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 f o in Table 3 for different walking mode. For the value of r 1, Ellis (2000) and Pretlove et al (1995) gave numerical values of r 1 for different pacing frequency f o in Table 3. Table 3 Values f o and r 1 for Different Mode of Walking Mode of Walking f o r 1 Ellis (2000) Pretlove at al (2008) 0.25 0.25 Slow Walk 1.7Hz 0.45 0.40 Normal Walk 2.0Hz 0.55 0.50 Fast Walk 2.4Hz (Source: Modified from Pan et al (2008), Ellis (2000) and Bachmann and Ammann (1987))

3.3.2.5 With these values of r 1 and f o, the force-time functions F(t) in Table 4 for different pacing frequencies are recommended: Table 4 Force-Time Function for Different Pacing Frequency Pacing Force-Time Function Frequency f o Ellis (2000) Pretlove at al (2008) 1.7Hz F (t ) = 700 + 175 cos10.68t F (t ) = 700 + 175 cos10.68t 2.0Hz

F (t ) = 700 + 315 cos12.56t F (t ) = 700 + 280 cos12.56t

2.4Hz

F (t )

= 700 + 385 cos15.07t F (t ) = 700 + 350 cos15.07t

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.


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(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 s ingle 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


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(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 952N Slow Walk 616N Normal Walk 700N Brisk Walk 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


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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 f o 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 f o 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


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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 − x x F i = F × ; F j = F × L L 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 Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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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 -1 of V ms over a simply-supported footbridge: a p

=

5W sl π 2V 2l 2

768 EI -2 where a p = peak vertical acceleration (in ms ) V = speed of the moving crowd ( Structures Design Manual for Highways and -1 Railways (2006) advises to be 3ms ) l = span of the footbridge (in m) W sl = unit live load on the footbridge (in kN/m) 2 E = modulus of elasticity (in kN/m ) 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 0.135 N ; 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.


Page 18 of 39


Number of People N

Magnification factor m Matsumoto et al (1978) Grundmann et al (1993) m=0.135 N m= N 1 3.2 4.5 5.5 6.3 7.1 10.0 14.1

1 10 20 30 40 50 100 200

0.135 1.4 2.7 4.1 5.4 6.8 13.5 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). 4

m r 3 o t c a F n o 2 i t a c i f i n g 1 a M 0 0

1

2

3

4

5

Fundamental Natural Frequency fn

Figure 8 Relationship between Magnification Factor m and the Fundamental Natural Frequency f n (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 2 before 2003, concluded that where the area of the floor structure is less than 37m , 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 2 cautioned that if the area of the floor structure is greater than 37m , 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 Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

Page 19 of 39


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 f n 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 1.62 2 1.8 person/m , and recommended that a value of 1 person/m 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 f n > 3Hz

Magnification factor

Floor area < 37m2

#

Floor area#

> 37m2 m = N or 0.135 N

m = 2

Same as building structures f n = 1~3Hz

m = 3

where N may be taken as 1×Area of the footbridge 2 deck (in m )

Note: * Designers should note that there may be cases i n 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


Page 20 of 39


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 t o be satisfied if the fundamental frequency f n 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 f n 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 f n(req’d) Usage Footbridge (vertical vibration) Footbridge (horizontal vibration)

Minimum Required Fundamental Natural Frequency f n(req’d) (Hz) 5 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 f o 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 f n is equal to or less than 5Hz, the maximum vertical acceleration of any part of the superstructure shall be limited as 0.5 f n ms −2 , where f n (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 -2 Railways (2006), in the order of 0.7ms (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 Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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has stated that the values proposed by Murray et al (1997) are based on ISO 26312, 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 f n 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


Page 22 of 39


critical number of pedestrians lateral vibration may not be a problem. The limiting number of people to avoid instability can be estimated from: 8π cf n M N L = k where N L = limiting number of people f n = natural frequency in the horizontal direction c = critical damping ratio = 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 f n (in Hz) and the span length l (in m): Concrete: Steel: Composite:

-0.77

f n = 39l -0.73 f n = 35l -0.84 f n = 42l

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 f n for single span, or two-or-three-span continuous, symmetric footbridge of constant cross-sec tion: f n

=

C 2

EIg

2π l 2 M 2 where g is the acceleration due to gravity (in m/s ) l is the length of the main span (in m) C is the configuration factor ( Table 9) 2 E is the modulus of elasticity (in kN/m ) 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)


Page 23 of 39


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 a p from the passage of a single-pedestrian loading with a pacing frequency equal to the fundamental frequency f n for single span, or two-or-threespan continuous, symmetric footbridge of constant cross-section:

a p

= 4π 2 f n 2 y s k ψ ------ Eqt. (3)

where f n is the fundamental frequency (in Hz) y s 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


Page 24 of 39


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 f n of the structure is greater than 4Hz, the calculated peak vertical acceleration a p 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 f n of the structure. Eqt. (1) by Murray et al (1997) further shows that the calculated peak acceleration decreases exponentially with the fundamental natural frequency f n 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 f n of the structure exceeds 5Hz, as BS 5400-2:2006 states that if the fundamental natural frequency of vibration f o of the structure exceeds 5Hz in the vertical direction, the vibration serviceability requirement is deemed to be satisfied. Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

Page 25 of 39


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: 3 mc = 2400 kg/m 2 Ac = 600000 mm 2 For a 28-day compressive strength f cu = 35 N/mm , 2 modulus of elasticity = 23.7 kN/mm (according to the Code of Practice for the Structural Use of Concrete 2004 ) 2 The dynamic concrete modulus of elasticity E c = 23.7 × 1.35 = 32 kN/mm 6 4 Ic = 1125 × 10 mm 838x292x194kg/m UB properties: ms = 193.8 kg/m 2 As = 24700 mm 2 Es = 205 kN/mm 6 4 Is = 2792 × 10 mm


Page 26 of 39


n

= Es/Ec = 205/32 = 6.41 600000 150

y

=

×

6.41

+ 24700 × (150 +

2 600000 6.41

840.7 2

)

+ 24700

= 178 mm I

= [1125 × 106

+ 600000 × (178 −

+ 2792 × 106

+ 24700 × (150 +

= 7763 × 10 mm 6

M C

150

) 2 ] / 6.41

2 840.7 2

− 178)2

4

= 2400 × 4 × 0.15 + 193.8 = 1634 kg/m = π for single span

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

f n

=

205 × 106 × 7.763 × 10 −3 × 9.81

π 2

ψ

2π (20) 2 1634 × 9.81 / 1000 = 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. Fl 3 0.7 × 20000 3 = = 48 EI 48 × 205 × 7763 × 10 6 = 0.073 mm = 1.0 for single span = 0.03 = 8.6

a p

= 4π 2 f n y s k ψ = 4

ys k

δ

2

= 0.373 ms

-2

× π2 × 3.882 × 0.073/1000 × 1.0 × 8.6

= 3.80 % g for a single-pedestrian loading

The acceptable peak acceleration limit is 0.5 × 3.88 =0.985ms (i.e. 10.04% g), and hence the footbridge is satisfactory for a single-pedestrian loading. -2

The footbridge deck area is 4 ×20=80m , which is greater than 37m , and the fundamental natural frequency is less than 3Hz. From Table 7, an appropriate 2

2

magnification factor m may be N (=8.94) or 0.135 N (=10.8). Therefore, with -2 crowd effect taken into account and take m=10.8, the peak acceleration 4.03ms (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 3 -1 Railways (2006) and with W sl=0.5×4=2.0kN/m=2.0×10 N/m and V =3ms , the peak -1 vertical acceleration due a crowd moving continuously at a speed of 3ms is -2 0.68ms (i.e. 6.93% g). Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

Page 27 of 39


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 3

mc = 2400 kg/m 2 For a 28-day compressive strength f cu = 35 N/mm , 2 modulus of elasticity = 23.7 kN/mm (according to the Code of Practice for the Structural Use of Concrete 2004 ) The dynamic concrete modulus of elasticity, 2 Ec = 23.7 × 1.35 = 32 kN/mm 6 4 I b = 12740 × 10 mm Self-weight of concrete (slab + beam) Supported weight w b

∆ b

=

5wb Lb

= [(0.15 ×4)+(0.6-0.15) ×0.4] ×24 = 18.72 kN/m

= 0.5×4+18.72 = 20.72 kN/m

4

384 E c I b

=

5 × 20.72 × 80004

384 × 32 × 103 × 12740 × 106 = 2.711 mm Beam 800 ×400 mc f cu

= 2400 kg/m 2 = 35 N/mm


3

Page 28 of 39


Ec Ig

= 23.7 × 1.35 = 32 kN/mm 6 4 = 29570 × 10 mm

2

= (0.8-0.15) ×0.4 ×24 = 6.24 kN/m

Self-weight of concrete (beam), w g

4

∆g

=

5w g L g

384 E c I g

=

3

+

wb Lb L g

48 E c I g

5 × 6.24 × 8000 4

384 × 32 × 103 × 29570 × 106 = 2.220 mm

+

20.72 × 8000 × 8000 3 48 × 32 × 103 × 29570 × 106

With supports of square columns size 600x600 and floor height of 3500mm and assume the stress in column is 7.5MPa, the elastic shortening, PLc 7.5 × 3500 ∆c = = = 0.820 mm E c Ac 32 × 103 Combined Mode Properties f n

= 0.18

g

∆ b + ∆ g + ∆ c

= 0.18

9810 2.711 + 2.220 + 0.820

= 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 P o = 0.29 kN W = 20.72 × 8+6.24 ×8 = 216kN a p P 0 exp(−0.35 f n ) 0.29 × e ( −0.35×7.43) = = = 0.00332 = 0.33 %g g β W 0.03 × 216 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: f n = 7.43 Hz Since f n>5Hz, use r n=0.1, and hence F =74.6. ω = 2π f n = 46.68 Hz β = 0.03 48 × 32 × 10 3 × 12740 × 10 6 48 EI k = = × 1000 = 38220000 L3 8000 3 Neff

L4 j + (9.0 × 10 ) = 0.49 + 34.2 S I t d e


−9

2

 L  − 0.00059 j  = 1.77  S 

Page 29 of 39


Pa

=

74.6 × w 2 2 β N eff k

= 0.04003 m/s

β wD

R 1

= 1− e

−

= 1 − e

V

−

2

0.03×46.68×8 1.57

= 0.999

1

R 2 a p

1

 β wD  8 =  0.03 × 46.68 × 8  8 = 0.879 =      20V   20 × 1.57  = Pa × R 1 × R 2 =0.04003 × 0.999 × 0.879 2

= 0.0352 m/s = 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 f n>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) f n (Hz) a p (%g) (Murray et al (1997))

5 10.74

6 9.62

7 8.50

8 7.43

9 6.46

10 5.60

0.15

0.19

0.25

0.33

0.43

0.53

a p (%g) (Ellis (2000))

0.18

0.25

0.31

0.36

0.38

0.39

a p (%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 Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

Page 30 of 39


Murray et al (1997)’s method: UB533×210×92kg/m Ms = 92.1 kg/m 2 Es = 205 kN/mm 2 For a 28-day compressive strength f cu= 35 N/mm , the dynamic concrete modulus of elasticity 2 Ec = 23.7 × 1.35 = 32 kN/mm n = Es/Ec = 205/32 = 6.41 1600 150 533.1 150 × × + 11700 × (150 + ) 6 . 41 2 2 = = 156 mm y 1600 150 × + 11700 6.41 6 2 3 I b = 552.3 × 10 + 11700 × (150-156+533.1/2) + (1600/6.41) ×150 /12 2 + (1600/6.41) ×150× (156-75) 6 4 = 1662 × 10 mm Self-weight of concrete slab+steel beam = 0.15 ×4×24+92.1×0.00981 = 15.3 kN/m Supported weight w b = 0.5×4+15.3 = 17.3 kN/m

∆ b

=

5wb Lb

4

384 E s I b

5 × 17.3 × 8000 4

=

384 × 205 × 10 3 × 1662 × 10 6 = 2.708 mm UB762×267×147kg/m Ms

= 146.9 kg/m 150 ×

y

=

1600 150 6.41

×

2

150 ×

Ig

1600

754 2

) = 226 mm

+ 18700 6.41 6 2 3 = 1685 × 10 + 18700 × (150-226+754/2) + (1600/6.41) ×150 /12 2 + (1600/6.41) ×150× (226-75) 6 4 = 4303 × 10 mm 4

∆g

+ 18700 × (150 +

= =

5w g L g

384 E s I g

3

+

wb Lb L g

48 E s I g

5 × 1.47 × 8000 4

384 × 205 × 10 3 × 4303 × 10 6 = 1.762 mm Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

+

17.3 × 8000 × 8000 3 48 × 205 × 10 3 × 4303 × 10 6

Page 31 of 39


With supports of square columns size 600x600 and floor height of 3500mm and assume the stress in column is 7.5MPa, the elastic shortening, PLc 7.5 × 3500 ∆c = = = 0.820 mm 32 × 10 3 E c Ac Combined Mode Properties f n

g

= 0.18

9810

= 0.18

= 7.75 Hz

2.708 + 1.762 + 0.820 ∆ b + ∆ g + ∆ c W = (17.3 × 8+1.47 ×8) = 150kN For office occupancy without full height partitions, β = 0.03 and P o = 0.29 kN a p P exp(−0.35 f n ) 0.29 × e ( −0.35×7.75) = 0 = g β W 0.03 × 150 = 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:

= 7.75 Hz f n Since f n>5Hz, use r n=0.1, and hence F =74.6. ω = 2π f n = 48.69 Hz β = 0.03 48 EI

k

=

Neff

L4 j + (9.0 × 10 ) = 0.49 + 34.2 S I t

Pa

L3

=

48 × 205 × 10 3 × 1663 × 10 6 8000 3 d e

=

74.6 × w 2 2 β N eff k

= 0.0515 m/s

β wD

R 1

= 1− e

−

= 1− e

V

1

R 2 a p

−9

−

× 1000 = 31960781 2

 L  − 0.00059 j  = 1.79  S 

2

0.03×48.69×8 1.57

= 0.999 1

 β wD  8 =  0.03 × 48.69 × 8  8 = 0.884 =      20V   20 × 1.57  = Pa × R 1 × R 2 =0.0515 × 0.999 × 0.884 2

= 0.0455 m/s = 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 f n>3Hz, Table 7 recommends that the magnification factor m is taken as 2, and the results are also given in the following table.


Page 32 of 39


Span (m) f n (Hz) a p (%g) (Murray et al (1997))

5 11.59

6 10.26

7 8.96

8 7.75

9 6.68

10 5.75

0.17

0.23

0.32

0.43

0.56

0.70

a p (%g) (Ellis (2000))

0.25

0.34

0.42

0.46

0.48

0.49

a p (%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: I = 2.243 x10 −3 m 4 w = 15.3kN/m E s

= 205kN/mm 2

5.5.2 Using the formula given in BS 5400-2:2006 (see Section 4), f n = 17.02Hz As the fundamental frequency f n 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.


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Using the formulae given in BS 5400-2:2006 (see Section 4), -5 -2 f n = 17.02Hz, y s = 1.62×10 m, and a p = 0.584ms In Section 4, it has been mentioned that BS 5400-2:2006 states that if the fundamental natural frequency f n of the structure is greater than 4Hz, the calculated peak vertical acceleration a p 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 54002:2006 does not provide the reduction factor. For a conservative design, take the reduction factor to be 70%, the calculated peak acceleration a p is 0.584 ×0.3= -2 0.175ms (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.


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0.1

) 0.05 s m ( 0 n o i t a r -0.05 e l e c c A -0.1

2 -

-0.15 0

2

4

6

8

10

12

span (m)

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

0.10

) 0.05 s m ( n 0.00 o i t a r -0.05 e l e c c A -0.10

2 -

-0.15 0

2

4

6

8

10

12

span (m)

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 -2 superimposed in Figure 18. The peak acceleration at mid-span is 0.077ms (i.e. 0.785% g) at t=2.5s as against 1.79% g calculated using the simplified formula in BS 5400-2:2006 .


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0.04

) 0.02 s m ( 0.00 n o i t -0.02 a r e l e -0.04 c c A-0.06 k a e P-0.08

2 -

-0.0769

-0.10 0

2

4

6

8

10

12

14

Time (sec)

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, 2 2 as the footbridge deck area is 12 ×3=36m , which is less than 37m 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 -2 at mid-span is 0.154ms (i.e. 1.55% g). Using the formula given in the Structures Design Manual for Highways and Railways (2006) and with W sl =0.5×3=1.5kN/m -1 and V =3ms , the peak vertical acceleration due a crowd moving continuously at a -1 -2 speed of 3ms is 0.121ms (i.e. 1.23% g). The following table summarises the computed results for Example 4: Peak acceleration a p at mid-span

Methods

0.175ms

-2

0.077ms

-2

Table 7

0.154ms

-2

Structures Design Manual for Highways and Railways (2006)

0.121ms

-2

BS 5400-2:2006 n a i r g - t n e i l s d e g d a Computer model using n e o i S p L SAP2000

g n d i w d o r a o C L


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6.

Design References

Allen, D.E. and Rainer, J.H. (1976), ‘Vibration Criteria for Long Span Floors,’ Canadian Journal of Civil Engineering, 3(2), pp. 165-73. American Society of Civil Engineers (2005), ASCE 7-05: Minimum Design Loads for Buildings and Other Structures (Reston, Va: American Society of Civil Engineers). Bachmann, H. and Ammann, W. (1987), Structural Engineering Document 3e: Vibrations in Structures Induced by Man and Machines (Zürich: International Association for Bridge and Structural Engineering). Blanchard, J., Davies, B.L. and Smith, J.W. (1977), ‘Design Criteria and Analysis for Dynamic Loading of Footbridges,’ Symposium on Dynamic Behaviour of Bridges, Transport and. Road Research Laboratory, Crowthorne Berkshire, UK (19 May 1977). British Standards Institution (2004), BS EN 1995-2:2004: Eurocode 5: Design of Timber Structures Part 2: Bridges (London: BSI). British Standards Institution (2006), BS 5400-2: 2006: Steel, Concrete and Composite Bridges Part 2: Specification for Loads (London: BSI). Brownjohn, J.M.W., Fok, P., Roche, M. and Moyo, P. (2004), ‘Long Span Steel Pedestrian Bridge at Singapore Changi Airport – Part 1: Prediction of Vibration Serviceability Problems,’ The Structural Engineer , 82(16), pp. 21-7. Brownjohn, J.M.W., Fok, P., Roche, M. and Moyo, P. (2004), ‘Long Span Steel Pedestrian Bridge at Singapore Changi Airport – Part 2: Crowd Loading Tests and Vibration Mitigation Measures,’ The Structural Engineer , 82(16), pp. 28-34. Buildings Department (2004), Code of Practice on Structural Use of Concrete 2004 (Hong Kong: Buildings Department). Buildings Department (2005), Code of Practice on Structural Use of Steel 2005 (Hong Kong: Buildings Department). Dallard, P., Fitzpatrick, A.J., Le Bourva, S., Low, A., Ridsdill-Smith, R. M. and Willford, M. (2001), ‘The London Millennium Footbridge,’ The Structural Engineer , 79(22), pp. 17-33. Ellis, B.R. (2000), ‘On the Response of Long-Span Floors to Walking Loads Generated by Individuals and Crowds,’ The Structural Engineer , 78(10), pp. 17-25. Ellis, B.R. (2003), ‘The Influence of Crowd Size on Floor Vibrations Induced by Walking,’ The Structural Engineer , 81(6), pp. 20-7. Hewitt, M. and Murray, T.M. (2004), ‘Office Fit-Out and Floor Vibrations,’ Modern Steel Construction, April, pp. 35-8. Structural Engineering Branch, ArchSD Guidelines on Walking Vibration Issue No./Revision No. : 1/-

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Hicks, S. (2006), NCCI: Vibrations (Ascot: SCI) (available: http://www.steelbiz.org/; accessed: 2 June 2009). Highways Department (2006), Structures Design Manual for Highways and rd Railways, 3 ed. (Hong Kong: Highways Department). ISO (1982), ISO 2631-2: Evaluation of Human Exposure to Whole-Body Vibration: Part 2: Continuous and Shock-Induced Vibration in Buildings (1 to 80 Hz) (Geneva: International Organisation for Standardization). Matsumoto, Y, Nishioaka, T., Shiojiri, H., and Matsuzaki, K. (1978), ‘Dynamic Design of Footbridges,’ IABSE Proceedings P-17/78, pp. 1-15. Murray, T.M. (1975), ‘Design to Prevent Floor Vibrations,’ Engineering Journal, 12(3), pp. 82-7. Murray, T.M. (1981), ‘Acceptability Criterion for Occupant-Induced Floor Vibrations,’ Engineering Journal , 18(2), pp. 62-70. Murray, T.M., Allen, D.E. and Ungar, E.E. (1997), Steel Design Guide Series 11: Floor Vibrations due to Human Activity (Chicago: American Institute of Steel Construction). Naeim, F. (1991), Steel Tips: Design Practice to Prevent Floor Vibrations (California: The Structural Steel Educational Council). Pachi, A. and Ji, T. (2005), ‘Frequency and Velocity of people Walking,’ The Structural Engineer , 83(3), pp. 36-40. Pan, T.C., You, X.T. and Lim, C.L. (2008), ‘Evaluation of Floor Vibration in a Biotechnology Laboratory Caused by Human Walking,’ Journal of Performance of Constructed Facilities, 22(3), pp. 122-30. Pimentel, R.L. and Waldron, P. (1997), ‘Guidelines for the Vibration Serviceability Limit State of Pedestrian Bridges,’ in Virdi, K.S., Garas, F.K., Clarke, J.L. and Armer, G.S.T. (eds.), Structural Assessment: the Role of Large and Full-Scale Testing (London: E. & F.N. Spon), pp. 339-46. Pretlove, A.J., Rainer, J.H. and Bachmann, H. (1995), ‘Pedestrian Bridges,’ in Bachmann, H. et al (1995), Vibration Problems in Structures: Practical Guidelines (Basel, Switzerland: Birkäuser Verlag ), pp. 2-10). Roberts, T.M. (2005), ‘Lateral Pedestrian Excitation of Footbridges,’ Journal of Bridge Engineering , 10(1), pp. 107-12. Silva, J.G.S., da S. Vellaso, P.C.G., de Andrade, S.A.L., de Lima, L.R.O., and Figueire, F.P. (2007), ‘Vibration Analysis of Footbridges due to vertical human loads,’ in Computer and Structures, 85(21-22), pp. 1693-703.


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Floor Vibration Guide

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