evaluation of floor vibration in a biotechnology laboratory caused by human walking

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Evaluation of Floor Vibration in a Biotechnology LaboratoryCaused by Human Walking

Tso-Chien Pan, M.ASCE1; Xuting You2; and Chee Leong Lim3

Abstract: A floor supported on long-span beams, which was designed to accommodate bio research instruments, is evaluated forvibration induced by people walking. First, a brief review in vibration criteria is given. The variation of force time histories imposed bypeople’s feet on supporting objects is also discussed. Both beam and floor finite-element models are then used to simulate the localwalking response of the floor mathematically. Footfall forces are applied to the finite-element models via triangular distribution function.A comparison of the time history analysis results with the vibration criteria shows that the floor performs well under people walking. Fieldmeasurements were also conducted after the completion of the construction. The measured results show a good correlation with thefinite-element analysis results. During the analyses, it was also found that as long as the local floor model covers a structural bay, theboundary conditions of the floor model do not affect the response much. Using an equivalent constant footfall force function can producesimilar results compared with those obtained using a more sophisticated force function.

DOI: 10.1061/�ASCE�0887-3828�2008�22:3�122�

CE Database subject headings: Vibration; Environmental issues; Floors; Human factors.

Introduction

Biotechnology research and development are being actively pro-moted by many governments as the next generation of drivingforce for economic development. Many biolaboratories are thusbeing constructed over the Singapore island. The strict require-ments on building vibration performance, where the highlyvibration-sensitive high-tech bioequipment are accommodated,pose new challenges on structural engineers. In one of the newdevelopments, the Biopolis project which includes seven build-ings, each with a height of about 40 m, has multistory laborato-ries that are supported on long-span reinforced concrete �RC�beams. The performance of the laboratory instruments under theexcitation induced by people walking within the laboratories is ofconcern. Thus, the vibration response of floors to the walkingexcitations is evaluated via time history analyses of finite-element�FE� models. The analysis results are compared with the vibrationcriteria as described in the next section for adequacy of the sup-porting floors.

1Professor and Director, Protective Technology Research Center,School of Civil and Environmental Engineering, Nanyang TechnologicalUniv., Nanyang Ave., 639798 Singapore �corresponding author�. E-mail:[email protected]

2Project Officer, Protective Technology Research Center, School ofCivil and Environmental Engineering, Nanyang Technological Univ.,Nanyang Ave., 639798 Singapore.

3Research Fellow, Protective Technology Research Center, School ofCivil and Environmental Engineering, Nanyang Technological Univ.,Nanyang Ave., 639798 Singapore.

Note. Discussion open until November 1, 2008. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on April 28, 2005; approved on August 23, 2005. Thispaper is part of the Journal of Performance of Constructed Facilities,Vol. 22, No. 3, June 1, 2008. ©ASCE, ISSN 0887-3828/2008/3-122–130/
$25.00.
122 / JOURNAL OF PERFORMANCE OF CONSTRUCTED FACILITIES © AS

J. Perform. Constr. Facil.

Vibration Criteria

After the first vibration problem in advanced technology facilitieswas experienced by Intel in their Livermore and Aloha facilitiesin late 1970s, with the rapid development of science and technol-ogy, the industries imposed more restrictive vibration criteria onstructures supporting high tech equipment which were normallyvibration sensitive. Over the last 2 decades, much research hasbeen done in this area, and many analysis methods and vibrationcriteria have been proposed �Pavic and Reynolds 2002�. If a spe-cific space was evaluated only for a particular piece of equipment,a very accurate response would be given by a total system ap-proach which includes both the support system and the equipmentin one single analytical model �Medearis 1995�. The responseresults of the equipment were then compared with the vibrationcriteria specified by the manufacturer’s installation requirements.However, sometimes the equipment has not yet been selected,such as the Biopolis project in this case, or for a more flexibleusage of the supporting structures, designers are more interestedin generic vibration criteria which would meet the needs of allequipment in a particular category.

Several well-known generic criteria are in use, includingspectrum-based generic and time domain. For the spectrum-basedgeneric criteria, some can be expressed in terms of discrete fre-quencies and others in terms of frequency bands. For spectrum-based generic criteria expressed in terms of discrete frequencies,one is the Design guide 11 �AISC 1997�. For those expressedin terms of frequency bands, one commonly adopted by the in-dustries is defined in terms of one-third octave bandwidth spectra.The one-third-octave band criteria are generally attributed toGordon and Ungar �Amick 1997�. The earliest presentations ofthese criteria were in 1983. Later, Ungar and Gordon made someamendments according to the new developments and republishedthem in 1990 and 1991 �Ungar et al. 1990; Gordon 1991�.The criteria were also adopted by the Institute of EnvironmentalSciences �IES� in both versions of IES-RP-CC012.1 published
in 1993 and 1998 �IES 1998�. The criteria were based on a review
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of a large number of equipment-specific criteria provided byequipment manufacturers. The one-third octave band was chosento be a reasonable approximation to the modal response band-widths of realistic components, avoiding too much detail or toolittle resolution.

The one-third octave band generic vibration criteria are shownin Fig. 1, which take the form of a set of root-mean-square �RMS�velocity spectra that are labeled as vibration criterion curvesVC-A–VC-E. Table 1 shows the application and interpretationof the generic vibration criteria �IES 1998�. For the laboratoryareas of the BioPolis project, it was recommended that the genericvibration criteria in velocity VC-A be adopted as the vibrationacceptance criteria for floor slabs.

Footfall Force Time History

Single Footfall Force Time History

Due to the presence of vibration-sensitive equipment within thelaboratory area, a major concern is the vibration caused by human

Fig. 1. Generic vibration criteria �Gordon 1991�

Table 1. Application and Interpretation of Generic Vibration Criteria

Criterion curveRMS amplitudea

��m�Detail size

��m�

Office �ISO� 400 N/Ab

VC-A 50 8

VC-B 25 3

VC-C 12.5 1

VC-D 6 0.3

VC-E 3 0.1

aAs measured in one-third octave bands of frequency.b
N/A�not available.
JOURNAL OF PERFORMANC


walking. Force time histories acting on supporting objects bypeople walking are complicated functions which consist of a se-ries of single footfall force time histories separated in time andspace �Fig. 2�. Therefore, there are two factors to consider whendetermining the force time histories. One is the impulse shape offorce time history induced by a single footfall, and the other is thedistribution of single footfall force time histories along time andspace. Both factors are in turn influenced by the pacing rate ofwalking. Thus, it is not surprising that many different formulasfor walking force time history have been proposed in the pub-lished literature �Bachmann and Ammann 1987; Ebrahimpouret al. 1996; Kappos 2002�. Generally, the pacing rate of humanbeings varies from 1.5 to 2.3 Hz, while the peak magnitude andduration of the forcing function depend on the pacing rate. Foot-steps of 1.5, 2.0, and 2.3 Hz are therefore simulated in this analy-sis to investigate the vibration response under different pacingrates.

Measured in the laboratory, the force time history of a singlefootstep is shown in Fig. 3. The measured force function wasbased on 2 Hz pacing rate. For the pacing rate of 1.5 and 2.3 Hz,the peak load magnitudes were scaled from that of the 2.0 Hzpacing rate by assuming a linear relationship between dynamicmagnification factor �Dy/St� and peak load magnitude. For thepacing rate of 1.5 and 2.3 Hz, the contact duration �i.e., durationof the load� was scaled from that of the 2.0 Hz pacing rate byassuming an inverse relationship between frequency and load du-ration. The magnitude scaling factors are based on the Fourierseries expression of walking force

Fg�t� = Wg�1 + �n=1

N

�n sin�2�ft + �n�� 1�

where Wg, �n, and f�people weight, Fourier series coefficients,and walking frequency, respectively. Normally, only the first term

Description of use

Perceptible vibration. Appropriate for offices and nonsensitive areas.

Adequate in most instances for optical microscopes to 400�,microbalances, optical balances, proximity and projection aligners, etc.

Appropriate standard for optical microscopes to 1,000�, inspectionand lithography equipment �including steppers� to 3 �m line widths.

A good standard for most lithography and inspection equipment�including electron microscopes� to 1 �m detail size.

Suitable in most instances for the most demanding equipment,including electron microscopes �TEMs and SEMs� and e-beamsystems, operating to the limits of their capability.

A difficult criterion to achieve in most instances. Assumed to beadequate for the most demanding of sensitive systems includinglong-path, laser-based small target systems, and other systemsrequiring extraordinary dynamic stability.

Fig. 2. Footfall forcing time histories, function of time, and space

E OF CONSTRUCTED FACILITIES © ASCE / MAY/JUNE 2008 / 123

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of the series is significant. Based on experimental observations,formulations for the first term of the series have been developedfor various pacing rates �Ebrahimpour et al. 1996� as follows:

�1 = 0.25 − 0.05 log�N� �for 1.5 Hz pacing� �2�



where N�number of people, taken as 1 for evaluating the singlefootfall force. The duration scaling factors are based on the workproposed by Bachmann and Ammann �1987�. The magnitudes,contact durations, and stride lengths for different pacing rates aresummarized in Table 2. Bachmann and Ammann �1987� havedetermined that the normal walking pace is about 2 Hz.

Application of Force Time History in FE Models

To simulate the dynamic loading of people walking mathemati-cally in FE models, the inputs of force time histories to FE mod-els are in terms of a series of forcing functions of single footstepsseparated in time and space. FE models consist of discrete ele-ments connected at nodal points. The location of footfall forceapplied depends on the stride length, and may fall within the spanof an element. However, it is more convenient to apply the foot-fall force at the nodal points of a FE model. Thus, for each pacingrate, the footfall force will be determined and the temporal varia-tion of the footfall force along the span of the supporting structurewould be determined based on the stride lengths corresponding tothe pacing rate �Table 2�. Footfall forces will therefore be distrib-uted linearly to the nodal points at the two ends �i & j� of anelement by triangular shape functions �Eq. �5�� when the foot fallswithin the span of the element

Fig. 3. Single footstep forcing time history for 2 Hz walking

Table 2. Properties of Footsteps

Freq. Peak Duration Stride length

�Hz� �Dy/St� �s� �m�

1.5 1.25 0.8 0.6

2.0 1.34 0.62 0.75

2.3 1.42 0.54 1



Fi�t� = F�t�L − x

L; Fj�t� = F�t�

x

L�5�

In Eq. �5�, F�t� and x�total force to be applied within the spanat time t, and the distance from nodal point i, respectively;L�element length; and Fi�t� and Fj�t��distributed forces at thetwo nodes �i and j� of an element at time t. For example, for a5 m long element with the foot falling 2 m away from the leftend, the distributed forces at each node in terms of time are asdescribed by Eq. �5� and shown in Fig. 4. A set of routines writtenin Matlab was used to determine the nodal loadings as functionsof time, as the walking load moves along the span of the support-ing structure. The example of loading time histories at nodalpoints for a beam model with footfalls at 2 Hz pacing rate isshown in Fig. 5. Because the beam model is divided into sixsubelements, there are seven nodal loading time histories asshown. The seven time histories can then be applied to their cor-responding nodal points. By doing this, the space variation of thewalking force function will be taken care of by the distance be-tween the nodal points, while the time variation will be taken careof by the different starting times of the individual nodal loadingtime histories.

Discussion for Multiperson Walking

The more critical situation may arise when multiple persons walktogether with the same pacing rate. In normal practice, the effectof multipeople walking can be taken care of by multiplying thesingle footfall force time history by an amplification factor. This

Fig. 4. Triangle distributions of footfall forcing time histories

Fig. 5. Nodal loading time histories for continuous 2 Hz walking

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factor can be calculated from Eqs. �2�–�4� by setting N equal tothe number of people. For example, for a case of four personswalking, a factor of 3.8 is needed. Because of the linear analysisused, the method is equivalent to multiplying the response of asingle person walking by the same factor. However, it is a veryconservative approach. Scaling the whole time history impliesmultiplying all the higher harmonic frequency contents by thesame factor, which leads to the underlining assumption thathigher harmonic frequencies of people walking are exactly thesame. While it is believed that, even for a single person, it isdifficult to keep the same pacing rate for a sufficiently long du-ration, it would be nearly impossible for multiple persons to keeptheir pacing rates under normal walking conditions. When a fewpersons walk together as a group, their walking frequencies mayappear to be the same, but their higher harmonic frequencies arelikely to be different from one another. Thus, it is proposed tomultiply only the first and perhaps the second harmonic frequencycomponents in the response time histories by the amplificationfactor for the case of multipeople walking.

Floor Structural Configuration

The building under study which accommodates a complex of biolaboratories is a RC structure designed and constructed accordingto the British building standards �BSI 1985�. In order to maximizethe usage of the facility, large open space is preferred. Therefore,the largest span between the supporting frames is about 14.25 m.Accordingly, the floor thickness and the floor beam section arearound 250 and 1,000 mm by 1,000 mm, respectively, so as toreduce the floor vibration effects. Because human walking is a

Table 3. Beam Midspan Velocity Response due to People Walking��m /s�

1.5 Hz 2.0 Hz 2.3 Hz

Model Max. RMS Max. RMS Max. RMS

Beam 112 31.6 116.7 38.8 79.8 25.6

Fig. 6. Structural layout of typical floor



relatively small vibration source and may affect the buildingstructure locally, a typical floor area is isolated from the structuralsystem to study the floor vibration response induced by peoplewalking. The typical floor layout is shown in Fig. 6. The mainbeams are running in the transverse direction of the building.

Beam Vibration Evaluation

Finite-Element Model

The main concern with the vibration problem is associated withthe long-span beam of 14.25 m. Thus, for a preliminary study, a14.25 m single span beam was modeled. The beam section usesan equivalent T-shape section with the effective flange width de-termined as recommended in the BS 8110 code �BSI 1985�. Tworotational springs are added to the ends of the beam to account forthe restraints by the columns of upper and lower floors and theadjacent beams. The mass carried by the beam is calculated basedon the tributary area, including all the superimposed dead loads

Fig. 7. Midspan velocity response of beam under 2 Hz walking

Fig. 8. Midspan displacement response of beam under 2 Hz walking


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and 20% of the superimposed live load. Damping ratio is assumedas 2%. The fundamental modal frequency of the long-span beamas determined is 8.6 Hz, indicating that the floor is relatively flex-ible compared to the recommended frequency of 10 Hz in normalpractice.

Comparison of Results with Criteria

Walking at 1.5, 2.0, and 2.3 Hz along the beam is simulated math-ematically via time history analyses as discussed in the previoussection. For example, for the 2 Hz walking excitation, the variousloading time histories as shown in Fig. 5 were applied at thecorresponding nodal points of the FE model. After extracting thevelocity response time histories from the FE analysis results, theRMS value over the whole duration can be calculated using thefollowing formula:

vrms = � 1

T�

0

T

v2�t�dt�1/2

�6�

where T�duration; and v�t��velocity time history. The maxi-mum vibration response at the midspan of the beam is shown inTable 3. The maximum peak vibration velocity is about117 �m /s, and the maximum RMS value over the whole durationis 39 �m /s. Both of the maximum values occurred for the case of2 Hz walking. Velocity and displacement response time histories

Fig. 9. RMS velocity spectra for beam under single people walking

Fig. 10. Plan view of floor model



from 2 Hz walking are also shown in Figs. 7 and 8, together withtheir Fourier spectral density plots. Because the excitation force isperiodical, the responses are mainly harmonic functions of theexcitation force. Fig. 9 shows the 1/3 octave spectra of the veloc-ity response time histories together with the VC-A curve for com-parison. The spectra show that the responses are the largestbetween frequencies 8 and 10 Hz, which are the common fre-quency region of the beam natural frequency and higher harmon-ics of walking frequencies �the fourth or fifth harmonics�. It canalso be seen that the 1/3 octave spectra are well below the criteriacurve and that the floor meets the vibration requirements ofVC-A.

Floor Vibration Evaluation

Finite-Element Model

For a better understanding of floor vibration and a more accuratevibration response evaluation, a floor FE model, including slabsand beams, was also built. The floor modeled is three bays by fourbays, as shown in Fig. 6, which is large enough to cover morethan one structural bay �Wyatt 1989�. A structural bay is definedas a floor area which is delimited by stiff lines of support or by afree edge where appropriate. In this case, the chosen area isbounded by one free edge on the one side and stiff support linesalong the other three sides. This boundary condition is termedrestrained. The FE model consisting of floor slabs and beams isshown in Fig. 10. The FE model assumed that the staircase is partof the floor slab. Because the slab is continuous over the threeedges of the model in the actual building, symmetrical boundarycondition is applied along the three supported edges. The firstfour natural mode shapes and frequencies of the floor model areshown in Fig. 11. In Fig. 11, the northwest edge of each vibrationmode shape corresponds to the column line 5/6 shown in Fig. 6.

Table 4. Floor Midspan Velocity Response due to People Walking��m /s�

1.5 Hz 2.0 Hz 2.3 Hz


Floor 156.1 38.9 170.8 55.4 203.1 64.4

Fig. 11. First four mode shapes of floor model with restrainedboundary condition

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The first mode frequency is 5.55 Hz along the beam span direc-tion. The second mode is 5.62 Hz in the perpendicular directionalong the slab span.

Comparison of Results with Criteria

Walking with three different frequencies, 1.5, 2.0, and 2.3 Hz, issimulated along both the longitudinal and transverse directions ofthe building, whereby the path of walking is shown by dottedlines on the floor plan in Fig. 10. However, the midspan vibrationresponses due to walking in both directions are similar. This maybe due to the similar dynamic characteristics along the two direc-tions; their first mode frequencies are compatible. The maximumvelocity responses are summarized in Table 4. The maximumRMS value of velocity response is about 64.4 �m /s with a pacingrate of 2.3 Hz. The velocity and displacement response time his-tories and their Fourier spectra induced by 2 Hz walking areshown in Figs. 12 and 13. Fig. 14 also shows a comparison of the

Fig. 12. Midspan velocity response of restrained floor under 2 Hzwalking

Fig. 13. Midspan displacement response of restrained floor under2 Hz walking



1/3 octave RMS velocity spectra with the VC-A criteria curve.The velocity vibration responses meet the criteria quite well.

As discussed in an earlier section, the velocity responses fromthe floor vibration under single person walking are multiplied by3.8 at the corresponding walking frequencies to account for fourpersons walking. The results are compared with the VC-A criteriacurve and shown in Fig. 15. The responses are generally accept-able and only slightly exceed the vibration criteria at 2.3 Hz.

Comparison of Results for DifferentBoundary Conditions

Two other different boundary conditions are applied to the threesupported edges of the floor FE model to investigate the effects ofadditional boundary conditions on the floor vibration. The firstboundary condition, termed free, has four pinned supports at thefour corners of the entire three by four bay floor slab while the

Fig. 14. RMS velocity spectra of restrained floor under single peoplewalking

Fig. 15. RMS velocity spectra of symmetrically restrained floorunder multiple people walking


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edges of the slab are free to translate and rotate. The secondboundary condition, termed fixed, has four pinned supports at thefour corners of the entire three by four bay floor slab while re-straining the edges of the slab against translation and rotation.Their first few natural mode frequencies and velocity responsesare summarized in Tables 5 and 6, respectively. From the com-parison, it can be seen that the results, including natural frequen-cies and velocity responses, are generally at the same order ofmagnitude for all the three different boundary conditions. Oneexception is the velocity response to 2 Hz walking on the fixed-edge floor. It may be due to the resonant response caused by thematching of higher walking harmonic frequency and the floornatural frequencies. However, people may not be able to keep aperfect pace long enough to develop resonance in a laboratoryarea. Based on the three by four bay floor slab studied, the com-parison of results suggests that the boundary condition only af-fects the midspan vibration response marginally and that peoplewalking only excites the floor significantly within a local struc-tural bay �Wyatt 1989�. The concept of a local structural bay isthus applicable in practice.

Comparison of Response of Static Moving Loadand Dynamic Moving Load

The dynamic response of a floor caused by people walking acrossit can be assumed to be assembled by two different types ofdynamic responses. One response is to the spatially varying forceinduced by feet contacting the supporting surface with a pacingrate as shown in Fig. 2, and the other response is to the temporalvariation of contact force where feet are in contact with the sup-porting surface, i.e., the force variation within the single footfallduration, as shown in Fig. 3. From the observation of the Fourierspectra of both beam and floor vibration results �Figs. 7, 8, 12,and 13�, it seems that the major response frequency componentsare the higher harmonics of the pacing frequency. This suggeststhat the dynamic response depends mainly on pacing rate, andthat the shape of the footfall force time history is less significant.In order to verify this, a constant footfall force which has anequivalent force over the contact time, as shown in Fig. 16, isapplied to the beam model. The displacement results from theequivalent constant footfall force are compared with those to thetemporal varying footfall forces as shown in Fig. 17. The constantfootfall force produces mainly the downward displacement, while

Table 5. First Six Natural Mode Frequencies of Floor Models withDifferent Boundary Conditions �Hz�

Model Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

Free 4.98 5.46 5.61 5.81 6.49 6.72

Fixed 5.62 6.1 6.71 7.45 8.11 8.43

Restrained 5.55 5.62 6.03 6.61 7.03 7.38

Table 6. Floor Midspan Velocity Response due to People Walking forDifferent Boundary Conditions ��m /s�

1.5 Hz 2.0 Hz 2.3 Hz


Free 107.7 31.1 141.8 54 194.1 64.2

Fixed 141.8 44.5 257.9 81.4 196.4 60.9

Restrained 156.1 38.9 170.8 55.4 203.1 64.4



the temporal variation in the footfall force induces the additionaloscillations about the deflected position due to the equivalent con-stant footfall force. Fig. 17�b� clearly shows the difference inresponses to the two types of footfall forces. Fig. 18 compares thevibration results from the two footfall force functions in the 1/3octave band scale. Except for the high-frequency region wherethe vibration is normally less concerned, the two results are rea-sonably comparable. Therefore, it should be acceptable to use aconstant footfall force instead of the actual variant footfall forceduring a preliminary study for vibration in the lower frequency.

The above discussion shows that it is the pacing rate that con-tributes to the main portion of the floor vibration. However, aconstant pacing rate is difficult to maintain in the real world. Onthe other hand, the exact walking frequency can be kept for a longduration in FE models. Thus, the walking simulated mathemati-cally is overly precise which makes the mathematical resonanceeasier than actually happens. This indicates the conservatism ofusing FE time history analysis in predicting walking response.

Fig. 16. Constant and variable single footfall forcing time histories

Fig. 17. Response difference between constant and variable footfallforcing time histories

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Nevertheless, using a constant footfall force function can producequite similar results compared with using a more sophisticatedforce function.

Field Measurement Results

The construction of the laboratory complex took 2 years. Aftercompletion of the construction work, the actual field measure-ments were conducted. Among them, the vibration for peoplewalking at 2 Hz pacing rate was recorded. Fig. 19 shows a com-parison between the measurement result and the FE result of theslab model in the 1/3 octave band scale. The dashed line repre-sents the FE analysis results, and the solid line is for the measuredresults. The distribution of velocity magnitude over the differentfrequency zones in octave band is generally comparable, espe-cially for the critical frequency range between 6 and 12 Hz. Inother words, the FE model gave a reasonably good prediction forthe critical frequency range.

Conclusions

A reinforced concrete building which accommodates a biotechresearch center has been evaluated for vibration induced bypeople walking in this paper. FE models of beam and floor slabswere separately subjected to vibration caused by human walkingfor different pacing rates through time history analyses. Presented

Fig. 18. Comparison of vibration from variable and constant footfallforce functions in octave scale

Fig. 19. Comparison of field measurement results and FE analysisresults for 2 Hz walking



in terms of velocity in one-third octave RMS spectrum, the resultsare compared with the well-accepted vibration criteria. The com-parison results show that the performance of the floor subjected topeople walking meets the criteria well, although the response tomultiperson walking may be slightly above the criteria around2.3 Hz. Field measurements were taken after completion of theconstruction. The field measured results correlate well with theFE numerical results.

The concept of using a structural bay is applicable in this case.The different boundary conditions do not affect the response sig-nificantly. Therefore, a local model is adequate for walking evalu-ation as long as a structural bay is included. The results also showthat a simplified model may be adequate for preliminary studies.For example, a well defined beam model may produce reasonablysimilar displacement results compared with those obtained from alocal floor model �Figs. 8 and 13�. By comparing the velocity ofthe simplified beam model and the refined floor model, in termsof one-third octave RMS spectrum for 2 Hz walking �Figs. 9 and14�, the former’s results are less than the latter’s. However, bothresults are well below the criteria curve. Thus, the simplifiedbeam model may serve as a preliminary tool during the designphase to determine if the velocity response is within the criteriacurve for 2 Hz walking. The equivalent constant footfall forcetime history can be used to replace the variable force functionduring the preliminary study.

From the series of mathematical analyses performed, it isfound that resonance in mathematical models, which may nothappen in reality due to the difficulty in keeping a pacing rateconstant, appears to be conservative. Thus, a better method ofsimulating walking, including the randomness in multipersonwalking effects in mathematical models, is worth studying inorder to obtain a better simulation.

Acknowledgments

The writers would like to express their thanks to Ms. W. Y. Maoand Mr. W. H. Cheong of the Jurong Consultants Pte Ltd. Withouttheir support, this project would not have been possible. The fielddata measured are due to the credit of Professor J. M. W. Brown-john of the University of Sheffield. The writers express theirgratitude.

Notation

The following symbols are used in this paper:Fg � force induced on supporting subject by people

walking;f � people walking frequency;L � length of element;N � number of people;T � duration of time history;

vrms � RMS value of velocity time history;Wg � people weight;

x � relative distance from one end of element; and�n � Fourier series coefficients.

References

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evaluation of floor vibration in a biotechnology laboratory caused by human walking

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