and Intrasubject Variabilities - Hindawi.com

Research ArticleProbabilistic Multiple PedestrianWalking Force Model includingPedestrian Inter- and Intrasubject Variabilities

Zandy O. Muhammad and Paul Reynolds

Vibration Engineering Section, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK

Correspondence should be addressed to Zandy O. Muhammad; [email protected]

Received 22 July 2019; Accepted 18 December 2019; Published 10 January 2020

Academic Editor: Harry Far

Copyright © 2020 Zandy O. Muhammad and Paul Reynolds. /is is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

A probabilistic walking load model that accounts for inter- and intrasubject variabilities has been developed to generate syntheticvertical load waveforms induced by pedestrians. /e mathematical model is based on a comprehensive database of continuouslyrecorded pedestrian walking forces on an instrumented treadmill, having a wide range of walking frequencies. /e proposedmodel is able to replicate temporal and spectral features of real walking forces, which is a significant advantage over conventionalFourier series models. /e load model results in more realistic force time histories than previous models, since it incorporatessignificant components of the spectra that are omitted in Fourier series approaches. /e proposed mathematical model can beimplemented in vibration serviceability assessment of civil engineering structures, such as building floors and footbridges, toestimate more realistically dynamic structural responses due to people walking.

1. Introduction

Contemporary civil engineering structures, such as slenderfloors, footbridges, manufacturing facilities, and operatingtheatres, occupied and dynamically excited by humanfootfall loading, require the critical design aspect of vibrationserviceability to be evaluated via prediction of vibrationresponses and subsequent performance assessment. In theparticular case of floors, present and contemporary guidancemethodologies [1–4] are often followed to assess vibrationserviceability. However, even if properly applied, the out-come may be unsatisfactory, leading to a knock-on effect onbuilding or facility owners and thus litigation [5]. /e keyreasons for this are (1) the lack of adequate and accuratedesign procedures in contemporary design guidelines [6],(2) a lack of appreciation of the importance of vibrationserviceability design dominance relative to other designparameters, such as strength and deflections, (3) the lack of aprobabilistic modelling strategy to account for variability ofexcitation source and hence representative footfall loadingmodel [7], and (4) the lack of appropriate assessment criteria

for subjective human perception [6, 8]./ese, by nature, leadto a major challenge in modern floor design, whereby theprediction of vibration responses under human-inducedfootfall remains demanding and uncertain [9].

Vibration analyses of floors, in design guidance docu-ments [1–3], are commonly addressed based on a frequencythreshold of first mode natural frequency. /is threshold,accepted as around 10Hz, results in two classes of floors,low-frequency floors (LFF) if below the threshold and high-frequency floors (HFF) if above the threshold, irrespective ofthe function and usage of the floor. Forced vibrations for LFFare assumed to be deterministic, even though CSTR43 AppG and CCIP-016 introduce the concept of 25% probability ofexceedance on DLFs. As such, the floor develops a resonantresponse by harmonic components of the force, whereasHFFs are assumed to undergo transient response underimpulsive footfall loading. However, these studies [5, 6, 10]have shown that design guidelines do not work in manycases and require major improvements in all aspects, par-ticularly in walking load models and design scenarios./is isowing to the fact that all the models assume walking as

HindawiAdvances in Civil EngineeringVolume 2020, Article ID 9093037, 14 pageshttps://doi.org/10.1155/2020/9093037

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https://orcid.org/0000-0003-3574-1485

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https://doi.org/10.1155/2020/9093037

deterministic. /e walking force is, however, not deter-ministic due to random variabilities inherent in real walking.

/ere have been a number of attempts to develop reliablesynthetic walking load models by a single pedestrian forvibration serviceability assessment, such as a stochastic loadmodel using a number of Gaussian curves by [11]. /ismodel relies on random parameters being drawn from anexperimental database, resulting in a detailed representationof a continuous walking force. However, access to the ex-perimental database is a prerequisite to implement the abovemodel, which is not available to the public domain.Zivanovic [12] proposed a probabilistic walking model forfootbridges considering a single pedestrian. /e forcingfunction was represented by a Fourier series and the pa-rameters were modelled using statistical values for the firstfive main harmonics based on the data of Kerr [13] andBrownjohn et al. [14]. However, it is reported that theFourier approach results in loss of information and intro-duction of inaccuracies for individual and multiple pedes-trians [11]. All of these walking load models tend to generatea continuous force time history rather than individual stepsfrom both footfalls. A walking model based on individualfootfalls might better describe the mechanism of walkingand its parameters. /erefore, the aforementioned loadmodels do not tend to reflect the true nature of pedestrianexcitation, which has potential for the variation in pacingfrequency at the same velocity or walking speed. A potentialload model can be established on the basis of right and left orstrong and weak legs, which could serve both single andmultiple pedestrian loading scenarios. /is in turn can beused as a unified load model for a wider frequency range./is could be achieved via a probabilistic framework thataccounts for the inter- and intrasubject variability in thewalking force modelling as well as the potential excitation atdifferent walking speed.

/e model developed in this paper advocates using astatistical approach for generating time domain walkingforces from individual steps (right and left footfalls) forindividual and multiple pedestrians based on pedestrianwalking speed. /e established model can be reproducedfrom the data and equations illustrated in the followingsections. In addition, the model can be used in discretefootfall analysis when individual steps are applied on dif-ferent structural components. It can be implemented in anyfinite element (FE) package for vibration serviceabilityanalysis. /is can facilitate the application of individualwalking step forces at sequential spatial positions along anywalking paths. /e frequency domain components of themodel have features of measured walking in exhibiting thenarrow band random process, which is vital for reliablevibration serviceability assessment. /e model is developedbased on a large database of continuously recorded walkingforces on an instrumented treadmill for significant numbersof individuals. /e measured database, to the authors’ bestknowledge, is the most comprehensive collection of con-tinuous walking forces available in the literature. /emodelling strategy is followed by extracting key points onthe shape of measured footfalls to develop statistical rela-tionships for right and left steps of an individual. Following

the introduction, Section 2 presents the measured databaseand Section 3 introduces key statistical parameters of in-dividual walking steps and their relationships. In Sections 4and 5, the continuous probabilistic walking force and itsvalidation are illustrated. Section 6 summarises overallprobabilistic model strategy based on left and right footfallsand discusses its limitations.

2. Continuous Measurement of Walking Dataon Instrumented Treadmill

Developing a realistic walking load model requires a widerange of continuously measured actual walking forces fordifferent pedestrians, so as to retain the essence of the in-herent variabilities of real walking. /e measured databaseused in this paper has previously been discussed and utilisedby [11], but some key points are mentioned in this section.

Right and left footfalls of each person were continuouslyand independently recorded on an instrumented split-belttreadmill sampled at 200Hz. /e acquisition of walkingrecords was not prompted by any stimuli such as a met-ronome but instead was controlled by the treadmill speed(i.e., constant treadmill speed), which started from 0.56m/sand increased in increments of 0.14m/s up to the maximumcomfortable walking speed. /is is an inherent limitation offorce measurement using instrumented treadmills. Walkingforces corresponding to ten different walking speeds werecollected for each person. Each person has different maxi-mum speed due to their overall height and leg length. Eachtest was completed when at least 64 successive footfalls wereacquired. In total, 852 vertical time histories of walkingforces were collected for 85 people./eir characteristic meanand standard deviation of body mass, height, and age are75.8± 15.2 kg, 174.4± 8.2 cm, and 29.8± 9.1 years, respec-tively. /e reader is referred to [5, 11] for more details. Inthis paper, only 600 time histories of walking forces wereused to develop the load model, for walking below 0.8m/sand walking above 2.2m/s have low probability [15]. Atypical measured walking force time history is illustrated inFigure 1.

3. Modelling Strategy of IndividualWalking Steps

/is section presents the concept of the modelling approachfor each footfall from analysis of the continuous measuredwalking. Right and left footfalls are considered separately toextract time and load components based on a single step.Establishment of these two components is based on theaforementioned measured data generated by a diverse rangeof pedestrians, which will provide statistical reliability in theconsideration of both inter- and intrasubject variabilities.

3.1. Key Parameters for the Walking Step

3.1.1. Walking Speed. Walking speed has a large effect ontemporal and spatial parameters of walking and hence isconsidered a significant parameter in this model for anumber of reasons. Firstly, numerous studies have

2 Advances in Civil Engineering

characterised relationships between walking speed andstride-length, step-length, step-width, and pacing frequency[16, 17]. Secondly, pedestrians naturally walk at differentvelocities that in an effortless way increase or decreasepacing frequency and spatial parameters [17]. /irdly, in-dividuals walking at the same speed have different excitationdynamic forces as well as different walking parameters,which could be hard to account for individually in anyforcing function. /erefore, walking speed tends to be aglobal parameter that is inherently capable of definingdistributions of several walking (temporal-spatial) param-eters, which are vital in producing walking forces. /ewalking load model in this paper considers walking speed asthe input parameter due to the aforementioned observa-tions, unlike any existing walking models. In this study, therange of walking speed is between 0.8m/s for slow walkingand 2.2m/s for fast walking. /ese values, based on theobservations in [16], correspond to the pacing frequency of1.4± 0.1Hz and 2.3± 0.1Hz, respectively.

3.1.2. Step Contact Time. Step contact time or stance time isthe time when a foot is in contact with the ground. /e stepcontact time itself depends on walking speed, as presented inthe next section. /is will later be used in deriving relationsof control points on a step. Figure 2 shows the shape of theforce of a single walking step, where there are five maincontrol points:/e start point, which has zero load and time./e first peak load, which is the first local peak amplitude ofthe force. /e second peak load, which is the second localpeak amplitude of the force on the shape of a walking step.First peak point is initiated with heel strike (heel contact)and second peak point is where the toe of the same foot hitsthe ground. /is phase is called stance phase or contact

phase. /e valley load, which is the trough or a low point onthe shape of a walking step between first peak and secondpeak. Also, the step contact time, which is the last point. Anypoints in between these main points are called intermediatepoints. /e intermediate points are those points that liebetween first peak point and valley point or valley point andsecond peak point, which will be discussed later. /e rightand left footfall shapes are represented in the followingsections using a set of intermediate and control points,which have different values, probabilistically generated usingmean and standard deviation, for each footfall of walking.

3.1.3. Overlap Time between Two Consecutive Steps.Overlap time or double support time is the period of timewhen both feet are in contact with the ground simulta-neously. /is period of time becomes proportionally shorteras the walking speed increases, and when it becomes zero itindicates the transition to running. /e overlap time

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Advances in Civil Engineering 3

depends on both walking speed and step contact time, basedon the analysis of the measured data discussed later.

3.2. Timing Component of a Step. /is section provides astatistical description of the timing components of thecontrol points on the shape of a walking step. /e relationsare derived between walking speed and step contact time aswell as the points described in Figure 2. Statistical rela-tionships among pedestrians were also formulated to ac-count for inter- and intrasubject variabilities in the form ofprobability distribution functions.

3.2.1. Step Contact Time Relationship. /ere is a linear re-lationship between the walking speed and step contact time(in seconds) based on the aforementioned measured data, asshown in Figure 3. Each measured data point corresponds toan individual at that particular walking speed. It can be seenthat the step contact time decreases with an increase in thewalking speed. /e measured data were extracted for eachperson’s footfall for the duration of measured walking usinggradient point (i.e., slope at that point) in MATLAB. /egradient of a line is a number that describes both the di-rection and the steepness of the line between two points. Amargin of error equal to 2 dt, where dt � 0.005 s, was in-troduced in the gradient point to achieve the lowest (ideallyzero) slope due to the effect of noisiness in the data, thusobtaining a better estimate of beginning and ending of a step.Both the beginning and ending of a footfall were obtainedseparately and the subtraction of them gives an estimate ofthe step contact time (in seconds)./emeasured data shownin Figure 3 are the mean values for each person for bothfootfalls (i.e., left and right footfalls). /e theoretical mean isthe best fit with a high value of R2, which indicates thegoodness of fit or degree of linear correlation of the model.

Subject variabilities can be observed from the measureddata and as such this study takes into account both inter- andintrasubject variabilities. Intersubject variations, which existbetween pedestrians, are represented by a normal distri-bution through mean (μ) and standard deviation (σ). /eμinter− subject contact (in seconds) is obtained by the theoreticalmean of Figure 3, whereas σinter− subject contact is computed foreach walking speed using both µ and σ. Since these twovalues are different, coefficient of variation (CoV) wascalculated, which is µ divided by σ. Averaging the CoV andrelating that to the mean will give the standard deviation./us, σinter− subject contact is, among individuals, an averagevalue of 6.61% of μinter− subject contact (in seconds). As far asintrasubject variations are concerned, which occur withinthe same pedestrian, µ and s are calculated for each personbased on measured data, having μintra− subject contact of 0.0 s andσintra− subject contact of 0.0138 s.

3.2.2. First Peak Time, Valley Time, and Second Peak Time.First peak time, valley time, and second peak time of rightand left footfalls (in seconds) are calculated as a function ofstep contact time as illustrated in Figures 4–6. Using stepcontact time for all timing components is utilised due to the

fact that first peak time, valley time, and second peak time (inseconds) are all proportions of a step contact time (inseconds). /e reason for both footfalls is to preserve theintrasubject variabilities that are innate in actual walking andas such each footfall has its own formula. With an increase inthe step contact time, the timing components increases,which is due to the slow walking. /ese timings are the keycontrol points in defining the intermediate points in betweenthem.

3.3. Loading Component of a Step. /e relationship betweenwalking speed and loading components of a step can beobserved from the measured data. Figure 7 shows that firstpeak load, normalised by body weight (NBW), is dependenton walking speed. Statistical relationships among pedes-trians were also formulated to account for subject vari-abilities in the form of probability distribution functions,which are assumed to follow normal distributions based onmeasurement observations. /erefore, μinter− subject 1st peak isgiven by the theoretical mean of Figure 7 for right and leftfootfalls, whereas σinter− subject 1st peak is computed with averagevalues of 0.0772 and 0.06464 for right and left footfalls,respectively. As far as intrasubject variations are concerned,µ and s are calculated, based on measurement, for right andleft footfalls as μintra− subject 1st peak of 0.02929 and 0.0339 andσintra− subjective 1st peak of 0.0218 and 0.01596, respectively.

Similar relationships can be developed for the valley load(NBW), as shown in Figure 8. Normal distribution wasassumed to govern the inter- and intrasubject variabilities,which was also in line with the measurement observation./erefore, μinter− subject valley is given by the theoretical mean ofFigure 8 for right and left footfalls, whereas σinter− subject valley iscomputed with an average value of 0.0505 and 0.0484 forright and left footfalls, respectively. As far as intrasubjectvariations are concerned, µ and s are calculated for right andleft footfalls as μintra− subject valley of 0.01923 and 0.0231 andσintra− subject valley of 0.013 and 0.0104, respectively.

As far as second peak load (NBW) is concerned, therelationship is such that the second peak load depends onboth walking speed and first peak load as shown in Figure 9.

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Measured dataTheoretical mean95% upper confidenceinterval

95% lower confidence intervalR2 = 0.776

Figure 3: Statistical relationship between walking speed and stepcontact time.


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Figure 4: Statistical relationship between step contact time and first peak time. (a) Right footfall; (b) left footfall.

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Figure 5: Statistical relationship between step contact time and valley time. (a) Right footfall; (b) left footfall.

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Second peak time = 0.6223 × contact time + 0.0773Second peak time = 0.641 × contact time + 0.09034Second peak time = 0.6037 × contact time + 0.0642Units: (s) = (–) × (s) + (s)




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Figure 6: Statistical relationship between step contact time and second peak time. (a) Right footfall; (b) left footfall.


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First peak load = 0.1468 × walking speed2 – 0.08314 × walking speed + 1.023First peak load = 0.1851 × walking speed2 + 0.0331 × walking speed + 1.107First peak load = 0.1085 × walking speed2 – 0.1994 × walking speed + 0.9391Units: (NBW) = (s2/m2) × (m2/s2) – (s/m) × (m/s) + (NBW)

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Measured dataTheoretical mean95% upper confidence interval


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First peak load = 0.1306 × walking speed2 – 0.0511 × walking speed + 1.014First peak load = 0.1646 × walking speed2 + 0.0499 × walking speed + 1.086First peak load = 0.0966 × walking speed2 – 0.1522 × walking speed + 0.9425Units: (NBW) = (s2/m2) × (m2/s2) – (s/m) × (m/s) + (NBW)

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Figure 7: Statistical relationship between walking speed and first peak load. (a) Right footfall; (b) left footfall.

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Figure 8: Statistical relationship between walking speed and valley load. (a) Right footfall; (b) left footfall.


/is is due to the goodness of fit, where poor relationshipswould result if the second peak load was derived only basedon walking speed (this is not shown here for clarity). /eintersubject variabilities have already been taken into ac-count because of dependence on the first peak load. /eintrasubject variabilities, on the other hand, follow a normaldistribution with µ and s calculated for right and left footfallsas μintra− subject 2nd peak of 0.0352 and 0.0368 andσintra− subject 2nd peak of 0.01895 and 0.01862, respectively. It isworth pointing out that this model assumes independentprobability distribution of individual parameters featuringin the model and human-structure interaction (HSI) is notincluded.

3.4. Model Development Methodology. /e control pointsestablished in the aforementioned sections are critical indefining the rest of the points. /e intermediate points’relationships are mainly dependent on the above maincontrol points. For example, points between beginning of astep and first peak point were estimated based on theproportion of the first peak load and/or first peak time./esecriteria were set after a significant number of trials (by theauthors) when visually checking the walking measurements.For instance, second point (P2) is at 1% of first peak load and1/7.5 of first peak time. /ird point (P3) is at 1.2% of firstpeak load and 1/5.1 of first peak time. Satisfying both of theseconditions was selected to extract the points. Similarly, for allthe other points, different conditions were set to extract therest of the points. /is was done to capture the changes onthe beginning of the step as the force value increases. Forpoints between first peak and valley point, the conditionswere as follows: point fifteen (P15) is at a fraction (1/1.2) oftime between the difference of first peak and valley point./e same methodology was followed for points betweenvalley and second point as well as second point and end of astep. Following extraction of these points, regression analysis[18] was used to develop the statistical relationships. Anumber of the relationships for the points had very small R2

(i.e., weak relationship); thus those with R2 < 0.5 were im-proved by relating to points with a better R2. /e rela-tionships are shown in Table 1, where time components arein seconds and load components are normalised by bodyweight (NBW). /ese intermediate points, which are 34points including the control points, tend to match the shapeof an actual walking step. Although more points could havebeen chosen, this may have led to a tedious process andbecome less effective. /erefore, spline interpolation inMATLAB is utilised to generate smooth walking steps for theabove points as shown in Figure 10.

A number of regenerated walking footfall time historyfor both right and left footfalls can be seen in Figure 10 interms of mean and individual steps, which matches thecorresponding measured walking step closely. Since actualwalking is a continuous process, continuity of walking isestablished via the overlap time between consecutive rightand left footfalls. /is was done by placing any consecutivestep at a specific time slot, which is computed from previousstep contact time subtracted from an overlap time. /e

overlap time is a function of both walking speed and stepcontact time as shown in Figure 11. /e theoretical meanvalue only will be used to construct a continuous walkingload time history, since the overlap time depends partly onstep contact time, which already has taken into account thenormal distribution.

4. Development of a Continuous ProbabilisticWalking Load Model

A continuous walking force time history can be synthesisedon the basis of individual footfall forces, with durationdepending on the number of steps (i.e., left and rightfootfalls) and its characteristics based on the statisticaldistributions of main control points described previously./e continuous walking algorithm illustrated in Figure 12shows the complete process of creating a synthetic walkingforce. For a specified walking speed and number of steps (i.e.,walking duration), the algorithm first estimates step contacttime, first peak load, second peak load, and valley load for aspecific person, taking into account intersubject variability.Following that for all number of walking steps, the load andtime component intrasubject variabilities are selected viacorresponding distribution functions, as mentioned earlier./ese result in the main control points for both right and leftfootfalls. Next, the intermediate points for left and rightfootfalls can be produced based on the time and loadcomponents from the aforementioned control points, suchthat, for each footfall and each component (i.e., time orload), the intermediate points are obtained, where each stephas its intravariabilities. At this step, the algorithm splits intotwo parallel actions, right footfall and left footfall.

/e overlap time is selected as a function of both walkingspeed and step contact time to combine any consecutivefootfalls. /e next step integrates everything generated so farto produce the synthetic dynamic load, which is body-weight-normalised. Pedestrian body mass (body weight), asa random parameter, can be generated via available data-bases of statistical models in the literature. Finally, a con-tinuous synthetic walking force time history is generatedafter it has been scaled by the body weight.

Figure 13 shows examples of generated walking timehistories at different pacing rates. A total of 64 successivefootfalls at sampling frequency of 200Hz were generated inthe synthetic model, where the first 20 seconds are shown inFigure 13 for clarity. A visual comparison of the signalsshows clear variations in walking steps and force amplitudes,as a result of the inherent inter- and intrasubject variabilitybuilt into the algorithm.

5. Model Validation

/e modelling strategy proposed in this work is validatedin the frequency domain at different pacing frequencies./is was done by comparing between measured Fourieramplitude spectra of pedestrian force time historiesat corresponding pacing rates and their synthetic coun-terparts. As mentioned, 64 successive footfalls weregenerated the same as the measurement data for


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Figure 9: Statistical relationship between walking speed and second peak load. (a) Right footfall; (b) left footfall.

Table 1: Intermediate points between the five control points.

Points Comp. Left footfall Right footfall R2

P2 Time (s) 0.134× P14 0.1374×P14 0.98Load (NBW) 0.3496× P3 − 0.01745 0.4914×P3 − 0.01233 0.51


P4 Time (s) 0.203× P14 0.203× P14 0.99Load (NBW) 0.8518× P6 − 0.02441 0.7774×P6 − 0.00468 0.86


P6 Time (s) 0.2413× P14 0.2395× P14 0.99Load (NBW) 0.6809× P7 − 0.002679 0.5926× P7+ 0.02623 0.65




comparison. Figure 13 shows typical time domain signalsof three pacing rates of synthetic walking to be used forvalidation.

Figures 14–16 illustrate Fourier spectra of measured andsynthetic time histories corresponding with three pacingrates (fp) of 1.8, 2.0, and 2.2Hz, representing typical likely

pacing rates for office floors. For the first four dominantharmonics, square root of sum of squares (SRSS) error in thearea under the graph between the measured and syntheticspectra over 10 different walking forces for each pacing rateis less than 12%. In addition, the measurement of spread ofenergy around the first four dominant harmonics (i.e., H

Table 1: Continued.

Points Comp. Left footfall Right footfall R2


P10 Time (s) 0.5887× P14 − 0.00894 0.5892×P14 − 0.0092585 0.99Load (NBW) 0.861× P14 − 0.126 0.8451×P14 − 0.1059 0.63

P11 Time (s) 0.6469× P14 0.6464× P14 0.99Load (NBW) 0.8339× P14 − 0.03597 0.826 × P14 − 0.0275 0.73

P12 Time (s) 0.7145× P14 0.7139×P14 0.99Load (NBW) 0.8368× P14 + 0.02732 0.8392× P14 + 0.02514 0.84

P13 Time (s) 0.8331× P14 0.8331×P14 0.99Load (NBW) 0.8991× P14 + 0.0529 0.8935× P14 + 0.05292 0.95

P14 Time (s) First peak time First peak timeLoad (NBW) First peak load First peak load

P15 Time (s) (0.8111× (P20 − P14) + 0.0093) − P20 (0.8178× (P20 − P14) + 0.00822) − P20 0.96Load (NBW) (0.9436× (P14 − P20) − 0.00708) + P20 (0.9398× (P14 − P20) − 0.00606) + P20 0.99





P20 Time (s) Valley time Valley timeLoad (NBW) Valley load Valley load

P21 Time (s) P25 − (1.026× (P25 − P20) − 0.03) P25 − (1.015× (P25 − P20) − 0.0282) 0.99Load (NBW) (0.0631× (P25 − P20) − 0.00098) + P20 (0.0631× (P25 − P20) − 0.00128) + P20 0.55




P25 Time (s) Second peak time Second peak timeLoad (NBW) Second peak load Second peak load

P26 Time (s) 1.071× P25 − 0.00321 1.088× P25 − 0.01337 0.95Load (NBW) 0.847× P25 + 0.1096 0.8612× P25 + 0.09151 0.88

P27 Time (s) 1.104× P25 − 0.00545 1.128×P25 − 0.01955 0.97Load (NBW) 0.6564× P25 + 0.2541 0.7104× P25 + 0.1869 0.63


P29 Time (s) 1.159× P25 − 0.00755 1.195×P25 − 0.02911 0.96Load (NBW) 1.073× P28 − 0.1533 1.073× P28 − 0.1561 0.89




P33 Time (s) 1.345× P25 − 0.01648 1.399×P25 − 0.0495 0.88Load (NBW) μ� 0.0634 s� 0.029 μ� 0.0617 s� 0.0237

P34 Time (s) Step contact time Step contact timeLoad (NBW) 0 0


harmonic number) in the aforementioned spectra wascomputed using area under each harmonic curve to rep-resent the spread of energy. /e values of upper frequencyand lower frequency were identified as 0.95Hfp and1.05Hfp, respectively, based on [12].

/e results illustrated in Table 2 demonstrate the extentof spread of energy around main harmonics in terms ofmean and standard deviations. Overall, the syntheticmodel is in agreement with measured spectra, despitesome errors which are acceptable in the context of human

00.20.40.60.8

11.21.4

Forc

e (no

rmla

lised

by

body

wei

ght)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (sec)

Synthetic stepsMean synthetic steps

Measured stepsMean measured steps

(a)

00.20.40.60.8

11.21.4

Forc

e (no

rmla

lised

by

body

wei

ght)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (sec)



(b)

00.20.40.60.8

11.21.41.6

Forc

e (no

rmla

lised

by

body

wei

ght)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (sec)



(c)

Figure 10: Comparison of synthetic walking steps against measured steps. (a) Pacing rate of 1.8Hz, (b) pacing rate of 2.0Hz, and (c) pacingrate of 2.2Hz.

0.1

0.15

0.2

0.25

0.3

0.35

Ove

rlap

time (

s)

Overlap time = (–0.07208 × walking speed + 0.33) × step contact time



Units: (s) = (–(s/m) × (m/s) + (--)) × (s)

0.8 1 1.2 1.4 1.6 1.8 2 2.2Walking speed (m/s)



Figure 11: Overlap time relationship.


Walking speed (m/s) Step contact time (s)

Overlap time (s)

First peak load (N)

Second peak load (N)

Valley load (N)

First peak time (s)

Second peak time (s)

Valley time (s)

pdf of body mass (kg)

Intermediate pointsfrom start to P14

Intermediate pointsfrom P25 to end

Intermediate pointsbetween P14 to P25

Intermediate pointsfrom start to P14

Intermediate pointsfrom P25 to end

Intermediate pointsbetween P14 to P25

Intra-subject via pdfsContinuous synthetic walking

Le� footfall

Right footfall

body weight (N)

Walking steps Intra-subject via pdfs

Intrasubject via pdfs

Intersubject via pdfs

Figure 12: Schematic flow chart describing the procedure for generating synthetic continuous walking.

0 2 4 6 8 10 12 14 16 18 200

5001000

Forc

e (N

)

0 2 4 6 8 10 12 14 16 18 200

5001000

Forc

e (N

)

0 2 4 6 8 10 12 14 16 18 200

1000500

1500

Forc

e (N

)

Synthetic walking at 1.8Hz


Time (s)

Time (s)

Time (s)


Figure 13: Synthetic continuous walking at three pacing frequencies.

0

50

100

150

200

Forc

e (N

)


0

50

100

150

200

Forc

e (N

)

Measured walking at 1.8Hz

Frequency (Hz)0 2 4 6 8 10 12 14 16 18 20

Frequency (Hz)0 2 4 6 8 10 12 14 16 18 20

Figure 14: Synthetic continuous walking and measured walking at 1.8Hz.


forcing function and mathematical modelling. All thisindicates good match in the frequency content between themeasured and synthesised walking force signals. /ere-fore, it is proposed that the synthetic forces generated bythis model can be utilised as a basis for vibration ser-viceability assessment of civil engineering structures, suchas floors and footbridges, to estimate more realistic vi-bration responses due to people walking than previouslyproposed deterministic models.

6. Conclusions

/is paper has presented a new probabilistic model to generatewalking force time histories for specific walking speeds. /efootfall forces of both left and right legs aremodelled separatelyand then combined with an overlap time to obtain a con-tinuous walking force. /e modelling strategy can account forspatial-temporal features of real vertical walking more realis-tically than conventional Fourier series-based deterministic

0

50

100

150

200

Forc

e (N

)


0

50

100

150

200

Forc

e (N

)


0Frequency (Hz)

2 4 6 8 10 12 14 16 18 20

0Frequency (Hz)

2 4 6 8 10 12 14 16 18 20


0

50

100

150

200

Forc

e (N

)


0

50

100

150

200

Forc

e (N

)


0Frequency (Hz)

2 4 6 8 10 12 14 16 18 20

0Frequency (Hz)

2 4 6 8 10 12 14 16 18 20


Table 2: Spread of energy in the spectra for the synthetic and measured walking.

Harmonic number Area synthetic walking Area measured walking % error1 μ� 0.0138; σ � 0.0034 μ� 0.014; σ � 0.003 − 1.432 μ� 0.0046; σ � 0.0012 μ� 0.0047; σ � 0.0027 − 2.123 μ� 0.0039; σ � 0.0009 μ� 0.0032; σ � 0.001 17.94 μ� 0.0032; σ � 0.0006 μ� 0.0034; σ � 0.0004 − 6.2


approaches. /e established probabilistic model has the fol-lowing advantages:

(1) A set of probabilistic walking steps, taking into ac-count intrasubject variability, is used to generate acontinuous walking force signal in the time domainbased on walking speed

(2) High-frequency components are inherently includeddue to variation in walking steps in both time and load.Hence, the load model can replicate actual walkingmore realistically than Fourier series approaches typ-ically based on low-frequency harmonics alone

(3) Variation between walking steps for both legs in eachinterval of consecutive steps is possible and as suchthe model demonstrates the narrow band randomphenomenon in frequency domain, showing theleaking of energy in the vicinity of the dominantFourier harmonics [11]. /is is a feature typical inmeasured pedestrian time histories

(4) Intersubject variability is taken into account via thestatistical distributions of physical characteristics ofentire populations of pedestrians

(5) /e walking model can be used to generate forcetime histories for both individual and multiple pe-destrians walking via superposition

(6) It is worth mentioning that this model does notaccount for human-structure interaction and as suchfurther investigation is required to include that as-pect. Also, the model could be improved by explicitlyintroducing a space variable

(7) /e model is amenable for use in Monte-Carlo sim-ulations of floor response, hence to provide statisticaldistributions of response to be used in probabilisticvibration serviceability assessment. It can also be usedto predict likely “vibration dose” over an extended timeperiod of occupant exposure to vibration

/is framework of probabilistic walking forces providesan opportunity to enhance current vibration serviceabilityassessment, which currently typically lacks appropriatestatistical perspective. /e established walking model can beused to predict realistic distributions of dynamic structuralresponses for assessment of civil engineering structuresdynamically excited by pedestrians such as building floorsand footbridges.

Data Availability

/e experimental database is confidential and thus cannot beprovided.

Conflicts of Interest

/e authors declare that they have no conflicts of interest.

Acknowledgments

/e financial support for this research was prdovided byQatar National Research Fund (QNRF; a member of the

Qatar Foundation) via the National Priorities ResearchProgram (NPRP) (Project no. NPRP8-836-2-353). /e au-thors would like to acknowledge Professor James Brownjohnand Professor Vitomir Racic for providing access to thedatabase of measured walking forces.

References

[1] T. M. Murray, D. E. Allen, E. E. Ungar, and D. B. Davis,Vibrations of Steel-Framed Structural Systems Due to HumanActivity, American Institute of Steel Construction (AISC),Chicago, IL, USA, 2016.

[2] A. Smith, S. Hicks, and P. Devine, Design of Floors for Vi-bration: A New Approach, Steel Construction Institute (SCI),Berkshire, UK, 2nd edition, 2009.

[3] M.Willford and P. Young,ADesign Guide for Footfall InducedVibration of Structures, Concrete Centre (CC), Surry, UK,2006.

[4] RFCS,Human Induced Vibrations of Steel Structures (HiVoSS)-Vibration Design of Floors: Background Documnet, EuropeanComission-RFCS, Brussels, Belgium, 2007.

[5] J. Brownjohn, V. Racic, and J. Chen, “Universal responsespectrum procedure for predicting walking-induced floorvibration,”Mechanical Systems and Signal Processing, vol. 70-71, no. 1, pp. 741–755, 2016.

[6] Z. O. Muhammad and P. Reynolds, “Vibration serviceabilityof building floors: performance evaluation of contemporarydesign guidelines,” Journal of Performance of ConstructedFacilities, vol. 33, no. 2, Article ID 01019012, 2019.

[7] Z. Muhammad, P. Reynolds, O. Avci, and M. Hussein,“Review of pedestrian load models for vibration serviceabilityassessment of floor structures,” Vibration, vol. 2, no. 1,pp. 1–24, 2018.

[8] A. F. Al-Anbaki, Footfall excitation of higher modes of vi-bration in low-frequency building floors, Ph.D. thesis, /eUniversity of Exeter, Exeter, UK, 2018.

[9] M. Bocian, J. H. G. Macdonald, J. F. Burn, and D. Redmill,“Experimental identification of the behaviour of and lateralforces from freely-walking pedestrians on laterally oscillatingstructures in a virtual reality environment,” EngineeringStructures, vol. 105, no. 12, pp. 62–76, 2015.

[10] S. Zivanovic and A. Pavic, “Probabilistic modeling of walkingexcitation for building floors,” Journal of Performance ofConstructed Facilities, vol. 23, no. 3, pp. 132–143, 2009.

[11] V. Racic and J. M. W. Brownjohn, “Stochastic model of near-periodic vertical loads due to humans walking,” AdvancedEngineering Informatics, vol. 25, no. 2, pp. 259–275, 2011.

[12] S. Zivanovic, Probability-based estimation of vibration forpedestrian structures due to walking, Ph.D. thesis, /e Uni-versity of Sheffield, Sheffield, UK, 2006.

[13] S. C. Kerr, Human induced loading on staircases, Ph.D. thesis,University College London, London, UK, 1998.

[14] J. M. W. Brownjohn, A. Pavic, and P. Omenzetter, “A spectraldensity approach for modelling continuous vertical forces onpedestrian structures due to walking,” Canadian Journal ofCivil Engineering, vol. 31, no. 1, pp. 65–77, 2004.

[15] R. W. Bohannon, “Comfortable and maximum walking speedof adults aged 20–79 years: reference values and determi-nants,” Age and Ageing, vol. 26, no. 1, pp. 15–19, 1997.

[16] H. V. Dang and S. Zivanovic, “Experimental characterisationof walking locomotion on rigid level surfaces using motioncapture system,” Engineering Structures, vol. 91, no. 3,pp. 141–154, 2015.


[17] A. J. J. Smith and E. D. Lemaire, “Temporal-spatial gait pa-rameter models of very slow walking,” Gait & Posture, vol. 61,no. 3, pp. 125–129, 2018.

[18] N. R. Draper and H. Smith, Applied Regression Analysis, JohnWiley & Sons, Inc., Hoboken, NJ, USA, 3rd edition, 1998.


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