Mathematical modelling of random narrow band lateral excitation of footbridges due to pedestrians walking

Computers and Structures 90–91 (2012) 116–130

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Mathematical modelling of random narrow band lateral excitation of footbridges due to pedestrians walking V. Racic ⇑, J.M.W. Brownjohn 1 Department of Civil & Structural Engineering, University of Sheffield, Sir Frederick Mappin Building, Sheffield S1 3JD, United Kingdom

a r t i c l e

i n f o

Article history: Received 2 May 2011 Accepted 4 October 2011 Available online 8 November 2011 Keywords: Vibration serviceability Human–structure dynamic interaction Footbridges Walking forces Dynamic loading

a b s t r a c t Motivated by the existing models of wind and earthquake loading, speech recognition techniques and a method of replicating electrocardiogram (ECG) signals, this paper presents a mathematical model to generate synthetic narrow band lateral force signals due to individuals walking. The model is fitted to a database comprising many directly measured walking time series, yielding a random approach to generating their artificial – yet realistic counterparts. This multi-disciplinary modelling strategy offers a radical departure from traditional Fourier-based representations of lateral walking loads towards more reliable and more realistic vibration serviceability assessment of footbridges. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction A few days after it was opened in June 2000, the London Millennium Bridge (LMB) was promptly closed to the public as large pedestrian crowds built up unexpected excessive lateral vibrations on the opening day [1]. Largely due to the publicity surrounding this iconic structure, and in part due to similar previously reported problems in other bridges [2,3], pedestrian-induced excitation of footbridges continues to attract multidisciplinary scientific research [4]. However, more than a decade after the LMB debacle, structural designers still have to manage significant uncertainties in lateral dynamic loads induced by pedestrian occupants due to crude guidance based on insufficient and fragmented science. The scale of the problem is reflected in the most recent measurements of the excessive lateral sway recorded on at least a dozen footbridges of different structural forms and sizes [5–9]. The inadequate guidance not only leads to economic costs but also results in safety concerns related to crowd panic due to unexpected and unfamiliar structural motion. A prime example is the stampede during festival celebrations in the Cambodian capital in November 2010 which began when, according to the BBC news, excessive bridge vibrations caused panic. At least 350 people were reported to have been killed and hundreds more injured in the crush.

⇑ Corresponding author. Tel.: +44 (0) 114 222 5790; fax: +44 (0) 114 222 5718. E-mail addresses: v.racic@sheffield.ac.uk (V. Racic), james.brownjohn@sheffield. ac.uk (J.M.W. Brownjohn). 1 Tel.: +44 (0) 114 222 5771; fax: +44 (0) 114 222 5700. 0045-7949/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2011.10.002

The focus of post-LMB research has been to simulate mathematically the lateral excitation mechanism in swaying footbridges, with early attempts assuming a tendency of pedestrians to synchronise pacing rate with the strong vibrations [10–15], a phenomenon generally known as ‘‘synchronous lateral excitation’’ (SLE) or ‘‘pedestrian lock-in’’. The most popular variant is the semi-empirical model presented by Arup [1], based on the analysis of the LMB dynamic response. It proposed the lateral force exerted by a pedestrian to maintain balance being directly proportional to velocity of lateral vibration, in fact resulting in a form of negative damping. The net damping, naturally existing in the structure occupied by people, would vanish as pedestrians reached a critical number, causing exponentially increasing vibrations. This approach is embodied in the UK National Annex to Eurocode EN1991-2:2003 [16], the HIVOSS guidelines [17] and the FIB [19] recommendations. However, some of the recent site measurements [7,9] have showed divergent lateral vibrations in spite of lack of evidence of synchronisation, indicating that this is not necessarily the cause and that greater attention should be paid to modelling of pedestrian gait. Based on studies on human body balance from the biomechanics field, Macdonald [19] argued convincingly that the lateral balance strategy during forward locomotion is all that matters. As in the case of normal walking on a rigid surface, pedestrians walking on a laterally oscillating bridge keep their body balance through lateral contact forces they exert to control lateral position, rather than the timing or positioning of footfalls. Averaging the product of these lateral forces with lateral bridge velocity over a large number of cycles, a pedestrian acts as a negative (or positive) damper to the bridge motion by extracting (or supplying) energy. The lateral

V. Racic, J.M.W. Brownjohn / Computers and Structures 90–91 (2012) 116–130

117

Nomenclature fw F(t) ai

lateral walking frequency force time history Fourier amplitudes ui Fourier phases ak scaling factors fs sampling rate cycle intervals T i ; T 0k si ; s0k normalised cycle intervals lT mean of Ti Ss ðf Þ; S0s ðfn Þ ASD of si and s0k As ; S0s ðfn Þ Fourier amplitudes of si Wj, Aj Gaussian heights (weights) balance is modelled as actively controlled swinging of an inverted pendulum, comprising a lumped body mass supported by a rigid massless leg. Hence, with instantaneous transfer of support from one foot to the other, the lateral force exerted against a bridge is a result of the body weight resolved along an inclined leg. A qualitative comparison of magnitude and shape against direct force measurements available in the literature suggests that the model generates reasonable lateral forces on rigid surfaces. However, Macdonald noted that the simulated vibration response of flexible bridges is very sensitive to small changes in modelling parameters and selection of balance strategies. The model is still in its infancy and further improvements need to include random variations of the modelling parameters for the general human population. In the case of the Solferino Bridge problem, Blekherman [20] proposed an excitation mechanism in which mainly vertical or torsional vibration modes, excited by vertical walking forces, can cause a nonlinear coupling (also called ‘‘autoparametric resonance’’) with mainly lateral modes. However, full-scale dynamic tests of other flexible bridges [7,9] showed a clear absence of such coupling. Piccardo and Tubino [21] proposed an alternative parametric resonance mechanism based on a harmonic lateral excitation characterised by a constant walking frequency and variable amplitude as a function of the bridge instantaneous lateral displacement. The model gives a reasonable representation of the observed behaviour of the LMB, when its first lateral mode at 0.5 Hz is excited by the lateral pedestrian forces at 1 Hz. However, the model is not applicable when a footbridge is sufficiently ‘‘stiff’’ in the lateral direction, i.e. its first lateral frequency is around 1 Hz. The main limitation of the current formulation was attributed to the values of the force parameters [21], which were adopted with the great uncertainty due to the lack of the actual lateral force data. In an attempt to address this lack of data, the most comprehensive dataset on lateral walking forcing functions available nowadays was recently collected by Ingólfsson et al. [22] as 71 adult Italians walked on a bespoke instrumented treadmill with variable lateral vibration amplitude and frequency [23]. This is a valuable resource that is limited by having all individual walking tests run at a single walking speed, pre-selected by each individual independently as their ‘‘comfortable’’ pacing rate. It could be argued that pedestrians in a group or crowd are often forced to maintain their speed at a certain level to stay with the flow, which can consequently affect the corresponding walking loads [24]. Hence, to improve the potential of this dataset for development of reliable mathematical characterisation of force signals in footbridge design, further lateral force data are required for a range of walking speeds. The database also needs to account for a wide diversity among the human population, as well to include time series for multiple pedestrians. The focus on the large-amplitude response has distracted researchers from the root of the problem, which is rudimentary

cj, tj, hkj b, dj, bk

Gaussian centres Gaussian widths angular frequency of rotation template cycle energy of weight normalised walking cycles energy of template cycle disturbance term coefficients of linear regression coefficients of linear correlation body weight

x0k

Z(t) Ek Etc DEk q0, q1

q BW

conceptualisation of pedestrian lateral forces such as limited and deterministic Fourier series. Realistic modelling of pedestrian forces, based on direct observation and measurement, is a prerequisite to predicting pedestrian-induced footbridge lateral vibrations whether or not they are perceptible to the pedestrian (i.e. perceived as lively). There is not yet even a reliable model for a stationary structure. Also, there is strong evidence that peripheral stimuli are an equally important factor influencing timing of the gait cycles and that resulting lateral forces are inevitably a narrow band random process [4]. Hence we focus on appropriate modelling of this process in order to build a framework that can generate the correct interface forces between the humans and the structure. As the first step towards achieving this ambition, the present study advances the field by bringing together a comprehensive database of continuously measured lateral walking loads on rigid surfaces (i.e. without the lateral bridge motion and hence discounting effects of human–structure interaction) and their mathematical characterisation which can simulate accurately the measurements. The proposed model could be applied to cases of incipient instability (i.e. when lateral motion is still not perceptible) and is ideally suited for considering the case of multiple pedestrians in a group or a crowd. 2. The state-of-the-art models of lateral walking force signals Measured continuous time histories are invariably nearperiodic, indicating their narrow band nature (Fig. 1). However, to simplify their mathematical representations suitable for hand calculations in bridge design, they are usually assumed perfectly periodic, deterministic and presentable via Fourier series:

FðtÞ ¼

m X

ai sinð2pifw t  ui Þ

ð1Þ

i¼1

Here, F(t) is the synthetic force, fw is the frequency of a full walking cycle (right step + left step), ai are harmonic amplitudes and ui are the corresponding phase angles, while parameter m determines the number of harmonics considered in a model. The simplest such model involving only the first dominant harmonic (i.e. a single sinusoidal function) can be found in Eurocode [16] and Setra [25] design guidelines for footbridges, while some models involve up to five dominant harmonics, as summarised in Table 1. To the best knowledge of the authors, values of the phase angles have never been publicised in detail and are usually ignored in Eq. (1). On the other hand, the Fourier amplitudes are commonly reported as body weight-normalised coefficients ai/BW, generally known as ‘‘dynamic load factors (DLFs)’’. Note that non-zero even harmonics in Table 1 indicate slight differences between the left and right footfalls due to the natural asymmetry of human gait [27]. In a time domain analysis procedure that uses the synthetic walking time series

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Force amplitudes/BW [-]

(a)

0.15 0.10 0.05 0 -0.05 -0.10 -0.15

(b)

0

5

10

15

20 Time [s]

25

30

35

40

Fourier amplitudes/ BW [-]

0.06

Fourier phases [rad]

(c)

Fourier harmonics dominant harmonics

0.05 0.04 0.03 0.02 0.01 0 0

1

2

3

4

5 6 Frequency [Hz]

7

8

9

10

0

1

2

3

4

5 6 Frequency [Hz]

7

8

9

10

4 3 2 1 0 -1 -2 -3 -4

Fig. 1. An example of measured (a) force–time history and the corresponding (b) Fourier amplitudes and (c) Fourier phases. Dominant harmonics at integer multiples of lateral walking frequency fw = 0.92 Hz are bold.

Table 1 DLF amplitudes of dominant harmonics suggested in the literature. Authors

Comment

DLF1

DLF2

DLF3

DLF4

DLF5

Bachmann and Ammann [2] Bachmann et al. [26] Setra [25] Eurocode [16] Ricciardelli and Pizzimenti [34] Ingólfsson et al. [22]

Pacing rates at 2 Hz only Pacing rates at 2 Hz only Any pacing rate [0.75–1.25] Hz lateral pacing rate Broad-band model [0.6–1.1] Hz lateral pacing rate at self-selected walking speed Broad-band model [0.6–1.1] Hz lateral pacing rate at self-selected walking speed

0.039 0.1 0.05 0.1 0.04 0.047

0.010 – – – 0.0077 0.007

0.043 0.1 – – 0.023 0.025

0.12 – – – 0.0043 0.005

0.015 – – – 0.011 0.011

(1), a footbridge is assumed to be linearly elastic and, using modal decomposition, the response of each vibration mode can be analysed separately using harmonic loads modulated by the appropriate mode shape to account for the moving load [28]. A number of studies have shown that the Fourier modelling approach leads to significant loss of information and introduction of inaccuracies during the data reduction process [29–33]. For instance, Brownjohn et al. [30] reported differences as high as 50% between vertical vibrations of pedestrian structures due the imperfect real walking forces and periodic Fourier-based simulations, which were related to neglecting the energy around dominant harmonics in actual narrow band forces (Fig. 1b). To account for this energy spread, Ricciardelli and Pizzimenti [34] and Ingólfsson

et al. [22] derived DLFs for perfectly periodic footfalls including Fourier amplitudes around the dominant harmonics (see Fig. 1b) – so called ‘‘broad-band model’’ (Table 1). However, a parametric study presented by Middleton [35] showed that simulated vibration response is not sensitive to changes in DLF amplitudes as much as to variations of pacing rate for successive footfalls in synthetic force time histories. This study clearly indicated the need for modelling walking as a near-periodic process, which has never been attempted to date for lateral pedestrian force signals. The Fourier concept given by Eq. (1) could describe nearperiodic walking accurately if there were reliable models representing each line in the amplitude and phase spectra of continuous walking (Fig. 1b and c). Although a quality model of the Fourier

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records is an essential component for development of a reliable random numerical generator presented in Section 6.

Number of occurrences [-]

250 200

3.1. Experimental setup

150 100 50 0 -4

-3

-2 -1 0 1 2 Lateral walking rate [Hz]

3

4

Fig. 2. Histogram of the Fourier phases (for harmonics up to 50 Hz), which portion is shown in Fig. 1c.

amplitudes could be obtained, e.g. as suggested for the vertical walking forces elsewhere [30,36], widely varying harmonic phase lags (Fig. 2) are very difficult to characterise analytically. If they are, however, assumed to be uniformly distributed in the range [p, p], the sum of Fourier harmonics normally does not match the actual force time history. This is clearly illustrated by comparison of Fig. 3 to Fig. 1a. Therefore, randomising phases is not the way forward. Their variation seems to be more subtle than a simple set of uniformly distributed random numbers. The authors’ long term research vision is to develop robust numerical generators of synthetic random walking force signals which can replicate both temporal and spectral features of actual force data, hence can be used in both time and frequency domain analysis of footbridge dynamics. As a part of this vision, this paper presents a data-driven analytical model of lateral pedestrian forces induced by individuals walking on stiff surfaces. Such a model is an essential prerequisite for future quality models of individual and multiple lateral pedestrian loading under a wider range of conditions. From the structural engineer’s point of view, guidance is still lacking on the proportion of pedestrians in a given situation who coordinate their body motion and the scale and character of the resulting net dynamic loads on the structure, with or without influence of the motion of the structure itself. In fact, perceptible motion of the structure is just one type of cue or stimulus among several others affecting pedestrian loads [4], such as different auditory, visual and tactile stimuli.

3. Experimental data acquisition

Force amplitudes/BW [-]

This section describes the experimental setup (Section 3.1) and test protocol (Section 3.2), so it is clear how the fundamental data were collected. A database of high-quality lateral walking force

All walking tests were carried out in the Light Structures Laboratory in the University of Sheffield. Continuously measured force records were collected for single pedestrians walking on the state-of-the-art instrumented treadmill ADAL3D-F [37], as illustrated in Fig. 4. All components of the treadmill, including two brushless servo motors equipped with internal velocity controllers, belts and secondary elements, are mounted on a rigid metal frame and mechanically connected to the supporting ground only through four Kistler 9077B three-axial piezoelectric force sensors [38]. These transducers have high stiffness to avoid any possibility of system dynamic characteristics affecting measurements. The whole system is mechanically isolated, i.e. the sensors measure only external (walking) forces, while the internal forces due to belt friction and belt rotation are not detected by the sensors [39]. A typical lateral force record is shown in Fig. 1a. Rotational speed of the treadmill belts is in the range 0.1– 10 km/h and can be controlled and monitored remotely either with a control panel or with bespoke software, run from the data acquisition PC. Similar to fitness treadmills, the remote control panel and the treadmill itself are equipped with a safety stop switch. 3.2. Test sequence Prior to the measurements, a test protocol approved by the Research Ethics Committee of the University of Sheffield required each participant complete a Physical Activity Readiness Questionnaire and pass a preliminary fitness test (by satisfying predefined criteria for blood pressure and resting heart rate) to check whether they were suited for physical activity required during the measurements. Also, measurements of the body mass, age and height were taken for each test subject who passed the preliminary test. All participants wore comfortable flat sole shoes. Participants who had no experience with treadmill walking were given a brief training prior to the measurements supervised by a qualified gym instructor. All test subjects did approximately 10 min long warming up, which included walking on the treadmill while the walking speed was varied randomly and controlled by the speed of rotation of the treadmill belts. This interval was reported in the literature [40] as long enough for pre-test habituation to treadmill walking, when all measured parameters between treadmill and over-ground walking vanish even for inexperienced test subjects. The acquisition of walking forces started at a speed of 2 km/h and continued in increments of 0.5 km/h up to the maximum walking speed, i.e. an ultimate self-selected speed at which jogging,

0.20 0.15 0.10 0.05 0 -0.05 -0.10 -0.15 -0.20

0

5

10

15

20

25

30

35

40

Time [s] Fig. 3. Force–time history generated from the Fourier amplitude spectrum shown in Fig. 1b and uniformly distributed phases in the range [p, p].

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variability might be affected by the measuring process. In the next section, the selected 34 cycles will be used to curve fit morphology of the force, a so called ‘‘template cycle’’, which can be scaled (stretched and compressed) along the synthetic time and amplitude axes to reflect inter-cycle variability present in the actual force–time history (Fig. 1a). This modelling strategy is motivated by an existing numerical generator of electrocardiogram (ECG) signals [41] which have similar near-periodic features as walking time series. Modelling variability of the cycle intervals and force amplitudes is presented respectively in Sections 4.2 and 4.3, while generation of synthetic force signals is outlined in Section 4.4. 4.1. Template cycle

Fig. 4. Experimental setup.

rather than walking, is more comfortable for a test subject. Pacing rate was not prompted by any stimuli such as a metronome, and it was determined only from subsequent analysis of the generated force signals. Each test was completed when at least 64 successive footfalls were recorded. Rests were allowed between successive tests. In total, 85 volunteers (57 males and 28 females, body mass 75.8 ± 15.2 kg, height 174.4 ± 8.2 cm, age 29.8 ± 9.1 years) were drawn from students, academics and technical staff of the University of Sheffield. On average, forces corresponding to ten different walking speeds were collected for each test subject depending on their maximum comfortable walking speed. All together they generated 850 lateral walking force time series of kind illustrated in Fig. 1a. Each walking time series was sampled at 200 Hz. The next section describes the concept of modelling individual lateral force records as a narrow band process. In Section 6 this model is fitted to all 850 force records, yielding a numerical generator of random walking time series. 4. Modelling of imperfect real walking As humans are not perfect robots, force measures fluctuate invariably with time from one walking cycle to the next. A walking cycle can be defined between any two nominally identical events of the total force–time history. In this study, the point of intersection between the force trace and time axis at which the first derivative of the force is positive was selected as starting (and completing) event (Fig. 5). From the 40 s long force signal given in Fig. 1 and yielding about 40 cycles, a window comprising 34 successive cycles was extracted from the middle of the signal. Three cycles at the beginning and end of the force record were cast aside as their natural

The 34 weight-normalised walking cycles have been temporarily displaced and resampled to the length of the longest cycle, as shown in Fig. 6. As a result, the modified cycles have the same length between the starting and ending events. Visual inspection does suggest that there is a common waveform of the force which distorts along time and amplitude axes on a cycle-by-cycle basis. Their average is considered an inadequate representative of the force morphology due to misalignments of the common events, such as positions of the local extreme values. Moreover, averaging would spoil frequency content. To line up the force cycles, a technique called dynamic time warping (DTW) had been applied before their average was computed. Originally developed for speech recognition, the DTW nonlinearly warps two discrete signals to align similar events and minimise the sum of squared differences [42]. This process is illustrated schematically in Fig. 7a. Similarly, wavelet analysis linearly shifts and stretches the original (also called ‘‘mother’’) wavelet. However, due to the apparent nonlinear misalignments of the common events in the force data, linear shifting and stretching is unsuitable here for determining the common force waveform. As the DTW is designed to warp only two trajectories at the same time, individual cycles are warped through iterations onto a ‘‘reference cycle’’ (Fig. 7a). It is one of the measured cycles in Fig. 6 that minimises the sum of point-by-point Euclidiean distances to their simple numerical point-by-point average [43]. The warped cycles and template cycle are shown in Fig. 7b. The underlying shape of the template cycle can be modelled mathematically as a sum of 29 equidistant Gaussian functions (Fig. 8):

ZðtÞ ¼

29 X



Aj e

ðttj Þ2 2d2

ð2Þ

j¼1

where Z(t) is the curve fit, the parameter Aj is the height (also called weight) of the jth Gaussian peak, tj is the position of the centre of the peak, and the common parameter d controls the width of the Gaussian bell functions. Aiming for the minimum number of Gaussians in the sum necessary to describe accurately the template cycle, values of parameters d, tj, and Aj were optimised using non-linear least-square curve fit [44]. As a result, the positions of Gaussian centres are optimally located in every eight sample (29 of them), the optimal common width is 14/fs, while the optimal heights are illustrated in Fig. 8a. 4.2. Variability of cycle intervals Variations of cycle intervals Ti (i = 1, . . ., 34) can be represented by a dimensionless set of numbers si:

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Force amplitudes/BW [-]

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0.15 0.10

Ei

0.05 0 -0.05 Ti-1

-0.10 -0.15

5

Ti

5.5

Ti+1

6.5

6

7

7.5

8

8.5

Time [s] Fig. 5. A portion of a force–time history shown in Fig. 1a after applying a fourth-order low-pass Butterworth digital filter with cut-off frequency 15 Hz and preserving the correct phases. The force is recorded at fixed treadmill speed of 4.5 km/h, while the estimated average lateral frequency is 0.92 Hz (Fig. 1b).

A2 ðfm Þ Ss ðfm Þ ¼ s ; 2 Df

Force amplitudes/BW [-]

0.15 0.10

0 -0.05 -0.10 -0.15 0.2

0.4

0.6 0.8 Time [s]

1

m ; 34

m ¼ 0; . . . ; 16

ð4Þ

where As(fm) is a single-sided discrete Fourier amplitude spectra and Df = 1/34 is the spectral line spacing (Fig. 9). The ASD ordinates do not depend on the number of discrete data points si, but it is coarse due to their limited number. More points might reveal a much richer structure but this requires more measured cycles, hence a longer walking force signal. However, concerning ethics in research, the test duration was limited to avoid fatigue and discomfort, which can also influence natural variability of the force records. Similarly to the template cycle (Section 4.1), the ASD Ss(fm) can be analytically described by a sum of Gaussian functions (Fig. 9):

0.05

0

fm ¼

1.2

Fig. 6. Resampled weight-normalised cycles.

S0s ðf Þ ¼

17 X



Wje

ðf cj Þ2

ð5Þ

2b2

j¼1

si ¼

T i  lT

lT

ð3Þ

lT ¼ meanðT i Þ

0.15 0.10

(b) 0.15

reference cycle measured cycle

Force amplitudes/BW [-]

(a) Force amplitudes/BW [-]

Taking si as a series having zero mean value, variance of si can be obtained as the integral of its auto spectral density (ASD) [45]. Assuming that the variation of si does not change for the given test subject, pacing rate and duration of walking (e.g. due to fatigue and physical discomfort), the aim here is to use the ASD of the actual si data to generate synthetic series s0k (k = 1, . . ., N) of arbitrary length (e.g. N  34) with the same statistical properties, such as standard deviation and ‘‘pattern’’ of variation between successive si values. Note that using an ordinary random number generator, such as a probability density function of si, would fail to model this pattern as it cannot capture the frequency content of si series. The ASD of si can be calculated as [45]:

0.05 0 -0.05 -0.10 -0.15

0

0.2

0.4

0.6 Time [s]

0.8

1

Here, parameter Wj is the height of the jth Gaussian peak, cj is the position of the centre of the peak, and b is the common width of the Gaussian function. The Gaussian centres cj are placed in each sample on the quasi-frequency axis to fit exactly the actual ASD (Fig. 9). For such fixed positions of cj and predefined widths b = Df, Gaussian heights Wj can be computed using the non-linear least-square method [44]. Generation of synthetic series s0k (k = 0, . . ., N) starts by calculating S0s ðfn Þ values using Eq. (5) at discretely spaced frequency points fn ¼ nDf 0 , where n = 0, . . ., N/2  1 and Df0 = 1/N. The synthetic ASD amplitudes are then used in conjunction with Eq. (4) to generate qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a set of Fourier amplitudes A0s ðfn Þ ¼ 2Df 0 S0s ðfn Þ. Finally, assuming random distribution of phase angles in the range [p, p], A0s ðfn Þ are used in inverse FFT algorithm to generate a set of N synthetic variations s0k . Different realisations of the random phases may be specified by varying the seed of the random number generator,

1.2

template cycle warped cycles

0.10 0.05 0 -0.05 -0.10 -0.15

0

0.2

0.4

0.6 Time [s]

Fig. 7. (a) Schematic illustration of dynamic time warping. (b) Template cycle.

0.8

1

1.2

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(a)

(b) 0.1

Gaussian weights [-]

100

0.08 0.06 0.04

50 0 -50 -100 0

0.2

0.4

0.6

0.8

1

1.2

0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1

Gaussian fit template cycle

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Time [s]

Fig. 8. Fitting the template cycle: (a) equidistant Gaussian functions and (b) their sum.

-4

x10

x10

-3

5.6 Gaussian fit actual ASD

Cycle energy Ei [Ns2]

ASD [-]

9 8 7 6 5 4 3 2 1 0

Gaussian centres 0

0.1

0.2 0.3 0.4 Quasi-frequency [-]

linear fit measured data

5.4 5.2 5 4.8 4.6 4.4 4.2 4 3.8 1.05

0.5

1.10

1.11

Fig. 11. Cycle energy vs. cycle intervals (q = 0.65).

Fig. 9. ASD of si and its fit.

z

1.11 measured

1.06 1.07 1.08 1.09 Cycle intervals Ti [s]

synthetic

Cycle intervals Ti [s]

1.10 1.09 1.08 1.07 1.06 1.05

34 0

5

10

15 20 25 Cycle number [-]

30

35

r

Fig. 10. Measured and an example of synthetic cycle intervals. Fig. 12. Template cycle in 3D space.

hence many different series s0k can be generated with the same spectral properties. According to Eq. (2), scaling s0k by lT and adding the offset value lT, results in a series of synthetic intervals T 0k (Fig. 10). Moreover, assuming that the test subject does not vary significantly their gait in a narrow range of (lateral) walking rates, lT value can be slightly changed in the scaling process to generate cycle intervals at rates close to 1/lT. Empirical evidence for this is presented in Section 6. 4.3. Variability of force amplitudes Energy of weight-normalised walking cycles Ei can be defined as the integral of squared weight-normalised force amplitudes over

the corresponding cycle intervals Ti (Fig. 5). The fairly linear trend between these two parameters (the correlation coefficient q = 0.65) makes it possible to describe energy transfer between successive cycles [46] using the following linear regression model (Fig. 11):

Ei ¼ q1 T i þ q0 þ DEi

ð6Þ

Here, q1 = 0.017 and q0 = 0.023 are regression coefficients and DEi is the subsequent error (also known as a disturbance term), which is a random variable and is usually modelled as a random Gaussian noise [47].

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Force amplitudes/BW [-]

(a)

0.15 0.10 0.05 0 -0.05 -0.10 -0.15 0

5

10

15

20

25

30

35

40

Time [s]

Fourier amplitudes/ BW [-]

(b)

0.06 Fourier harmonics dominant harmonics

0.05 0.04 0.03 0.02 0.01 0 0

1

2

3

4

5

6

7

8

9

10

Frequency [Hz] Fig. 13. (a) An example of synthetic time series and (b) the corresponding discrete Fourier amplitude spectrum.

x10 Acceleration/BW [1/kg]

6

-3

(a)

4 2 0 -2 -4 -6

-3

Acceleration/BW [1/kg]

x10 5 4 3 2 1 0 -1 -2 -3 -4 -5

(b)

-3

x10

Acceleration/BW [1/kg]

1.5

(c)

1 0.5 0 -0.5 -1 -1.5

0

5

10

15

20 Time [s]

25

30

35

40

Fig. 14. Dynamic responses of 1000 kg oscillator with 0.5% damping and natural frequencies of (a) 0.92 Hz (b) 1.84 Hz and (c) 2.76 Hz to real and synthetic force time series of walking at mean lateral rate fw = 0.92 Hz. Envelope of responses corresponds to measured walking while the grey curve corresponds to synthetic walking.

Given a set of synthetic cycle intervals T 0k (k = 1, ..., N) generated as explained in Section 4.2, the corresponding synthetic series E0k

can be calculated using Eq. (6). These energies can be assigned to k template cycles by scaling their amplitudes by factors ak:

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120

1.6

100

1.4

Lateral walking rate [Hz]

Number of occurrences [-]

124

80 60 40 20 0 0.4

0.6

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Gaussian centres tj in Eq. (2) now correspond to fixed angles hkj ¼ x0k tj around the unit circle, as illustrated in Fig. 12.

sffiffiffiffiffiffi E0k ak ¼ Etc

ð7Þ

where Etc is the energy of the template cycle. Arranged in a sequence along the time axis, the scaled template cycles reflect the smooth variations of the force amplitudes on a cycle-by-cycle basis, as presented in the next section. 4.4. Generation of synthetic forces The continuous walking could be better visualised when the template cycle is ‘‘wrapped’’ around the surface of a circular cylinder (Fig. 12). Now the first and the last sample of the template overlap, thus the corresponding trajectory becomes a closed orbit in a three-dimensional (3D) space with coordinates (r, h, z). A synthetic time series is generated by retracing fit Z(t) around a circle of unit radius in the (r, h) plane, where each revolution corresponds to one walking cycle. Variations of the successive cycle intervals T 0k (Section 4.2) can be incorporated by varying angular frequency of rotation x0k ¼ 2p=T 0k , while the variations of force amplitudes can be reflected by scaling Z(t) by ak factors (Section 4.3). The equations to generate a series of N successive cycles are therefore given by a set of three coupled equations:

r k ðtÞ ¼ 1 hk ðtÞ ¼ x0k t Z k ðhk Þ ¼ ak

measured data linear fit

29 X



Aj e

4.5. Model verification Some validation of the approach is provided by visual comparison of Fig. 13a to Fig. 1a and Fig. 13b to Fig. 1b, which shows apparent similarity between a measured signal and an example artificial force signal in both time and frequency domain. Also, there is little to choose between responses for 1000 kg SDOF systems with 0.5% damping and natural frequencies f of 0.92, 1.84 and 2.76 Hz (Fig. 14). These frequencies correspond exactly to the fundamental, third and fifth dominant harmonics of the simulated lateral walking excitation. The shape of the responses indicates that perfect resonance is difficult to maintain due to narrow band nature of walking – the effect being particularly prominent for higher harmonics (Fig. 14b and c). Further parametric studies can demonstrate much greater divergence from resonant build-up with increase of damping of the SDOF systems. Due to the randomness of the modelling parameters T 0k and DE0k , two identical synthetic force signals can be generated only by chance. However, a family of synthetic forces share several properties inherited from the measured signal: (1) Shapes of the walking cycles are drawn from the same template cycle. (2) Cycle intervals T 0k are statistically equivalent as they share the same ASD. (3) The statistical equivalence of T 0k series reflects directly equivalence of E0k energies according to Eq. (6).

ð8Þ

ðhk hkj Þ2 2b2 k

j¼1

where k = 1, . . ., N is the label of a cycle, t is the time vector sampled at fs P 200 Hz and bk ¼ x0k d in radians. The time positions of the

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The next section shifts focus of the study from intra-subject variability to inter-subject differences between human abilities to

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Fig. 16. (a) Normal probability plot and (b) normal probability density function of lateral walking frequencies fw.

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generate lateral walking loads, thereby justifying the random modelling approach presented in Section 6. 5. Inter-subject randomness of lateral forces Lateral walking frequency, the corresponding Fourier amplitude scaled by body weight (i.e. fundamental DLF1), walking speed and body weight are extracted from each force record in the database established in Section 3 for further analysis of their random nature and mutual correlations.

The reasonably symmetric bell shape of the histogram shown in Fig. 15 suggests that the lateral walking frequencies are approximately normally distributed. The hypothesis is validated by the so called ‘‘normal probability plot’’ shown in Fig. 16a, where the actual data are plotted against a theoretical normal distribution [48]. The graph looks fairly straight, at least when few extreme values are ignored, indicating that the data do follow the normal distribution (Fig. 16b). Estimates of mean and standard deviation are l = 0.91(0.90–0.92) Hz and r = 0.17(0.16–0.18) Hz, respectively, where the values in brackets are 95% confidence intervals.

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DLF1 [-] Fig. 21. Histogram of DLF1 in a narrow range of three characteristic walking speeds (a) slow 3 km/h (0.83 m/s), (b) moderate 4.5 km/h (1.25 m/s), and (c) fast 6 km/h (1.67 m/ s).

The normal probability distribution was also observed by Ingólfsson et al. [22] but the reported statistics (l = 0.86 Hz and r = 0.08 Hz) are considerably different. One reason for this could be discrepancies between the populations of test subjects measured. In the present study, tests subjects were drawn from a multi-ethnic group of students and staff members in the University of Sheffield, whereas the latter study focused on a group of adult Italians only. Another reason can be a wider range of walking speeds measured in the present study (see Section 3.2), as opposed to a narrow range of speeds 1.29 ± 0.21 m/s studied by Ingólfsson et al.

Apart from the linear trend between walking speed and lateral walking frequency (q = 0.9), Fig. 17 highlights a large scatter of the frequencies at fixed walking speeds, and vice versa. Moreover, taking 4.5 km/h (1.25 m/s) as an example of moderate walking speeds, along with 3 km/h (0.83 m/s) and 6 km/h (1.67 m/s) as representatives of slow and fast walking, histograms in Fig. 18 illustrate a great degree of inter-subject variability of lateral walking frequencies at fixed walking speeds. This observation can play a key role in developing realistic load case scenarios due to heavy pedestrian traffic (e.g. in busy city centres), where the walking speed of

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input data: frequency and duration of walking from the corresponding cluster select a set of the force parameters by chance

number of cycles N random number seed

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]

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individuals can be controlled by a steady pedestrian flow. The corresponding models available in the contemporary design guidelines [17,18,25] assume wrongly that all individuals in a group/ crowd walk at equal rate. Further visual inspection of the histograms suggests that distribution of the frequencies at slow and fast walking speeds cannot be approximated by any known probability density function. However, a normal distribution (l = 0.93 Hz and r = 0.07 Hz) appears a suitable model of the distribution of lateral walking frequencies in a narrow range of the moderate walking speed. A positive linear trend between DLF1 and lateral walking frequency (Fig. 19a) indicates that the force amplitudes generally increase as people increase their pacing rate. However, the large scatter of the DLF amplitudes from the simple linear regression suggests that the data cannot be described reliably as a simple function of the walking rate, as suggested elsewhere for vertical pedestrian loads [49,50]. DLFs are statistically random and most likely uniformly distributed in a certain range (Fig. 20). Similar statistical randomness can be also observed in the case of fixed walking speeds (Fig. 21). However, plot of DLF1 across a range of walking speeds (Fig. 19b) does not show a generally increasing trend as in Fig. 19a between DLF1 and lateral walking frequency. The trend is observed only for slow and moderate speeds, whereas the force amplitudes stay flat with further speed increments. A similar effect is observed for vertical walking forces elsewhere [24]. Finally, Fig. 22 proves that the force amplitudes and body weight are uncorrelated variables. As a result, an arbitrary body weight can be assigned to a synthetic force signal of the kind shown in Fig. 13a. Moreover, values of body weight can be

generated artificially and randomly using a probability density function, as suggested by Hermanussen et al. [51] for German, Austrian and Norwegian citizens. The next section integrates the numerous database of individual force records established in Section 3, their analytical characterisation outlined in Section 4 and the knowledge on their inter-subject variability from Section 5, to create a numerical generator of random narrow band walking time series. 6. Development of random numerical generator of force signals Each of the 850 force records in the database (Section 3) was processed using the concept described in Section 4. Multiple sets of information, such as parameters of the template cycle Zk(t), the ASD S0s ðf Þ, disturbance term DEi(t) and the regression coefficients q1 and q0, were extracted and stored in MATLAB structure files [52]. These are called ‘‘mat files’’ in the remaining part of the paper. The mat files were classified into 16 categories (clusters) with respect to the lateral walking frequency, as shown by the histogram in Fig. 15. As each bar in the histogram is only 0.06 Hz wide, it can be assumed that any mat file in a cluster can be used to generate synthetic walking time series at any frequency within the cluster’s narrow frequency range. As already mentioned in Section 4.2, this means that the modelling parameters extracted from real walking at a certain rate can be used to synthesise walking forces at close rates in the same cluster. This is a key feature of the random modelling strategy adopted in this section. The flow chart in Fig. 23 outlines the algorithm of creating synthetic force signals. For specified lateral walking frequency and

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(a)

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duration of walking, the algorithm first calculates the total number of walking cycles N in the given walking period. Then, from a cluster corresponding to the walking frequency, it selects randomly and equally likely a mat file. Based on the information on the ASD function S0s ðf Þ (Section 4.2) stored in the file, the algorithm generates cycle intervals T 0k . These are further used to calculate the corresponding scaling factors ak and angular frequencies x0k , as presented in Sections 4.3 and 4.4, respectively. At this stage, the algorithm runs the dynamic model given by the set of coupled Eq. (8). The template cycle Z(t) is stretched and/or compressed along time and amplitude axes, yielding weight normalised walking forces of a kind shown in Fig. 13a. Finally, these become equivalent walking force time histories when their amplitudes are additionally multiplied by a random body weight. For the same set of input parameters – lateral walking rate 1 Hz and walking period 40 s, Fig. 24 illustrates examples of force

signals generated after the model was run twice in succession. A visual comparison of the time series provides convincing evidence that the model does account for the inter-subject variability. On the other hand, the ability to generate different degrees of the intra-subject variability is more obvious from comparison between the corresponding frequency spectra (Fig. 25). The broader spread of energy around dominant harmonics in Fig. 25a relative to Fig. 25b indicates that walking of the first ‘‘virtual’’ pedestrian is characterised by greater intra-cycle variability. 7. Conclusions A random data-driven mathematical model has been developed to generate synthetic lateral pedestrian force signals with realistic temporal and spectral features for healthy individuals walking on stiff surfaces. Therefore, pedestrian excitation of excessively

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swaying bridges, when human gait and the corresponding ground reaction forces can be dramatically different from normal, is beyond the scope of the paper. The fundamental data are collected using a state-of-the-art force measuring treadmill (Section 3), transferred and adapted from the biomedical field of human locomotion, while the modelling strategy is motivated by the existing models of wind and earthquake loading, human electrocardiogram (ECG) signals and speech recognition. The ability to simulate reliably dynamic response of footbridges occupied by individual pedestrians gives this model a definite advantage over the lateral walking force models available in the relevant contemporary design guidelines. Therefore, it can serve as a framework for a more realistic dynamic performance assessment which can be adopted in everyday design practice. The first refinement of the current design procedures would address the near-periodic nature of walking loads. To simplify dynamic analysis, walking is conservatively assumed perfectly periodic and therefore presentable as a series of identical footfalls replicated at precise intervals. However, this gives rise to serious inaccuracies in predicted dynamic response. To simulate reality better, the proposed model accounts successfully for variations of intervals, amplitudes and energy of successive walking cycles. The second refinement addresses inter-subject variations of walking loads. In addition, walking is considered random rather than deterministic process, commonly adopted in the current design guidelines of footbridges. This approach is similar to other random excitation of civil engineering structures, such as due to wind, waves or earthquakes. Therefore, the unique numerical generator of walking loads presented here offers an opportunity to radically change the current philosophy behind vibration serviceability assessment of structures due to pedestrian-induced excitation. The modelling strategy is numerically too complex to generate artificial forces manually, but it can be coded and distributed to structural designers as computer software. Hence, the third refinement would address modernisation of vibration serviceability assessment from traditional hand calculations towards fully computerised design. In fact, modal properties are commonly predicted nowadays with support of comprehensive FE software, structural elements are designed using a vast variety of computer packages and technical drawings are prepared with CAD programs. Moreover, as the model generates synthetic force signals in a fraction of a second even on a standard office PC configuration, the software would enable efficient and cost-effective structural design. Finally, this framework can be extended further to random walking loads due to groups and crowds. At present, individual synthetic forces can be summed with random phase lags as suggested elsewhere [53]. However, there are indications that this is not what is happening in reality [4] and more knowledge into synchronisation between people walking is needed. Also, to account for human–structure dynamic interaction, variations in the force amplitudes, timing and shape for successive walking cycles should be modelled as a function of bridge lateral dynamic response and human perception to lateral vibrations. However, experimental data needed to meet this challenge is currently non-existent.

Acknowledgements The authors would like to acknowledge the financial support provided by the UK Engineering and Physical Sciences Research Council (EPSRC) for grant reference EP/E018734/1 (‘‘Human Walking and Running Forces: Novel Experimental Characterisation and Application in Civil Engineering Dynamics’’) and to thank all test subjects for participating in the data collection.

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