Simplified procedures for vibration serviceability analysis of footbridges subjected to realistic walking loads

Computers and Structures 87 (2009) 890–903

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Simplified procedures for vibration serviceability analysis of footbridges subjected to realistic walking loads Giuseppe Piccardo *, Federica Tubino DICAT, Department of Civil, Environmental and Architectural Engineering, University of Genoa, Via Montallegro 1, 16145 Genova, Italy

a r t i c l e

i n f o

Article history: Received 30 July 2008 Accepted 16 April 2009 Available online 9 May 2009 Keywords: Vibration serviceability Walking Moving loads Footbridges Closed-form solutions Probabilistic load models

a b s t r a c t This paper deals with the vibration serviceability analysis of footbridges subjected to realistic pedestrian traffic conditions, based on a probabilistic characterization of pedestrian-induced forces. The dynamic response to three different loading conditions is analysed through a non-dimensional approach, which permits the identification of the essential non-dimensional parameters governing the dynamic behaviour. Two simplified procedures are then proposed, founded on the sound definition of two coefficients, the Equivalent Amplification Factor and the Equivalent Synchronization Factor, which allow the evaluation of the vibration serviceability without requiring numerical analyses. Final applications confirm the efficiency of the proposed methods. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Pedestrians walking on a footbridge exert dynamic loadings in the vertical and horizontal directions which can be schematized as periodic through a Fourier series. The first main harmonic is characterized by a frequency corresponding to the pedestrian pacing rate in the vertical direction, while horizontal loading is characterized by a fundamental frequency component at a half of the pacing rate. The frequency of the dominant harmonic is included between 1.6 and 2.4 Hz in the vertical direction, and between 0.8 and 1.2 Hz in the lateral direction [1,2]. Modern footbridges are very slender structures, often characterized by natural frequencies in the vertical and lateral directions within the range of the dominant walking harmonics; thus, they can be very sensitive to walking-induced vibrations. The serviceability assessment is based on a comparison between the pedestrian-induced acceleration and a suitably defined limit value [3]. Until the 1990s, standard codes [4–6] required the analysis of vertical pedestrian-induced vibrations of footbridges considering a loading model constituted by a single pedestrian, modelled as a resonant moving harmonic load. Such a scenario was analysed from a theoretical point of view by the authors of this paper, who proposed a closed-form solution for the dynamic response of a footbridge excited by a sinusoidal resonant load [7,8]. * Corresponding author. Tel.: +39 010 353 2970; fax: +39 010 353 2292. E-mail addresses: [email protected] (G. Piccardo), federica.tubino@ unige.it (F. Tubino). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.04.006

However, the loading model proposed by old standard codes is conventional and is not representative of real pedestrian traffic conditions. During its design working life, depending on the pedestrian traffic, a footbridge is expected to be crossed by several pedestrians at the same time [9,10], and it is scarcely probable that a pedestrian walks in perfect resonance with the footbridge. Based on the classical paper by Matsumoto et al. [11], ISO [10] and FIB [3] require the analysis of a loading scenario in which streams of pedestrians occupy the footbridge, taking into account the partial correlation among pedestrians by multiplying the maximum dynamic response due to a single resonant pedestrian by the square root of the number of pedestrians crossing the bridge. A similar approach is proposed by the new EC5 where, however, simplified expressions for the estimate of the maximum structural accelerations are directly furnished [12]. From a general point of view, real pedestrian traffic conditions should be modelled probabilistically [13,14], considering several sources of randomness among which pedestrian arrivals, walking frequencies and velocities, force amplitudes and pedestrian weights. This randomness of walking parameters can be named inter-subject variability [15], being related to the fact that different pedestrians can induce different dynamic loads. Moreover, even the same pedestrian exerts a walking force that can differ with each step: this particular source of randomness can be called intra-subject variability [15]. The vibration serviceability analysis of footbridges subjected to a realistic walking load scenario is a very topical subject in the international research panorama and it is being analysed from different

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

points of view. On the one hand, sophisticated load models, representative of a single pedestrian, have been studied characterizing their inherent randomness by means of both inter-subject and intra-subject variability in a multi-mode approach [15]: in this way, the probability of an assigned acceleration level and cumulative probability for accelerations can be obtained simulating a sufficiently high number of single pedestrians, aiming to improve the rough estimation furnished by classical codes [4–6] assuming a single resonant pedestrian characterized by deterministic parameters. On the other hand, simplified load models have been proposed with the aim of simulating, in an approximate way, groups of pedestrians walking in a natural (i.e. random) way on a footbridge, taking into account only inter-subject variability: this approach has originated models characterized by a single resonant moving force suitably amplified by an equivalent number of resonant pedestrians [16], models made up of equivalent distributions of uniform resonant loads [17,3] and models based on response spectra as functions of pedestrian stream density [18] and of return period of peak acceleration [19]. The serviceability analysis is then based on the comparison of the footbridge peak acceleration with suitable comfort limits, which generally depend on bridge location [17]. This paper belongs to the latter class of models, dealing with two different simplified procedures to evaluate the maximum dynamic response of footbridges to a realistic loading scenario. It is based on the sound proposal of two coefficients: the Equivalent Amplification Factor (EAF), defined as the ratio between the maximum dynamic response to a realistic loading scenario and the maximum dynamic response to a single resonant pedestrian, and the Equivalent Synchronization Factor (ESF), introduced as the ratio between the maximum dynamic response to a realistic loading scenario and the maximum dynamic response to uniformly-distributed resonant pedestrian loadings. Particular emphasis is given to the non-dimensional expressions of the dynamic response, which permits the identification of the non-dimensional parameters governing the dynamic behaviour. A full probabilistic characterization of all the non-dimensional parameters involved is considered, taking into account the inter-subject variability. Analyses are performed based on the simplifying assumption of a sole vibration mode affecting the structural dynamic response, which is induced by low-density pedestrian streams. Furthermore, vertical vibrations are analysed, but the same procedure could be adopted to study lateral vibrations in ordinary conditions, when synchronization between footbridge and pedestrians does not occur (i.e. absence of pedestrian–structure interaction). After illustrating the loading models adopted (Section 2), the dynamic response to a resonant pedestrian group, to uniformly-distributed resonant pedestrians and to a probabilistically-modelled group of pedestrians is analysed, identifying the non-dimensional parameters governing the excitation mechanisms (Section 3). The general tendencies of the EAF and of the ESF as functions of the relevant non-dimensional parameters previously identified are analysed (Section 4), carrying out a comparison with analogous expressions proposed by international guidelines. The maximum dynamic response of an ideal simply-supported bridge and of a real cable-stayed bridge are then evaluated (Section 5). Finally, some conclusions and prospects are reported in Section 6.

2. Load modelling The actions induced by walking involve dynamic loadings in the vertical, lateral and longitudinal directions. Concerning vertical actions and taking into account only the main harmonics in the spectrum of forces induced by walking, the classical pedestrian loading model can be introduced, consisting of a periodic force described by the following Fourier series [1–3,17,18]:

fp ðtÞ ¼ G þ

n X

Gak sinð2pkfp t  /k Þ

891

ð1Þ

k¼1

where t is the time, G is the person’s weight, k is the order number of the main harmonic, ak are Fourier’s coefficients, called Dynamic Load Factors, DLFs, of the kth main harmonic, fp is the walking rate (Hz), /k is the phase-shift of the kth main harmonic, n is the total number of contributing main harmonics of walking force. The model (1) was originally introduced in the deterministic field [1,2]; further studies, mainly developed in the last decade, have highlighted that all the parameters in Eq. (1) should be modelled from a probabilistic point of view (e.g. [15]). In this Section, three simplified load models based on Eq. (1), taking into account only one term in the summation, are considered. Section 2.1 considers a model able to describe the effect of resonant pedestrian groups; Section 2.2 analyses a continuous resonant pedestrian stream. Then, a probabilistic-based model is introduced (Section 2.3) in order to give a more realistic representation of pedestrian streams. The proposed models can be applied in case of low pedestrian density (i.e. less than 0.6 persons/m2 [3]), which does not imply any restriction on people’s freedom of movement. 2.1. Resonant pedestrian group From the technical point of view, a periodic loading in resonance with a structural natural frequency gives origin to the largest response of the system. Simplified expressions can be deduced from Eq. (1) with only one term in the summation, corresponding to the walking harmonic in resonance with a natural structural mode (in general the first one). Thus, the dynamic effect of a resonant pedestrian group can be approximately modelled as a sinusoid of circular frequency X, moving across the bridge with speed c:

f ðx; tÞ ¼ cF 0 sinðXtÞdðx  ctÞ

ð2Þ

where x is the abscissa describing a one-dimensional structure, d is the Dirac delta function, F0 = amGm is the force amplitude, am and Gm being the mean value of the DLF and of the person’s weight, respectively. The coefficient c is an amplification factor able to describe the effect of a group with a limited number of unsorted walking people (generally less than 20). Up to the end of the 20th century, standard codes [4–6] assumed a loading model related to a single moving resonant pedestrian (Eq. (2) with c = 1 and X = 2pnj, nj being the jth natural frequency of the footbridge, generally the first one). Considering a crowd with Np individuals all at the same frequency with a random phase distribution, the coefficient cpcan ffiffiffiffiffiffi be conservatively estimated by the classical expression c ¼ N p [11,3,10]. More recently, alternative simplified expressions for the amplification factor c have been proposed, as a function of the bridge natural frequency [3,16]. 2.2. Uniformly-distributed resonant pedestrians When pedestrians walk in a group, they tend to synchronize their walking speed and frequency; a reference condition corresponding to such a situation is to assume a uniformly-distributed load per-unit-length, having a frequency equal to a natural footbridge frequency (usually the fundamental one):

f ðx; tÞ ¼ k

F 0 Np sinðXtÞ L

ð3Þ

where L is the structural span, X = 2pnj is the load circular frequency resonant with the natural footbridge frequency nj (usually j = 1), k < 1 is a synchronization factor which considers that only a

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small percentage of pedestrians (generally around 20%) can walk in perfect synchronization, as shown by laboratory and free-field tests [20,21]. Recent international guidelines [3,17] propose equivalent values for the synchronization factor k; in particular, on the basis of digital simulations based on a probabilistic model of walking load, Sétra [17] proposed an expression of k valid for a sparse or pffiffiffiffiffiffiffiffiffiffiffiffi dense crowd, k ¼ 10:8  nj =N p , nj being the damping ratio of the structural vibration mode under consideration. 2.3. Probabilistically-modelled group of pedestrians A more realistic loading scenario can be considered in which pedestrians arrive in a random way and are able to move undisturbed, each of them with his own characteristics (in terms of loading amplitude, frequency, velocity and phase); in such a case, the force induced by Np pedestrians can be expressed as:

f ðx; tÞ ¼

Np X

fi ðx; tÞ

ð4Þ

i¼1

where fi(x, t) is the force exerted by the ith pedestrian and it is given by:

8 0 t < si > < L fi ðx; tÞ ¼ F i sinðXi ðt  si Þ þ Wi Þd½x  ci ðt  si Þ si  t  si þ ci > : 0 t > si þ cL i

ð5Þ where Fi (=aiGi), Xi, Wi, ci and si are, respectively, the force amplitude, the walking circular frequency, the phase-angle, the walking speed and the arrival time of the ith pedestrian, while ai and Gi are the DLF and the weight of the ith pedestrian. All these quantities have to be probabilistically modelled in order to consider the intersubject variability in walking forces induced by different pedestrians. In particular, the walking circular frequency X, the pedestrian weight G, the DLF a and the walking velocity c can be considered as Gaussian random variables. In the literature, some measurements have been carried out in order to provide a statistical characterization of walking frequencies [11,14,15]; in general, it depends on people’s characteristics and differs between different countries. Table 1 summarizes some values deduced from classical and recent papers for the mean walking frequency fpm, for its standard deviation rfp and its coefficient of variation Vfp. The discussion concerning the dynamic load factors a is more complicated. First of all, it has been experimentally observed that the mean value of the dynamic loading factor is a function of both the walking frequency and the frequency of motion of the structure [13]. However, in the literature, only expressions defining a as a function of the walking frequency can be found, e.g. [13,15]; this dependence is particularly pronounced for the first main harmonic. Then, for a particular walking frequency, the distribution of the DLFs around their mean value can be obtained multiplying the relevant mean value by a suitable coefficient of variation (e.g. Va = 0.16 for the first harmonic [15]).

Table 1 Statistical moments of the pedestrian walking frequency. References [11] [14] [15]

fpm Mean [Hz]

rfp Standard deviation [Hz]

Vfp Coefficient of variation

2 1.82 1.87

0.173 0.12 0.186

0.09 0.07 0.1

Very few data appear in the literature about people’s weight: in [18] the mean weight has been assumed Gm = 744 N, with a standard deviation rG = 130 N, corresponding to a coefficient of variation VG = 0.17. The description of walking speed, walking frequency and step length deserves some further comments. Recent experiments have pointed out that walking frequency and step length can be approximately considered as two independent random variables described by Gaussian distributions [15]; a knowledge of both these parameters permits the calculation of the time required to cover a determined distance (that is, of the pedestrian velocity). In the proposed formulation, Eqs. (4) and (5), attention is directly focused on the pedestrians’ walking velocity c that can be, in turn, approximately described as a Gaussian random variable; to the best of authors’ knowledge, the sole direct statistical characterization of the pedestrian velocity available in the literature was carried out by [14] (mean velocity cm = 1.37 m/s, with a standard deviation rc = 0.15 m/s, corresponding to a coefficient of variation Vc = 0.11). Furthermore, walking frequency and velocity are probably correlated, but no information concerning their correlation can be found in the literature; thus, they will be assumed as uncorrelated in the following. Concerning the phase-angle W among different pedestrians, few measurements are available in the literature as regards crowded conditions [20]; since in this paper low-density pedestrian streams are considered, it may reasonably be assumed that the phase-angle among pedestrians is a random variable uniformly-distributed in the interval [0, 2p]. A similar assumption seems to be made in [17]. Finally, the arrival time s of the ith pedestrian can be expressed P as si ¼ ik¼1 s0k , where s0k is the time-lag between the arrival of the kth and (k  1)th pedestrian. If pedestrian arrivals are considered as Poisson events [11], then the time-lags s0k between their arrivals are represented by an exponentially-distributed random variable with an average return period T = Tpm/Np, where Tpm = L/cm is the footbridge mean crossing time for a single pedestrian: in such a way, on average, Np pedestrians cross the bridge at the same time.

3. Dynamic response Modelling the footbridge as a linear mono-dimensional classically-damped dynamical system, under the simplified hypothesis that one structural mode (namely the jth mode) mainly contributes to the footbridge liveliness [2], the structural displacement can be expressed as:

qðx; tÞ ¼ uj ðxÞpj ðtÞ

ð6Þ

where uj(x) is the jth mode of vibration and pj(t) is the corresponding principal coordinate (having the same physical dimension of the Lagrangian displacement q). In case of footbridges with closelyspaced natural frequencies, more than one mode of vibration can contribute to the structural response; thus, Eq. (6) may be unsafe and many modes of vibration have to be accounted for in the evaluation of the dynamic response. If the footbridge natural frequencies are suitably separated, and higher modes are excited by higher walking harmonics, the contribution of each structural mode to the dynamic response of the footbridge can be evaluated separately through the procedure proposed in the following, and the total response can be calculated by the superposition principle. The equation of motion of the jth principal coordinate pj is given by [22]:

€j ðtÞ þ 2nj xj p_ j ðtÞ þ x2j pj ðtÞ ¼ p

1 F j ðtÞ Mj

ð7Þ

where nj is the modal damping ratio, xj is the circular natural frequency, Mj is the modal mass, Fj(t) is the modal force:

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F j ðtÞ ¼

Z 0

L

f ðx; tÞuj ðxÞdx

ð8Þ

f(x,t) being the generalized pedestrian-induced force per-unitlength (Section 2). Eq. (7) can be re-written in the following non-dimensional form:

€~j ð~tÞ þ 2n p _ ~ ~j ð~tÞ ¼ F~j ð~tÞ p j ~ j ðtÞ þ p

ð9Þ

~j and F~j are given by: where the non-dimensional quantities ~t; p

~t ¼ xj t;

~j ¼ p

pj M j x2j pj ¼ ; pjs F0

Fj F~j ¼ F0

ð10Þ

pjs being the static response to the force F0 = amGm (Section 2). At first, the dynamic response of a generic footbridge to a resonant pedestrian group is tackled (Section 3.1), then the effect of a continuous resonant pedestrian stream is dealt with (Section 3.2). Finally, a bridge loaded by a probabilistically-modelled group of pedestrians is considered (Section 3.3). The aim of these analyses is to derive closed-form solutions when possible but, above all, to determine the non-dimensional parameters governing the maximum dynamic response in these three loading cases. The reference quantities in all these evaluations are structural accelerations since limit values for accelerations are directly linked to pedestrian comfort, and thus to bridge vibration serviceability, in international standards [3].

ð11Þ

~ and X ~ c are non-dimensional quantities: where X

X

xj

;

~c ¼ X

c

xj L

c expðnj~tÞ ~j ð~tÞ ¼  p 2

t

0

~ c sÞ expðnj sÞds cosð~tÞ uj ðX

ð16Þ

~j can be expressed in the following meaningful way: Then, p

~j ð~tÞ ¼ Aj ð~tÞ cosð~tÞ p

ð17Þ

where Aj ð~tÞ is the amplitude, which can be interpreted as a slowvarying quantity [7,8]:

Aj ð~tÞ ¼ 

c

"Z

2

~t 0

# ~ c sÞ expðnj sÞds expðnj~tÞ uj ðX

ð18Þ

~j From Eq. (17), it may be deduced that the maximum value of p coincides with the maximum of the amplitude Aj. Furthermore, ~j can be approximately expressed as follows: the acceleration of p

€~j ð~tÞ ffi Aj ð~tÞ cosð~tÞ p

ð19Þ

€ ~ Thus, also the   value  maximum  of pj occurs when the amplitude Aj is € ~j max  ¼ Aj max  . maximum p € ~j due to a single resFrom Eqs. (18) and (19) the maximum of p € ~ onant pedestrian (c = 1), pjmax , can be written as a function of the SP following quantities: SP

ð20Þ

€ðtÞ are finally obThe structural displacement q(t) and acceleration q tained substituting Eqs. (17) and (19) into the expression (6). For simple structural schemes analytical expressions are available for the modal shape uj [7]: in such cases, the integrals in Eqs. (13) and (16) can be solved analytically and closed-form solutions for the maximum response, and thus for the DAF b can be introduced [7,8]; Appendix A illustrates the closed-form solution for simply-supported beams. 3.2. Uniformly-distributed resonant pedestrians

ð12Þ

From Eqs. (9) and (11), it is evident that the non-dimensional prin~j is a function of the non-dimensional parameters cipal coordinate p ~ c; X ~ , nj. ~t; c; X ~ ffi 1Þ, Eq. (9) with F~j given by Eq. In case of quasi-resonance ðX (11) can be solved in closed-form through a perturbation tech~j can be expressed as follows [8]: nique; in such a case, p

9 8h R i ~t > ~ sÞ expðnj sÞds cosð~tÞ > ~ c sÞ cosðDX = <  u ð X ~ j 0 c expðnj tÞ ~j ð~tÞ ¼ hR ~ i p t > 2 ~ sÞ expðnj sÞds sinð~tÞ > ~ c sÞ sinðDX ; : þ 0 uj ðX ð13Þ ~  1 is the (small) non-dimensional detuning be~ ¼X where DX tween the frequency of the load and the natural frequency of the structure. Let us define, now, the Dynamic Amplification Factor (DAF) b as the ratio between the maximum dynamic response and the maximum static response. From the definition of the non-dimensional principal coordinate (10) and the expression of the non-dimensional modal force (11), it may be deduced that:

 ~j max c¼1 b¼p

~ ¼ 0Þ, Eq. (13) reIn the particular case of perfect resonance ðDX duces to: "Z ~ #

SP

Substituting Eq. (2) into Eq. (8), taking into account Eq. (10) and expressing the generic modal shape uj as a function of a nondimensional abscissa f = x/L, the non-dimensional modal force due to a resonant pedestrian group crossing the bridge with a walking circular frequency X is given by:

~ ¼ X

ð15Þ

€~j €~j ~ c ; nj ; u Þ p ¼p ðX j max max

3.1. Resonant pedestrian group

~ ~tÞu ðX ~ c~tÞ F~j ð~tÞ ¼ c sinðX j

~ ; nj ; u Þ ~ c ; DX b ¼ bðX j

Considering a uniformly-distributed resonant load condition, the non-dimensional modal force is obtained substituting Eq. (3), with X = xj, into Eq. (8) and taking into account Eq. (10):

F~j ð~tÞ ¼ kN p

Z

1

0



uj ðfÞdf sinð~tÞ

ð21Þ

~j is simply given by: The maximum value of p

~jmax ¼ p

kNp

R1 0

uj ðfÞdf

2nj

ð22Þ

Furthermore, from Eqs. (9) and (21), it may be deduced that € ~jmax . Thus, the absolute value of the maximum acceleration ~jmax ¼ p p caused by a group of uniformly-distributed Np resonant pedestrians € ~jmax U is given by Eq. (22) with k = 1; it follows that: p

€~j €~ j ðn ; Np ; u Þ p ¼p j j max max U

U

ð23Þ

When analytical expressions of the modal shape uj are available, the integral in Eq. (22) can be solved analytically and closed-form expressions for the maximum response to uniformly-distributed resonant pedestrians can be easily determined. 3.3. Probabilistically-modelled group of pedestrians

ð14Þ

Thus, in general, the factor b can be expressed as a function of the following quantities:

Substituting Eq. (4) into Eq. (8), and taking into account Eq. (10), the non-dimensional modal force due to a probabilistically-modelled group of pedestrians is given by:

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F~j ð~tÞ ¼

Np X

F~ji ð~tÞ

ð24Þ

SP

i¼1

where F~ji ð~tÞ is the contribution of the ith pedestrian to the jth modal force, and is obtained substituting Eq. (5) into Eq. (8):

8 0 > > < ~ i sinðX ~ i ð~t  s ~ ci ð~t  s ~iG ~i Þ þ Wi Þuj ðX ~i ÞÞ F~ji ð~tÞ ¼ a > > :0

~t < s ~i s~i  ~t  s~i þ

1 ~ X

ci

~t > s ~i þ ~1 X

ci

ð25Þ where the following non-dimensional parameters have been introduced:

a~ i ¼

ai ~ i ¼ Gi ; X ~ i ¼ Xi ; X ~ ci ¼ ci ; s ~ i ¼ xj s i ; G am Gm xj xj L

ð26Þ

It should be noted that, considering the step length as an input independent random variable (Section 2.3) instead of the walking velocity, would have involved the use of another non-dimensional quantity (defined as the step length divided the bridge span) in ~ ci . place of X In Eq. (25), the non-dimensional random variables ~ i; X ~ ci and s ~i can be expressed as: a~ i ; G~ i ; X

a~ i ¼ 1 þ V a N1 ð0; 1Þ ~ i ¼ 1 þ V G N2 ð0; 1Þ G ~i ¼ X ~ m ð1 þ V X N3 ð0; 1ÞÞ X 1

s~i ¼ ~ Xcm Np

ð27Þ

Ek ð1Þ

k¼1

~ m and X ~ cm are given by Eq. (26) with Xi = Xm and ci = cm, where X N‘(0, 1) (‘ = 1,2,3,4) are independent normal random variables with zero mean and unit variance, and Ek(1) are exponential random variables with unit return period, while Ve (e = a, G, X, c) represents the coefficient of variation of the quantity e. ~j can be obtained The non-dimensional principal coordinate p through the convolution integral [22]:

~j ð~tÞ ¼ p

Z

~t

ð28Þ

0

where hj ð~tÞ is the non-dimensional impulse response function:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  n2j ~t

Thus, from Eqs. (24)–(29), it may be deduced that the non-dimensional maximum acceleration due to a probabilistically-modelled € ~jmax PM , can be expressed as a function of the group of pedestrians, p following quantities:

€~ €~ ~ ~ p jmax PM ¼ p jmax ðnj ; N p ; V a ; V G ; V X ; V c ; Xm ; Xcm ; uj Þ PM

~ m; X ~ cm ; u Þ ceq ¼ ceq ðnj ; Np ; V a ; V G ; V X ; V c ; X j

ð30Þ

Using this approach, the closed-form solution providing the dynamic response to a moving harmonic load (Section 3.1) could still be used to evaluate the structural dynamic response to the ith pe~ ci ), ~i þ 1=X ~i < ~t < s destrian when he is crossing the footbridge (s but it has to be replaced by the free-vibration equation when the ~ ci ). For this reason the di~i þ 1=X pedestrian leaves the bridge (~t > s rect use of the convolution integral appears preferable from a computational point of view since the dynamic response can be directly evaluated in both forced and free regimes by the MatlabÒ convolution function. Let us define the Equivalent Amplification Factor (EAF), ceq, as the ratio between the maximum acceleration caused by a group

ð32Þ

Let us introduce, now, the Equivalent Synchronization Factor (ESF), keq, as the ratio between the maximum acceleration caused by a € ~jmax PM , and the group of probabilistically-modelled pedestrians, p maximum acceleration due to uniformly-distributed resonant € ~jmax U : pedestrians, p

€~ p jmaxPM €~jmax p U

ð33Þ

Substituting k = keq into Eq. (3), the ESF allows one to analyse a uniformly-distributed load condition giving rise to the same value of the maximum dynamic response as the one provided by the realistic loading scenario considering a group of probabilistically-modelled pedestrians. The synchronization factor estimated here is called ‘equivalent’ since it does not take into account the real effects of synchronization among pedestrians when walking in groups or the phenomenon of crowd-structure interaction: actually, such effects could lead to different (generally higher) values of the synchronization factor. From Eqs. (23) and (30), it may be deduced that:

ð34Þ

From the definitions (31) and (33), it is straightforward to determine a relationship between the EAF ceq and ESF keq:

ceq ð29Þ

ð31Þ

Substituting c = ceq into Eq. (2), the EAF permits one to evaluate the maximum dynamic response to a probabilistically-modelled group of pedestrians through the analysis of an equivalent resonant pedestrian group (characterized by a mean speed cm and a mean force amplitude ceq  amGm), that is able to lead to the same value of the maximum dynamic response (in terms of acceleration) achieved by the realistic loading scenario, in which each pedestrian is free to move completely undisturbed. The amplification factor is called ‘equivalent’ since it is introduced for serviceability purposes only, and it does not take into account the real effects of synchronization among pedestrians when they are walking in low-density streams. From Eqs. (20) and (30), it may be deduced that:

~ m; X ~ cm ; u Þ keq ¼ keq ðnj ; Np ; V a ; V G ; V X ; V c ; X j

F~j ðsÞhj ð~t  sÞds

1 hj ð~tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðnj~tÞ sin 1  n2j

€~ p ceq ¼ €jmaxPM ~jmaxSP p

keq ¼

~ ci ¼ X ~ cm ð1 þ V C N4 ð0; 1ÞÞ X i X

€ ~jmax PM , and the maxiof probabilistically-modelled pedestrians, p € ~jmax : mum acceleration due to a single resonant pedestrian, p

keq

¼

R1 €~jmax Np 0 uj ðfÞdf p U ¼ €~jmax 2nj b p SP

ð35Þ

~ c ; nj and uj. Therefore, setting where the DAF b depends, in turn, on X an analytical expression for the mode shape uj, it is possible to deduce closed-form expressions for the ratio (35) as functions of Np, ~ c and nj. X To sum up, the definition of EAF and ESF enables the evaluation of realistic levels of acceleration on footbridges through simple analyses based on conventional loading schemes, like those reported in Eqs. (2) and (3), for which a closed-form solution of the dynamic response can be deduced and successfully handled (paragraphs 3.1 and 3.2). The effective sensitivity of the structure to dynamical vibrations can finally be checked comparing the obtained maximum acceleration with suitable limit values; a complete overview of limit values for accelerations according to international standards can be found in [3]. 4. General behaviour of the maximum dynamic response In this section, the general tendency of the maximum dynamic response of footbridges due to the different scenarios proposed in

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

Section 2 is analysed, as a function of the non-dimensional parameters identified in Section 3. A simplified footbridge model, schematized as a simply-supported beam, is considered, assuming that the dynamic response is governed by the first mode of vibration (the modal shape is given by Eq. (36) in Appendix A, for j = 1). However, the procedures introduced in Section 3 could be applied to any structural example having a generic vibration mode uj(x): analyses carried out considering different modal shapes, and not reported here for the sake of brevity, have shown that, from a qualitative point of view, the general behaviour of the maximum dynamic response is scarcely affected by the modal shape assumed. In confirmation of this fact, Appendix B presents some results concerning ESF for the classical case of clamped beams. Section 4.1 refers to the closed-form solution of the dynamic response to a single pedestrian, modelled as a single moving harmonic force. This case can be seen as a particular resonant pedestrian group (Section 2.2) with amplification factor c set equal to 1. Diagrams providing the DAF b as a function of the non-dimen~ cj , of the non-dimensional detuning DX ~ and of sional frequency X the damping ratio nj are provided. In Section 4.2, the dynamic response to a group of pedestrians with random arrival times, walking frequencies, walking velocities, weight, DLF and phase-lags is analysed. Diagrams plotting the equivalent amplification factor ceq and the equivalent synchronization factor keq as functions of the non-dimensional parameters Np, ~ m and X ~ cm , are provided. The coefficient of variation of the nj, X walking frequency VX has been set as 0.09 (a mean value among the Vfps available in the literature and reported in Table 1), the coefficients of variation of the pedestrians’ weight VG, of the DAF Va and of the walking speed Vc have been imposed according to the values proposed in the literature and reported in Section 2.3. It should be noted that, using the modelling framework proposed here, analyses could easily be performed adopting different values for the coefficients of variation, affecting the results from a quantitative point of view only. Moreover, thanks to the selected nondimensional random parameters (27), all the analyses in this section are performed without the need of fixing particular structural and loading characteristics (e.g. it is not necessary to set the span and the natural frequency of the bridge, or the mean values of the DLF am, of the persons’ weight Gm, of the walking circular frequency Xm and of the walking speed cm). 4.1. Single pedestrian Fig. 1 shows the DAF b for different values of the damping ratio, in case of a resonant, Fig. 1a, and non-resonant single pedestrian, Fig. 1b. Fig. 1a plots b as a function of the non-dimensional param~ cj ¼ 0:01 has been set and the variation ~ cj ; in Fig. 1b a value X eter X ~  1 is analysed. ~ ¼X with the small non-dimensional detuning DX As expected, Fig. 1 points out that the dynamic amplification is strongly reduced with the increase of the damping ratio nj, and of ~ cj and DX ~. the non-dimensional parameters X Fig. 1a, obtained from Eq. (39) in Appendix A, allows one to easily determine the maximum dynamic response of a bridge, which can be schematized as a simply-supported beam, to a single resonant harmonic moving load. Starting from the value of the span length, of the pedestrian walking speed and of the natural fre~ cj is evaluated; as a funcquency, the non-dimensional frequency X tion of the damping, the curve is chosen; then, the DAF b is estimated. The maximum dynamic displacement is easily evaluated multiplying the DAF b by the maximum static response; finally, the maximum acceleration (in dimensional form) can be obtained multiplying the maximum dynamic displacement by the square of the circular natural frequency, x2j . Furthermore, as will be shown in the next paragraph, this solution constitutes the basis of a simplified method to estimate in closed-form the vibra-

895

tion serviceability of a footbridge subjected to a realistic pedestrian loading, if a reliable value of the EAF c = ceq is determined. 4.2. Group of pedestrians The maximum response to a probabilistically-modelled group of pedestrians is a random variable; therefore, it is characterized by a probability density function. Many simulations have been carried out, considering different structural schemes and different values of the relevant non-dimensional parameters previously identified. Such simulations have shown that, in general, the probability density function of the maximum dynamic response is not very narrow (i.e. its coefficient of variation is not small). To a first approximation, trying to work on the completely safe side, the mean value plus 2 standard deviations of the maximum structural acceleration will be considered in the rest of the paper as a representative value of the footbridge dynamic response; such a choice, in case of normal distributions, corresponds to a 2% exceedance probability. As an introductory case-study, the results concerning a simply~ cm ¼ 0:005, crossed ~ m ¼ 1 and X supported footbridge, assuming X by Np = 5 pedestrians, for three values of the damping ratio (nj = 0.005, nj = 0.01, nj = 0.05), are shown, based on 1000 simulations for each of the three damping values examined. Fig. 2 and Table 2 show, respectively, the probability density function pp€~max and the statistical moments (mean value, standard deviation and coefficient of variation) of the non-dimensional maximum acceleration (nj = 0.005 Fig. 2a, nj = 0.01 Fig. 2b, nj = 0.05 Fig. 2c). It can be observed that the coefficients of variation are not very small, especially for low values of the damping ratio nj; thus, this introductory case-study confirms that the maximum response cannot be conservatively characterized by its mean value only. In order to analyse the general tendencies of the dynamic response to a probabilistically-modelled group of pedestrians, several Monte Carlo simulations have been performed on simplysupported beams, varying the four non-dimensional parameters ~ cm ,Np, nj). For each test-case, ~ m; X governing the phenomenon (X 1000 simulations have been carried out generating every time ran~ i , walking fre~ i , weights W dom values of non-dimensional DLFs a ~ ci , phases Wi and pedestrians ~ i , walking velocities X quencies X ~i ; therefore, a representative value of the maximum arrival times s € ~j max PM ¼ lp€~ þ 2rp€~j max PM , has been extracted dynamic response, p j max PM for each case, from which the EAF ceq and the ESF keq can easily be ~ cm , for fixed values of Np ~ m; X deduced and plotted as functions of X and nj. Fig. 3 shows the results concerning the case Np = 5, nj = 0.005: 3D surface plots are presented in order to identify the qualitative ~ cm . Fig. 3 ~ m and X behaviour of the dynamic response on varying X points out a regular behaviour of these surfaces. In particular, in Fig. 3a, ceq < 1 means that a single resonant pedestrian causes a dynamic response greater than that of the realistic stream of pedestrians; this generally happens for footbridges with a natural ~m frequency far enough from the mean loading frequency (i.e. X far from 1). On the contrary, ceq > 1 identifies situations where the conventional single resonant pedestrian gives rise to underes~ m ffi 1, especially for high valtimated responses; this occurs for X ~ cm . Furthermore, from Fig. 3b, since keq is always less (or ues of X much less) than 1, the importance of a correct estimation of the equivalent synchronization factor appears in all its evidence. Also assuming a conservative constant value keq (for instance, equal to 0.4 in the present example, with Np = 5, nj = 0.005) can lead to puni~ m far from 1, or structures tive loading in many situations (i.e. X ~ cm ). characterized by high values of X Figs. 4 and 5 provide contour plots of ceq and keq, respectively, for different pedestrian numbers (diagrams on the first line Np = 5, diagrams on the second line Np = 10, diagrams on the third

896

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

60

β

40

ξj=0.005

ξj=0.005

ξj=0.01

ξj=0.01

ξj=0.025

ξj=0.025

ξj=0.05

ξj=0.05

20

0

0.02

0.04

(a)

0.06 ∼ Ωcj

0.08

0.1

0.02

0.04

(b)

0.06 ∼ ΔΩ

0.08

0.1

~ cj ¼ 0:01Þ. Fig. 1. Dynamic amplification factors: (a) resonant, (b) non-resonant pedestrian ðX

0.1

pp∼..max

0.08 0.06 0.04 0.02 0 0

..80 p∼max

(a)

160

0

..80 p∼max

(b)

160

0

..80 p∼max

(c)

160

€ ~max for the non-dimensional dynamic response to a group of 5 pedestrians: (a) nj = 0.005, (b) nj = 0.01, (c) Fig. 2. Probability density function pp€~max of the maximum value p nj = 0.05.

Table 2 Statistical moments of the maximum non-dimensional dynamic response for a simply-supported beam to a group of five pedestrians. nj

lp€~max

rp€~max ðm=s2 Þ

V p€~max

0.005 0.010 0.050

41.6 35 16.5

18.3 13.5 4.7

0.44 0.39 0.28

line Np = 20) and different values of the damping ratio (diagrams in the left-hand column nj = 0.005, diagrams in the central column nj = 0.01, diagrams in the right-hand column nj = 0.05); the high value of damping ratio nj = 0.05 may be representative of a footbridge equipped with external damping devices (e.g. tuned mass dampers). An overall inspection of Figs. 4 and 5 confirms the general qualitative behaviour already observed in Fig. 3 with reference to

the particular case Np = 5, nj = 0.005; however some quantitative additional considerations can be introduced. From Fig. 4, the comparison between the diagrams corresponding to the same Np and different nj (i.e. diagrams on the same line) points out that, on increasing nj, ceq also increases involving a greater equivalent correlation in the motion of the group of pedestrians. A similar comment can be made when Np increases, nj being constant (i.e. comparing diagrams in the same column): ceq increases and becomes much higher than 1, leading to a remarkable growth with regard to the conventional [3–6] analysis carried out with a single resonant pedestrian (corresponding to c = 1). From Fig. 5, a comparison between the diagrams corresponding to the same Np and different values of nj (e.g. diagrams on the same line), shows that, for any value of the damping ratio, keq is always lower than one, pointing out that only a limited portion of the uniformly-distributed resonant load is effectively synchronized (and

γ , N =5, ξ =0.005 eq

p

λ , N =5, ξ =0.005

j

eq

p

j

0.4

3 2

0.2 1 0 1.2

(a)

0 1 ∼ Ω

m

0.8

2

4 ∼ Ω

6

8 x 10

cm

-3

(b)

1 ∼ Ω

m

0.8

2

4 ∼ Ω

6

cm

Fig. 3. 3-D behaviour of (a) EAF ceq and of (b) ESF keq (Simply-supported beam, Np = 5, nj = 0.005).

8 x 10

-3

897

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

N =5, ξ =0.005 p

1.2

0.8

1

1.15

N =5, ξ =0.05

j

p

j

2

1.2 1.4

1.2 1.4

2.2 2.4

1.6

1.6

2.6

1.8

1.8

1.05

∼

p

1

1.1 Ωm

N =5, ξ =0.01

j

2.8

2

2

1 0.95 0.9 0.85 0.8 N =10, ξ =0.005 p

1.2 1.15 1.1

1.2 1.4 1.6 1.8

2

m

1.05

∼

Ω

N =10, ξ =0.01

j

p

1.4 1.6 1.8 2 2.2 2.4

2.6

N =10, ξ =0.05

j

p

j

2.8 3.2 2.2 2.4

2.8

4

2.6

3.8 3.6

3

3.4 3.8

2.8

1 0.95 0.9 0.85 0.8 N =20, ξ =0.005 p

N =20, ξ =0.01

j

1.15 1.1

∼

Ω

m

1.05 1

N =20, ξ =0.05

j

p

j

4.2

3.8

4.6 3

3.8

3.4

3

p

1.8 2.2 2.6

1.2

5

5.4

3.4 3.8

2.6

0.95 0.9 0.85 0.8 1

2

3

4

∼

5

Ω

cm

6

7

8 9 -3 x 10

1

2

3

4

5

∼

Ω

cm

6

7

8 9 -3 x 10

1

2

3

4

∼

5

Ω

cm

6

7

8 9 -3 x 10

Fig. 4. EAF ceq for simply-supported beams.

this portion becomes smaller as damping decreases, i.e. keq becomes smaller and farther from 1 for the left-hand figures). Fixing nj and changing Np (e.g. comparing diagrams in the same column), it can be observed that keq decreases on increasing Np; thus, when the pedestrian density increases, always leaving people free to move undisturbed (no-crowd conditions), the equivalent number of synchronized people diminishes in ratio, and the ESF keq is consequently reduced. The diagrams provided in Figs. 4 and 5 allow one to estimate the footbridge maximum dynamic response to a realistic loading scenario very easily: given the structural properties, the non-dimen~ cm are evaluated; assuming a ~ m and X sional parameters X damping ratio and a number of pedestrians, the value of ceq and keq are given by diagrams; finally, the maximum dynamic response to a group of pedestrians can be obtained, without any numerical analysis, either amplifying the response to a single pedestrian € € ~j max s ; p ~j max s being the maximum static re~jmax ¼ ceq bp ~jmax ¼ ceq p (p PM SP sponse) or reducing the response due to a uniformly-distributed € ~jmax ¼ resonant load having the same pedestrian density ðp PM R1 € ~ ~ keq pjmax ¼ ½keq N p 0 uðfÞdf=ð2nj Þpj max s Þ. U Fig. 6 provides a comparison between the EAF ceq obtained for ~ m ¼ 1 (i.e. assuming that the mean walking frequency coincides X

with the natural frequency of the footbridge), nj = 0.005 and three different pedestrian numbers, Np = 5, 10, 20 (thick lines), and the pffiffiffiffiffiffi corresponding value N p proposed by Matsumoto [11] and adopted by ISO [10] and FIB [3] (thinplines, Section 2.1). It may ffiffiffiffiffiffi be deduced that the assumption ceq ¼ N p always provides a conservative estimate of EAF; however, such an approximation may be very conservative if the footbridge is characterized by low values of ~ cm . Moreover, the approximation obviously becomes even more X conservative if the natural frequency of the footbridge does not fall ~ m canwithin the range of possible mean walking frequencies (i.e. X not be assumed equal to 1). ~ m ¼ 1 (thick lines) for X Fig. 7 compares the ESF keq obtained pffiffiffiffiffiffiffiffiffiffiffiffi with the corresponding value 10:8  nj =N p proposed by Sétra [17] (thin lines; Section 2.2): in Fig. 7a Np = 20 has been fixed and different values of the damping ratio are considered, while Fig. 7b analyses the variation with Np, assuming nj = 0.005. From a direct inspection of Fig. 7, it can be deduced that the equivalent pedestrian number proposed by Sétra is conservative, especially ~ cm , for notable pedesfor bridges characterized by high values of X trian density; such a criterion can become slightly non-conserva~ cm in case of light pedestrian tive only for very low values of X traffic (Np = 5).

898

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

N =5, ξ =0.005 p

N =5, ξ =0.01

j

p

N =5, ξ =0.05

j

p

0.15 0.3

m

0.7

0.65

0.6

0.8

0.4

0.25

∼ Ω

0.75

0.35 0.45

0.3

1

0.

0.25

0.2

1.1

j

0.85

0.9 0.8

N =10, ξ =0.005 p

N =10, ξ =0.01

j

p

1.2 0.12

1.1 1.05 m

p

0.1

1.15

∼ Ω

N =10, ξ =0.05

j

0.

0.2 0.22

1

0.18 0.16

0.14 0.32

0.24 0.26

0.28

0.3

0.2 0.22

j

0.1 0.16 0.18

0.4 0.45 0.5 0.55 0.6

0.95 0.9 0.85 0.8 N =20, ξ =0.005 p

N =20, ξ =0.01

j

p

1.2

0.12

0.08

1.1

0.1

1.05 m

p

0.06

1.15

∼ Ω

N =20, ξ =0.05

j

0.14 0.42 0.4 0.44

0.2

0.14

1

0.16

0.18

0.12

0.1

j

0.28 0.3 0.32 0.34 0.38 0.36

0.95 0.9 0.85 0.8 1

2

3

4

5

∼Ω

6

7

cm

8 9 -3 x 10

1

2

3

4

5

∼

Ω

cm

6

7

8 9 -3 x 10

1

2

3

4

5

∼ Ω

cm

6

7

8 9 -3 x 10

Fig. 5. ESF keq for simply-supported beams.

per, are presented. Section 5.1 considers an ideal simply-supported beam, Section 5.2 studies the dynamic response of a real cablestayed bridge. Their response is analysed with the procedures proposed in Section 4. In particular, the first example evaluates the maximum dynamic response to a probabilistically-modelled group of pedestrians through the application of the EAF, starting from the response to a single resonant pedestrian; in the second example the maximum dynamic response to a realistic scenario is dealt with using the ESF for a real bridge characterized by a generic modal shape. The average force amplitude F0 = amGm = 0.4  700 N = 280 N and the average walking velocity c = cm = 1.37 m/s (Section 2.3) are considered in the following.

5

Np=5 Np=10

4

γeq

Np=20 3

2

1

0.002

0.004

0.006 ∼ Ωcm

0.008

Fig. 6. EAF ceq for simply-supported beams: comparison with Matsumoto’s ~ m ¼ 1, nj = 0.005). criterion [11] (X

5. Applications In this Section, two applications showing the dynamic response of structural examples to the loading scenarios analysed in this pa-

5.1. Ideal simply-supported beam In this section, an ideal simply-supported bridge with span length L = 40 m, mass per unit length m = 1000 kg/m, natural frequency n1 = 2 Hz, damping ratio nj = 0.01, is considered. Fig. 8 plots the time-histories of the dynamic response in the footbridge mid-span to a single pedestrian (Eq. (2) with c = 1). Fig. 8a shows the response to a resonant pedestrian (X = 2pn1) (Eq. (6) for x = L/2, with pj(t) given by Eqs. (10), (17), and (38) in

899

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

1 0.8

Np=5

ξj=0.01

Np=10

ξj=0.05

Np=20

λeq

0.6

ξj=0.005

0.4 0.2 0

0.002

0.004

(a)

0.006 ∼ Ωcm

0.002

0.008

0.004

(b)

0.006 ∼ Ωcm

0.008

q(L/2,t) (m)

(b)

(c)

.. q (L/2,t) (m/s2)

-0.004

0.8

0.004

-0.004

-0.8 0.8

0.004

-0.004 0

Table 3 Maximum dynamic response of a simply-supported beam to a single pedestrian. ~ DX

b

qmax SP ðmÞ

€max SP ðm=s2 Þ q

0 0.01 0.1

39 33 5

0.0035 0.0032 0.00048

0.55 0.50 0.092

-0.8

.. q (L/2,t) (m/s2)

q(L/2,t) (m)

(a)

0.8

0.004

.. q (L/2,t) (m/s2)

q(L/2,t) (m)

~ m ¼ 1): (a) Np = 20; (b) nj = 0.005. Fig. 7. ESF keq for simply-supported beams: comparison with Sétra’s criterion [17] (X

10

t (s)

20

30

-0.8

Fig. 8. Dynamic response of a simply-supported beam to a single pedestrian: (a) ~ ¼ 0, (b) DX ~ ¼ 0:01, (c) DX ~ ¼ 0:1. DX

Appendix A); Fig. 8b and c shows the response to a non-resonant pedestrian (Eq. (6) for x = L/2, with pj(t) given by Eq. (37) in Appen~ ¼ 0:01 (Fig. 8b) and DX ~ ¼ 0:1 (Fig. 8c), dix A), considering DX respectively. As expected, the dynamic response due to a non-resonant pedestrian rapidly decreases as the detuning between the load and the structural frequency increases. The maximum dynamic displacement can be directly obtained multiplying the maximum static response (qmax s = p1max s = F0/M1x12 = 8.865 5  10 m) by the DAF b, obtained from Fig. 1; then, the maximum acceleration is simply obtained multiplying the maximum displacement with the square of the circular natural frequency x2j . In the resonant case, the DAF is obtained from Fig. 1a, setting ~ cj ¼ pc=xj L ¼ 0:009; in the two non-resonant cases, the DAF X can approximately be obtained from Fig. 1b, setting ~ ¼ 0:01 and DX ~ ¼ 0:1, respectively. Table 3 summarizes the reDX sults obtained using this approach: a comparison between the maxima of the time-histories in Fig. 8 and the numerical values in Table 2 shows that the maximum dynamic response is perfectly estimated through the analytical procedure. Fig. 9 plots a sample function of the dynamic response to a group of Np = 5 pedestrians. It was obtained performing a Monte

Carlo simulation of the non-dimensional random variables according to the probabilistic characterization of Eq. (27). Fig. 9a plots the time-histories concerning the contribution to the modal force of each pedestrian, together with the overall modal force; Fig. 9b plots the contributions to the principal coordinate due to the different pedestrians, together with the principal coordinate pj(t), which coincides with the mid-span displacement q(L/2, t). From Fig. 9a, the randomness of pedestrian arrival times can be observed, since they are not equally-spaced; from Fig. 9b it can be noted that, depending on the action randomness related to the generic ith pedestrian, the contribution of the different pedestrians to the response varies significantly. The time-histories in Fig. 9 are representative of the results of only one simulation and do not furnish any valuable information about the serviceability of the footbridge. Without performing a burdensome Monte Carlo simulation, the maximum value of the dynamic response to Np = 5 pedestrians can easily be obtained from the combined use of the results concerning the single reso€max SP ¼ ð2pnj Þ2  qmax s  b ¼ 0:55m=s2 , Table 3), nant pedestrian (q together with the diagrams in Fig. 4. Starting from the values of ~ c ð¼ 0:0027Þ, the ~ m ð¼ 1Þ and X the non-dimensional frequencies X damping ratio nj = 0.01 and Np = 5, ceq = 1.82 is estimated from the top-centre picture in Fig. 4 (ceq represents the equivalent synchronized portion of Np = 5 pedestrians crossing the bridge). The maximum acceleration that can be expected from the passage of €max SP ¼ 1 m=s2 . €max PM ¼ ceq  q a group of 5 pedestrians is thus q 5.2. Real cable-stayed bridge Let us consider a real cable-stayed footbridge, which was recently built in Italy. The bridge, characterized by a span length L = 59.37 m, has been modelled by a finite-element code (Fig. 10a), and a modal analysis has been performed (Fig. 10b). The first mode of vibration (n1 = 1.3931 Hz) is flexural in the lateral direction, the second (n2 = 1.7354 Hz), third (n3 = 2.9347 Hz) and fourth (n4 = 4.2616 Hz) modes of vibration are flexural in the vertical direction. These four modes of vibration are all susceptible to pedestrian excitation. In particular, the second mode of vibration is characterized by a natural frequency close to the mean pedestrian walking frequency in the vertical direction; thus, in first

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

pj 5

Fj5

pj4

Fj4

pj3

Fj3

pj 2

Fj2

pj1

Fj1

900

(a)

0

20

t (s)

40

1.2 .. q (m/s2)

Fj

q (m)

0.008 0

-0.008 0

60

20

(b)

40

t (s)

60

-1.2

Fig. 9. Sample functions of the dynamic response of a simply-supported beam to a group of Np = 5 pedestrians: (a) modal forces due to single pedestrians and total modal force, (b) contributions to response of single pedestrians and total response.

Fig. 10. Real cable-stayed bridge: (a) finite-element model, (b) modal shapes.

.. q(xm,t) (m/s2)

q(xm,t) (m)

0.4 0.002

-0.002 0

10

20 t (s)

30

-0.4

40

Fig. 11. Dynamic response of a cable-stayed bridge to a single resonant pedestrian.

.. q(xm,t) (m/s2)

approximation, analyses have been carried out assuming that only the second mode of vibration (i.e. the first one in the vertical direction) contributes to the dynamic response. On the safe side, since

the natural frequency n2 is close to the range of possible mean ~ m ¼ 1 has been set; furtherwalking frequencies fpm (Table 1), X more, a modal damping nj = 0.005 is considered. Fig. 11 shows the vertical dynamic response to a single pedestrian (Np = 1, i.e. Eq. (2) with c = 1) crossing the bridge in resonance with the second mode of vibration. The mid-span acceleration (thin solid line), obtained from Eq. (17), is plotted together with the amplitude (18) (thick solid lines): the slow-varying amplitude represents a perfect envelope of the solution. The maximum dynamic response arising from a single resonant pedestrian is €max SP ¼ 0:39 m=s2 . q Monte Carlo simulations have been performed in order to estimate the dynamic response of the bridge to groups of pedestrians

0.8

-0.8 0

20

40 t (s)

60

80

0

20

40 t (s)

60

80

0

20

40 t (s)

60

80

Fig. 12. Sample functions of the dynamic response of a cable-stayed bridge to a group of pedestrians: (a) Np = 5, (b) Np = 10, (c) Np = 20.

901

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

3

..

..

.. pQmax

2 1 0 0

0.8 .. qmax PM (m/s2)

(a)

1.6

0

0.8 .. qmax PM (m/s2)

(b)

1.6

0

0.8 .. qmax PM (m/s2)

(c)

1.6

€max PM for a cable-stayed bridge to a group of pedestrians: (a) Np = 5, (b) Np = 10, (c) Np = 20. Fig. 13. Probability density function pQ€ max of the maximum dynamic response q

(Np = 5, 10, 20). Fig. 12 plots sample functions of the dynamic response of the footbridge to Np = 5 (Fig. 12a), Np = 10 (Fig. 12b), Np = 20 (Fig. 12c) pedestrians, at the position xm where the second modal shape, which is not perfectly symmetric, presents a maximum. The dynamic response obviously increases on increasing the average number of pedestrians crossing the footbridge. However, quantitative information concerning the actual sensitivity of this footbridge to pedestrian-induced vibrations cannot be extracted directly from Fig. 12, representing only the result of one

Table 4 Maximum dynamic response of a real cable-stayed bridge to different loading scenarios. Np

€max PM ðm=s2 Þ q

€max U ðm=s2 Þ q

keq

€max ðm=s2 Þ q

5 10 20

0.61 0.82 1.1

1.60 3.20 6.40

0.36 0.245 0.165

0.58 0.78 1.06

N =5, ξ =0.005 p

N =5, ξ =0.01

j

p

1.2

0.3

0.45

0.5

0.8

0.75 0.7

0.65

0.9

0.4

0.35

1

0.85

0.35

0.25

1.05

j

0.3

0.2

1.1 m

p

0.25

0.15

1.15

∼ Ω

N =5, ξ =0.05

j

1

0.95

0.55

0.95 0.9 0.85 0.8 N =10, ξ =0.005 p

N =10, ξ =0.01

j

p

N =10, ξ =0.05

j

0.1 0.12

1.15 1.1

∼ Ω

m

1.05 1

0.24 0.26

0.18 0.22 0.2

p

0.1

1.2

j

0.45

0.2 0.25

0.14 0.16

0.6

0.3

0.55

0.5

0.65

0.35

0.7

0.95 0.9 0.85 0.8 N =20, ξ =0.005 p

N =20, ξ =0.01

j

0.1

1.1

0.12

m

1.05

0.16

0.2

0.14

0.24

1

N =20, ξ =0.05

j

0.08

1.15

∼ Ω

p

0.06

1.2

p

0.1

0.14 0.16 0.18

j

0.12 0.46 0.5 0.48

0.22

0.44

0.42

0.4

0.34 0.38 0.36

0. 0.32

0.95 0.9 0.85 0.8 1

2

3

4

5

∼

Ω

cm

6

7

8 9 -3 x 10

1

2

3

4

5

∼ Ω

6

7

cm

Fig. 14. ESF keq for clamped beams.

8 9 -3 x 10

1

2

3

4

5

∼Ω

cm

6

7

8 9 -3 x 10

902

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

simulation. A statistical analysis of several simulations is needed instead: Fig. 13 plots the probability density function pQ€ max of the maximum dynamic response of the footbridge to Np = 5 (Fig. 13a), 10 (Fig. 13b), 20 (Fig. 13c) pedestrians obtained from 1000 simulations. A comparison between the maximum values of accelerations in Fig. 12 and the corresponding probability density functions in Fig. 13 points out the incompleteness of results deriving from a single simulation: the maximum acceleration of the sample time history in Fig. 12a (Np = 5) is rather below the average, whereas the maximum values of the sample functions in Fig. 12b and c (Np = 10,20) are close or above the corresponding average values. The results of this analysis are finally summarized in Table 4, listing the maximum dynamic response to a probabilistically€max PM , and to a uniformly-distribmodelled group of pedestrians q €max U (Eq. (3) with k = 1). The increasing values uted resonant load q €max PM on increasing Np in the first column of Table 4 point out of q the amplification c, compared to the single resonant-pedestrian re€max SP ¼ 0:39 m=s2 ), because of the presence of a group of sponse (q pedestrians walking in a realistic way. However, the use of an equivalent synchronization factor keq allows one to avoid a burdensome Monte Carlo simulation: the maximum dynamic response can easily be determined from Eq. (22), using a suitable value of k = keq deduced from the diagrams. Since the modal shape of the footbridge can be assumed similar to that of a clamped beam (even though slightly non-symmetric), values of ESF keq for clamped beams (Fig. 14, Appendix B) have been estimated in correspondence ~ c ð¼ 0:0021Þ. ~ m ð¼ 1Þ and X of the non-dimensional frequencies X Comparing the results provided in Table 4, it can be concluded that the equivalent synchronization factor keq provides maximum dy€max almost coincident with those obtained from namic responses q €max PM , confirming the effective reliability Monte Carlo simulations q of the methods proposed in this paper.

6. Conclusions and prospects In this paper, the dynamic response of footbridges to vertical pedestrian-induced vibrations has been analysed through three simplified load models: resonant pedestrian groups (including the classical condition of single pedestrian as a particular case), uniformly-distributed resonant pedestrian streams and probabilistically-modelled groups of pedestrians. The most relevant outcomes of the present study, compared to the current state-of-art of the literature, can be summarized as follows:  the characterization of the essential non-dimensional parameters governing the dynamic response to the three loading models, which has permitted a complete probabilistic description of these variables based on the experimental data currently available;  the sound definition of the Equivalent Amplification Factor, ceq, and of the Equivalent Synchronization Factor, keq, as functions of the identified non-dimensional parameters. Such definitions are based on the comparison between the maximum dynamic response to realistic loading conditions and, respectively, the conventional analysis concerning a single resonant pedestrian, or the response induced by a uniformly-distributed resonant pedestrian stream;  the introduction of simplified procedures to evaluate realistic levels of peak accelerations and, therefore, to assess the vibration serviceability of footbridges in closed-form, without any step-by-step analysis or Monte Carlo simulations. Using the EAF, this appears possible thanks to the closed-form solution of the footbridge dynamic response to a single resonant moving load, proposed by the authors [7,8]. Concerning the ESF, the

closed-form solution of the problem is even simpler, requiring the easy evaluation of the response to a uniformly-distributed resonant load;  a first comparison of the proposed procedures with analogous methods advised by codes and guidelines in terms of amplification and synchronization factors, has pointed out that such suggested loading conditions are generally conservative (and, frequently, even too conservative) and can become (slightly) non conservative in particular cases. The modelling framework proposed here is easily open to new experimental data and, in fact, the identification of the non-dimensional parameters highlights some gaps in experimental data available in the literature for the probabilistic modelling of pedestrianinduced forces. First of all, more experimental data are necessary concerning the probabilistic distribution of pedestrian speeds, and their real correlation with walking frequencies and step lengths. Furthermore, the introduction of expressions for the DLF as a function of the ratio between walking and natural frequencies (i.e. as a function of a non-dimensional frequency), rather than depending on the walking frequency only, would be desirable; this lack in current experimental data is due to the fact that experiments are often more easily performed on rigid treadmills. Another point deserving further investigation is the study of the probabilistic density function of the maximum dynamic response, treated here in an approximate way only. A final prospect concerns the experimental check of the reliability of the proposed simplified procedures through a comparison of the estimated dynamic responses with levels of acceleration measured during the free movement of groups of pedestrians on real footbridges. Such experimental tests would also allow one to verify whether the motion of a significant pedestrian stream can modify the level of structural damping, which remains one of main parameters governing footbridge dynamic behaviour. Acknowledgements This work has been partially supported by the Italian Ministry of University and Research (MIUR) through a national PRIN co-financed programme, and by the University of Genoa. Appendix A. Closed-form solution for simply-supported beams Let us consider a bridge that can be modelled as a simply-supported beam. Its jth modal shape is given by the following expression: 

uj ðxÞ ¼ sin jp

x L

ð36Þ

The dynamic response of a simply-supported beam, crossed by a quasi-resonant harmonic load, is determined substituting Eq. (36) into Eq. (13); in such a case, the integrals in Eq. (13) can be solved in closed-form [8] and the jth principal component is given by:

2  3 9 8 ~ DX ~ ~ þDX ~ X X cj cj > > ~ > > cosð tÞþ þ  > ~ Þ2 þn2 ~ Þ2 þn2 ~ DX ~ þDX 6 7 > ðX ðX > > cj cj j j > > 6 7 ~ > >  þ tÞ expðn > > j 4 5 > > > > n n j j > > ~ > >  ðX~ DX~ Þ2 þn2 þ ðX~ þDX~ Þ2 þn2 sinðtÞ > > > > cj cj j j > > > > 2 3 > > ~ Þ cosððDX ~ X ~ X ~ DX ~ Þ~tÞþn sinððDX ~ Þ~tÞ > > ð X cj cj j cj = < þ  c ~ Þ2 þn2 ~ DX X ð cj 6 7 ~j ð~tÞ ¼ j p ~ cosð tÞþ  4 5 ~ ~ ~ ~ ~ ~ > > 4> ðX þDXÞ cosððXcj þDXÞ~tÞþnj sinððXcj þDXÞ~tÞ > > > þ cj > > ~ Þ2 þn2 ~ þDX > > ðX cj > > j > > 2 3 > > ~ ~ ~ ~ ~ ~ ~ ~ > > ðXcj DXÞ sinððDXXcj ÞtÞþnj cosððDXXcj ÞtÞ > > > > þ 2 2 > > ~ ~ ðXcj DXÞ þnj > > 6 7 > > ~ > > þ sinð tÞ 4 5 > > ~ þDX ~ þDX ~ þDX ~ Þ sinððX ~ Þ~tÞþn cosððX ~ Þ~tÞ ðX > > cj j cj ; :  cj ~ 2 2 ~ ðXcj þDXÞ þnj

ð37Þ ~ cj ¼ jpX ~ c. where X

G. Piccardo, F. Tubino / Computers and Structures 87 (2009) 890–903

Considering the simply-supported beam crossed by a perfectlyresonant sinusoidal load, the response can be obtained imposing ~ ¼ 0 in Eq. (37); thus, it can be expressed as a harmonic with DX slow-varying amplitude, Eq. (17), the amplitude Að~tÞ being given by [7,8]:

Að~tÞ ¼

h

c ~ 2 þ n2 Þ 2ðX j cj

i ~ cj expðnj~tÞ þ X ~ cj cosðX ~ cj~tÞ  nj sinðX ~ cj~tÞ X ð38Þ

Furthermore, the dynamic amplification factor b can be expressed ~ cj and nj: as a function of two non-dimensional parameters, X

"

~ ~ cj exp nj t max X b¼ 2 2 ~ cj ~ X 2ðXcj þ nj Þ ! ~tmax ¼ arctan  nj þ p ~ cj X 1

!

#

~ cj cosð~tmax Þ  nj sinð~tmax Þ þX

ð39Þ Appendix B. ESF for clamped beams Let us consider a bridge that can be modelled as a clamped beam. Its first modal shape can be approximated by the following expression:

uj ðxÞ ¼

 xi 1h 1  cos 2p 2 L

ð40Þ

Actually, the exact expression for the first modal shape of clamped beams is more complicated, involving hyperbolic trigonometric functions too. However, the influence of such minor approximation on the results of the present study is negligible. Following the same procedure adopted for simply-supported beams (Section 4), Fig. 14 shows the ESF keq for clamped beams. From an inspection of Fig. 14, it may be deduced that the Equivalent Synchronization Factor for clamped beams is similar to the one obtained for simply-supported beams (Fig. 5), in terms of both qualitative dependence on the non-dimensional parameters and numerical values (which are slightly larger, in any case less than 20%). Analogous graphs can be obtained for different modal shapes of technical interest (e.g. for skew-symmetric modes having a central node).

903

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