Shock and Vibration

Volume 2018, Article ID 1917629, 26 pages

https://doi.org/10.1155/2018/1917629

## Deterministic and Probabilistic Serviceability Assessment of Footbridge Vibrations due to a Single Walker Crossing

^{1}College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China^{2}DICDEA, Università della Campania Luigi Vanvitelli, Via Roma 9, 81031 Aversa, Italy

Correspondence should be addressed to Alberto Maria Avossa; ti.ainapmacinu@assova.airamotrebla

Received 9 July 2017; Revised 5 September 2017; Accepted 12 November 2017; Published 17 January 2018

Academic Editor: Hugo Rodrigues

Copyright © 2018 Cristoforo Demartino et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a numerical study on the deterministic and probabilistic serviceability assessment of footbridge vibrations due to a single walker crossing. The dynamic response of the footbridge is analyzed by means of modal analysis, considering only the first lateral and vertical modes. Single span footbridges with uniform mass distribution are considered, with different values of the span length, natural frequencies, mass, and structural damping and with different support conditions. The load induced by a single walker crossing the footbridge is modeled as a moving sinusoidal force either in the lateral or in the vertical direction. The variability of the characteristics of the load induced by walkers is modeled using probability distributions taken from the literature defining a Standard Population of walkers. Deterministic and probabilistic approaches were adopted to assess the peak response. Based on the results of the simulations, deterministic and probabilistic vibration serviceability assessment methods are proposed, not requiring numerical analyses. Finally, an example of the application of the proposed method to a truss steel footbridge is presented. The results highlight the advantages of the probabilistic procedure in terms of reliability quantification.

#### 1. Introduction

Vibrations of footbridges due to human loading have recently received much attention, due to the increasing number of vibrations incidents occurring worldwide. The main cause of these large vibrations is the low stiffness and damping of recently built footbridges. As a matter of fact, if only static dead and live loads are considered in the design process, footbridges can prove unable to meet serviceability requirements against vibrations.

In particular, the significant amount of research produced in the last one and a half decades has been triggered by the two vibration incidents of the Paris Passerelle Solferino on December , , and of the London Millennium Bridge on July , , that played a similar role in the public opinion and in the scientific community to that played by the collapse of the Tacoma Narrows bridge in November 1940 [1]. Nevertheless, crowd-related failures of bridges have occurred over the centuries with much more devastating effects, the first of which being documented is probably that of the bridge over river Ouse in England, in [2]. In spite of the long series of failures, to the authors’ knowledge the first paper to have appeared dealing with the effects of human movements on structural loading is that of Tilden [3]. This is a pioneering work where many aspects of the human loading of structures seem to have already been recognized, though not quantified.

The vibrations are generated by the “quasi” harmonic load induced by walkers and joggers. If the central frequency of the load and a natural vibration frequency of the footbridge are similar, resonant vibrations can occur. The latter is a rather common condition [4]. However, this apparently simple mechanism is in fact not easy to quantify. First of all, walkers do not induce a perfectly periodic load due to the intrasubject variability gait, and this load differs from one subject to another (intersubject variability). Finally, two forms of feedback can take place: (i) interwalker interaction and (ii) walker-structure interaction [5]. The first refers to gait modifications due to the presence of neighboring walkers, whereas the second is the adjustment of gait as an effect of the floor vibration.

Standards and guidelines have been developed to help the designer in the evaluation of the vibration serviceability based on simplified loading models, simulating different possible scenarios [5, 6]. In broad terms, the design of footbridges against pedestrian-induced vibrations requires the knowledge of [7] (i) the characteristics of the pedestrian action, (ii) a response evaluation method, and (iii) a comfort criterion. The interested reader is referred to [5, 6, 8, 9]. Design guidelines outline procedures for vibration serviceability checks, but it is noticeable that most of these assume that the action is deterministic [10], yet this is stochastic, and it would be reasonable to incorporate its variability in the loading models. However, it should be mentioned that other loads on bridges should be considered in the design procedures such as seismic, wind, and impact loads (e.g., [11–16]).

Different authors have tried to characterize the randomness of the pedestrian action in either time or frequency domain, considering both intrasubject and intersubject variability. Brownjohn et al. [17] for the vertical direction and Pizzimenti and Ricciardelli [18] and Ricciardelli and Pizzimenti [19] for the lateral direction gave Power Spectral Density Functions (PSDFs) of the load induced by a walker, for use in the evaluation of the stationary response to a stream of walkers, including intrasubject and intersubject variability of gait.

Subsequently, Butz [20] presented a spectral approach for the evaluation of the peak acceleration induced by unrestricted pedestrian traffic. The PSDFs of the modal force were approximated through a Gaussian function, fitting data coming from Monte Carlo simulations. These were carried out for different bridge geometries and for four different pedestrian densities. Step frequency, pedestrian mass, force amplitude, and pedestrian arrival time were randomly selected from given probability distributions. The PSDF of the acceleration was evaluated from the PSDF of the modal force, and the 95th fractile peak modal acceleration was derived as the product of the RMS acceleration and a peak factor. The latter was evaluated to be around . All the coefficients required for the application of the procedure were expressed in a parametric form. This model has then been incorporated into HIVOSS guidelines [21].

Živanović et al. [22] presented a multiharmonic force model for calculation of the multimode structural response to a crossing, accounting for inter- and intrasubject variability in the walking force. The model is again based on Monte Carlo simulations, with pedestrian characteristics also selected from given distributions. The intrasubject variability was accounted for describing the force in the frequency domain and then converting it to the time domain. No parametric form of the peak response as a function of the different parameters and no procedure for the serviceability assessment are given.

Ingólfsson et al. [23] proposed a Response Spectrum approach inspired by earthquake engineering. Through Monte Carlo simulations, they evaluated a reference vertical acceleration to the action of a flow of with probabilistically modeled characteristics, to which empirical correction factors are applied to account for return period, modal mass, mean arrival rate, structural damping, footbridge span, and mode shape. A total of windows, 300 s long, were used to establish the peak acceleration Generalized Extreme Value (GEV) distribution parameters. Two reference populations were considered, with step frequencies of 1.8 and 2.0 Hz, with STD of 0.1 Hz and Poisson distributed arrivals. The input force was modeled as a harmonic load and the intersubject variability was considered varying the characteristics of each pedestrian according to given distributions. All the parameters contained in the procedure are given in parametric form, and the GEV distribution of the peak acceleration is found to tend to a Gumbel distribution (i.e., the shape parameter tends to 0).

Piccardo and Tubino [24] studied the vertical vibration serviceability of footbridges, based on a probabilistic characterization of pedestrian-induced forces taking into account intersubject variability and considering only one mode of vibration. Only the 95th fractile peak modal acceleration was derived and expressed in two nondimensional forms: (a) the Equivalent Amplification Factor, that is, the ratio between the maximum dynamic response to a realistic loading scenario and the maximum dynamic response to a single resonant pedestrian; (b) the Equivalent Synchronization Factor, that is, the ratio between the maximum dynamic response to a realistic loading scenario and the maximum dynamic response to uniformly distributed resonant pedestrians. Comparison of their procedure with similar methods contained in standards and design guidelines has pointed out that the latter are generally conservative (often largely conservative) and can become only slightly nonconservative in particular cases. They concluded that further investigations on the evaluation of the PDF of the maximum dynamic response are required.

Živanović et al. [25] reviewed different time-domain design procedures for vibration serviceability assessment of footbridges exposed to streams of pedestrians and evaluated their performance in predicting the vertical vibration response of two existing footbridges. They compared the procedures contained in Eurocode 5 [26], ISO 10137 [27], Sétra [28], BSI [29], Brownjohn et al. [30], Butz [20], Ingólfsson et al. [23], and Živanović et al. [22]. They found some discrepancies between the predicted and measured vibration levels and discussed their potential causes, among which are interwalker and walker-structure interaction.

Pedersen and Frier [10] evaluated the effect of the probabilistic modeling of the parameters describing walking loads: step frequency, stride length, Dynamic Load Factor (DLF), and walker weight. A literature review revealed a variety of probability distributions through which the walking parameters can be modeled, and these were used for exploring the sensitivity of the 95th fractile of midspan acceleration. They observed that the step frequency distribution can have a strong influence, whereas the DLF, the walker weight, and stride length have a much lower influence. No interactions were considered in this study.

Ingólfsson and Georgakis [31] presented a probabilistic lateral load model in which the forces are given as the sum of an external component and a frequency and amplitude-dependent self-excited component; the latter is quantified through equivalent pedestrian damping and mass coefficients measured from experiments. They found that the peak response of a footbridge to a pedestrian flow is very sensitive to the selection of the pacing rate distribution.

Piccardo and Tubino [32] introduced an equivalent spectral model for the analysis of the dynamic response of footbridges to unrestricted pedestrian traffic (i.e., no interwalker interaction) using a complete probabilistic representation of pedestrians. They provided simple closed-form expressions for the evaluation of the maximum dynamic response for use in vibration serviceability analyses, similarly to classical procedures adopted in wind engineering. These expressions are based on the definition of the peak factor found by Davenport [33].

Recently, Ricciardelli and Demartino [5] compared background hypotheses, fields of applicability, and results obtained through a number of different loading and response evaluation models. In particular, they compared single walker models, multiple walkers models, interaction models (interwalker and walker-structure), and instability models, together with current design procedures incorporated into standards and guidelines. Avossa et al. [34] applied the design procedures to various steel footbridges highlighting the large differences that they bring in the results. They concluded that a critical revision of design procedures is needed as these, even though inspired by the same principles and applying the same rules, show different results; this should also be done through validation with the available full-scale data. Finally, Avossa et al. [35] evaluated through Monte Carlo simulations the probability distribution of the footbridge peak acceleration to single and multiple crossing walkers for two specific footbridge configurations.

In combination with a probabilistic definition of the load, criteria for the probabilistic definition of the structural capacity must be set. For instance, Eurocode 0 [36] requires that a structure is designed to have adequate (i) resistance, (ii) serviceability, and (iii) durability. In particular, the limit state of vibrations causing discomfort to people and/or limiting the functional effectiveness of the structure must be considered. Moreover, it establishes that when the structure is prone to significant acceleration, dynamic analyses must be performed. Similarly, ISO 2394 [37] specifies general principles for the reliability assessment of structures subjected to known or foreseeable types of actions, providing more or less similar requirements for safety, serviceability, and durability. In Annex E of ISO 2394 [37], principles of reliability-based design are given specifying the requirements in terms of probability of failure for different limit states.

ISO 10137 [27] contains structural acceleration limits for different situations. ISO 10137 recognizes the vibration source, path, and receiver as three key elements which require being identified when dealing with vibration serviceability. In the context of walking-induced vibrations in footbridges, the walkers are the vibration source, the footbridge is the path, and the walkers are again the receivers. According to ISO 10137, an analysis of the response requires a calculation model that incorporates the characteristics of the source and of the transmission path, which must be solved for the vibration response of the receiver; in doing so, the dynamic action of one or more walkers can be described as force time histories. This action varies in time and space as the walkers move on the footbridge. It is recommended that the following scenarios are considered: (i) one person walking across the bridge, (ii) an average pedestrian flow (group size of 8 to 15 walkers), (iii) streams of walkers (significantly more than 15 walkers), and (iv) occasional festive choreographic events (when relevant).

However, although many authors have derived probabilistic models to describe the vibration response induced by pedestrian loads, a fully probabilistic procedure for the serviceability assessment of footbridge vibrations due to a single walker crossing and a comparison with deterministic approaches is not yet available. In particular, the studies reviewed above do not allow for variation of the reliability levels, as they take as demand parameter the 95th fractile of the peak acceleration response. It is important to notice that although the research interest is nowadays mainly oriented towards the multipedestrian case, the need for analyzing the single pedestrian case stems from at least three different reasons: (i) this case can induce the largest acceleration, especially for short low-damped footbridges, (ii) many standards and codes of practices refer to this load scenario, and (iii) vibration assessment procedures for multipedestrian loading are often derived from the single pedestrian case.

This study presents criteria for the deterministic and probabilistic vibration serviceability assessment of footbridges to the crossing of one walker. In Section 2, the load induced by a single walker is modeled as a moving harmonic force having lateral and vertical components, whose characteristics derive from a Standard Population (SP) of walkers. The latter is defined based on data available in the literature, concerning the probabilistic distribution of walker characteristics and gait parameters. In Section 3, the dynamic characteristics of a single span footbridge (span length, natural frequencies, mass, structural damping, and support conditions) are defined and a modal dynamic model is presented. In Section 4, numerical analyses of the transient response to a moving harmonic load are presented, through which the peak response is evaluated in both a deterministic and probabilistic way. In Section 5, closed-form deterministic and probabilistic vibration serviceability methods are proposed, whose applications do not require numerical analyses. These incorporate the acceleration limits of ISO 10137 [27] and the required reliability level of ISO 2394 [37], leading to a method which also complies with Eurocode [36]. As an example, in Section 6, the deterministic and probabilistic methods are applied to a prototype truss steel footbridge. Finally, some conclusions and prospects are drawn (Section 7).

#### 2. Single Walker Behavior

Ground Reaction Forces (GRFs) are defined as the forces induced on the ground by walkers. The measurement of GRFs has advanced considerably over the recent decades. It first became a useful clinical tool starting from the pioneering work of Beely [38] and Elftman [39]. Nowadays, observational gait analysis is regularly performed by physical therapists to determine treatment goals and is used as an evaluation tool during rehabilitation [40]. In Medical Sciences, the main goal is the identification of the kinematic characteristics of a subject. Differently, in civil engineering it is of interest to characterize GRFs, with the final aim of evaluating the structural response for comfort assessment [9, 41].

GRFs are characterized by different magnitudes and frequency content in the vertical, lateral, and longitudinal directions. Based on the existing knowledge of GRFs, several loading models have been developed for footbridges, some of which consider the crossing of a single pedestrian [5]. A common approach is that of periodic loading, assuming that a walker generates identical footfalls with constant frequency neglecting intrasubject variability. In this case, the dynamic part of the GRF is expanded in Fourier series:where the subscript indicates the vertical or lateral direction (the longitudinal component is neglected) and where is the weight of the walker, is the th Dynamic Load Factor (DLF), that is, the th harmonic load amplitude normalized by the body weight, when and when , being the step frequency, and is the phase lag of the th harmonic. Moreover, in the following, only the first harmonic will be retained, and accordingly subscript will be omitted. This representation of the load is consistent with different standards such as UK Annex to EC1 [29] and ISO 10137 [27].

When the walker crosses a footbridge of span , the modal load associated with the first bending mode iswhere is first mode shape (Section 3), is the Dirac Function, defines the position of the walker on the bridge, is the Heaviside function, and is the crossing time, being the walking speed, the span length, and the step length.

##### 2.1. Standard Population of Walkers

The definition of a Standard Population (SP) of walkers is needed to characterize intersubject variability probabilistically. This is not trivial due to the large scatter of the data available in the literature, and one must be aware of the fact that changing the population will lead to a different vibration response [10].

The parameters governing the excitation generated by a walker are (i) the walking speed , (ii) the step frequency , (iii) the Dynamic Load Factors and , (iv) the weight of the walker , and (v) the phase angles and . The data mainly come from the Biomechanics and Transportation fields, although recent results have also been published in the area of structural engineering. The SP defined in this section is based on research developed in European countries.

Humans can walk up to [42], but the speed of roughly represents a natural transition from walking to running [43, 44]. In spite of this, the walking speed is usually considered as normally distributed, and a large scatter in the mean value is found in the literature. This is due to physiological and psychological factors, such as biometric characteristics of the walker (body weight, height, age, and gender), cultural and racial differences, travel purpose, and type of walking facility [45]. In the following, the walker speed is assumed as [46]

Equation (3) is truncated at as smaller values lead to negative STDs in (4).

It is agreed that walking occurs at an average step frequency of approximately (e.g., [28]). Biomechanics studies established that walkers tend to adjust their step frequency and therefore step length, , so to minimize energy consumption at a given walking speed [47]. The step length (and its double, the stride length) varies with the physical characteristics of the subject (height, weight, etc.) and from one country to another due to the different traditions and lifestyle. Accordingly, the correlation between the walking speed and the step frequency has been reported in literature with a large scatter [45, 48–50]. Many researchers have considered the step frequency as normally distributed [41, 48, 51]. In this study, it is assumed that the mean and STD of the normal distribution of are linearly dependent on the walking speed [52]:Negative values are truncated as meaningless. The mean value of associated with the mean walking speed (i.e., ) is , and the standard deviation is .

Values of the DLFs have been reported in many publications and have been incorporated into design guidelines. These are usually derived from force measurements on instrumented floors or treadmills [53]. In this study, only fixed floor conditions are considered since walker-structure interaction effects are neglected. This interaction is significant in the lateral direction [54] and less in the vertical direction. DLFs measured on a rigid floor are therefore assumed to be the same as those that would be measured on a moving floor if the displacements are small. Moreover, DLFs can also be derived using analytical models that are usually inspired by biomechanics as the inverted pendulum model [55, 56]. Živanović et al. [8] reported a review of the DLFs used in single walker force models. The mean values of are approximately in the range of [57] to [58]. Generally, the dynamic part of the vertical GRF is found to be dependent on . The first study revealing this issue is that of Kajikawa [59] (reported in [60]). It is now widely accepted that increases with up to a maximum of approximately [61, 62]. Accordingly, in this study, the SP is described through a depending on (see (4)) as [62]

According to (5), is described by a normal distribution with mean equal to and STD .

On the other hand, the mean values of are approximately in the range of [63] to [58]. Accordingly, in this study, the SP is assumed to have a described through a normal distribution [52]:

In (5) and (6), negative values are truncated as meaningless. Moreover, in (5) an upper bound was set at .

The body weight is very much dependent on height; therefore, in medical applications, it is preferred to refer to the Body Mass Index (BMI), that is, the body mass divided by the square of the height [64]. For each country, Walpole et al. [65] used available data on BMI and height distribution to estimate average adult body mass. In particular, they reported the average body mass by world regions as in 2005. The average body mass ranges between for Asia and for North America. Indeed, load models (e.g., (1)) require the definition of the walker weight, ; that is, the body weight increased by the weight of clothing and other items carried by the walker. The walker weight is taken equal to by many loading models (e.g., [28, 57]). In this study, the weight of the SP is assumed to be normally distributed as in HIVOSS [21]:The mean value in (7) is larger than that of corresponding to the average body mass reported by Walpole et al. [65] for Europe since the latter lacks clothing and other items carried by the walker. Negative values are truncated as meaningless.

Finally, the distribution of phase lags, , between walkers is a measure of the correlation of the forces they exert. For a continuous PDF of walking frequencies, phase lags are characterized by the PDF of the phase spectrum. If this is uniformly distributed between 0 and , then the walkers and the walking forces are uncorrelated. Correlation increases when the PDF of the phase spectrum is peaky around a given value as this value approaches ; the forces tend to be in phase. In the case of a single walker, all this loses its meaning and will be neglected in the following.

In Figure 1, the Probability Density Functions (PDFs) and the Cumulative Distribution Functions (CDFs) of the random variables defining the SP previously described are shown.