A frequency-domain approach to model vertical crowd-structure interaction in lightweight footbridges

Journal of Sound and Vibration 557 (2023) 117750 Available online 28 April 2023 0022-460X/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi A frequency-domain approach to model vertical crowd-structure interaction in lightweight footbridges Christian Gallegos-Calderóna,∗, Javier Naranjo-Péreza,b, Carlos M.C. Renedoa, Iván M. Díaza aE.T.S.I. Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Calle del Profesor Aranguren 3, 28040, Madrid, Spain bE.T.S. Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Seville, Spain ARTICLE INFO Keywords: Human-structure interaction Crowd-structure system Lightweight footbridges FRP pedestrian structure Frequency domain ABSTRACT Load models that account for Human-Structure Interaction (HSI) may be preferable to accurately predict the dynamic response of lightweight footbridges subjected to pedestrian actions. Representing each person within a crowd may not be practical in engineering design calculations as time-variant models with a large number of degrees of freedom have to be managed. In addition, high computational time may be required to achieve the steady-state response. In this sense, this paper proposes a novel approach to calculate the vertical steady-state response of footbridges from a time-invariant coupled crowd-structure system. Considering the model of the structure and a feedback model of the crowd, a total closed-loop Transfer Function (TF) of the coupled system is derived. Based on this frequency-domain interacting methodology, a step-by-step procedure is set to asses the vibration serviceability of lightweight footbridges due to harmonic excitations through simple algebraic operations. The proposal is used to study a Fibre Reinforced Polymer footbridge subjected to two streams of walking pedestrians. For this structure, a good compromise between experimental and numerical results is obtained in terms of vertical vibrations and TFs. To further validate the proposed approach, a pre-stressed concrete laboratory facility is also analysed, obtaining a satisfactory agreement between the experimental and numerical TFs. Thus, the proposed approach allows to evaluate lightweight footbridges under crowd-induced loads considering HSI in a simple and accurate manner, which is clearly geared to practice. 1. Introduction In recent years, the adoption of innovative construction techniques and novel materials has led to slender and lightweight pedestrian structures [1–3]. Thus, a lively dynamic response may be expected in these footbridges when subjected to human actions. To assess a lightweight bridge at Vibration Serviceability Limit State (VSLS), guidelines, such as SETRA [4] or HIVOSS [5], presented pedestrian load models based on experimental and numerical studies carried out in the first years of the current century. Since these models represent the pedestrians as external forces, neglecting the influence of the human body on the dynamic behaviour of the footbridge, they can be classified as non-interacting load models. The new proposals for Eurocodes [6,7] have incorporated the load models from the mentioned design guidances to evaluate the dynamic performance of footbridges. Additionally, modern experimental measurement techniques [8–10] are available to better characterize the forces exerted by people on lightweight pedestrian structures. ∗Corresponding author. E-mail address: [email protected] (C. Gallegos-Calderón). https://doi.org/10.1016/j.jsv.2023.117750 Received 17 October 2022; Received in revised form 17 April 2023; Accepted 24 April 2023 Journal of Sound and Vibration 557 (2023) 117750 2 C. Gallegos-Calderón et al. An issue related to the pedestrian load models available in existing guidelines [4,5] is that only low harmonics of human actions are considered to compute the response of a lightweight bridge. For walking, this assumption is based on the concept that a structure can be significantly excited in the vertical direction just with the first or second harmonic of the human action. Nevertheless, higher harmonics have been found to contribute considerably to the structural acceleration of lightweight pedestrian structures [11,12]. Another issue of applying non-interacting load models when assessing the response of lightweight footbridges is the large differences between the experimental and numerical vibration levels. The main reason for the discrepancies is due to the omission of Human- Structure Interaction (HSI) phenomenon [13,14]. If vibration response of a footbridge could be predicted adequately by accounting for HSI, over-dimensioning structural elements could be avoided. In addition, the inertial mass of a vibration control device for a pedestrian structure may be reduced when the interaction phenomenon is considered [15]. Among the different proposals to account for HSI in the vertical direction [16–19], Single-Degree-of-Freedom (SDOF) systems, defined by the parameters of the human body, acting on the structure have been employed to represent a pedestrian. To model the effects of a continuous stream of pedestrians walking on a footbridge, while considering HSI, several moving SDOF systems to represent the individuals in a crowd have been used [20]. For instance, a SDOF Mass–Spring–Damper (MSD) system plus an external harmonic force have been used to describe active people [21–23]. A SDOF Mass–Spring–Damper–Actuator (MSDA) system has also been employed to represent humans exciting a structure [24,25]. The aforementioned systems, which may be classified as interacting models, are equivalent and lead to the same results when computing the response of a lightweight footbridge under the action of a single walking person [26]. One of the drawbacks of using interacting load models is the high computational time required to achieve a steady-state response when the bridge is analysed under the action of a stream of pedestrians. Another approach to indirectly account for HSI while still using non-interacting load models recommended in design guidelines is the modification of the modal parameters of the bare structure due to the pedestrians. In Ahmadi et al. [27], an empirical expression is given to calculate the contribution of the pedestrians to the damping of a simply supported structure. Recently, a simplified procedure to consider vertical interaction between a crowd and a footbridge has been proposed by Van Nimmen et al. [28]. Charts in terms of the pedestrian-to-structure mass ratio (up to 0.30) and the modal parameters of the empty structure are employed in this methodology to define an equivalent SDOF system with an effective natural frequency and a damping ratio. As low-frequency structures (<6 Hz) were considered, the proposed charts may not be applicable to very lightweight footbridges, where higher harmonics of pedestrians actions may excite the structure significantly. Furthermore, pedestrian-to-structure mass ratios can be surpassed in structures comprised of novel materials, such as Fibre Reinforced Polymers (FRPs) or aluminium alloys. Even though non-interacting and interacting load models are usually employed in time-domain analyses, frequency-domain methods are preferable and more useful for assessing the response of a footbridge due to a continuous harmonic excitation. Thus, a general approach, based on the closed-loop Transfer Function (TF) of a coupled crowd-structure system is proposed to predict the steady-state response of a structure subjected to a continuous pedestrian stream. The fundamental vibration mode of the structure, represented as a SDOF MSD system, and the crowd, depicted as a distributed MSDA system, are used in this frequency-domain methodology. Based on this approach, a simple step-by-step procedure is set to calculate the bridge response due to a walking crowd just through algebraic operations while accounting for HSI. The proposed procedure is employed to assess a 10-m long FRP footbridge under two crowd scenarios at the VSLS. Weak (0.2 pedestrians/m2) and dense (0.5 pedestrians/m2) load cases are considered. Experimental results and numerical predictions, using non-interacting and interacting approaches, are compared in terms of the vertical steady-state response. Additionally, experimental TFs obtained from controlled force tests performed with pedestrians walking on the structure are contrasted with the numerical closed-loop TFs. This comparison is also carried out on a pre-stressed concrete structure to demonstrate that the proposal presents no limitations neither the natural frequency of the structure nor the pedestrian-to-structure mass ratio. After this introduction, the current pedestrian load models are briefly described in Section 2. The theoretical basis to set the frequency-domain approach together with the VSLS assessment procedure are explained in Section 3. In Section 4, the FRP footbridge is presented, and the experimental response due to streams of walking pedestrians is described. In Section 5, the numerical response of the FRP structure is computed considering non-interacting load models and the proposed interacting approach. In addition, a sensitivity analysis of the parameters of the coupled crowd-structure system is carried out in this section. In Section 6, a comparison between experimental and numerical TFs of the FRP structure is discussed, and a pre-stressed concrete footbridge is also studied employing the proposed approach. Finally, the main conclusions and future steps are presented in Section 7. 2. Current pedestrian load models The assessment of the dynamic response of a footbridge at VSLS has evolved from considering a single person walking or jogging to scenarios where pedestrian streams that account for different traffic classes (different pedestrian densities) crosses the structure. Current guidelines [4,5] and codes [7] adopt a moving force travelling along the structure to model a single pedestrian. Whilst to consider a stream of pedestrians walking over a footbridge, a distributed harmonic load acting on the bridge deck may be considered (Fig. 1). For this case, CEN [7] suggests to employ the following non-interacting load model to compute the resonant response of a structure 𝑞ℎ(𝑡) = 280 cos (2𝜋𝑓𝑎𝑠𝑡)𝑛′𝜓𝑤(N/m2)(1) being 𝑛′=10.8√𝜁𝑠𝑛 𝑆for𝑑 < 1pedestrian/m2(2) Journal of Sound and Vibration 557 (2023) 117750 3 C. Gallegos-Calderón et al. 𝑛′=1.85√𝑛 𝑆for𝑑≥1pedestrian/m2(3) where 𝑓𝑎𝑠 is the gait frequency assumed to be equal to the footbridge natural frequency under consideration, 𝑛′is the number of individuals who walk perfectly synchronized within a crowd of 𝑛pedestrians, 𝜓𝑤is a reduction coefficient that accounts for the probability that the gait frequency approaches the critical range of natural frequencies under consideration, 𝑆is the loaded surface of the footbridge, 𝜁𝑠is the damping ratio of the structure and 𝑑is the pedestrian density. Fig. 1. Non-interacting load model for a stream of walking pedestrians on a footbridge. The value of 280 in Eq. (1) is obtained by multiplying the static weight of a pedestrian, taken as 700 N, and the Dynamic Load Factor (DLF) that corresponds to the first harmonic of the action (DFL1= 0.40). The second harmonic of walking action is considered through the reduction of 𝜓𝑤from 1 to 0.25. Harmonics higher than 4.8Hz are not considered in CEN [7]. However, when dealing with very lightweight footbridges, the last consideration may not be applicable since high harmonics of pedestrian actions can contribute to the resonant response. Besides, the interaction phenomenon may be a key factor that affects the dynamic response. A more sophisticated approach which accounts for HSI is to represent a stream of pedestrians by using as many SDOF systems as individuals within the crowd [14]. In Fig. 2, several MSDA systems moving at a velocity 𝑣are displayed as an example. To address the effect that not all pedestrians walk synchronously, a phase shift angle between the human driving forces (𝐹𝑎𝑖) of these SDOF systems should be considered [20]. This type of modelling leads to time-variant coupled dynamic systems, which are computationally expensive and have to be solved by direct numerical integration methods. Additionally, extensive Monte Carlo simulations have to be performed to account for the effect of the variation of pedestrians’ properties on the bridge dynamic behaviour [22], and high computational time may be required to achieve the steady-state response. Therefore, representing each person of a crowd may not be practical in engineering design calculations since a problem with a large number of DOFs have to be managed, requiring expertise and advance modelling skills from the users. This approach may not be suitable for future versions of current codes, such as CEN [7]. Fig. 2. Representation of a crowd employing several SDOF systems. To overcome the aforementioned drawbacks, a linear time-invariant (LTI) system may be an alternative. Hence, a general approach in the frequency domain to model Crowd-Structure Interaction (CSI) on lightweight footbridges is proposed hereof. This approach is aligned with current standards, so the steady-state bridge response is computed for a scenario where just few pedestrians within the crowd walk perfectly synchronized. Thus, 𝑛′(Eqs. (2)–(3)) is considered for a conservative case of (near-) resonant harmonic excitation. 3. Proposed interacting model The theoretical basis for the proposed model of the crowd-structure system is presented in this section. Also, a step-by-step procedure is set to estimate accurately and fast the vertical dynamic response of lightweight footbridges subjected to streams of walking pedestrians. 3.1. Crowd-structure system Considering that 𝑛pedestrians are walking freely and unrestricted over the deck of the structure, it can be assumed that all the individuals form a uniformly distributed MSDA system (Fig. 3). Journal of Sound and Vibration 557 (2023) 117750 4 C. Gallegos-Calderón et al. Fig. 3. Representation of a crowd employing a distributed MSDA system. Based on the dynamic parameters of the human body and the total number of pedestrians (𝑛) acting on the structure, the parameters that define the crowd system (Fig. 3) can be defined as follows 𝑚𝑐𝑟 ℎ= 𝑛 ∑ 𝑖=1 𝑚ℎ𝑖 (kg)(4) 𝑐𝑐𝑟 ℎ= 𝑛 ∑ 𝑖=1 𝑐ℎ𝑖 (Ns/m)(5) 𝑘𝑐𝑟 ℎ= 𝑛 ∑ 𝑖=1 𝑘ℎ𝑖 (N/m)(6) where 𝑚ℎ𝑖 (kg) is the mass of the 𝑖th pedestrian, 𝑘ℎ𝑖 =𝜔2 ℎ𝑖 𝑚ℎ𝑖 (N/m) is the stiffness of the 𝑖th human, 𝜔ℎ𝑖 = 2𝜋𝑓ℎ𝑖 (rad/s) is the 𝑖th pedestrian’s angular natural frequency, 𝑓ℎ𝑖 (Hz) is the natural frequency of the 𝑖th individual, 𝑐ℎ𝑖 = 2𝜔ℎ𝑖 𝑚ℎ𝑖 𝜁ℎ𝑖 (Ns/m) is the viscous damping of the 𝑖th person, and 𝜁ℎ𝑖 is the damping ratio of the 𝑖th pedestrian. The superscript ‘𝑐𝑟’ in the previous and upcoming equations refers to the crowd system. The distributed MSDA system also presents a crowd harmonic driving load associated to the number of pedestrians who walk synchronously (𝑛′), and it may be expressed as follows 𝑞𝑐𝑟 𝑎(𝑡) = 𝑊ℎGLF𝑟cos (2𝜋𝑓𝑎𝑠𝑡)𝑛′𝜓𝜔(N/m2)(7) where 𝑊ℎis the weight of a pedestrian and GLF𝑟is the Generated Load Factor associated to the r𝑡ℎ harmonic. Assuming that the gait frequency is always in the critical range of the natural frequencies under consideration, 𝜓𝜔= 1 can be adopted conservatively. Note that the GLF is different from the DLF due to the consideration of HSI. From the distributed load, the crowd driving force (𝐹𝑐𝑟 𝑎) can be computed as follows 𝐹𝑐𝑟 𝑎(𝑡) = 𝑞𝑐𝑟 𝑎(𝑡)𝑆(N).(8) Fig. 4a displays the subsystems, crowd and structure, that comprise the coupled system and the forces that act on them. The crowd driving force per unit length, 𝐹𝑐𝑟 𝑎∕𝐿(N/m), is assumed to affect both subsystems simultaneously. Similarly, the distributed springs and dampers are considered to generate loads per unit length, 𝐹𝑐𝑟 𝑘∕𝐿(N/m) and 𝐹𝑐𝑟 𝑐∕𝐿(N/m) that act on the crowd system and the structure system. The first equation of motion can be derived from the loads acting on the structure system (bottom part of Fig. 4a) as follows: 𝑚𝑠𝑦𝑠+𝑐𝑠𝑦𝑠+𝑘𝑠𝑦𝑠= − ∫𝐿 0 𝐹𝑐𝑟 𝑎(𝑡) 𝐿𝜙(𝑥)d𝑥+∫𝐿 0 𝐹𝑐𝑟 𝑐(𝑡) 𝐿𝜙(𝑥)d𝑥+∫𝐿 0 𝐹𝑐𝑟 𝑘(𝑡) 𝐿𝜙(𝑥)d𝑥(9) If a sinusoidal form for the unit-normalized fundamental mode shape is assumed 𝜙(𝑥) = sin (𝜋𝑥 𝐿)(10) for the case of a simply supported structure, then, ∫𝐿 0 𝜙(𝑥)d𝑥=2 𝜋(11) and Eq. (9) can be rewritten as follows 𝑚𝑠𝑦𝑠+𝑐𝑠𝑦𝑠+𝑘𝑠𝑦𝑠=2 𝜋(−𝐹𝑐𝑟 𝑎(𝑡) + 𝐹𝑐𝑟 𝑘(𝑡) + 𝐹𝑐𝑟 𝑐(𝑡))(12) where 𝑦𝑠(m) is the modal displacement of the structure (upper dots indicating time derivatives), 𝑚𝑠(kg) is the mass associated to a vibration mode of the structure, 𝑐𝑠= 2𝜔𝑠𝑚𝑠𝜁𝑠(Ns/m) is the viscous damping of the structure, 𝜔𝑠= 2𝜋 𝑓𝑠(rad/s) is the structure’s angular frequency, 𝑓𝑠(Hz) is the natural frequency, and 𝑘𝑠=𝜔2 𝑠𝑚𝑠(N/m) is the stiffness of the structure. At the bottom of Fig. 4b displays the free body diagram of the simply supported structure considering one vibration mode. Accounting for the crowd harmonic force and the corresponding forces from distributed springs and dampers, a total resulting force on the structure is obtained as follows 𝐹𝑐𝑟 ℎ(𝑡) = −𝐹𝑐𝑟 𝑎(𝑡) + 𝐹𝑐𝑟 𝑘(𝑡) + 𝐹𝑐𝑟 𝑐(𝑡).(13) Journal of Sound and Vibration 557 (2023) 117750 5 C. Gallegos-Calderón et al. Fig. 4. Coupled crowd-structure system: (a) Forces acting on the crowd system and the structure system, and (b) Free body diagram of the 2 DOFs. Taking the Laplace transform of Eq. (12), the following expression is derived 𝑌𝑠(𝑠)(𝑚𝑠𝑠2+𝑐𝑠𝑠+𝑘𝑠)= 2∕𝜋 𝐹 𝑐𝑟 ℎ(𝑠)(14) where 𝑠=𝑗𝜔 is the Laplace variable, 𝜔is the angular frequency, 𝐹𝑐𝑟 ℎ(𝑠)is the Laplace transform of the resulting force acting on the structure, and 𝑌𝑠(𝑠)is the Laplace transform of the modal structure displacement. Multiplying both sides of Eq. (14) by 𝑠2and rearranging the equation, the TF between the structure acceleration and the resulting force can be obtained as follows 𝑠2𝑌𝑠(𝑠) = 2∕𝜋⋅ 𝑠2 𝑚𝑠𝑠2+𝑐𝑠𝑠+𝑘𝑠 𝐹𝑐𝑟 ℎ(𝑠)(15) where 𝑠2𝑌𝑠(𝑠)is the Laplace transform of the acceleration of the footbridge. The TF of the structure is identified as 𝐺𝑆(𝑠) = 𝑠2 𝑚𝑠𝑠2+𝑐𝑠𝑠+𝑘𝑠 .(16) Hence, Eq. (15) can be written as follows 𝑠2𝑌𝑠(𝑠) = 2∕𝜋⋅𝐺𝑆(𝑠)𝐹𝑐𝑟 ℎ(𝑠).(17) A second equation of motion based on the loads acting on the crowd system (top part of Fig. 4a) can be derived as follows 𝑚𝑐𝑟 ℎ𝑦𝑐𝑟 ℎ+𝐹𝑐𝑟 𝑘(𝑡) + 𝐹𝑐𝑟 𝑐(𝑡) = 𝐹𝑐𝑟 𝑎(𝑡)(18) where 𝑦𝑐𝑟 ℎ(m) is the displacement of the distributed MSDA system. The free body diagram of the crowd system is presented at the top of Fig. 4b. Since the footbridge movement influences the crowd motion (and vice versa), it is assumed that the acceleration of the structure affects the distributed MSDA system through the value of 𝜙(𝑥)along the bridge length. For the case of a simply supported structure, the value is 2∕𝜋. This is graphically illustrated in Fig. 5. Thus, the forces associated to the springs and dampers can be expressed as follows 𝐹𝑐𝑟 𝑐(𝑡) = 𝑐𝑐𝑟 ℎ(𝑦𝑐𝑟 ℎ−2 𝜋𝑦𝑠)(19) 𝐹𝑐𝑟 𝑘(𝑡) = 𝑘𝑐𝑟 ℎ(𝑦𝑐𝑟 ℎ−2 𝜋𝑦𝑠)(20) Accounting for Eqs. (19)–(20) and Eq. (18), the following expression for the crowd system is achieved 𝑚𝑐𝑟 ℎ𝑦𝑐𝑟 ℎ+𝑐𝑐𝑟 ℎ(𝑦𝑐𝑟 ℎ−2 𝜋𝑦𝑠)+𝑘𝑐𝑟 ℎ(𝑦𝑐𝑟 ℎ−2 𝜋𝑦𝑠)=𝐹𝑐𝑟 𝑎(𝑡).(21) Journal of Sound and Vibration 557 (2023) 117750 6 C. Gallegos-Calderón et al. Fig. 5. Mean value of the modal coordinates of 𝜙(𝑥). Taking the Laplace transform of Eq. (21) and rearranging the equation, the following expression is derived 𝑌𝑐𝑟 ℎ(𝑠)(𝑚𝑐𝑟 ℎ𝑠2+𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ)− 2∕𝜋 𝑌𝑠(𝑠)(𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ)=𝐹𝑐𝑟 𝑎(𝑠)(22) where 𝑌𝑐𝑟 ℎ(𝑠)is the Laplace transform of the displacement of the crowd system, and 𝐹𝑐𝑟 𝑎(𝑠)is the Laplace transform of the crowd driving force (Eq. (8)). Multiplying both sides of Eq. (22) by 𝑚𝑐𝑟 ℎand 𝑠2and rearranging the equation, the following expression is achieved 𝑚𝑐𝑟 ℎ𝑠2𝑌𝑐𝑟 ℎ(𝑠) = 𝑚𝑐𝑟 ℎ𝑠2 𝑚𝑐𝑟 ℎ𝑠2+𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ 𝐹𝑐𝑟 𝑎(𝑠) + 2∕𝜋⋅ 𝑚𝑐𝑟 ℎ(𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ) 𝑚𝑐𝑟 ℎ𝑠2+𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ 𝑠2𝑌𝑠(𝑠)(23) where 𝑠2𝑌𝑐𝑟 ℎ(𝑠)is the Laplace transform of the acceleration of the distributed MSDA system. From Eq. (23), the TFs associated to the crowd system and the interaction phenomenon are identified, respectively, as follows 𝐺𝑐𝑟 𝐻(𝑠) = 𝑚𝑐𝑟 ℎ𝑠2 𝑚𝑐𝑟 ℎ𝑠2+𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ (24) 𝐺𝑐𝑟 𝐻𝑆𝐼 (𝑠) = 𝑚𝑐𝑟 ℎ(𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ) 𝑚𝑐𝑟 ℎ𝑠2+𝑐𝑐𝑟 ℎ𝑠+𝑘𝑐𝑟 ℎ .(25) Finally, replacing Eq. (13) into Eq. (18) and taking the Laplace transform, the expression presented below is obtained 𝑚𝑐𝑟 ℎ𝑠2𝑌𝑐𝑟 ℎ(𝑠)=−𝐹𝑐𝑟 ℎ(𝑠)(26) resulting in 𝐹𝑐𝑟 ℎ(𝑠)=−𝐺𝑐𝑟 𝐻(𝑠)𝐹𝑐𝑟 𝑎(𝑠) − 2∕𝜋⋅𝐺𝑐𝑟 𝐻𝑆𝐼 (𝑠)𝑠2𝑌𝑠(𝑠).(27) Eq. (27) indicates that the resulting force on the structure is the sum of the driving force transmitted through the crowd system (𝐺𝑐𝑟 𝐻) and the interacting force due to the structure response. Based on Eqs. (17) and (27), a block diagram of the crowd-structure system, mainly characterized by a feedback loop associated to HSI, can be assembled as presented in Fig. 6. This diagram employs TFs related to the structure system (𝐺𝑆(𝑠)), crowd system (𝐺𝑐𝑟 𝐻), and interaction phenomenon (𝐺𝑐𝑟 𝐻𝑆𝐼 ). Fig. 6. Block diagram of the crowd-structure system. From Fig. 6, the 2-DOF system (𝑌𝑐𝑟 ℎ,𝑌𝑠) can be merged in a total closed-loop TF between the acceleration of the structure (𝑠2𝑌𝑠(𝑠)) and the crowd driving force (𝐹𝑐𝑟 𝑎(𝑠)). This TF is derived through algebraic operations with the blocks, resulting in the following expression 𝐺𝐻𝐶𝐿(𝑠) = 𝑠2𝑌𝑠(𝑠) 𝐹𝑐𝑟 𝑎(𝑠)=(2∕𝜋)⋅𝐺𝑆(𝑠)𝐺𝑐𝑟 𝐻(𝑠) 1 + (2∕𝜋)2⋅𝐺𝑆(𝑠)𝐺𝑐𝑟 𝐻𝑆𝐼 (𝑠)with 𝑠=𝑗𝜔. (28) To compute the steady-state acceleration response of the structure, Eqs. (7)–(8) and the closed-loop TF from Eq. (28) are employed. Thus, the following expression can be used 𝑦𝑠(𝑡) = |||𝐺𝐻𝐶𝐿(𝑗𝜔)|||𝜔=2𝜋𝑓𝑎𝑠 ⋅(𝑊ℎGLF𝑟𝑛′𝜓𝜔𝑆)cos (2𝜋𝑓𝑎𝑠𝑡).(29) Journal of Sound and Vibration 557 (2023) 117750 7 C. Gallegos-Calderón et al. The maximum response due to the resonant harmonic of the pedestrian action may be further simplified through an algebraic multiplication, using the following expression apeak = max 𝑗𝜔 |||𝐺𝐻𝐶𝐿(𝑗𝜔)||| ⋅(𝑊ℎGLF𝑟𝑛′𝜓𝜔𝑆)(30) where the maximum value of |||𝐺𝐻𝑒𝑞 𝐶𝐿(𝑗𝜔)|||is the 𝐻∞norm of the total closed-loop TF presented in Eq. (28). Given this approach considers the resonant response of the bridge, the Maximum Transient Vibration Value (MTVV) may be employed for the assessment of a footbridge at VSLS, as suggested by ISO [29]. The MTVV can be calculated as MTVV =apeak √2 .(31) 3.2. Step-by-step procedure Considering the explanation in Section 3.1 and summarizing, the following steps may be used to predict the dynamic response of a footbridge while considering CSI: 1. Define the parameters of the considered vibration mode of the structure (𝑚𝑠,𝑓𝑠, and 𝜁𝑠). 2. Set the number of pedestrians walking on the structure (𝑛). 3. Define the parameters of the human body (𝑚ℎ,𝑓ℎ, and 𝜁ℎ). 4. Determine the parameters of the crowd system (𝑚𝑐𝑟 ℎ,𝑓𝑐𝑟 ℎ, and 𝜁𝑐𝑟 ℎ) through Eqs. (4)–(6). 5. Obtain the total closed-loop TF of the crowd-structure system (𝐺𝐻𝐶𝐿(𝑗𝜔)) using Eq. (28). 6. Compute apeak employing Eq. (30) for evaluating the structure at VSLS. Note that if a comparison between measurements and numerical results is required, the maximum value calculated from the experimental 1s-running Root-Mean-Square acceleration may also be employed. Hence, MTVVs obtained from a test and a simulation can be contrasted. 4. FRP footbridge and experimental tests This section presents the FRP footbridge which is later employed to apply the proposed frequency-domain procedure. Additionally, the acceleration response of the bridge due to the action of walking pedestrians is presented. 4.1. Description of the structure The footbridge is a simply supported structure of 10.0 m long by 1.5 m wide, as displayed in Fig. 7a. Pultruded Glass-FRP (GFRP) elements and Carbon-FRP (CFRP) strips, manufactured by Fiberline Composites A/S [30], are employed to assemble the bridge. GFRP panels (Plank HD) placed onto three GFRP stringers (1 I 300 ×150 ×15 and 2 U 300 ×90 ×15) comprise the superstructure. Additionally, GFRP crossbeams (I 160 ×80 ×8) restrained laterally the stringers, and CFRP strips (1 E 139∕150∕4.9 and 2 E 139∕90∕4.9) are adhesively bonded to the flanges of the stringers along their length. For the handrails, GFRP profiles (SHS 60 ×60 ×5) and stainless-steel cables are used. The connections among the aforementioned GFRP elements are achieved employing stainless-steel bolts class A2-50 and GFRP profiles (L 75 ×75 ×8). At the bridge ends, concrete blocks are placed for the installation of pinned and roller supports (Fig. 7a). A detailed description of the structural design of the FRP footbridge, whose linear mass is about 80 kg/m, can be found in Gallegos-Calderón et al. [31]. Fig. 7. FRP footbridge: (a) Built structure, (b) Accelerometer at midspan, and (c) Shaker in a fixed body mode. Prior the tests involving pedestrians, an Operational Modal Analysis (OMA) was performed to identify the vibration modes of the structure [26]. The first vibration mode of the FRP bridge (Fig. 8a) is a bending vertical mode at 7.66 Hz, whereas the second Journal of Sound and Vibration 557 (2023) 117750 8 C. Gallegos-Calderón et al. and third vibration modes are lateral torsional modes at 10.96 and 15.01 Hz, respectively. Then, controlled force experiments concentrated on the first vibration mode of the structure were carried. The acceleration collected from a piezoelectric sensor model 393A03 (sensitivity 1 V/g and 0.00001 g RMS wide band resolution) [32] attached at the bottom of the central stringer (Fig. 7b) and a controlled force applied from an electrodynamic shaker APS 400 [33] (Fig. 7c) were used to obtain experimentally the TF between both magnitudes. Since the electrodynamic device was set-up in a fixed body mode, the force generated by the device was obtained by monitoring its instantaneous current during the tests. A stainless-steel rod connected the shaker to a crossbeam near the edge of the midspan cross-section. The input signal to the shaker was a chirp waveform whose instantaneous frequency increases slowly and linearly from 6 to 9 Hz during 5 min, ensuring that the sweeping at each frequency was sufficiently slow to obtain a satisfactory identification of the TF of the first vertical vibration mode of the bridge. Three tests considering the bare footbridge were conducted varying the shaker force level and the corresponding TFs were computed through the 𝐻1estimator. Thus, a representative (mean) TF was obtained, as shown in Fig. 8b. In this graph, the width of the coloured band corresponds to the minimum and maximum values among the three identified TFs. A reasonable linear behaviour can be seen. From the mean TF, a natural frequency (𝑓𝑠) of 7.47 Hz, a damping ratio (𝜁𝑠) of 1.40%, and an effective modal mass (𝑚𝑠) of 405 kg are identified. The difference between the value of the fundamental frequency from the OMA and the experimental modal analysis is explained by the activation, or not, of the roller support. The FRP footbridge appears to be slightly stiffer when negligible excitation (OMA test) is applied. Fig. 8. Experimental characterization of the first vibration mode of the bare FRP footbridge: (a) Mode shape from the OMA, and (b) TF. 4.2. Dynamic response assessment The dynamic response of the FRP footbridge under the action of two crowd scenarios is presented herein. For the tests, the piezoelectric accelerometer shown in Fig. 7b was employed to collect the response of the bridge at midspan. Fig. 9 displays the experiments carried out. Three and eight people walking continuously for 5 min over the deck were considered to represent weak and dense Traffic Classes (TCs), respectively. The duration of each test was also selected to obtain a representative value of the steady-state response of the footbridge. In Table 1, the total mass of the people involved in each test and the pedestrian-to-structure mass ratio, accounting for the mass of the FRP footbridge, are presented. Table 1 Mass of the pedestrians involved in the experiments. Traffic Number of 𝑑Total mass of Mass class pedestrians (pedestrians/m2) pedestrians (kg) ratio Weak 3 0.20 218.8 0.27 Dense 8 0.53 585.7 0.73 First, the weak TC was analysed, so three pedestrians were asked to walk comfortably and freely along the full length of the bridge (Fig. 9a). Neither the distance among individuals nor the gait frequency were set during the test, and overtaking was allowed. The response was recorded with a sampling frequency of 1000 Hz using the sensor shown in Fig. 7b. The measured raw data is shown in Fig. 10a, where the maximum acceleration is 4.82 m∕s2and the MTVV is 1.23 m∕s2. Then, the signal was processed using a zero-phase 4th order band-pass Butterworth filter with an upper and lower cut-off frequencies at 1 Hz and 20 Hz, respectively. The resulting signal is displayed in Fig. 10b, where the peak response is 2.14 m∕s2and the MTVV is 1.22 m∕s2. As expected, the maximum acceleration decreases significantly since values associated to high-frequency components are disregarded once the filter is applied. Nevertheless, the computed MTVVs remain almost identical. Hence, this parameter is assumed to be a representative and robust value for the VSLS assessment. Journal of Sound and Vibration 557 (2023) 117750 9 C. Gallegos-Calderón et al. Fig. 9. Experiments carried out considering: (a) weak TC, and (b) dense TC. Fig. 10. Response of the FRP footbridge due to a weak TC: (a) Raw data, and (b) Processed signal with a zero-phase 4th order band-pass Butterworth filter between 1 and 20 Hz. Second, the dense TC was studied. Eight pedestrians walked in the same manner as previously explained (Fig. 9b). The recorded data was processed employing the same zero-phase 4th order band-pass Butterworth filter. Fig. 11 displays the response, where the maximum acceleration is 2.42 m∕s2and the MTVV is 1.20 m∕s2. In comparison with the weak TC, the peak response increased 13% whilst the MTVV slightly decreased 2%. To determine the harmonic (𝑟) of the walking action that is mainly exciting the structure, the Fast Fourier Transform of the processed signals were computed. In Fig. 12, the spectrum of the footbridge response is presented for the two crowd scenarios. Considering the peaks in the graphs, it is appreciated that the structural acceleration is mainly associated to the fourth harmonic. Journal of Sound and Vibration 557 (2023) 117750 16 C. Gallegos-Calderón et al. Fig. 19. Sheffield footbridge: (a) Structure, and (b) Closed-loop TFs varying 𝜁ℎ. •The experimental and numerical TFs of coupled crowd-structure systems derived on two footbridges, an FRP bridge and a pre-stressed concrete structure, have shown good agreement. Thus, the proposed frequency-domain procedure is suitable for different pedestrian structures, regardless the material employed for their construction. •Adopting lower values regarding the parameters of the human body may be preferable for a conservative evaluation of the response of a lightweight footbridge subjected to crowd loads. Values between 10% and 20% are recommended for 𝜁ℎ. In the future, other type of lightweight structures considering different crowd scenarios will be studied to further validate the proposed framework. Also, synchronization among pedestrians will be investigated to determine if provisions established in existing guidelines can be directly applied to models that account for HSI. Accomplishing both tasks, a robust and easy-to-apply procedure to model CSI on footbridges could be proposed for inclusion in design guidelines. CRediT authorship contribution statement Christian Gallegos-Calderón: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization. Javier Naranjo-Pérez: Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization. Carlos M.C. Renedo: Methodology, Investigation, Writing – review & editing. Iván M. Díaz: Conceptualization, Methodology, Resources, Supervision, Writing – review & editing, Project administration, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgements The authors acknowledge the grant PID2021-127627OB-I00, funded by Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación and 10.13039/501100011033 FEDER, European Union. Christian Gallegos-Calderón expresses his gratitude to Secretaría de Educación Superior, Ciencia, Tecnología e Innovación de Ecuador (SENESCYT) for the financial support. Javier Naranjo-Pérez thanks Ministerio de Universidades for the grant funded through the program ‘European Union - NextGenerationEU’. Journal of Sound and Vibration 557 (2023) 117750 17 C. Gallegos-Calderón et al. References [1] R.A. Votsis, T.J. Stratford, M.K. Chryssanthopoulos, E.A. 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