1. Introduction
Pedestrian bridges are critical structures in urban and rural infrastructure, allowing the safe transit of pedestrians over obstacles, such as rivers or vehicular roads. However, these bridges exhibit high dynamic sensitivity owing to their intrinsic characteristics such as low mass, stiffness, and damping [
1]. These characteristics result in low natural frequencies, generally below 10 Hz, which can coincide with the anthropic excitation frequencies generated by pedestrian traffic [
2,
3]. This coincidence of frequencies is known as resonance, which generates excessive vibrations that compromise the serviceability, safety, and comfort of pedestrians [
4]. The most relevant cases of this problem have been observed in Millennium and Solferino pedestrian bridges, where vibrations induced by the synchronised movement of pedestrians generated significant concerns [
5,
6].
Following these events, research on Human-Structure Interaction (HSI) has grown significantly, and models have been proposed to evaluate the serviceability of these structures. The most widely used method is to represent a human as a periodic force that traverses the structure at a constant velocity [
7]. In this case, it is convenient to model the anthropic loads by means of Fourier Series as a function of the frequency of passage, weight of the person, and Dynamic Load Factors (DLFs), which have been defined by experimental tests [
8,
9,
10]. However, studies have shown that this approach generates an overestimation of up to 4 four times in the structural response compared to the experimental tests in the resonance condition [
11]. This estimation error is attributed to the assumption of periodicity in anthropic loading and the omission of pedestrian interaction effects with structural vibrations [
10,
11].
The Mass Spring Damper (MSD) model has been proposed as a better representation of the HSI. In this model, dynamic loads are represented by modelling the pedestrian as a single-degree-of-freedom (SDoF) system moving with constant velocity along the structure and taking a single point of contact [
12]. This type of model has been widely studied and refined by adding degrees of freedom [
13,
14,
15,
16] and subsequently calibrated with experimental tests [
14,
17]. Nevertheless, one of its limitations is associated with the poor accuracy of the mass, stiffness, and damping parameters assumed for pedestrian systems. Several models for predicting these parameters have been proposed [
13,
18]; however, the high inter- and intra-subject variability makes the reliability of the results unfeasible [
7]. On the other hand, the simplification of the MSD model into an sDoF means that the study of the effect of structural vibrations on the properties and gait patterns, known as structure to human interaction (S2HI), cannot be evaluated and are ignored in other studies [
19]. The second method used by the authors consists of representing human gait by means of a mechanical inverted pendulum (IP) model coupled to the structural system [
20], which has been sophisticated by incorporating the lateral swing condition and MSD systems for each component limb [
11,
21,
22,
23,
24]. However, the limitation of this model is associated with the large number of assumptions required for its simulation, such as the control force regulating the stability for constant running added to the initial energy input which reduces the reliability of the prediction of GRFs [
7].
In contrast to traditional gait models, in recent years, the authors have developed complete biomechanical models that include more Degrees of Freedom (DoF) based on specific anthropometric characteristics [
25,
26] shown that a realistic representation of human gait based on anthropometric characteristics improves the prediction of GRFs [
27]. Anderson and Pandy [
28] developed a biomechanical model of human gait based on optimization using the direct dynamics method. However, its limitation is the high CPU processing requirement with a simulation time of approximately 10,000 h. For this reason, the inverse dynamics method has become the main option for the development of human gait models based on optimization techniques added to the progress of novel numerical algorithms and robust hardware, positioning this method as a reliable, affordable, and cost-effective alternative for the simulation of human gait [
26,
29].
Many authors have restricted their studies on gait prediction to models in the sagittal plane and simplified the human body into fewer segments, owing to the complexity and high computational demands [
25]. Some authors have utilised three-dimensional models, such as those in [
25,
26], which predict certain body movements based on experimentally known ones. Subsequently, [
30] limited his model to the assumption of alternating leg contact and restriction of foot rotation in the coronal plane.
Although increasingly robust biomechanical models of human gait have been developed, such as [
26], research has been limited to modeling human gait on static surfaces. For that reason, Aux et al. (2024) [
31] developed a predictive model of vertical HSI composed of a seven-element biomechanical model and a simplified footbridge, analyzing, on the one hand, the kinematics and kinetics of gait including HSI phenomena and on the other hand the dynamic response of the footbridge. Nevertheless, this model was limited to gait in the sagittal plane and therefore the HSI analysis was only developed in the vertical direction. Consequently, it is evident that there is still a need to develop robust three-dimensional models capable of estimating the kinematics and kinetics of human gait under the effects of S2HI that allow further evaluation of anthropic loads and their implementation in HSI studies in lateral and vertical directions, which is a topic that is little addressed in the literature.
To address these gaps, this paper proposes a novel coupled HSI predictive model composed of a complete three-dimensional biomechanical model of the human body and a pedestrian bridge represented as a simply supported Euler-Bernoulli beam. The main novelties and contributions of this work are as follows: (1) The incorporation of anthropic loads defined by a realistic three-dimensional human gait model as a source of excitation for HSI, based on inverse dynamics, and able to represent the changes in gait resulting from surface vibrations (S2HI). (2) A simplified method is proposed to evaluate the three-dimensional HSI that considers the changes in the dynamic properties of the structure under anthropic excitation under HSI effects. A description and formulation of the three-dimensional human gait model are presented in
Section 2, followed by the proposed structural system in
Section 3.
Section 4 describes the simplified HSI formulation, and
Section 5 presents the optimization scheme for the coupled HSI based on the minimization of the total mechanical energy of the pedestrian moving through the structure. Finally,
Section 6 presents and discusses the results.
4. HSI Formulation
A fourth-order partial differential equation (PDE) describes the deflection
at a point on the
x axis of the beam at a given instant
t (Equation (
8)), where
represents the load generated by a pedestrian traversing the structure,
represents the mass of the structure, and
c represents the coefficient of the viscous damper.
Equation (
8) was solved using the separation-of-variables method, which incorporates modal decomposition, as shown in Equation (
9). In this context, the vibration modes are described by sinusoidal functions, as expressed in Equation (
7). Equation (
10) describes the motion (
) of the
vibration mode (
) in modal coordinates, where the natural frequency (
) and damping ratio (
) are defined using the parameters listed in
Table 4. The mass (
) and modal loads (
) were determined using Equations (
11) and (
12), respectively.
Drawing from the equations that detail the kinematics and kinetics of human gait, as outlined in
Section 2, the human body and the structure interact and hecne are dynamically connected. This connection was established by incorporating the force generated by walking as a function of time and position in the human system, as shown in Equation (
13). The transition of the left and right foot forces,
, is defined as a function that is normalized to a range of 0 to 1. In single-stance zones, where only one foot is in contact with the ground, function
D(
t) is equal to 1 during the Begin (BGN) to Heel Contact Right (HCR) and Toe Off Left (TOL) to Heel Contact Left (HCL) periods. However, in the double-stance zones, specifically in the shaded region between the HCR to TOL and HCL to TOR,
changes linearly from 0 to 1 and from 1 to 0, respectively.
In Equation (
13), the left and right feet are represented by
and
, respectively, and the pressure centers for the left and right feet are represented by
and
, respectively, in Equation (
14). The accelerations induced by the structure owing to the human gait are represented by Equation (
15), where
is calculated based on the positions of the feet and the vertical modes of vibration. The applied force of human walking in modal coordinates was obtained by combining Equations (
12) and (
13), resulting the Equation (
16).
where,
Here,
and
represent the forces generated by walking on the left and right feet, respectively, without considering the influence of acceleration on the structure. The total mass of the human system is denoted as
. By substituting Equation (
16) into Equation (
10), the modal coordinate equation for solving the HSI model, given by Equation (
17), can be expressed in matrix form, as shown in Equation (
18). In this equation,
represents the modal mass,
C represents damping, and
K represents the stiffness of the human structural system.
7. Conclusions
A three-dimensional model of the HSI was developed that consists of a whole-body biomechanical model of human gait and a structural system represented as a simple supported beam. The proposed model was used in an optimization algorithm that minimizes the mechanical energy expenditure required by a pedestrian to walk across the structure. The structural response, considering HSI, was implemented by combining inverse dynamics to predict the kinematics and kinetics of human gait and direct dynamics to solve the differential equations of the coupled system.
Multibody models such as that of Salehi et al. (2016) [
26] demonstrate that predictive gait models using inverse dynamics have up to 8 times faster execution speed compared to commercial multibody models using direct dynamics. For the model presented in this research, the primary factor contributing to the computational expense is the solution of the differential equations governing the structural system and HSI. Nevertheless, the integration of direct and inverse dynamics efficiently reduces the overall time required for each optimization iteration. The computational efficiency of the model was evidenced in the simulation times obtained during the study. Each gait pattern was simulated in approximately 12 s on the computer described in
Section 5, demonstrating the feasibility of the model for practical analysis compared to other approaches that require significantly longer processing times [
28].
The predictions of body movements and ground reaction forces considering HSI were based solely on three simple gait characteristics: average walking speed (
), frequency (
), and step length, as well as the anthropometric data of the person and physical characteristics of the structure. It should be noted that none of the human body movements were predefined for a specific motion, and their behaviors were determined solely by the optimization process. The only constraints that were imposed corresponded to physically realizable gait patterns. The results suggest that human gait is a biological process of energy minimization, as other authors have suggested [
42].
The proposed model effectively simulates the kinematics and kinetics of human gait and accurately captures the dynamic response of structures subjected to HSI effects. It is important to note that the vertical force pattern (see
Figure 5 and
Figure 6) exhibits a different number of peaks compared to the existing force patterns reported in the literature. However, this discrepancy is commonly observed in models with high leg stiffness at low walking speeds, where a greater number of force peaks is generated as the walking speed decreases [
43].
The results of the simulation of the HSI demonstrate the potential of this approach for modelling HSI effects pedestrian bridges. As such, it is a viable methodology for the evaluation and understanding of the impact of HSI. A comparison of the results of simulations of a person walking on a rigid surface and walking on a flexible bridge revealed significant differences in the contact forces induced by pedestrians traversing the structure. In particular, differences of 63% and 23% for vertical and lateral forces respectively, highlight the influence of HSI effects on contact forces and their implications for structural response.
Finally, as shown in
Figure 15, pedestrian bridges that are not expected to have excessive vibrations may experience frequency synchronization problems if high occupancy levels are reached and the natural frequency of the coupled system decreases. This highlights the importance of taking this factor into account during the design and construction phases in order to prevent serviceability issues arising from HSI phenomena.
Limitations and Future Research
The model presented does not enable the assessment of HSI with people running or jumping. This limitation is linked to one of the main restrictions of normal walking implemented in this model, which requires that at least one of the pedestrian’s feet is in contact with the ground, contrary to the activity of running, where there may be moments where both feet are in flight. Therefore, future work will extend this study by eliminating restrictions on the duration and length of double support such that activities other than walking can be considered. Another potential area of research is to consider other types of load associated with humans crossing pedestrian bridges, such as those due to bicycles or strollers.
Biomechanical models of human gait that use reverse dynamics have been validated in previous research by proving their feasibility in predicting the kinematics and kinetics of gait (Movements and Forces) [
29,
31]. However, the validation of these models considering vibrating surfaces in real environments has not been addressed. The collection of experimental data where the complete body movements of the gait are recorded under HSI conditions and simultaneously structural vibrations in real bridges are configured as the next step to validate and comprehensively evaluate the approach proposed in this research. In addition, a detailed statistical study of anthropometric characteristics is just as important. This analysis will allow defining key anthropometric parameters that can be integrated into the proposed human-structure interaction models, such as the one proposed in this study, and performing comprehensive comparisons and validations, reducing the uncertainty of the physical parameters of the human system.
Finally, it is known that anthropic loads are not the only dynamic loads to which pedestrian bridges are subjected, in that sense the integration of this model with wind or earthquake effects becomes an interesting field of research that is worth exploring.
A future key aspect to be perfected within the model presented in this research is related to the kinematic model assumed for the body segment of the foot (
Section 2.1.1). This is due to the biological variability of the dynamics of the foot, where the kinematics and kinetics are different according to the anthropometric conditions of the person, such as the anatomical structure of the foot, body weight, body mass index (BMI), joint range of ankle motion, or age. In this sense, a robust model that can be implemented in this model that reduces the uncertainty gap of biological variability is of great importance.