Innovative Experimental Assessment of Human–Structure Interaction Effects on Footbridges with Accurate Multi-Axial Dynamic Sensitivity Using Real-Time Hybrid Simulation

Abstract

This study evaluates the dynamic performance of a reference footbridge under human–structure interaction (HSI) effects using real-time hybrid simulation (RTHS). The footbridge, designed with precise multi-axial dynamic sensitivity, is tested under pedestrian gait velocities of 1.20, 1.50, and 1.80 $\mathrm{m}·{\mathrm{s}}^{-1}$. The RTHS framework involves an analytical continuous model of the footbridge as a numerical substructure and real human gait loads as the experimental substructure. The results reveal significant dynamic coupling between pedestrian-induced loads and the responses of the structure. Lateral vibrations exhibit a fundamental frequency of approximately 1.0 Hz, whereas vertical vibrations peaked near 2.0 Hz. Dynamic synchronization, particularly at higher gait velocities, amplified the structural vibrations, with lateral loading increasing by up to 300% in the middle span. Vertical loads show substantial amplification and attenuation depending on gait velocity and footbridge location. Lateral accelerations display a dispersion of approximately 15.0%, whereas vertical accelerations showed higher variability, with dispersions reaching up to 20%. The RTHS technique demonstrates high fidelity and accuracy, with global errors below 2.95% and delays of less than 2.10 ms across all evaluated directions. These results emphasize the critical importance of accounting for HSI effects in the design of pedestrian footbridges because human-induced vibrations can significantly impact structural serviceability and user comfort. This study offers important insights into optimizing footbridge design to mitigate the risks of excessive vibrations and ensure both safety and functionality under typical pedestrian loads.

1. Introduction

In contemporary urban development, there is growing emphasis on prioritizing pedestrian traffic, leading to the creation of civil structures designed specifically for walkable use. This shift aims to enhance walkability and promote sustainable urban living [1]. However, the dynamic interaction between human-induced activity loads and the response of civil structures frequently limits serviceability owing to the development of excessive structural vibrations. Dynamic coupling between gait pedestrians and structural responses has been extensively discussed in the literature concerning civil structures, such as stairs [2,3,4], grandstands [5,6,7,8,9], slabs [10,11,12], and footbridges [13,14,15,16,17,18,19,20,21,22,23,24]. The latter type of structure has become the most affected urban civil structure regarding vibration serviceability limits due to the increase in technical and aesthetic requirements, their condition of spanning large distances, and the ongoing integration of advanced materials that are more robust and lighter during and after the construction process. Grouping of these conditions allows for design optimization, resulting in relatively low mass and damping. Consequently, footbridges are becoming increasingly slender and flexible, increasing their dynamic sensitivity to excessive vibrations owing to human activity [13,17,25]. Normally, vandalism-induced human loads have the greatest dynamic effect on the response of footbridges [18,26]. However, owing to their intensity and wide frequency range, human-induced gait loads have the highest recurrence and potential to limit the structural serviceability due to the vibrations generated. In addition, the human gait frequency content is typically close to the dynamic properties of these types of structures, which poses a latent risk of synchronization phenomena affecting both footbridges and pedestrian traffic.

The adaptive [17,27,28], random [29,30], and nonstationary conditions [17,29,31] of human-induced gait loads have generated an interrelation phenomenon with structural vibrations known as structure-to-human interaction (S2HI) [17,32,33,34]. Concurrently, the dynamic properties of footbridges are influenced by the presence of these anthropic loads, and the participatory mass and damping provided by human loads are significant for these dynamically sensitive structures; this relationship is known as human-to-structure interaction (H2SI) [23,35,36,37]. In addition, dynamic effects between the gait pedestrians themselves have been observed and are denoted as human-to-human interaction (HHI) [38,39,40]. These interaction aspects are referred to in the literature as human–structure interaction effects (HSI) [13,18,20,27,41,42,43,44]. These effects on footbridges represent an advanced case of dynamic synchronization and have become one of the most intriguing topics within the structural scientific community in recent years.

Traditionally, human-induced gait load effects on footbridge dynamic responses have been numerically evaluated by assuming the behavior of these anthropic loads on vibrating surfaces without complete interrelation effects with the structural response, which underestimates the dynamic coupling effects and leads to a tendency to condition serviceability due to the low estimation of structural vibrations [21,31,45,46,47,48,49]. Likewise, S2HI effects have been assessed on surfaces with lateral movement using instrument treadmills, which allows unidirectional or multidirectional analysis of the vibration effects on human-induced gait loads [17,32,33,34,50]. Although experimental tests allow better measurement of the human gait response due to controlled conditions and predefined protocols, limitations have generally been reported in the development of variability conditions under HHI effects and individual human gait disturbances [9,14,17,51,52]. In order to reduce the impact of laboratory conditions, in the last two decades, several footbridge benchmarks have been built, evaluated, and described in state-of-the-art structural engineering to submit benchmarking problems of international reference, which have been highlighted for conditioning the structural design to fundamental frequencies close to the frequency content of the human gait [25,41,42,49]. However, the construction process and variability of the materials used have led to shifts in the target fundamental frequencies, limiting their effectiveness in studying the HSI effects. In addition, the cost-effectiveness, maintenance, and obsolescence of this type of footbridge development after the research has been completed represent another important constraint.

Studies on functional urban footbridges are ideal study prototypes because of their natural geometry, large-scale dimensions, actual service loads induced by pedestrian flows, and environmental perception around the structure, which affect pedestrian gait behavior. Fujino et al. [53] conducted a pioneering study on a large-scale footbridge under typical pedestrian loading conditions (Toda Park Footbridge, Japan). They quantified the influence of HSI lateral effects for more than 20.0% of pedestrians, observing coupling with the fundamental frequency of the structure (≈0.91 Hz) and subsequent serviceability limitations due to structural vibrations. Similarly, excessive lateral vibrations caused by human-induced loads have been reported during the openings of the Millennium Bridge (London), Solferino Bridge (Paris), and Changi Airport Bridge (Singapore). Large-scale tests on these structures have gradually demonstrated dynamic alterations in the lateral structural vibration behavior due to HSI effects, as extensively documented by Dallard et al. [54], Živanović et al. [13], Dziuba et al. [55], Ingólfsson et al. [27], and Brownjohn et al. [56]. Numerous cases have been reported in the literature on HSI effects, modeling, and anthropic load testing related to footbridges, including the Wuhan Yangtze Bridge, China [57], and the Le Ponte del Mare, Pescara, Italy [58], among others. Additionally, review studies have focused on the phenomena of dynamic interaction between pedestrians and structures in the vertical [59,60] and lateral [14,17,27,33,36,43,46,50,61,62] directions. Van Nimmen et al. [24,41] and Colmenares et al. [15] conducted the most comprehensive studies, evaluating large-scale urban footbridges: the Eeklo Bridge-Belgium and the Folke Bernadotte Bridge, respectively. These structures are subjected to human loads and have become international benchmark datasets. However, measurements of the structural responses and behavior of human-induced loads are difficult to acquire experimentally under in situ conditions, which limits their widespread implementation.

In recent years, an innovative cyber-experimental technique known as real-time hybrid simulation (RTHS) or hardware-in-the-loop (HIL) simulation has been used to assess the dynamic behavior of complex structural systems [63,64,65,66], in particular to assess the influence of structural control devices on building seismic response [16,67,68,69,70]. However, this methodology has recently been implemented to evaluate non-conventional dynamic systems in structural engineering [71,72,73]. RTHSs are an experimental evaluation technique characterized by real-time scales [74,75,76,77,78], comprising a physical component that includes inertial effects in constant interaction with a numerical component [79,80,81,82,83,84,85,86,87,88,89,90,91], which have related communication through transfer systems (TSs) that normally implement hydraulic actuators and instrumentation sensor systems. The potential, cost-effectiveness, versatility, and relative accuracy of the RTHS technique allow the concentration of resources on the assessment of the relevant components of interest within a dynamic system, which leads to reduced experimental setup and logistic cost, decreasing execution time of the tests, and increased productivity and efficiency compared to traditional large-scale testing.

In this study, HSI effects on a footbridge with precise sensitivity in its dynamic properties affected by human-induced gait loads were evaluated using RTHS techniques. The implementation of this innovative methodology in the field of the experimental evaluation of HSI effects on footbridges allowed the division of the dynamic system into three principal components: a footbridge modeled using a fourth-order partial differential equation as the numerical substructure (NS), pedestrian walking as the experimental substructure (ES), and a multi-axial test framework developed and described by Castillo et al. [28,84] as the transfer system (TS). This proposal allowed the assessment and measurement of the loads induced by human gait under laboratory conditions and simulated the complete states of the footbridge dynamic responses through a continuous model, which had an interaction in real-time to create a dynamic system with continuous feedback. The remainder of this paper is organized as follows. Section 2 provides details of the evaluated reference footbridge, a description of the general information of the test subjects (TSs), and a test description. In Section 3, the dynamic substructuring of the evaluation of the effects of human–structure interaction on a pedestrian bridge with precise dynamic sensitivity, the transfer systems used, and the experimental setup. A complete analysis of the dynamic performance of the reference footbridge under HSI effects of crowd-induced loads is presented and discussed in Section 4. In addition, the performance of the RTHS technique is determined. Finally, Section 5 and Section 6 present the future works and conclusions of the study.

2. Main Dynamic System

2.1. Footbridge with Precise Dynamic Sensitivity

A reference footbridge (RF) with precise dynamic sensitivity in its fundamental frequency content, close to the human gait frequency (lateral and vertical frequency of 1.0 Hz and 2.0 Hz, respectively), was designed and proposed as an analytical reference structure. This structure employed structural steel with $f_y=210.0 \mathrm{G}\mathrm{P}\mathrm{a}$ and ${\rho }_g=7500.0 \mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}$ and an equivalent cross-section, with an effective area $A_g=533.40 {\mathrm{c}\mathrm{m}}^2$ and inertial moment (I) specified for each direction in

Table 1. Properties of the reference footbridge.

Analysis Direction	$\mathbf{Inertial}\mathbf{Moment}\left[{\mathbf{m}}^4\right]$	Structural Damping [%]	Numerical Frequency $\left[\mathbf{H}\mathbf{z}\right]$	FE Model $\mathbf{Frequency}\left[\mathbf{H}\mathbf{z}\right]$	$\mathbf{Maximum}\mathbf{Frequency}\mathbf{Error}\left[∆\%\right]$
Lateral	$1.00\times {10}^{-4}$ Axis (y-y)	1.00	$f_{l-N,1}=1.04$	$f_{l-FEM,1}=1.06$	$2.04\%$
			$f_{l-N,2}=4.15$	$f_{l-FEM,2}=4.22$
			$f_{l-N,3}=9.35$	$f_{l-FEM,3}=9.48$
Vertical	$3.60\times {10}^{-4}$ Axis (x-x)	1.00	$f_{v-N,1}=2.00$	$f_{v-FEM,1}=2.03$	$1.45\%$
			$f_{v-N,2}=8.00$	$f_{v-FEM,2}=8.07$
			$f_{v-N,3}=18.00$	$f_{v-FEM,3}=17.91$

. The dynamic properties of the RF were evaluated using two methods of analysis: numerical evaluation and a finite element (FE) model. Both dynamic assessments used the same cross-sectional properties, physical properties of the materials, and structural damping of $1.0\mathrm{\%}$ for all modes. The numerical evaluation used Equation (1) to calculate the natural frequencies of the RF based on their properties, where $E=f_y$, $I$^(*), $m_l=400.05 \mathrm{k}\mathrm{g}·{\mathrm{m}}^{-1}$, and $l=36 \mathrm{m}$ represent the elastic modulus, inertial module (specified in Table 1 for each analysis direction), mass per unit length, and length of the structure, respectively. In addition, simple support conditions were considered. Similarly, the FE analysis modeled the dynamic behavior of the RF using SAP2000 V.25.3.0. The natural frequencies of the RF are reported in Table 1 for both analysis cases, which generated a maximum frequency error of less than $2.04\mathrm{\%}$ in all directions assessed.

$$f_{n,i}={(n\times \pi )}^2·\sqrt{{\displaystyle \frac{{\mathit{f}}_{\mathit{y}}·I}{m_l·l^4}}}$$

(1)

In addition, the first three dynamic modal shapes were determined by the FE model analysis in each assessment direction, as shown in Figure 1. These principal modal shapes reached a cumulative modal mass of more than 95.0%, developing a representative dynamic performance of the RF in all analysis directions. The dynamic behavior of the RF is closely related to the dynamic performance of the human gait, which positioned this structure with a precise dynamic sensitivity to the influence of human-induced gait loads as an international benchmark to evaluate HSI dynamic effects.

2.2. Human-Induced Gait Loads: Test Subjects (TSs)

Fifteen volunteer test subjects (TSs) (6 females and 9 males aged $24.11\pm 4.23\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}$, $167.16\pm 7.09 \mathrm{c}\mathrm{m}$ in height, and $68.10\pm 10.51 \mathrm{k}\mathrm{g}$ of mass) were evaluated in this study. The general information is presented in

Table 2. General information. Identification number (ID), age, height, aproximal center mass (CM), body mass (BM), and sex (S, f→female or M→male) for each test subject.

$\mathbf{I}\mathbf{D}$	$\mathbf{Age}\left[\mathbf{y}\mathbf{r}\right]$	$\mathbf{Height}\left[\mathbf{c}\mathbf{m}\right]$	CM [cm]	$\mathbf{BM}\left[\mathbf{k}\mathbf{g}\right]$	$\mathbf{S}$
1	22.0	169.0	100.0	68.3	M
2	25.0	170.0	95.0	85.6	M
3	27.0	153.0	97.0	66.5	f
4	33.0	170.0	110.0	65.2	M
5	22.0	165.0	97.0	68.6	M
6	23.0	175.0	99.0	81.5	M
7	22.0	175.0	103.0	90.9	M
8	24.0	183.0	110.0	78.3	M
9	24.0	173.0	101.0	75.3	M
10	34.0	162.0	92.0	54.7	f
11	19.0	165.0	95.0	51.1	f
12	20.0	167.0	100.0	62.2	f
13	21.0	157.0	98.0	65.7	f
14	22.0	168.0	102.0	61.6	M
15	21.0	158.0	90.0	62.4	f

. All TSs were free of injuries for at least 1.0 month, had a body mass index (BMI) of less than $25.89 \mathrm{k}\mathrm{g}·{\mathrm{m}}^{-2}$ (average level), and were aged between 19 and 34 years. All tests involving human TSs were performed in accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki) for experiments involving humans and the Universidad del Valle Ethics Committee (Deed of Endorsement Number: 011-2023).

3. Real-Time Hybrid Simulation (RTHS)

3.1. Numerical Substructure (NS): Reference Footbridge Model

The time-history dynamic response of the reference footbridge, described in Section 2.1, under interaction with human-induced gait loads, was assessed using an analytical continuous model represented by a fourth-order partial differential equation (PDE), Equation (2), which constitutes the numerical substructure (NS). In this equation, $u\left(x,t\right)$ represents the deflection of any point along the x-axis of the structure for all the simple times evaluated. Likewise, $p\left(x,t\right)$ includes human-induced gait loads as dynamic perturbations in the system that estimates the reference footbridge, where $m\left(x\right)$ and $c\left(x\right)$ are constants that represent the mass and viscous damping coefficient of the dynamic system, respectively.

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}\left[EI\left(x\right){\displaystyle \frac{{\partial }^{2}u\left(x,t\right)}{\partial x^2}}\right]+m\left(x\right){\displaystyle \frac{{\partial }^{2}u\left(x,t\right)}{\partial x^2}}+c\left(x\right){\displaystyle \frac{\partial u\left(x,t\right)}{\partial t}}=p\left(x,t\right)$$

(2)

The variable separation method was used to resolve the posed equation, which describes the dynamic behavior of a reference footbridge under interaction with human gait. This solution technique incorporates modal decomposition, as shown in Equation (3), which employs vibration models represented by sinusoidal functions as shown in Equation (4). The dynamic motion ($Y_{\widehat{n}}\left(t\right)$) of the n-th in modal coordinates (${\varphi }_{\widehat{n,i}}\left(x\right)$) is determined using Equation (5), where the natural frequencies (${\omega }_{\widehat{n},i}$) and structural damping ratio (${\xi }_i$) correspond to the parameters described in Section 2.1 (Table 1). The mass ($M_{\widehat{n}}$) and modal loads ($P_{\widehat{n}}\left(t\right)$) were calculated using Equations (6) and (7), respectively.

$$u\left(x,t\right)=\sum _{\widehat{n}=1}^{\infty }{\varphi }_{\widehat{n}}\left(x\right)·Y_{\widehat{n}}\left(t\right)$$

(3)

$${\varphi }_{\widehat{n}}\left(x\right)=\mathit{sin}\left({\displaystyle \frac{\widehat{n}\pi }{L}}x\right)$$

(4)

$${\ddot{Y}}_{\widehat{n}}\left(t\right)+2·\xi {·\omega }_{\widehat{n}}{\dot{·Y}}_{\widehat{n}}\left(t\right)+{{\omega }_{\widehat{n}}}^2·Y_{\widehat{n}}\left(t\right)={\displaystyle \frac{P_{\widehat{n}}\left(t\right)}{M_{\widehat{n}}}}$$

(5)

$$M_{\widehat{n}}={\int }_0^L{\left[{\varphi }_{\widehat{n}}\left(x\right)\right]}^2·m\left(x\right)dx$$

(6)

$$P_{\widehat{n}}\left(t\right)={\int }_0^L{\varphi }_{\widehat{n}}\left(x\right)·p\left(x,t\right)dx$$

(7)

$$p\left(x,t\right)=\left[\sum _{i=1}^n{m}_i·\left({\ddot{a}}_i\left(t\right) -g+{\ddot{a}}_{s,j}\left(t\right) \right)\right]·\mathsf{\delta }\left(x-x_{fb}\left(t\right) \right)$$

(8)

Human-induced gait loads in generalized coordinates ($p\left(x,t\right)$) were formulated as the sum of the inertial load of the n-th human body segment affected by the dynamic interaction of the footbridge structural acceleration effects (${\ddot{a}}_{s,j}\left(t\right)$), which act in the $x_{fb}\left(t\right)$ position on the reference structure for all times evaluated, as expressed in Equation (8). In this equation, $g$, $x$, and $x_{fb}$ represent the gravity acceleration (only in the y-axis), longitudinal axis of the reference structure, and current position of the anthropic loads on the footbridge, respectively. The multi-axial structural accelerations generated by the human-induced gait loads were calculated in modal coordinates using Equation (9), which was based on the position of the anthropic load on the structure and n-th vibration modes assessed. The human-induced gait loads expressed in modal coordinates were obtained by replacing Equations (4) and (8) in Equation (7) in order to obtain Equation (10), which depends on the human-induced gait load ($F_{human}\left(t\right)$), modal information, crowd modal mass, and multi-axial structural acceleration (${\ddot{a}}_{s,j}\left(t\right)$). Finally, the modal coordinate equation for solving the dynamic interaction between the structural response and anthropic-induced loads, denoted and proposed by [44] as a general HSI model, is extensively described by Equation (11) and is expressed in matrix notation by Equation (12). In these equations, $S_{\widehat{n}}\left(t\right)$ is equivalent to dynamic modal coordinates (${\varphi }_{\widehat{n}}\left(x\right)$) proposed in Equation (4), where x is the temporal position of the anthropic load on the reference footbridge ($x_{fb}\left(t\right))$.

$${\ddot{a}}_{s,j}\left(t\right)=\sum _{\widehat{n}=1}^n\left[\mathit{sin}\left({\displaystyle \frac{\widehat{n}·\pi ·x_{fb}\left(t\right)}{L}}\right)\right]{·\ddot{Y}}_{\widehat{n}}\left(t\right)$$

(9)

$$P_{\widehat{n}}\left(t\right)=-\left[F_{human}\left(t\right)·\mathit{sin}\left({\displaystyle \frac{\widehat{n}·\pi ·x_{fb}\left(t\right)}{L}}\right)\right]-\left[m_{crwd}·sin\left({\displaystyle \frac{\widehat{n·}\pi ·x_{fb}\left(t\right) }{L}}\right)\right]·{\ddot{a}}_{s,j}\left(t\right)$$

(10)

$${\ddot{Y}}_{\widehat{n}}\left(t\right)+{\displaystyle \frac{m_{crwd}}{M_{\widehat{n}}}}·S_{\widehat{n}}\left(t\right)·\left[\sum _{m=1}^n{S}_m\left(t\right){·\ddot{Y}}_{\widehat{n}}\left(t\right)\right]+2{·\zeta }_{\widehat{n}}{·\omega }_{\widehat{n}}·{\dot{Y}}_{\widehat{n}}\left(t\right)+{{\omega }_{\widehat{n}}}^2·Y_{\widehat{n}}\left(t\right)=-{\displaystyle \frac{1}{M_{\widehat{n}}}}\left[F_{human}\left(t\right)·\mathit{sin}\left({\displaystyle \frac{\widehat{n}·\pi ·x_{fb}\left(t\right)}{L}}\right)\right]$$

(11)

$$\left[M+m_{crwd}·S^T\left(t\right)·S\left(t\right)\right]·\ddot{Y}\left(t\right)+C\dot{·Y}\left(t\right)+K·Y\left(t\right)=-\mathsf{\Phi }\left(x_{fb}\left(t\right)\right)F_{human}\left(t\right)$$

(12)

3.2. Experimental Substructure (NS)

Human gait is a complex biomechanical process characterized by pseudo-cyclic movement of the human body as it progresses through space, which has highly variable behavior. It involves the three-dimensional motion of multiple body segments, which interact through a series of joints with varying degrees of freedom. This dynamic system is driven by muscular forces and influenced by gravitational, external, and inertial effects. In addition, gait pseudo-cycles are typically divided into two primary phases: the stance phase, where one foot is in contact with the ground, providing support and propulsion, and the swing phase, where the opposite leg moves forward to prepare for the next step.

Human gait has been extensively modeled using different approaches in order to represent the precise and natural dynamic behavior of human activity [36,38,44,85]. However, the randomness, variability, and spatiotemporal performance of the human gait on rigid surfaces is difficult because of the lack of capacity and parameterization to represent biological and biomechanical conditions computationally. Therefore, many studies have simplified the dynamic performance of human gait, which in turn reduces the reliability and accuracy of human-induced loads, particularly on vibrating surfaces. In addition, there are few models that include the dynamic interaction effects between human gait-induced loads and flexible surfaces, typically with lightweight structures, such as footbridges, thin slabs, and stairs. This dynamic feedback condition increases the requirements of analytical models, which generates greater uncertainty in the estimation of human gait-induced loads and dynamic structural responses.

In fact, human gait analytical models have limitations in simulating and evaluating the dynamic behavior of the HSI effects on footbridges. Therefore, in this study, each TS gait was assumed to be an experimental substructure with active anthropic loads ($F_{k,j}$, k→analysis direction {vertical (blue), lateral (red), or friction (green)}, and j→selected foot {left or right}), which were induced in the reference footbridge, as detailed in Section 3.5. Likewise, the dynamic structural response (${\ddot{a}}_{s,total}$, total structural acceleration) feedback of the experimental TS gaits, which modifies human-induced gait loads on the structure, generates a dynamic interaction in real-time, as schematically shown in Figure 2. This methodology allows the measurement and incorporation of real human gait-induced loads under dynamic HSI effects, which increases the accuracy of the structural response simulation and acquires anthropic loads. In addition, each TS used a safety lifeline to ensure the integrity and sense of security during the gait tests and was instrumented with a pair of insole pressure sensors (Stridalyzer Insight developed) by ReTiSense [17]. Moreover, active motion markets were positioned in the body joints to measure the tridimensional displacements using two motion capture systems: sagittal and coronal planes, as experimentally shown in Figure 3 (top). This indirect and direct measurement allows to generate a comprehensive dataset about HSI effects on reference footbridge.

3.3. Transfer System: Human–Structure Interaction Multi-Axial Test Framework (HSI-MTF)

Transfer systems (TrSs), as components in real-time hybrid simulations, are intended to accurately and consistently exchange information between the experimental and numerical substructures within the simulation loop. This system ensures that the dynamic interaction between the physical specimen (anthropic load induced by crowd gait, $F_{k,j}$) and the computational model (dynamic response of the reference footbridge, ${\ddot{a}}_{s,total}$) is synchronized in real-time. In order to ensure the fidelity of the real-time synchronization requirements in the simulation, this study adopted the human–structure interaction multi-axial test framework (HSI-MTF) as the TrS prototype, which was used in previous studies of human gait on surfaces with lateral movements [17] and completely described its development by Castillo et al. [28].

The HSI-MTF is located in the Laboratory of Seismic Engineering and Structural Dynamics at the Universidad del Valle, Cali, Colombia, and was developed to assess the multi-axial dynamic loads induced by pedestrians walking on surfaces with lateral and vertical movements. Castillo et al. [28] discomposed the HIS-MTF, shown and detailed in Figure 3, into five principal components: the unidirectional shake table to induce lateral movements with displacement of $\pm 80.0 \mathrm{m}\mathrm{m}$, the lateral support system to support the dynamic loads generated, a sole brand treadmill model F65 with an effective action area of 72.0 cm $\times$ 150.0 cm and operating speeds ranging from 0.50 to 10.0 $\mathrm{m}·{\mathrm{s}}^{-1}$, a load acquisition system that was developed to measure the tridimensional anthropic loads generated by the human gait, and the vertical hydraulic system to induce vertical movements with displacement of $\pm 75.0 \mathrm{m}\mathrm{m}$. In addition, the upper section of Figure 3 shows how the human gait is developed on HSI-MTF, and the TSs were instrumentalized to measure the vertical load wearing pressure insole and joint displacement using active motion markers of a low-cost motion capture system.

The lateral movement system implements a robust model control type $H_{\infty }$ coupled with a Kalman filter, and the vertical movement system incorporates a proportional-integral-derivative (PID) model and an anticipatory function control. Both were integrated to ensure the fidelity, stability, and accuracy of multi-axial displacement tracking. These requirements are necessary for the proper development and stabilization of the simulation in real-time.

3.4. Experimental Setup

The experimental and numerical substructures and transfer system were interrelated according to the schematic diagram shown in Figure 4 to develop the RTHS technique. The analytical substructure was affected by the input signal, which corresponds to the human-induced measured gait loads, $F_{human,m}(x_p,t)$. Simultaneously, the numerical footbridge responds to this spatiotemporal load excitation according to its dynamic properties, which generates the dynamic response $U_r(x_p,t)$. Structural accelerations were multi-axially reproduced using HSI-MTF, ${\ddot{U}}_m(x_p,t)$, and were induced in the experimental test subjects, as described in Section 3.5. The reaction forces exerted by human gait under dynamic HSI effects were measured by an instrumentation system coupled to the HSI-MTF, which is composed of multi-axial cell loads. The human-induced gait loads, $F_{human,r}(x_p,t)$, represent the total feedback force, $F_{human,m}(x_p,t)$, of the numerical substructure, which closes the dynamic loop. This process was programmed in Simulink software (V.21.0) and executed with a SpeedGoat processor using a host computer with the Xpc protocol of MATLAB (V.21.b) and a sampling frequency of 1024 Hz. SpeedGoat has a high processing capacity to guarantee the real-time evaluation of the hybrid simulation.

3.5. Experimental Input: Evaluation of HSI Effects

Gait tests were carried out in this study for each TS, which consisted in the dynamic evaluation of HSI effects due to crossing of crowd-induced gait loads over the reference footbridge to three constant typical gait velocities: $1.20 \mathrm{m}·{\mathrm{s}}^{-1}$, $1.50 \mathrm{m}·{\mathrm{s}}^{-1}$, and $1.80 \mathrm{m}·{\mathrm{s}}^{-1}$. In addition, each TS-induced experimental gait load acted on the centroid of the reference footbridge, which constituted the numerical prototype, using HSI-MTF as the transfer and instrumentation system, as described in Section 3.3.

4. Results and Discussion

4.1. Human-Induced Gait Load Performance on the Reference Footbridge

Dynamic interactions between human activities and lightweight structural responses have been reported and have significantly increased in recent years, which constitutes a risk to condition the structural serviceability due to the coupling of dynamic changes in human gait behavior and structural vibrations. Therefore, in this section, the evaluation of pedestrian-induced gait loads in direct connection with reference footbridge responses is divided into two main assessment directions: lateral and vertical loads.

4.1.1. Lateral Loads

The lateral gait loads induced by all TSs on a rigid surface were evaluated for 30.0 s. The induced lateral load ($L_L$) was normalized with respect to the weight ($W_0$) of each ST, yielding an average normalized lateral load of approximately 2.0% and a maximum of 2.9%. These human gait loads induced by TSs are presented in Figure 5 (top left) for a 3.0 s range and calculate the average normalized gait loads (${\displaystyle \raisebox{1ex}{$\overline{L_L}$}\!\left/ \!\raisebox{-1ex}{$W_0$}\right.}$) with the respective standard deviation ($\pm \sigma$). In addition, the lateral frequency response assessment of the anthropic loads on rigid surfaces determined a fundamental frequential contend of $f_1=1.0 \mathrm{H}\mathrm{z}$, with relevant frequential content in $f_2=2.0 \mathrm{H}\mathrm{z}$, as shown in Figure 5 (top right). However, the lateral frequency response assessment of pedestrian-induced gait loads on the reference footbridge showed a slight increase in the damping of the first fundamental frequency of the gait load compared to the increase in the gait velocity of the TSs (green band, Figure 5, middle). Moreover, a frequential content was developed by TSs close to the second fundamental frequency, denoted in this study as $f_{HSI}\cong 1.8 \mathrm{H}\mathrm{z}$ (purple band, Figure 5, middle), which exerts a stabilizing action on the ST during the crossing of the reference structure, proportional to the test speed. Likewise, the lateral pedestrian-induced gait loads on the reference structure by TS₁ had a similar behavior, initiating and terminating the crossing of the structure, as shown in Figure 5 (bottom), for all gait velocities of the TSs evaluated. However, an increase in lateral normalized pedestrian-induced gait loads was identified in the middle third of the reference footbridge due to dynamic HSI effects, which increased the magnitude of the walking loads by more than 300% out of phase compared to the lateral structural response at the action point of the pedestrian. Finally, the peak lateral pedestrian-induced gait loads were determined for the three gait velocities evaluated, which reached overage-normalized gait loads and standard deviations of 9.86% and 1.33%, respectively, as presented in Figure 6 (left).

4.1.2. Vertical Loads

Similarly, the vertical gait loads induced by all TSs on a rigid surface were evaluated for 30.0 s. The induced lateral load ($L_L$) was normalized with respect to the weight ($W_0$) of each ST, yielding an average normalized lateral load of approximately 105.0% and a maximum of 125.0%. These pedestrian-induced gait loads by TSs are presented in Figure 7 (top left) for a 1.5 s range and the average normalized gait loads (${\displaystyle \raisebox{1ex}{$\overline{L_L}$}\!\left/ \!\raisebox{-1ex}{$W_0$}\right.}$), with the respective standard deviation ($\pm \sigma$). In addition, the vertical frequency response assessment of the gait loads on the rigid surface determined a fundamental frequency band around $f_1=1.75 \mathrm{H}\mathrm{z}$, as shown in Figure 7 (top right). However, the vertical frequency response assessment of pedestrian-induced gait loads on the reference footbridge demonstrated that a frequential content was developed by TSs close to the first fundamental frequency, called $f_{HSI}\cong 1.8 \mathrm{H}\mathrm{z}$ (purple band, Figure 7, middle), which allows a stabilizing action on the ST during the crossing of the reference structure, increasing its effects according to the test velocity. On the other hand, changes in vertical normalized pedestrian-induced gait load by TS₁ were identified at the different points of the reference footbridge due to dynamic HSI effects, which decreased ($\approx -20.0\mathrm{\%}$) or increased ($\approx +50.0\mathrm{\%}$) the anthropic loads in direct relation with the vertical structural response at the action point of the pedestrian compared to the overall gait-induced loads on the rigid surface. Finally, the peak vertical pedestrian-induced gait loads were determined for the three gait velocities evaluated, which reached overage-normalized gait loads and standard deviations of 156.56% and 15.58%, respectively.

4.2. Reference Footbridge Performance, Including HSI

The spatio-temporal dynamic behavior structural responses of the reference footbridge under HSI effects due to human-induced gait loads, described in Section 2.1, were experimentally evaluated using RTHS as a testing methodology, as described in Section 3. In order to show the complete structural response during the crossing of pedestrians (TSs), the structural displacement and acceleration under gait loads induced by TS₁ at the contact point between the pedestrian and the footbridge and in the middle span were determined and processed, as shown in Figure 8 and Figure 9. Lateral structural accelerations at the contact point for the three gait velocities, evaluated in time history and frequency response (1.20 $\mathrm{m}·{\mathrm{s}}^{-1}$—orange, 1.50 $\mathrm{m}·{\mathrm{s}}^{-1}$—blue, and 1.80 $\mathrm{m}·{\mathrm{s}}^{-1}$—green), determined an inverse relation between structural response times and gait velocity of the TSs. Likewise, the fundamental frequency that composes the structural response was close to 1.0 Hz, as shown in Figure 8 (top left). A similar behavior was observed when evaluating the lateral structural displacement at the contact point of the ST₁ and its frequency response (Figure 8, top right). In addition, the lateral structural displacement and acceleration in the structural middle span exhibited frequential behavior similar to the fundamental frequency around 1.0 Hz, and a free vibration behavior of the footbridge was identified once the pedestrian left the structure, allowing the identification of the structural damping, as presented in Figure 8 (bottom). It is important to note that while an approximately constant step length was maintained during the experiments, this step length was not restricted and could vary from step to step. This variability in step length arises due to the coupling between the test subjects and the structure, influenced by the effects of human–structure interaction (HSI). Such interaction can result in longer or shorter step lengths depending on the synchronization and adaptation of the subject to the movement of the footbridge.

Similarly, vertical structural accelerations at the contact point for the three gait velocities were evaluated in time history and frequency response, which determined a direct relation between structural response times and gait velocity of the TS₁ (1.20 $\mathrm{m}·{\mathrm{s}}^{-1}$—orange, 1.50 $\mathrm{m}·{\mathrm{s}}^{-1}$—blue, and 1.80 $\mathrm{m}·{\mathrm{s}}^{-1}$—green). Likewise, the fundamental frequency that composes these acceleration structural responses was close to 2.0 Hz, as shown in Figure 9 (bottom left). A similar behavior was observed when evaluating the vertical structural displacement at the contact point of ST₁ and its frequency response (Figure 9, bottom right). Additionally, the vertical structural displacement and acceleration in the structural middle span exhibited frequential behavior similar to the fundamental frequency around 2.0 Hz, with some content frequency due to dynamic HSI effects (close to 1.80 Hz) and harmonic behavior (around 4.0 Hz). The free vibration behavior of the footbridge was identified once the pedestrian left the structure, which allowed the identification of the structural damping to be assigned, as presented in Figure 9 (bottom). It is also important to note that the vertical structural response was found to be directly proportional to the gait velocity. Significant differences in both accelerations and displacements were observed, with vertical accelerations increasing by more than 70% and displacements exceeding 100% when comparing tests conducted at different gait velocities. These findings underscore the influence of gait velocity on the dynamic response of the footbridge, reinforcing the critical importance of considering such effects when evaluating structural performance under pedestrian-induced loads.

The assessment of the peak dynamic response of the reference footbridge was carried out using a hybrid spider–violin plot for each gait velocity evaluated by TSs, as shown in Figure 10. This schematic analysis allows the identification, for $n$ independent events (each branch), of the normal probability density function (PDF), using Equation (13), of the peak dynamic responses ($P_{SR}$) normalized by the maximum value in the respective event. This equation uses the average (${\mu }_{P_{SR}}$) and standard deviation (${\sigma }_{P_{SR}}$) of the normalized peak structural response generated during the gait tests on the reference footbridge, including the dynamic HSI effects using the RTHS technique.

$${PDF}_{P_{SR}}=f\left(P_{SR}|{\mu }_{P_{SR}},{\sigma }_{P_{SR}}\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}{e}^{{\displaystyle \frac{-{(P_{SR}-{\mu }_{P_{SR}})}^2}{2{{\mu }_{P_{SR}}}^2}}},\mathrm{for}\forall P_{SR}\in \mathbb{R}$$

(13)

Each branch presents the respective PDF shape with punctual events that were composed, and the average structural response was identified. In addition, the colored shadow represents the first quantile of the standard deviation of the data points that compose the specific event. This approach allowed the identification of a higher dispersion in the vertical displacement of the middle span of the structure and contact point of the TSs compared to the same response in the lateral direction, with percentage differences of mean normalized values of less than 10.0% for all gait velocities, as shown in Figure 10 (top). Likewise, a similar behavior was identified with the lateral and vertical accelerations; however, the dispersion of the vertical acceleration in the middle span of the structure and contact point of the TSs was significant because of the dynamic HSI effects calculated in the frequency response. Moreover, the percentage differences of the mean normalized values of accelerations in the middle span of the structure and contact point of the TSs were approximately 15.0% for all gait velocities, which were always major for the lateral direction evaluated, as shown in Figure 10 (bottom).

4.3. RTHS Performance

RTHS introduces errors due to factors inherent to its methodology, such as discrepancies between the numerical simulation (NS) model and the physical structure, electronic noise in the experimental signals of the experimental substructure (ES), and the dynamics of actuators within the transfer systems, among others [86,87]. Consequently, various indices have been proposed to assess the accuracy of RTHS, accounting for potential deviations between the reference and experimental signals. Many of these evaluation metrics emphasize the local response during hybrid simulations (HS), particularly focusing on the physical component and its interaction with its numerical counterpart. The evaluation of local performance was based on the synchronization of the boundary conditions between the numerical response ($u_{i,d}^{*}$) sent to the actuator and the displacement achieved in the physical component ($u_{i,m}^{*}$). The normalized root mean square in the experiment (NRMSE) can be used, according to Equation (14), to obtain a single value representing the difference between the actuator signals in the time domain. In addition, frequency domain comparison, Equations (15)–(19) are used, corresponding to the frequency evaluation index (FEI), which compares the fast Fourier transform (FFT) of the desired signal $u_{i,d}^{*}$ with that of the measured $u_{i,m}^{*}$ at the actuator, the dominant frequency during RTHS $f^{eq}$, the generalized amplitude $A_0$, and the delay $\delta$, respectively [88].

$${NRMS}_{error}=\sqrt{{\displaystyle \frac{\sum _{j-1}^N{[u_{i,d}^{*}\left(j\right)-u_{i,m}^{*}(j\left)\right]}^2}{\sum _{j-1}^N{\left[u_{i,d}^{*}\left(j\right)\right]}^2}}}$$

(14)

$$FEI={\sum }_{j=1}^N\left({\displaystyle \frac{u_{i,m}^{*}\left(j\right)}{u_{i,d}^{*}\left(j\right)}}·{\displaystyle \frac{{‖u_{i,d}^{*}\left(j\right)‖}^l}{\sum _{j=1}^P{‖u_{i,d}^{*}\left(j\right)‖}^l}}\right)$$

(15)

$$f^{eq}={\displaystyle \frac{\sum _{j-1}^N\left({‖u_{i,d}^{*}\left(j\right)‖}^l·f_i\right)}{\sum _{j-1}^N{‖u_{i,d}^{*}\left(j\right)‖}^l}}$$

(16)

$$A_0=‖FEI‖$$

(17)

$$\varnothing =arctan\left[{\displaystyle \frac{Im\left(FEI\right)}{Re\left(FEI\right)}}\right]$$

(18)

$$\delta =-{\displaystyle \frac{\varnothing }{2·\pi ·f^{eq}}}$$

(19)

In addition, the global indices consider the interaction between the numerical and experimental substructures, and they were evaluated in the time domain, quantifying the largest difference between the calculated and measured displacements using $e_{DM}$ and $e_{MD},$ as shown in Equations (20) and (21) [85].

$$e_{DM}={\displaystyle \frac{\left|{\left|u_{i,d}^{*}\right|}^{max}-{\left|u_{i,m}^{*}\right|}^{max}\right|}{{\left|u_{i,m}^{*}\right|}^{max}}}$$

(20)

$$e_{MD}={\displaystyle \frac{{\left|u_{i,d}^{*}-u_{i,m}^{*}\right|}^{max}}{{\left|u_{i,m}^{*}\right|}^{max}}}$$

(21)

All indexes were calculated for all test subjects and all gait velocities evaluated. The mean values per each velocity are listed in

Table 3. RTHS evaluation indexes: vertical and lateral transfer systems.

Test	${\mathit{N}\mathit{R}\mathit{M}\mathit{S}}_{\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}$ [%]	$\mathit{F}\mathit{E}\mathit{I}$	${\mathit{f}}_{\mathit{e}\mathit{q}}$ [Hz]	${\mathit{A}}_0$	${\mathit{e}}_{\mathit{D}\mathit{M}}$ [%]	${\mathit{e}}_{\mathit{M}\mathit{D}}$ [%]	$\mathit{\delta }$ [ms]
Vertical Transfer System
$1.20\mathrm{m}·{\mathrm{s}}^{-1}$	2.910 $\pm$0.80	0.985–0.012i $\pm$0.01	2.065 $\pm$0.03	0.985 $\pm$0.01	0.540 $\pm$0.30	2.930 $\pm$0.30	2.010 $\pm$0.10
$1.50\mathrm{m}·{\mathrm{s}}^{-1}$	2.900 $\pm$0.11	0.987–0.013i $\pm$0.01	2.063 $\pm$0.04	0.987 $\pm$0.01	0.560 $\pm$0.20	2.940 $\pm$0.30	2.045 $\pm$0.10
$1.80\mathrm{m}·{\mathrm{s}}^{-1}$	2.850 $\pm$0.16	0.989–0.012i $\pm$0.01	2.065 $\pm$0.03	0.989 $\pm$0.01	0.560 $\pm$0.30	2.880 $\pm$0.30	2.085 $\pm$0.10
Lateral Transfer System
$1.20\mathrm{m}·{\mathrm{s}}^{-1}$	1.450 $\pm$0.43	1.005–0.019i $\pm$0.01	1.034 $\pm$0.01	1.005 $\pm$0.01	0.310 $\pm$0.28	1.860 $\pm$0.40	1.700 $\pm$0.30
$1.50\mathrm{m}·{\mathrm{s}}^{-1}$	1.380 $\pm$0.36	1.004–0.011i $\pm$0.01	1.045 $\pm$0.01	1.004 $\pm$0.01	0.310 $\pm$0.29	1.770 $\pm$0.37	1.600 $\pm$0.20
$1.80\mathrm{m}·{\mathrm{s}}^{-1}$	1.350 $\pm$0.19	1.002–0.011i $\pm$0.01	1.059 $\pm$0.02	1.002 $\pm$0.01	0.240 $\pm$0.15	1.670 $\pm$0.18	1.600 $\pm$0.20

. All tests developed a high level of horizontal tracking, with a generalized amplitude $A_0$ around 1.0 and global errors lower than 2.95%, which allowed us to infer that the vertical and lateral transfer system, integrated in HSI-MTF, has high performance and allows performing RTHS with a high level of accuracy, which demonstrates its optimal performance during the RTHS to simulate general HSI effects.

5. Future Works and Perspectives

The integration of structural health monitoring (SHM) into real-time hybrid simulations (RTHS) of human–structure interaction (HSI) presents a promising avenue for future research. Although current SHM technologies allow for continuous monitoring of footbridges, several aspects remain underdeveloped. Specifically, the future development of SHM in RTHS should focus on improving sensor accuracy, expanding sensing coverage, and developing real-time feedback loops that can more effectively capture the dynamic effects induced by pedestrian loads. These improvements would enhance the fidelity of RTHS in capturing the complex, nonlinear behavior of footbridges under varying HSI scenarios [89,90].

Moreover, an essential aspect that future research must address is the recovery of missing measurement data, a common issue in SHM systems. While various methods, such as statistical techniques and machine learning algorithms, have been developed to mitigate the impact of data loss, further advancements are necessary. Future research should explore the application of deep learning models and advanced Kalman filtering methods tailored to the unique demands of RTHS applications. These approaches could offer higher precision in data recovery and ensure that missing data do not compromise the accuracy of the simulation results [91,92].

Additionally, the long-term objective of integrating SHM and data recovery methods into HSI studies should be the development of autonomous systems capable of real-time damage detection and self-healing mechanisms for data recovery. This would allow for real-time decision-making during RTHS, enabling proactive interventions to mitigate excessive vibrations or structural damage. The continuous improvement of SHM and data recovery technologies will be instrumental in ensuring the resilience, safety, and sustainability of footbridge structures as urban environments continue to evolve.

6. Conclusions

In this study, the dynamic performance assessment of a reference structure with accuracy multi-axial dynamic sensitivity including human–structure interaction effects was performed using the RTHS technique for the three gait velocities of the pedestrian: 1.20, 1.50, and 1.80 $\mathrm{m}·{\mathrm{s}}^{-1}$. The main conclusions are as follows.

(1)

The assessment of human-induced gait loads on the reference footbridge revealed significant interactions between pedestrian activities and structural responses. Both lateral and vertical loads exhibited dynamic coupling, with frequency content closely aligned to the footbridge’s natural frequencies, particularly in the range of 1.0 Hz and 1.8 Hz. This dynamic synchronization, particularly during higher gait velocities, amplified structural vibrations, especially in the middle span, leading to increased loading by up to 300.0% in lateral directions. Vertical loads showed similar variation, with substantial amplification and attenuation depending on the gait velocity and footbridge position. These findings highlight the critical role of human–structure interaction (HSI) in dynamically sensitive structures, underscoring the importance of accounting for these interactions in the design and evaluation processes to ensure serviceability and safety under regular pedestrian use.

(2)

The performance analysis of the reference footbridge under human–structure interaction (HSI) effects demonstrated significant dynamic responses in both the lateral and vertical directions, particularly at higher pedestrian velocities. Lateral accelerations and displacements exhibited a fundamental frequency around 1.0 Hz, whereas vertical responses peaked near 2.0 Hz. The coupling of pedestrian-induced loads with footbridge dynamic properties led to increased vibration amplitudes, especially in the middle span. Lateral accelerations showed a dispersion of approximately 15.0%, whereas vertical accelerations exhibited higher variability, with dispersions reaching up to 20.0%. Moreover, the percentage difference between lateral and vertical displacements was less than 10.0% for all evaluated gait velocities. These findings underscore the critical importance of integrating HSI considerations into footbridge design because human activity can induce substantial variations in structural performance, potentially affecting serviceability and user comfort.

(3)

The experimental assessment was performed using the RTHS technique, taking an analytical continuous model (through a fourth-order partial differential equation) of the reference structure with accurate multi-axial dynamic sensitivity as a numerical substructure and the gait of the test subjects as an experimental substructure. All cases had high-level tracking for the horizontal and vertical transfer systems, which developed a generalized amplitude A₀ around 1.0 and global errors lower than 2.95%. Furthermore, HSI-MTF had delays of less than 2.10 ms in all directions evaluated with an NRMS of less than 2.95%, employing a robust H_∞ type controller in the lateral axis and a PID setup with a phase anticipator in the vertical axis. The above results indicate that the RTHS exhibits high fidelity and accuracy.