Assessment of Vertical Dynamic Responses in a Cracked Bridge under a Pedestrian-Induced Load

Abstract

Cracks, common indicators of deterioration in bridge frameworks, frequently stem from wear and rust, leading to increased local flexibility and changes in the structure’s dynamic behavior. This study examines how these cracks affect the dynamics of footbridges when subjected to loads generated by walking individuals. The pedestrian is modeled as a linear oscillator, while the cracked bridge is represented by a simply supported beam following Euler–Bernoulli’s theory. The use of the Dirac delta function allows for the precise representation of the localized stiffness reduction at the crack location, facilitating the calculation of analytical expressions for the beam’s vibration modes. The research suggests that the presence of cracks minimally affects the bridge’s mid-span displacement. However, with a limited depth of cracks, the appearance of cracks notably amplifies the mid-span acceleration amplitude of the bridge, leading to a pronounced concentration of energy at the third natural frequency of the bridge in the acceleration spectrum. As the depth and number of cracks increase, the acceleration amplitude continues to decrease, but the corresponding spectrum remains almost unchanged. The study’s outcomes enhance the comprehension of how cracks affect the performance of bridge structures when subjected to loads from pedestrians, offering insights for the monitoring and evaluation of the condition of cracked footbridges.

1. Introduction

Infrastructure such as pedestrian bridges, walkways, bleachers, and stairways frequently experience dynamic forces due to everyday activities like strolling, sprinting, and leaping [1,2]. Advances in construction materials and methodologies have enabled the creation of more streamlined and elongated civil infrastructures, characterized by lower stiffness and reduced mass. Consequently, the inherent frequency of these contemporary structures often falls below the 5 Hz mark, aligning with the spectrum of frequencies associated with human motion, potentially leading to significant vibration challenges [3,4]. To address the risk of vibrations triggered by human activities, it is crucial to precisely determine the natural frequencies and the resultant acceleration levels [5,6]. Moreover, numerous investigations have highlighted the value of assessing the interaction between humans and structures to accurately gauge the impact of vibrations [7,8].

The dynamic relationship between pedestrians and the bridges they traverse is a critical aspect of human–structure interaction, emphasizing the reciprocal influence between foot traffic and the structural integrity of footbridges. This focus is on understanding the effects of pedestrian activities, including walking and running, on the bridge’s dynamic behavior. This interaction is vital as it influences the safety, comfort, and functionality of the bridge [9,10,11,12,13,14]. Historically, research has often assumed bridges to be in pristine condition. However, in practice, bridges frequently develop cracks due to wear and corrosion, which can alter the bridge’s stiffness and its response to vibrations. Given that cracked bridges are a common reality, examining how these compromised structures react to the forces exerted by pedestrians is essential for practical engineering considerations.

The initial exploration into the impact of localized imperfections, particularly on the dynamic characteristics of structural elements, was spearheaded by Kirmsher [15] and Thomson [16]. Their pioneering studies established approaches for simulating cracks caused by fatigue and for determining the resonant frequencies of beams with such defects. Building on this, Friswell and Penny [17] assessed a range of crack simulation methods, ultimately determining that straightforward models that used beam elements to depict crack-induced flexibility offered adequate precision. These cracks can be envisioned as elastic joints that reconnect the disjointed parts of a beam, a concept further elaborated in the subsequent literature [18,19,20,21].

The scholarly discourse on beam structures encompasses various techniques for modeling cracks, which are typically categorized into three main approaches: reducing local stiffness [22], using spring models or elastic hinges [23,24], and applying finite element models [25,26]. Researchers have explored the vibrational characteristics of beams with open cracks under moving loads, considering the crack as a rotational spring that connects two undamaged parts of the beam [27]. That research has shown that the presence of an open crack can increase beam deflections and cause a pronounced change in the slope of the beam’s deflection curve at the crack location. Further studies [20] used rotational springs to represent beam damage, with their compliance determined by the principles of linear elastic fracture mechanics. These studies employed the transfer matrix method to calculate the structural response, taking into account the actual mass distribution along the beam. The results indicated that experimental responses often exceed the values predicted by theoretical calculations. Inhomogeneous beams with open cracks under axial force and moving loads have also been studied [28], revealing that while the crack significantly affects the natural frequencies, the dynamic deflection is less influenced by the crack and more by the axial compression. Real-world cracks can open during the tensile phase and close during the compressive phase, leading to variations in structural stiffness as the material undergoes cyclic deformation [29]. This cyclic behavior results in abrupt changes in structural stiffness when the crack opens and closes. A semi-analytical method for analyzing the nonlinear vibrations of a beamlike bridge with such cyclically closing cracks under the load of a moving vehicle was introduced [30]. This method showed that the bridge’s vibration amplitude lay between the conditions of having no cracks and the cracks being fully open.

Despite the extensive research on the dynamic response of cracked bridges under moving mass loads, the response of cracked bridges under pedestrian loads has received little attention. Pedestrian-induced loads differ from moving mass loads in that they generate periodic varying loads not only vertically but also laterally and longitudinally on the structure. The magnitude of these loads depends on the pedestrian’s step frequency, walking speed, and stride length. Experiments revealed that the vertical excitation force generated during normal walking by pedestrians had two peak values and one minimum value, and both the stride and the force peak values increased with an increase in speed [31]. This highlights the complexity of pedestrian loads and their dependence on many factors. Controlling any one variable among step frequency, walking speed, or stride length will result in different relationships between that variable and the others. Moreover, as the walking speed increases, the variability of the vertical and lateral forces increases. Currently, there is limited understanding of how cracks affect the dynamic behavior of pedestrian bridges. The specific impact of cracks on the vibrations of footbridges under pedestrian loads is not extensively covered in the current body of research.

This study investigates the effects of cracks on the vertical dynamic behavior of a bridge under the forces generated by pedestrians. The pedestrian’s influence is depicted through a linear oscillator model [5,32], and the bridge, marred by cracks, is conceptualized as a simply supported beam following Euler–Bernoulli’s theory. It is assumed that the cracks remain open under all stress conditions. These localized imperfections alter the beam’s bending rigidity, which is quantitatively captured using the Dirac delta function. The core equation for the interaction between the pedestrian and the bridge with cracks is derived to assess how the location, depth, number of the cracks, and velocity of the pedestrian influence the bridge’s dynamic characteristics. The objective of this research is to discern the variations in the bridge’s response when compared to an uncracked condition. The research findings contribute to the understanding of the impact of cracks on the structural performance of bridges under pedestrian loads and provide references for the health monitoring and assessment of cracked pedestrian bridges.

The structure of the subsequent parts of this paper is outlined as follows: In Section 2, we establish and solve the coupled dynamic equations of the pedestrian–crack–bridge system; in Section 3, the correctness of the pedestrian–crack–bridge coupled equations is verified by comparing with results from the existing literature; in Section 4, the impact of crack location, depth, and quantity on the bridge’s dynamic behavior is discussed through numerical calculations; conclusions and applications are provided in Section 5.

2. Dynamic Equation of the Pedestrian–Cracked Bridge Interaction System

The pedestrian–cracked bridge coupling system is shown in Figure 1. The model of the bridge with cracks is depicted as a simply supported beam following Euler–Bernoulli’s theory, spanning a length of L and featuring a rectangular cross-section with dimensions of width b and height h. The pedestrian is modeled using a linear oscillator model [5,32], in which the pedestrian’s body mass is divided into two parts: the sprung mass $m_a$ and the unsprung mass $m_s$. The parameters $k_p$ and $c_p$ represent the spring constant and the damping coefficient of the pedestrian’s equivalent mechanical system, respectively. For analytical convenience, it is assumed that the dynamic load, designated as $F_R$, is exclusively a result of the pedestrian’s vertical displacement. This implies that horizontal and longitudinal movements of the body do not contribute to the vertical force exerted on the ground. The pedestrian moves at a consistent speed, $v_p$, which is constant from one step to the next.

The equations governing the pedestrian-cracked bridge interaction system are derived using D’Alembert’s principle, which can be succinctly stated as:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^2}{\partial x^2}}\left(EI\left(x\right){\displaystyle \frac{{\partial }^{2}y(x,t)}{\partial x^2}}\right)+m{\displaystyle \frac{{\partial }^{2}y(x,t)}{\partial t^2}}+c{\displaystyle \frac{\partial y(x,t)}{\partial t}}=F_R\left(t\right)\delta (x-v_{p}t),\hfill \\ \hfill & F_R\left(t\right)=F_W-[m_s{\ddot{z}}_s+c_p({\dot{z}}_s-{\dot{z}}_a)+k_p(z_s-z_a)],\hfill \\ \hfill & m_a{\ddot{z}}_a+c_p({\dot{z}}_a-{\dot{z}}_s)+k_p(z_a-z_s)=0,\hfill \end{array}$$

(1)

where $y(x,t)$ represents the vertical displacement of the cracked bridge, the parameters m and c denote the mass per unit length and the damping coefficient of the bridge, respectively. The expression $F_R\left(t\right)$ and the Dirac delta function $\delta (x-v_{p}t)$ indicate the dynamic load applied by the pedestrian and its position along the bridge at any given time, respectively. In addition, $EI\left(x\right)$ is the flexural stiffness of the cracked beam, which has the following form [33]:

$$EI\left(x\right)=E{I}_0\left[1-\sum _{i=1}^n{\widehat{\kappa }}_i\delta \left(x-x_{0i}\right)\right],$$

(2)

where the term $E{I}_0$ denotes the uniform bending stiffness of the pristine beam, while ${\widehat{\kappa }}_i$ signifies a parameter that captures the severity of the ith crack. The function $\delta (x-x_{0i})$ represents the Dirac delta function, which models the localized reduction in bending stiffness at the position of the ith crack, located at the abscissa $x=x_{0i}$. The parameters ${\widehat{\kappa }}_i$ that are multiplied by the Dirac deltas correspond to the rotational stiffness of a hypothetical internal spring that mimics the crack’s behavior. Details on how Equation (2) can be used to describe the flexural stiffness of a cracked beam are briefly provided in Appendix A.

Additionally, the symbols $z_a$ and $z_s$ are used to denote the displacements of the sprung and unsprung masses, respectively. The notation of a dot above $z_a$ and $z_s$ signifies the rate of change of these displacements with time t. The relationship $z_s\left(t\right)=y(x-v_{p}t,t)$ always holds true when the pedestrian is walking on the bridge. $F_W$ indicates the force exerted by the pedestrian due to walking, which is characterized according to the definitions provided in the referenced literature [34]:

$$F_{\mathrm{W}}=(m_{\mathrm{a}}+m_s)\mathrm{g}\sum _{j=1}^4{\beta }_{\mathrm{w}j}sin\left(2\mathrm{j}\pi f_{w}t-{\vartheta }_{\mathrm{w}j}\right),$$

(3)

where the symbols ${\beta }_{\mathrm{w}j}$ and ${\vartheta }_{\mathrm{w}j}$ represent the dynamic amplification factor and the phase difference for the jth harmonic component, respectively. The term $f_w$ denotes the frequency associated with the pedestrian’s walking motion. The letter g signifies the acceleration due to gravity.

By employing a dimensionless coordinate $\xi ={\displaystyle \frac{x}{L}}$ in Equation (1), and incorporating the flexural stiffness model from Equation (2), the dynamic differential equation for a bridge with multiple cracks subjected to a pedestrian’s load can be formulated as:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^2}{\partial {\xi }^2}}\left(\left[1-\sum _{i=1}^n{\kappa }_i\delta \left(\xi -{\xi }_{0i}\right)\right]{\displaystyle \frac{{\partial }^{2}y(\xi ,t)}{\partial {\xi }^2}}\right)+{\displaystyle \frac{m{L}^4}{E{I}_0}}{\displaystyle \frac{{\partial }^{2}y(\xi ,t)}{\partial t^2}}\hfill \\ \hfill & +{\displaystyle \frac{c{L}^4}{E{I}_0}}{\displaystyle \frac{\partial y(\xi ,t)}{\partial t}}={\displaystyle \frac{L^4}{E{I}_0}}{F}_R\left(t\right)\delta (x-vt),0\le \xi ={\displaystyle \frac{x}{L}}\le 1,\hfill \end{array}$$

(4)

where ${\xi }_{0i}=x_{0i}/L$, ${\kappa }_i={\widehat{\kappa }}_i/L$ and $v=v_p/L$.

Utilizing the modal superposition technique, the cracked bridge’s vibration can be articulated as:

$$y(\xi ,t)=\sum _{j=1}^{\infty }{u}_j\left(t\right){\varphi }_j\left(\xi \right),$$

(5)

where the jth mode shape is represented by ${\varphi }_j\left(\xi \right)$, and the jth normal coordinate is denoted by $u_j\left(t\right)$. By inserting Equation (5) into Equation (4), a differential equation describing the modal displacements is derived. With some basic algebraic manipulations, this equation can be transformed into the subsequent format:

$${\varphi }_j^{⁗}\left(\xi \right)-{\eta }_j^4{\varphi }_j\left(\xi \right)=D_j\left(\xi \right),j=1,2,3,\cdots$$

(6)

where ${\eta }_j^4={\omega }_j^{2}m{L}^4/\left(E{I}_0\right)$, ${\omega }_j$ is the jth circle frequency of the cracked beam; The prime symbol denotes the derivative with respect to $\xi$, and

$$D_j\left(\xi \right)=\sum _{i=1}^n{\kappa }_i[{\varphi }_j^{⁗}\left(\xi \right)\delta (\xi -{\xi }_{0i})+2{\varphi }_j^{‴}\left(\xi \right){\delta }^{\prime }(\xi -{\xi }_{0i})+{\varphi }_j^{″}\left(\xi \right){\delta }^{″}(\xi -{\xi }_{0i})].$$

(7)

The formulation of ${\varphi }_j\left(\xi \right)$ is derived from the resolution of Equations (6) and (7), as detailed in Appendix B. By substituting expression (5) into the dynamic Equation (4) and then scaling each term of Equation (4) by ${\varphi }_j\left(\xi \right)$, followed by integrating over the span of the beam, we obtain:

$$\begin{array}{cc}\hfill & \sum _{j=1}^{\infty }{u}_j\left(t\right){\int }_0^1{\varphi }_i\left(\xi \right){\displaystyle \frac{d^2}{d{\xi }^2}}[\left(1-\sum _{i=1}^n{\kappa }_i\delta \left(\xi -{\xi }_{0i}\right)\right){\displaystyle \frac{d^2{\varphi }_j\left(\xi \right)}{d{\xi }^2}}]d\xi \hfill \\ \hfill & +{\displaystyle \frac{m{L}^4}{E{I}_0}}\sum _{j=1}^{\infty }{\ddot{u}}_j\left(t\right){\int }_0^1{\varphi }_i\left(\xi \right){\varphi }_j\left(\xi \right)d\xi +{\displaystyle \frac{c{L}^4}{E{I}_0}}\sum _{j=1}^{\infty }{\dot{u}}_j\left(t\right){\int }_0^1{\varphi }_i\left(\xi \right){\varphi }_j\left(\xi \right)d\xi \hfill \\ \hfill & ={\displaystyle \frac{L^4}{E{I}_0}}{\int }_0^1{F}_R\left(t\right){\varphi }_i\left(\xi \right)\delta (\xi -vt)d\xi \hfill \end{array}$$

(8)

The notation of a dot above $u_j\left(t\right)$ signifies the time derivative with respect to t. Utilizing the orthogonality characteristics of the mode shape in Equation (8) results in:

$${\ddot{u}}_j\left(t\right)+2{\zeta }_j{\omega }_j{\dot{u}}_j\left(t\right)+{\omega }_j^2{u}_j\left(t\right)={\displaystyle \frac{F_j}{M_j}},j=1,2,3,\cdots$$

(9)

where

$$F_j=F_R\left(t\right){\varphi }_j\left(vt\right),M_j=m{\int }_0^1{\varphi }_j^2\left(\xi \right)d\xi .$$

The variables ${\zeta }_j$ and ${\omega }_j$ represent the damping factor and the angular frequency for the jth vibrational mode, respectively. Equation (9) is very similar to the motion equation of a dynamic system with only one degree of freedom. When it is solved together with the final two equations from Equations (1) and (3), it calculates the modal coordinate $u_j\left(t\right)$ for the jth mode. The overall dynamic reaction of the bridge system affected by pedestrian traffic and cracks is calculated by solving the first N separate equations for these modal coordinates as per Equation (9). Thereafter, the individual modal responses are synthesized using the formulation provided in Equation (5).

3. Numerical Verification of the Correctness of the Solution to the Pedestrian–Cracked Bridge Coupling Equation

In this section, the correctness of the coupled Equation (1) is verified by numerically solving Equations (5)–(7) and (9), and comparing the computational results with those from the existing literature. The verification process is divided into two parts. The first part involves using the cracked beam parameters from reference [24] with Equation (A21) to calculate the natural frequencies of the cracked beam. The results are then compared with those from reference [24] to validate the correctness of Equation (A21), thereby indirectly validating the accuracy of the mode shape functions presented in Appendix B. The second part utilizes the cracked beam and moving-mass parameters from reference [27] with Equations (5)–(7) and (9) for calculations. The mid-span displacement response of the beam is extracted and compared with the results from reference [27], thus verifying the correctness of coupled Equation (1).

We begin by validating the accuracy of frequency Equation (A21). Employing the beam dimensions specified in reference [24]: the beam spans a length of $L=2$ m and has a cross-sectional width of $b=0.05$ m and a height of $h=0.1$ m. We examined scenarios with cracks at depth ratios of $5\%$, $50\%$, and $25\%$, situated at distances of 0.466 m, 0.466 m, and 1 m from the beam’s left support for cases 1, 2, and 3, respectively. In Rubio’s study [24], the natural frequencies were determined by employing a model that simulated the crack using a torsional spring. This methodology entailed imposing boundary conditions on two distinct beam segments that were connected via the spring. The perturbation technique was applied to derive the natural frequencies for the beam with a crack.

Table 1. The comparison of natural frequencies from Equation (A21) and reference [24] (unit: Hz).

Cases		First-Order		Second-Order		Third-Order
		Reference	Equation (A21)	Reference	Equation (A21)	Reference	Equation (A21)
Uncracked		427.9	427.9	1711.7	1711.7	3851.4	3851.4
Case 1		427.9	427.9	1709.4	1711.1	3847.9	3848.0
Case 2		397.9	397.9	1499.5	1511.7	3597.6	3610.3
Case 3		415.1	415.7	1711.7	1711.7	3741.5	3746.5

displays a comparison between the natural frequencies documented in the literature [24] and those calculated using Equation (A21). The frequencies from both sources are found to be in close alignment, confirming the precision of Equation (A21) for determining the eigenfrequencies of the bridge.

Next, we utilized the parameters for a simply supported cracked beam and moving mass provided in reference [27] and applied Equations (5)–(7) and (9) to calculate the displacement response of the cracked beam. By comparing these results with those from reference [27], we verified the correctness of the coupled Equation (1). The bridge parameters used in Equations (5)–(7) and (9) were as follows: length $L=50$ m, width $b=0.5$ m, height $h=1.0$ m, Young’s modulus $E_0=2.1\times {10}^{11}$ N/m², and mass per unit length $m=3930$ kg/m. In addition, the moving mass was $20\%$ of the total beam mass. In reference [27], based on a torsional spring model, the dynamical response of the cracked beam was derived using an iterative modal analysis method, considering six vibration modes and dividing the beam into 100 segments. It was discovered that the outcomes were identical to those achieved by incorporating nine vibration modes with the beam segmented into 400 parts.

Figure 2 displays the mid-span displacement response of the beam, as calculated by both the referenced study [27] and the current approach, for moving speeds of 10 m/s and 20 m/s. The depicted displacements were normalized to the static mid-span deflection that would result from the load on an uncracked beam. The figure demonstrates a significant correlation between the displacement time-histories of both the cracked and uncracked beams, as obtained from this study and reference [27]. It indicates that the pedestrian–cracked bridge coupling equations obtained in this paper can be effectively used for subsequent parameter discussions.

4. Case Study

In this section, we utilize Equations (5)–(7) and (9) to calculate the dynamic response of a cracked bridge under pedestrian load and conduct an in-depth analysis of the impact of the cracks’ location, depth, and quantity on the bridge. In Section 4.1, we introduce the bridge and pedestrian parameters used. Section 4.2 evaluates the mid-span displacement and acceleration time-history curves for the bridge without damage under the influence of pedestrian loading, along with the associated spectral analysis. Section 4.3, Section 4.4, Section 4.5 delve into the impact of varying crack quantities on the previously derived time-history curves and spectral data. Given the rarity of encountering extremely deep cracks in practical engineering applications, we established a threshold of 0.1 or less for the crack depth ratio in our calculations of the bridge’s mid-span dynamic response.

4.1. Parameter Specification for the Bridge and Pedestrian

The bridge specifications were taken from [35] as follows: length $L=12$ m, width $b=1.2$ m, height $h=0.16$ m, flexural rigidity $E{I}_0=1.579\times {10}^7$ N/m², linear mass density $m=150.8$ kg/m, and damping ratio $\zeta =0.003$. Utilizing these parameters, the initial three eigenfrequencies for the unblemished beam were determined to be $3.5298$, $14.1191$, and $31.7680$ Hz.

Table 2. The first three natural frequencies of the cracked bridge under different crack conditions (unit: Hz).

Crack Depth Ratio	$\mathit{\phi }=0.05$	$\mathit{\phi }=0.1$	$\mathit{\phi }=0.2$
One crack located at $1/5$th of the span	3.5294	3.5284	3.5245
	14.1153	14.1046	14.0642
	31.7593	31.7354	31.6456
One crack located at $1/2$ of the span	3.5287	3.5258	3.5146
	14.1191	14.1191	14.1191
	31.7584	31.7320	31.6323
One crack located at $7/10$th of the span	3.5291	3.5272	3.5198
	14.1153	14.1046	14.0644
	31.7671	31.7646	31.7551
One crack located at $9/10$th of the span	3.5297	3.5294	3.5283
	14.1176	14.1136	14.0980
	31.7617	31.7444	31.6783
Three cracks located at $1/4$, $1/2$,	3.5276	3.5218	3.4996
$3/4$ of the span	14.1106	14.0871	13.9984
	31.7488	31.6958	31.4957
Five cracks located at $1/6$, $1/3$,	3.5266	3.5178	3.4848
$1/2$, $2/3$, $5/6$th of the span	14.1063	14.0711	13.9392
	31.7392	31.6600	31.3630

outlines the initial three natural frequencies for the bridge with cracks under diverse cracking conditions. It is evident that the presence of cracks consistently resulted in a decrease in the bridge’s natural frequency. The placement, severity, and count of cracks all influenced the degree of this frequency reduction. For a fixed number of cracks, deeper cracks resulted in a lower natural frequency. Similarly, for a consistent crack depth, an increased number of cracks correlated with a further decrease in natural frequency.

Following the recommendation of Smith et al. [34], Equation (3) can be elaborated upon as follows:

$$\begin{array}{cc}\hfill F_{\mathrm{W}}& =(m_{\mathrm{a}}+m_s)\mathrm{g}\left[0.436(f_w-0.95)sin\left(2\pi f_{w}t\right)+0.006(2f_w+12.3)sin(4\pi f_{w}t-{\displaystyle \frac{\pi }{2}})\right.\hfill \\ \hfill & \left.+0.007(3f_w+5.2)sin(6\pi f_{w}t+\pi )+0.007(4f_w+2.0)sin(8\pi f_{w}t+{\displaystyle \frac{\pi }{2}})\right]\hfill \end{array}$$

(10)

Measurements of walking activities have shown that the range of possible pace frequencies that may occur is from 1.5 Hz to 2.5 Hz [36], but the range of probable pace frequencies is much narrower, and, as such, the following pace frequencies should be used for design: 1.8 Hz $\le f_w\le 2.2$ Hz [34]. Additionally, Bachmann and Ammann [37] presented a relationship between pace frequency and velocity of walking that can be approximated by the following equation:

$$v_p=1.67f_w^2-4.83f_w+4.50,1.7\mathrm{Hz}\le f_w\le 2.4\mathrm{Hz}.$$

(11)

Based on references [38,39,40,41], the following parameter ranges were set in Equation (1):

$$\begin{array}{cc}\hfill & {\displaystyle \frac{m_a}{m_a+m_s}}=80–90\%,{\xi }_p={\displaystyle \frac{c_p}{4\pi m_a{f}_p}}=25–60\%,\hfill \\ \hfill & f_p={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{k_p}{m_a}}}=1–6\mathrm{Hz},\hfill \end{array}$$

(12)

where ${\xi }_p$ is the equivalent damping ratio, and $f_p$ is the natural frequency of the body. In this study, we used $m_a$:$m_s$ = 9:1, ${\xi }_p$ = 0.36, and $f_p$ = 3.7 Hz [38]. The pedestrian’s weight was assumed to be 60 kg. In accordance with Equation (12), the parameters for the pedestrian were set as $m_a$ = 54 kg and $m_s$ = 6 kg for the masses, $k_p=2.9185\times {10}^4$ N/m for the spring constant, and $c_p$ = 9.04 ×${10}^2$ N· s/m for the damping coefficient. The selected walking velocities were 1.22, 1.52, and 1.96 m per second, which yielded corresponding walking frequencies of 1.8, 2.0, and 2.2 Hz, respectively, as derived from Equation (11). Equation (9) was numerically integrated using the fourth-order Runge–Kutta technique, with a time increment of 0.001 s.

4.2. Dynamic Response of an Intact Beam under Pedestrian Load

Figure 3 illustrates the time-varying displacement and acceleration profiles at the bridge’s mid-span in the absence of cracks, along with their respective Fourier amplitude spectra at various walking velocities. The findings indicated a slight reduction in both displacement and acceleration at the mid-span of the uncracked beam with an increase in the pedestrian’s walking speed.

The spectra for both displacement and acceleration exhibited the walking frequency and its higher harmonic components. The difference lay in the fact that when the pedestrian speed was lower, the spectral energy was mainly concentrated at twice the walking frequency, and as the pedestrian speed increased, the spectral energy was mainly concentrated at the fundamental walking frequency.

4.3. The Case of a Single Crack

Figure 4 and Figure 5, respectively, illustrate the impact of a crack with depth ratios of 0.05 and 0.1 located at a quarter of the span from the left end on the mid-span displacement and acceleration of the bridge. Figure 6 and Figure 7, on the other hand, demonstrate the effects of a crack with depth ratios of 0.05 and 0.1 located at the mid-span on the mid-span displacement and acceleration of the bridge.

The appearance of a minor crack had little effect on the mid-span displacement of the bridge, only adding a very small peak in the displacement spectrum corresponding to the third natural frequency of the bridge.

The emergence of a minor crack led to a significant increase in the amplitude of the bridge’s mid-span acceleration, with a greater increase in amplitude when the crack was located at mid-span compared to when it was not. As the depth of the crack increased, the amplitude of the acceleration noticeably decreased. This indicated that both the location and depth of the crack had a significant impact on the amplitude of acceleration. The presence of a crack had a very clear effect on the acceleration spectrum, with the spectral energy concentrated at the position corresponding to the third natural frequency of the bridge, unaffected by changes in the crack’s location and depth.

4.4. The Impact of Three Cracks

Next, we assumed that there were three cracks located at the quarter, half, and three-quarter span of the bridge. Figure 8 and Figure 9 demonstrate the impact of the cracks on the mid-span displacement and acceleration time-history curves and spectra of the bridge when the crack depth ratios were 0.05 and 0.1, respectively.

From Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, minor cracks had a negligible impact on the bridge’s mid-span displacement, with the sole addition being an insignificant peak in the displacement spectrum that matched the bridge’s third natural frequency. The increase in the number and depth of cracks led to a reduction in the mid-span acceleration amplitude of the bridge, but it had little effect on the acceleration spectrum.

4.5. The Impact of Five Cracks

Suppose the bridge has five cracks positioned at the one-sixth, one-third, one-half, two-third, and five-sixth points of the span. Figure 10 and Figure 11 illustrate the impact of the cracks on the mid-span displacement and acceleration time-history curves and spectra of the bridge at different pedestrian speeds for two cases with crack depth ratios of 0.05 and 0.1, respectively.

From Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 the presence of minor cracks barely influenced the bridge’s mid-span displacement, merely resulting in a minor peak in the displacement spectrum corresponding to the bridge’s third natural frequency. As the number of cracks increased from one to three, and then to five, the mid-span acceleration amplitude of the bridge consistently decreased, with the spectrum remaining essentially unaffected.

5. Conclusions

This paper investigated the dynamic behavior of a bridge with cracks when subjected to the load of a pedestrian. The pedestrian was represented by a linear oscillator model, while the bridge, which included cracks, was modeled as a simply supported beam following Euler–Bernoulli’s theory. The cracks’ impact on the beam’s bending stiffness was captured using generalized functions. An in-depth analysis was conducted to assess the effects of variations in crack location, depth, and number, as well as pedestrian velocity, on the mid-span displacement and acceleration of the bridge. Based on the study’s results, the following conclusions are drawn:

• The appearance of cracks consistently results in a decrease in the bridge’s natural frequency. The more numerous and deeper the cracks, the greater the reduction in frequency.
• The emergence of cracks has little impact on the mid-span displacement of the bridge, only causing a very small peak to appear at the third natural frequency of the bridge in the displacement spectrum. The increase in the number and depth of cracks has almost no effect on the mid-span displacement of the bridge.
• The emergence of cracks significantly increases the mid-span acceleration amplitude of the bridge, causing almost all the energy in the acceleration spectrum to be concentrated at the bridge’s third natural frequency. For a single crack, the acceleration amplitude is greatest when it appears at the mid-span compared to other locations on the bridge. As the number and depth of cracks increase, the acceleration amplitude continues to decrease, but the impact on the acceleration spectrum is minimal.

The research presented in this paper indicates that if the depth of cracks on a bridge is relatively small, we can assess the presence of cracks under pedestrian loads by monitoring the mid-span acceleration amplitude and spectrum of the bridge. When a noticeable peak appears at the position corresponding to the bridge’s third natural frequency in the mid-span acceleration spectrum, accompanied by a significant increase in acceleration amplitude, it suggests that cracks may have formed on the bridge. During further monitoring, if there is a marked decrease in the mid-span acceleration amplitude of the bridge while the spectrum remains almost unchanged, it implies that the number of cracks or the depth of the cracks has increased.

Additionally, this study suggests that assessing the presence of minor cracks on a bridge by monitoring its natural frequencies may be challenging, as the changes in the bridge’s natural frequencies caused by the emergence of cracks are minimal. Moreover, the study in this paper assumed that the cracks were always open, which differs from real-world scenarios where cracks may open and close, complicating their effect. This cyclic behavior can lead to minor variations in the bridge’s frequency, potentially complicating the assessment of the bridge’s health through frequency monitoring. Furthermore, the bridge’s natural frequencies and mode shapes can be influenced by the presence of pedestrians, which adds another layer of complexity.

The dynamics of cracked bridges under pedestrian loading have not been extensively studied and warrant further investigation for effective bridge condition monitoring and for understanding the implications of cracks.