Coupled Vibration of a Vehicle Group–Bridge System and Its Application in the Optimal Strategy for Bridge Health Monitoring

Abstract

The accuracy of bridge performance monitoring and evaluation is easily affected by unfavorable factors such as vehicle coupling and test noise. In order to accurately evaluate the dynamic response and health monitoring threshold of the bridge under different operating conditions, a time-varying dynamic vehicle group model including the main uniform mass and the coupling mass was established, and the influence of road roughness was considered in the coupling equation. A bridge monitoring strategy considering signal noise ratio and vehicle–bridge interaction was proposed, and the effectiveness of the monitoring strategy was verified by taking a simple supported beam as an example. The results showed that the proposed time-varying dynamic vehicle group model could accurately consider the influence of road roughness and estimate the threshold of health monitoring, and the proposed bridge monitoring strategy could filter out a large amount of low signal-to-noise ratio or meaningless data, thus saving computing resources and realizing the lightweight safety monitoring of bridges.

1. Introduction

In civil engineering, bridges constitute a crucial component of the national transportation system, with reinforced concrete bridges being the most common form in highway infrastructure. Over their extended service life, these bridges undergo material deterioration and damage accumulation due to the combined impact of vehicular load and environmental effects such as wind load, carbonation, and chloride ion erosion [1]. Ultimately, their reliability does not meet the requirements of their use. Different types of sensors are arranged on important bridges to monitor their health status in real time or in a timely manner through a health monitoring system, to identify different degrees of damage, which can not only ensure the operation safety of bridges and improve the management level but also allow the understanding of the evolution law of bridge performance deterioration and promote the development of bridge engineering. Therefore, bridge health monitoring has become an important research direction in bridge engineering. However, bridge health monitoring is a multidisciplinary and comprehensive technology, which involves bridge engineering, sensing technology, testing technology, signal analysis, computer technology, network communication technology, and other research areas. Bridge health monitoring has shortcomings in sensor performance and signal transmission, massive data mining and processing, monitoring and safety assessment theory, and software development, and does not comprehensively consider the environmental factors of bridges, so it is in urgent need of improvement and development. It is necessary to develop a fast and accurate new method for bridge safety state assessment [2,3].

Axle dynamics theory is the basic framework for calculating the dynamic response of bridges, and it is the basic way to accurately calculate and understand the dynamic performance of bridge structure. When a vehicle passes across a bridge deck, a dynamic response amplification occurs, necessitating consideration in response analysis and performance assessment [4,5,6,7]. With the growth of China’s economy and traffic transportation, the widespread use of heavy trucks places a significant burden on bridges. The increased dynamic loads contribute to a continuous decline in bridge reliability, underscoring the increasing need for regular maintenance and monitoring [8,9]. As a result, the dynamic analysis of coupled vehicle–bridge systems has attracted significant attention in recent years [10,11,12,13,14,15]. The aim of these studies is to investigate and simulate the structural behavior of bridges subjected to moving vehicles, as well as the ride comfort of vehicles crossing bridges [16,17,18].

The predominant focus in the study of vehicle-bridge interaction lies in closed-form or analytical expressions, especially in simplified cases such as the consideration of a simply supported Euler–Bernoulli beam model for the bridge and a moving load model for the vehicle. When representing the vehicle as a moving mass or oscillator, the vehicle–bridge interaction problem requires the simultaneous solution of a set of coupled partial differential equations governing the motion of both the bridge and the vehicle [17,19]. In contrast to the modal superposition technique, the finite element method provides a more versatile approach for handling complex bridge–vehicle models in the dynamic interaction analysis [20,21]. It was found that the most important parameters influencing the separation are the sliding speed and the mass ratio between the moving mass and the beam [22]. On the other hand, the road roughness of the bridge deck is an inherent physical phenomenon present in almost all bridges, due to construction quality and maintenance capacity. Several scholars have observed that road roughness amplifies the dynamic response due to the moving action of the mass, potentially compromising the safety and serviceability of the structure [23,24]. Existing studies exhibit certain deficiencies, primarily in the assumption that vehicles on the bridge are typically regarded as one or several separate moving load models. However, in reality, especially in high-traffic scenarios for busy bridges, actual traffic vehicles should be conceptualized as a continuous group. Hence, it is important to establish a coupled bridge–vehicle group model that accurately represents the interaction under conditions of intensive traffic flow. Moreover, the investigation should delve extensively into crucial factors such as coupling masses, the speed of the corresponding vehicle group, and the effects of road roughness.

In view of this, in order to accurately evaluate the dynamic response and health monitoring threshold of the bridge under different operating conditions, a time-varying dynamic vehicle group model including the main uniform mass and the coupling mass was established, and the influence of road roughness was considered in the coupling equation. A bridge monitoring strategy considering the signal–noise ratio and vehicle–bridge interaction was proposed, and the effectiveness of the monitoring strategy was verified by taking a simple supported beam as an example.

2. Dynamic Equations for Vehicle Group–Bridge Coupled Motion

Before activating the health monitoring system, it is essential to thoroughly investigate and analyze the structural dynamic characteristics, traffic loads, and vehicle–bridge coupling characteristics of the bridge. This ensures a comprehensive understanding of the bridge’s behavior before undertaking damage detection.

While the theory of vehicle–bridge coupled vibration has been studied extensively in recent years, the development of new dynamic models that consider more influencing factors is still necessary. In this study, the vehicle group is represented as a time-varying dynamic model with body parts and distributed coupling mass. The dynamic equilibrium equation and analytical expressions pertaining to the interaction between the bridge structure and the moving vehicle group are established.

Typical bridge structures exhibit bending deformation primarily under the action of vertical vibrations, with the shear deformation and rotary inertia of the central shaft section being negligible. This assumption is illustrated in Figure 1, where the cross-sectional area of the bridge is considered constant along its length, resulting in a constant flexural stiffness (D).

Under the assumption that the vehicle group is a uniform moving mass load, the coupling effect and dynamic response of the vehicle group and the bridge can be effectively analyzed directly solving the relevant differential equations.

The external excitation acting on the bridge results from a group of moving vehicles. This group is assumed to consist of a main body with uniform mass density, body stiffness, and body damping. If we denote the dynamic deflection of the beam as $y(x,t)$, the displacement of the main body in the vehicle group as $z(t)$, the distribution length as $l_p$, and the constant speed of the vehicle group as $v$, the dynamic equation for the vehicle group is expressed as follows:

$${\overline{\mu }}_1{l}_p\ddot{z}(t)+{\overline{k}}_1\left(z(t)-y(x,t)\right)+{\overline{c}}_1\left(\dot{z}(t)-\partial y(x,t)/\partial t\right)=0$$

(1)

The dynamic equation governing the bridge response under the moving excitation of a vehicle group is formulated as follows:

$$D{\partial }^{4}y(x,t)/\partial x^4+\overline{m}{\partial }^{2}y(x,t)/\partial t^2+c\partial y(x,t)/\partial t=p(x,t)$$

(2)

The dynamic loads p(x, t) exerted on the bridge by the vehicle group involve the gravity load resulting from the mass of the vehicle group, represented as $p_G(x,t)=({\overline{\mu }}_0+{\overline{\mu }}_1)g$; the inertial force, $p_I(x,t)=-{\overline{\mu }}_0{\partial }^{2}y(x,t)/\partial t^2$; the reaction force arising from the stiffness of the vehicle group, $p_S(x,t)={\overline{k}}_1\left[z(t)-y(x,t)\right]$; and the reaction force due to the damping of the vehicle group, $p_D(x,t)={\overline{c}}_1\left[\dot{z}(t)-\partial y(x,t)/\partial t\right]$. Hence, p(x, t) can be expressed as follows:

$$p(x,t)=[p_G(x,t)+p_I(x,t)+p_S(x,t)+p_D(x,t)]\left\{\mathrm{H}[x-(Vt-l_p)]-\mathrm{H}(x-Vt)\right\}$$

(3)

where H(t) is the unit step function. The above equation can be solved using the mode-superposition method.

The corresponding transformation expression is shown as:

$$y(x,t)={\displaystyle \sum _{i=1}^{\infty }{q}_i(t)\cdot {\varphi }_i(x)}$$

(4)

where q_i(t) is the generalized modal coordinates and ${\varphi }_i(x)$ is the mode function.

Substituting Equation (4) into Equation (2) and multiplying each term by the nth-order mode function, the integral along the beam is calculated, accounting for modal orthogonality. The modal decomposition method is then applied, combining with Equation (1), to derive the coupled dynamic equations for the bridge and vehicle group system. Utilizing the stepwise integration method, the nth-order modal response under the influence of the vehicle group is computed.

In the case of multi-span continuous bridges or other intricate structures, such as suspension and cable-stayed bridges, the corresponding modes are highly complex, necessitating the use of the finite element method for obtaining exact mode shapes and frequencies. Conversely, for simply supported beams, the equation can be analytically solved by the following method.

The nth-order mode shape of a simply supported beam is denoted by ${\varphi }_n=\sin \frac{n\pi v}{L}(t-\frac{l_p}{2v})$. By integrating it along the bridge length, the subitem integration result of $p(x,t)$ is expressed as follows:

$$p_G(x,t)={\displaystyle {\int }_{vt-l_p}^{vt}({\overline{\mu }}_1+{\overline{\mu }}_2)g\sin \frac{n\pi x}{L}}dx=\frac{2({\overline{\mu }}_0+{\overline{\mu }}_1)gL}{n\pi }\sin \frac{n\pi l_p}{2L}\sin \frac{n\pi v}{L}(t-\frac{l_p}{2v})$$

(5)

$$p_I(x,t)=-{\displaystyle \sum _{i=1}^{\infty }{\overline{\mu }}_0{\ddot{q}}_i(t)}{\displaystyle {\int }_{vt-l_p}^{vt}\sin \frac{i\pi x}{L}\sin \frac{n\pi x}{L}dx}=-\frac{{\overline{\mu }}_{0}L}{\pi }{\displaystyle \sum _{i=1}^{\infty }{\ddot{q}}_i(t)}{\Phi }_{in}$$

(6)

where ${\mathrm{\Phi }}_{in}=\frac{1}{i-n}\left[\cos \frac{(i-n)\pi v}{L}(t-\frac{l_p}{2v})\sin \frac{(i-n)\pi l_p}{2L}\right]-\frac{1}{i+n}\left[\cos \frac{(i+n)\pi v}{L}(t-\frac{l_p}{2v})\sin \frac{(i+n)\pi l_p}{2L}\right]$.

$$p_S(x,t)=\frac{2{\overline{k}}_{1}Lz(t)}{n\pi }\sin \frac{n\pi l_p}{2L}\sin \frac{n\pi v}{L}(t-\frac{l_p}{2v})-\frac{{\overline{k}}_{1}L}{\pi }{\displaystyle \sum _{i=1}^{\infty }{q}_i(t)}{\mathrm{\Phi }}_{in}$$

(7)

$$p_D(x,t)=\frac{2{\overline{c}}_{1}L\dot{z}(t)}{n\pi }\sin \frac{n\pi l_p}{2L}\sin \frac{n\pi v}{L}(t-\frac{l_p}{2v})-\frac{{\overline{c}}_{1}L}{\pi }{\displaystyle \sum _{i=1}^{\infty }{\dot{q}}_i(t)}{\mathrm{\Phi }}_{in}$$

(8)

Let us define ${\omega }_n=n^2{\pi }^2\sqrt{EI/\overline{m}}/L^2$ as the nth-order circular frequencies of the structure and $c_n=2\overline{m}{\xi }_n{\omega }_n$ as the corresponding damping. By dividing both sides of Equation (2) by $\overline{m}L/2$ and rearranging the equation, the dynamic equilibrium equation for the nth-order mode beam–slab structures under vehicle group excitation can be expressed in standard form as:

$$\begin{array}{l}[{\ddot{q}}_n(t)+\frac{2{\overline{\mu }}_0}{\overline{m}\pi }{\displaystyle \sum _{i=1}^{\infty }{\ddot{q}}_i(t){\mathrm{\Phi }}_{in}}]+\left[2{\xi }_n{\omega }_n{\dot{q}}_n(t)+\frac{2{\overline{c}}_1}{\overline{m}\pi }{\displaystyle \sum _{i=1}^{\infty }{\dot{q}}_i(t)}{\mathrm{\Phi }}_{in}\right]+\left[{\omega }_n^2{q}_n(t)+\frac{2{\overline{k}}_1}{\overline{m}\pi }{\displaystyle \sum _{i=1}^{\infty }{q}_i(t)}{\mathrm{\Phi }}_{in}\right]\\ -\frac{4}{n\overline{m}\pi }\sin \frac{n\pi l_p}{2L}{\varphi }_n\left[{\overline{k}}_{1}z(t)+{\overline{c}}_1\dot{z}(t)\right]=\frac{4({\overline{\mu }}_0+{\overline{\mu }}_1)g}{n\overline{m}\pi }\sin \frac{n\pi l_p}{2L}{\varphi }_n\end{array}$$

(9)

For a simply supported girder bridge, if the generalized coordinates are taken as n items, the overall degree of freedom is reduced from an infinite number to n. Consequently, the dynamic equation for the nth order system is transformed from the expression of Equation (1) to the following:

$${\overline{\mu }}_1\ddot{z}(t)+{\overline{c}}_1\dot{z}(t)+{\overline{k}}_{1}z(t)-{\overline{c}}_1{\displaystyle \sum _{i=1}^n{\dot{q}}_i(t)\cdot {\varphi }_i(x)}-{\overline{k}}_1{\varphi }_n{\displaystyle \sum _{i=1}^n{q}_i(t)\cdot {\varphi }_i(x)}=0$$

(10)

Combining Equations (9) and (10) allows for the derivation of simultaneous equations describing the coupled system for the bridge and the vehicle group. For a simply supported girder bridge, adopting the displacement series involving n items results in a generalized degree of freedom of n. Adding the freedom associated with the main body of the vehicle group, the system equation can be expressed as an n + 1 order matrix:

$$M\ddot{U}+C\dot{U}+KU=F(t)$$

(11)

where the generalized displacement vector is denoted as $U={[q_1, q_2, ... , q_n, z]}^T$. The corresponding generalized mass matrix M, damping matrix C, stiffness matrix K, and force vector F(t) are given as Equation (12) to Equation (15), respectively:

$$M=\left[\begin{array}{ccccc}1+{\rho }_M{\mathrm{\Phi }}_{11}& {\rho }_M{\mathrm{\Phi }}_{12}& \cdots & {\rho }_M{\mathrm{\Phi }}_{1n}& 0\\ {\rho }_M{\mathrm{\Phi }}_{21}& 1+{\rho }_M{\mathrm{\Phi }}_{22}& \cdots & {\rho }_M{\mathrm{\Phi }}_{2n}& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\rho }_M{\mathrm{\Phi }}_{n1}& {\rho }_M{\mathrm{\Phi }}_{n2}& \cdots & 1+{\rho }_M{\mathrm{\Phi }}_{nn}& 0\\ 0& 0& \cdots & 0& {\overline{\mu }}_1\end{array}\right]$$

(12)

$$C=\left[\begin{array}{ccccc}2{\xi }_1{\omega }_1+{\rho }_C{\mathrm{\Phi }}_{11}& {\rho }_C{\mathrm{\Phi }}_{12}& \cdots & {\rho }_C{\mathrm{\Phi }}_{1n}& -{\rho }_C{\varphi }_1\\ {\rho }_C{\mathrm{\Phi }}_{21}& 2{\xi }_2{\omega }_2+{\rho }_C{\mathrm{\Phi }}_{22}& \cdots & {\rho }_C{\mathrm{\Phi }}_{2n}& -{\rho }_C{\varphi }_2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\rho }_C{\mathrm{\Phi }}_{n1}& {\rho }_C{\mathrm{\Phi }}_{n2}& \cdots & 2{\xi }_n{\omega }_n+{\rho }_C{\mathrm{\Phi }}_{nn}& -{\rho }_C{\varphi }_n\\ -{\overline{c}}_1{\varphi }_1& -{\overline{c}}_1{\varphi }_2& \cdots & -{\overline{c}}_1{\varphi }_n& {\overline{c}}_1\end{array}\right]$$

(13)

$$K=\left[\begin{array}{ccccc}{\omega }_1^2+{\rho }_K{\mathrm{\Phi }}_{11}& {\rho }_K{\mathrm{\Phi }}_{12}& \cdots & {\rho }_K{\mathrm{\Phi }}_{1n}& -{\rho }_K{\varphi }_1\\ {\rho }_K{\mathrm{\Phi }}_{21}& {\omega }_2^2+{\rho }_K{\mathrm{\Phi }}_{22}& \cdots & {\rho }_K{\mathrm{\Phi }}_{2n}& -{\rho }_K{\varphi }_2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\rho }_K{\mathrm{\Phi }}_{n1}& {\rho }_K{\mathrm{\Phi }}_{n2}& \cdots & {\omega }_n^2+{\rho }_K{\mathrm{\Phi }}_{nn}& -{\rho }_K{\varphi }_n\\ -{\overline{k}}_1{\varphi }_1& -{\overline{k}}_1{\varphi }_2& \cdots & -{\overline{k}}_1{\varphi }_n& {\overline{k}}_1\end{array}\right]$$

(14)

$$F(t)={[{\rho }_F{\varphi }_1, {\rho }_F{\varphi }_2, ... , {\rho }_F{\varphi }_n, 0]}^T$$

(15)

where ${\rho }_M=\frac{2{\overline{\mu }}_0}{\overline{m}\pi }$, ${\rho }_C=\frac{2{\overline{c}}_1}{\overline{m}\pi }$, ${\rho }_K=\frac{2{\overline{k}}_1}{\overline{m}\pi }$, ${\rho }_F=\frac{4({\overline{\mu }}_0+{\overline{\mu }}_1)g}{n\overline{m}\pi }\sin \frac{n\pi l_p}{2L}$, ${\mathrm{\Phi }}_{nm}={\varphi }_n{\varphi }_m$, and ${\varphi }_n=\sin \frac{n\pi v}{L}(t-\frac{l_p}{2v})$.

Due to the movement of the vehicle group on the bridge, the coupling mass and the coefficient Φ_mn of the system dynamic equations continuously varies. Consequently, the equations evolve into higher-order differential equations of the time-varying system, requiring a stepwise integration method for resolution. From Equation (11), it is evident that the key factors significantly influencing the dynamic response of the bridge include the mass, stiffness, distribution length, and speed of the vehicle group. For actual bridge structures, the assumption that the distribution length of the vehicle group l_p is approximately equal to the length of the bridge is made, and this assumption is conservative. Therefore, this case is primarily studied in later studies.

3. Vibration Induced by Road Roughness

When vehicles traverse a bridge, the weight, speed, and dynamic characteristics of the vehicles, as well as the road roughness, collectively influence bridge vibrations and ride comfort. The vibrations induced by road roughness can have a detrimental effect on both the bridge and the vehicles.

In general, when a vehicle moves at a constant speed, the road roughness can be modeled as a stationary random process in the spatial domain. For bridges with uneven surfaces, as shown in Figure 1, the road roughness curve is represented as r(x). Additional displacements need to be incorporated into the original dynamic system, resulting in the updated bridge dynamic deflection as follows:

$$y_r(x,t)=y(x,t)+r(x)$$

(16)

Equation (16) is substituted into Equations (1) and (2) while ignoring the derivative of the road roughness with respect to time, since the additional displacement r(x) is time-independent until the vehicle group reaches a specific position. Consequently, the resulting equations are established as the coupling equations described above. After integration, the form of the simultaneous equations for the coupled system considering road roughness takes the same form as Equation (11), and the supplementary equations include Equations (12)–(14). However, the expression of Equation (15) is modified as follows:

$$F(t)={[{\rho }_{Fr}{\varphi }_1, {\rho }_{Fr}{\varphi }_2, ... , {\rho }_{Fr}{\varphi }_n, {\rho }_r]}^T$$

(17)

where ρ_Fr equals $\frac{4({\overline{\mu }}_0+{\overline{\mu }}_1)g}{n\overline{m}\pi }\sin \frac{n\pi l_p}{2L}-{\rho }_{k}r(x)$, and ρ_r equals k₁r(x).

If a vehicle travels at a constant speed, road roughness becomes a stationary random process in the time domain due to the relationship between the distance traveled and the time. For a bridge experiencing vibrations from vehicles moving at a constant speed, this stationary displacement can be utilized as an input. Under normal conditions, the surface roughness of the bridge is considered a Gaussian random process with zero mean and ergodic states. The power spectral density function of a given road roughness is used to obtain the bridge deck roughness sequence.

There are various methods available to obtain the roughness of the bridge deck according to the given power spectral density. The triangle series superposition method has been widely used due to its simplicity and high simulation accuracy. This method involves representing a stochastic process as the sum of several cosine functions. According to the calculation principle of the triangle series superposition method, the road roughness can be obtained as follows:

$$r(x)=\sqrt{2}{\displaystyle \sum _{k=1}^N\sqrt{G_d(n_k)\Delta n}\cos (2\pi n_{k}x+{\phi }_k)}$$

(18)

where r(x) is the simulated road roughness sample; G_d(n_k) is the power spectral density function of surface displacement, as defined by the standard; n_k is defined as the discrete spatial frequency in the range of [n_l, n_u]; Δn is the frequency interval in the frequency space; φ_k is a set of random numbers uniformly distributed in the range of [0, 2π]; and N is the total number of sampling points.

Δn and n_k can be calculated by Equations (19) and (20):

$$\Delta n=(n_u-n_l)/N$$

(19)

$$n_k=n_l+(k-1/2)\Delta n$$

(20)

where n_u and n_l are the upper and lower limits of the effective spatial frequency of the road, respectively.

Sun (2003) [25] elucidated the process of generating roughness using the triangular series superposition method. Road roughness is primarily characterized by the road power spectral density. In 1984, the International Standard Organization (ISO) released the Draft Standard for Road Roughness [26], and in 1986, China also promulgated the national standard, GB7031-86, regarding road roughness [27]. Both documents recommend a fitting expression for the road power spectral density, expressed as:

$$G_q\left(n\right)=G_q\left(n_0\right){\left(\frac{n}{n_0}\right)}^{-\omega }$$

(21)

where n is the spatial frequency, defined as the reciprocal of the wavelength with unit of m⁻¹; n₀ is the spatial reference frequency, set at n₀ = 0.1 m⁻¹; G_q(n₀) is called the road roughness coefficient, denoting the power spectral density at the spatial reference frequency n₀, with unit of m⁻³; and ω, called the frequency exponent, is the slope in double logarithmic coordinates that determines the frequency structure of the road power spectral density. For reinforced concrete bridges in China, it is recommended to simulate road roughness according to GB7031-86, with the triangular series superposition method being the preferred choice.

4. Monitoring Strategy Considering Noise Ratio and Coupled Vibration

The precision of the monitoring signal is a key factor determining the accuracy of the monitoring when calculating the general range of dynamic responses of the bridge and the vehicle group based on the theory of coupled bridge–vehicle vibration. Accurate data obtained through health monitoring systems are essential for precise results. Measured signals, received through sensors and data acquisition instruments, can be broadly categorized into three types. The first type is the structural vibration signals generated by traffic loads, dominating in amplitude and fully representing the dynamic characteristics of the structure. The second type is ambient vibration signals caused by natural environment factors such as wind and earthquakes, exhibiting small amplitudes and wide frequency bandwidths with characteristics of both randomness and structural dynamics. Often, such signals need to be suppressed and separated. The third type is electromagnetic interference signals or noise produced by natural sources and electrical equipment. Depending on the circumstances, this signal type has characteristics such as sudden, instantaneous, and periodic, exhibiting sudden, instantaneous, and periodic characteristics.

In traditional structural detection and monitoring, performance parameters such as instrument range, sensitivity, resolution, and scope are generally guaranteed to meet specified requirements. Among these properties, resolution refers to the minimum value that a test instrument can effectively distinguish, indicating its ability to discern signals from noise. The minimum decibel of the measured signal to noise ratio is typically specified as 20 lg(S/N) ≤ 5 dB, where S represents the signal level of the measured signal, and N is the noise level. The measured signal amplitude satisfying this equation is called the lowest measured vibration level, at which point the signal level is about 1.77 times the noise level. Considering the current general performance of monitoring instruments and the actual dynamic characteristics of the bridge, the above signal-to-noise ratio requirements are assumed to be conservative, making them fundamental prerequisites for bridge health monitoring.

One of the important elements of structural health monitoring is the removal of electromagnetic interference signals or noise signals to enhance vibration signals. Under normal conditions, various filtering techniques can partially suppress electromagnetic interference signals, but it is challenging to completely eliminate noise because electromagnetic interference and vibration signals often overlap in the frequency spectrum. In addition, general filtering techniques struggle to achieve real-time filtering and are computationally intensive. Hence, in actual data monitoring and analysis, it is necessary not only to optimize monitoring efficiency through filtering techniques but also to process massive data and optimize monitoring strategies.

The specific signal-to-noise ratio requirements are derived based on the actual situation, monitoring requirements and capabilities of the bridge, combined with the results of the vehicle–bridge coupling dynamic analysis.

Based on the above analysis, an optimization strategy for bridge health monitoring is proposed, taking into account both signal-to-noise ratio and vehicle–bridge interactions. The objective of this strategy is to ensure structural safety throughout the entire life cycle. It involves assessing the structural conditions, scale, form, and economic considerations to select suitable monitoring instruments and software systems. This approach aims to seamlessly integrate health monitoring with routine bridge inspections and combine real-time monitoring with timely assessments.

In adherence to the performance parameters requirement, particularly the signal-to-noise ratio, the dynamic characteristics of the bridge are estimated through ambient vibration and traffic load data provided by the monitoring system. This includes information on traffic flow magnitude, vehicle weight, and speed. Multilevel thresholds are defined for different traffic scenarios and safety states, serving as criteria to initiate signal filtering and damage detection. Upon detecting damage, the analysis model and monitoring strategy can be regulated, leading to the proposal of a maintenance scheme. The monitoring strategy flowchart is depicted in Figure 2. This approach offers the advantage of filtering out numerous signals with low signal-to-noise ratio or inaccessible data, thereby conserving computing resources. Ultimately, it enables purposeful monitoring based on theoretical analysis.

5. Application Analysis

5.1. Monitoring System and Sensor Layout

We applied the dynamic analysis and monitoring optimization strategies outlined above to a reinforced concrete bridge located in Beijing. A section of the bridge is presented in Figure 3. The structure is a simply supported T-girder beam bridge with a span of 30 m. The girder has a width of 12.50 m and a height of 1.80 m, featuring an equivalent cross-sectional moment of inertia of 0.80 m⁴. The number of traffic lanes is two. The equivalent elastic modulus is 3.53 × 10⁶ MPa, and the reduced density of the main beam is 3.66 × 10³ kg/m³. The monitoring is carried out in normal traffic conditions.

This bridge is equipped with a health monitoring system, specifically designed to observe bearing capacity and durability. Five items are considered in the health monitoring and evaluation system, namely sensing and data acquisition, data interrogation, modeling and simulation, correlation analysis and data management. Sensing and data acquisition consists of the following eight parts: monitoring strategies for damage prognosis, types of data for acquisition, required sensory systems and their properties in terms of measurable bandwidth and sensitivity, calibration and stability, sensor durability, typical numbers of sensors and locations of sensors, data acquisition/transmission/storage system and data cleaning. Data interrogation consists of the following six parts: data validation, feature extraction, data normalization, characterization of feature distribution, statistical inference for damage prognosis, and prediction modeling for future loading estimates. Modeling and simulation consists of the following two parts: types of models (numerical models, statistical models, etc.) and uncertainty analysis (environment variability, parametric variability and modeling variability). Correlation analysis consists of the following three parts: correlation of analysis-to-analysis for model verification and calibration, correlation of measurement-to-measurement for features extractions, defects detection and correlation of analysis-to-measurement for model verification and updating, performance monitoring, etc. Data management consists of the following four parts: communications among different hardware and software platforms, security, visualization and accessibility.

Five servo-controlled accelerometers and five inclinometers were symmetrically installed on the midspan and other symmetrical positions to collect and analyze the dynamic response. The accelerometer and inclinometer are shown in Figure 4 and Figure 5, respectively.

The sampling rate of accelerometer is 100 Hz. The sampling rate of inclinometer is 10 Hz. Two piezoelectric thin film sensors, spaced 8 m apart, were installed in the bridge deck to collect traffic signals, as shown in Figure 6.

The sampling rate of piezoelectric thin film sensor is 100 Hz. In addition, fifteen fiber Bragg grating strain sensors were installed on the bending bottom and inside the beam to monitor the load-bearing capacity of important components, as shown in Figure 7. The arrangement of the instruments was shown in Figure 8. The sampling period was 4 months.

5.2. Measured Responses and Analysis

During the 4-month-long monitoring period, data on vehicle speed and weight were obtained through piezoelectric thin film sensors, totaling 18,820 valid samples. The frequency histogram and probability density function for vehicle speed and weight are shown in Figure 9 and Figure 10, respectively. Apparently, the maximum vehicle speed does not exceed 30 m/s, and the maximum vehicle weight does not exceed 50 tons, meaning that the maximum value of the effective uniform load hardly exceeds 10 ton/m. Therefore, the analysis of the coupled vibration of the vehicle group–bridge system considers both normal and extreme conditions of vehicle speed and weight. The corresponding results will provide effective support for the optimization strategy of bridge health monitoring. In addition, based on the strain data from the midspan compression zone, the equivalent three-dimensional stress spectrum is calculated by using the rain-flow counting method. The results are presented in Figure 11, providing valuable insights for the optimal strategy.

The power spectrum of the measured acceleration is shown in Figure 12. The compare between the actual expected bridge frequency and the obtained measured frequency is shown in

Table 1. Comparison of measured and calculated values.

Number	FEM Frequency/Hz	Measured Frequency/Hz	Error/%	Description
1	5.56	5.76	3.60	1st vertical bending
2	18.10	16.50	8.84	1st longitudinal bending (symmetric)
3	24.44	26.11	6.83	2nd longitudinal bending of flange(anti-symmetric)
4	41.48	41.02	1.11	2nd vertical bending

. As shown in Table 1, the measured frequency is basically consistent with the FEM frequency.

Using the numerical analysis method proposed in this work for vehicle–bridge coupling analysis, we investigate the influence of traffic loads on the dynamic characteristics of the bridge. The coupling ratio of the vehicle group, defined by the mass density V_cm, is the ratio of the coupling uniform mass density to the total mass density of the vehicle group. In the initial theoretical analysis, V_cm is assumed as 0.3 or 0.15, the main body has a uniform mass density of 10 ton/m, and the vehicle speed is 15 m/s. The acceleration responses, with or without consideration of the coupling mass of the vehicle group, are shown in Figure 13. The results show that the dynamic response of the bridge, considering the coupling mass of the vehicle group, is slightly reduced, but the difference is not significant. Based on experience and monitoring data, the actual coupling mass ratio can be set to 0.15 for normal analysis.

With the gradual deterioration of the pavement layer, road surface roughness becomes a crucial consideration in dynamic analysis and the estimation of response range. According to the Chinese national standard, there are eight grades for road roughness, with grade C assumed to be suitable for this bridge according to visual inspection, indicating good surface condition. The average value of the displacement power spectral density coefficient G_q(n₀) is 256 × 10⁻⁶ m⁻¹. Based on the triangular series superposition method, the simulated road roughness curve is obtained, as shown in Figure 14, assuming a vehicle group speed of 10 m/s. Figure 15 presents the fitted results of the simulated spatial power spectral density, indicating a strong level of fitness. Next, based on the simulation results, we discuss the influence of road roughness on dynamic analysis. Assuming a uniform mass density of 10 ton/m and a speed of 10 m/s or 20 m/s for the vehicle group, we calculate the corresponding dynamic responses of the vehicle and the bridge using Equations (10)–(16), considering the impact of road roughness. Figure 16 and Figure 17 illustrate the dynamic response of the vehicle group at different speeds. The comparison of these figures reveals that the increase in speed reduces the response of the vehicle, but the effect is considerably less than that of road roughness. Figure 15 demonstrates that when road roughness is considered, the dynamic responses of the vehicle group, in terms of displacement, velocity and acceleration, significantly increase. Passengers in the vehicle may experience discomfort at certain points in time, as the vertical acceleration approaches the uncomfortable threshold of 0.5 m/s². Therefore, it is crucial to consider the precise degree of road roughness and its impact on the vehicle, especially when studying driving comfort.

On the other hand, the responses of the bridge under different vehicle speeds exhibit periodic fluctuations, as shown in Figure 18 and Figure 19. The dynamic responses of the bridge significantly amplify as the speed of the vehicle group increases. The impact of road roughness on the dynamic responses of the bridge is only slightly greater than that without roughness effects, given the bridge’s sufficient stiffness and mass to resist the impacts and disturbances caused by road roughness.

To investigate the combined effect of uniform mass and road roughness on the dynamic response of the bridge, the displacement and acceleration history is calculated assuming a vehicle group of 10 m/s, a coupling mass ratio of 0.15, and mass densities of the main body of 5 ton/m and 10 ton/m. The results are shown in Figure 20 and Figure 21. It is observed that the response amplitudes are mainly determined by the uniform mass of the vehicle group, and heavy vehicle groups may induce excessive deflection and stress, emphasizing the need for the strict monitoring and control of overloading. The rate of increase in structural acceleration caused by surface irregularity generally does not exceed 7%.

Existing studies have shown that the coupled vibration between different types of vehicles and bridges is complex, and the response does not vary linearly. Therefore, it is necessary to study the typical maximum amplitude under vehicles with different frequencies and speeds to estimate the effective monitoring threshold values.

Figure 22 and Figure 23 show family curves of the maximum displacement and acceleration amplitude of the bridge with a uniform mass density of 10 ton/m but different vertical frequencies and speeds. The results indicate that the coupling effect between the vehicle group and the bridge tends to stabilize when the vertical frequency of vehicles is greater than 0.4 Hz. As the speed of the vehicle group increases, the displacement and acceleration of the bridge increase significantly, but the fluctuation is also more noticeable. Therefore, the coupling effect of vehicle–bridge cannot be ignored, and the driving speed is an important factor for comprehensive analysis.

Figure 24 illustrates the bridge acceleration signals collected by the health monitoring system with or without vehicle effect. The total amplitude of the measured ambient vibration signal and electromagnetic interference signal typically falls in the range of ±1.0 × 10⁻³ m/s², while the acceleration amplitude caused by a vehicle is generally in the range of ±0.07 m/s², consistent with the results of the numerical analysis.

Considering the monitoring needs, pavement flatness, environmental factors, along with the preliminary monitoring results, the bridge health monitoring threshold is generally determined as follows: If the acceleration amplitude is below ±5.0 × 10⁻³ m/s² or the displacement amplitude is less than ±1.0 × 10⁻⁴ m, the modal analysis function is not initiated. Otherwise, suitable data are selected, and modal analysis is carried out by using the stochastic subspace method to obtain the frequencies, modal shapes, and damping ratios of the first three order modes. If the acceleration amplitude is less than ±1.0 × 10⁻² m/s² or the displacement amplitude is less than ±1.8 × 10⁻⁴ m, the damage detection program is not initiated. Otherwise, appropriate data are selected for damage detection, determining the possible damage location and degree, and the corresponding important data are stored for long-term analysis.

Following this monitoring strategy, only essential data are recorded and analyzed, effectively saving computational resources while ensuring the effectiveness of health monitoring.

For example, in a scenario where the health monitoring system operates continuously for 24 h, the memory space in the original database is 504.15 megabyte. However, after the implementation of the monitoring strategy proposed in this paper, the actual memory space is reduced to only 57.06 megabyte. The valid data used for the actual analysis aligns more closely with the conclusions.

6. Conclusions

In this study, a dynamic vehicle group model incorporating main uniform parameters and coupling mass is established, and the dynamic coupling equations for the vehicle group–bridge system are proposed. Analytical expressions are derived based on the modal superposition method. In addition, improved equations, considering the influence of road roughness, are incorporated into the coupling equations.

The effective selection, denoising, analysis, and storage of monitoring data in bridge health monitoring systems are crucial in engineering applications. The proposed monitoring strategy involves signal component analysis, bridge structure establishment, dynamic equilibrium equation formulation for moving vehicle groups, and analytical expression derivation. This strategy has the advantage of filtering out a large amount of data with a low signal-to-noise ratio or data that lack significance, thereby saving computing resources.

Analyzing simulation results in comparison with actual monitoring outcomes reveals that, when considering road roughness, the bridge responses decrease with an increase in coupling mass ratio, but dynamic values increase. Bridge response variations are also influenced by vehicle frequencies. However, the most influential parameters affecting dynamic amplitudes of a bridge are vehicle load and speed.

For vehicle groups, the critical parameters affecting responses and comfort are vehicle speed and road roughness. Combining the theory of vehicle–bridge coupling dynamics with actual monitoring results allows for the determination of more definite monitoring thresholds. Appropriate analysis techniques can then be adopted for different types of data. Therefore, it is crucial to conduct an in-depth study based on vehicle–bridge coupling dynamics theory and actual monitoring results. The combination of theoretical analysis and experimental results can establish more explicit monitoring thresholds, enhancing the effectiveness of the health monitoring strategy to improve the cost–benefit ratio and monitoring quality.