Dynamic response prediction of long-span cable-stayed bridges by integrating Bi-LSTM and physical modeling

Abstract

Accurately predicting the vibration response of large-span cable-stayed bridges is essential for assessing their structural safety. Traditional physical models based on vehicle-bridge interaction techniques can effectively simulate bridge vibrations but may yield results that deviate from actual observations. Meanwhile, data-driven methods employing neural network surrogate models excel in prediction accuracy due to their robust nonlinear fitting capabilities but often require extensive datasets for training. These models may struggle with generalization when faced with limited or inadequately labeled data. This study introduces a novel approach that integrates physical models with data-driven methods to predict the dynamic response of long-span cable-stayed bridges. This hybrid method leverages the interpretability and robustness of physical models while enhancing predictive accuracy through data-driven techniques. Field tests on bridges validated the method’s applicability and effectiveness, demonstrating an 85% improvement in prediction accuracy for bridge vibration responses.

Keywords

cable-stayed bridge vibration response prediction vehicle-bridge interaction physical and data-driven fusion neural network

Introduction

Vehicular loads are important active loads on long-span bridges and have a profound effect on the service safety of bridges (Li et al., 2021; Yi et al., 2010; Yu and Cai, 2019). This effect has become particularly evident in the complex dynamic system of vehicle-bridge interaction (VBI) (González et al., 2023; Lu et al., 2020; Yang et al., 1997). In this system, bridge and vehicle vibrations are complexly coupled, especially under conditions of varying road surface roughness (Feng et al., 2023; Huang et al., 2023; Xia et al., 2022; Zhang et al., 2025). The response of this coupled dynamic system is difficult to reasonably model and predict (Abbas et al., 2020; Malekjafarian et al., 2022; Mohammadi et al., 2019). Therefore, an effective prediction method is needed for dynamic response prediction of long-span bridges.

The dynamic prediction of the VBI system in long-span bridges is an important problem that has been studied for many years (Jian et al., 2022; Piao et al., 2023; Zuo et al., 2024). Many scholars have made significant progress in this field (Yin et al., 2023b). For instance, Camara et al. (2019) proposed an entire framework of wind-vehicle-bridge interaction for predicting the dynamic response of the long-span cable-stayed bridge. In this framework, the vehicle is modeled as a three-dimensional (3D) mass-spring-damper system with multiple degrees of freedom (DOFs). To improve the computational efficiency, Eshkevari et al. (2020) introduced a simplified VBI model to simulate the random traffic loads, in which the vehicle model is simplified to a two-dimensional (2D) mass-spring-damper system with one DOF. Li and Feng (2023) comparatively investigated the difference in the bridge dynamic response between 2D and 3D vehicle models, and the results indicated that the response of the 2D vehicle model for cable-stayed bridges is greater than that of the 3D vehicle model.

Despite the widespread use of mass-spring-damper systems in both 2D and 3D vehicle modeling, the vibration effect of bridges is still amplified by the point contact pattern of the vehicle model with the pavement (Greco et al., 2020; Kreslin et al., 2024; Xiao and Ren 2019). developed a 3D refined finite element (FE) vehicle model by using the LS-DYNA software. Compared to the conventional point contact, the FE vehicle model considered a tire-pavement contact area, which makes the simulated dynamic response more realistic (Chen et al., 2018; Kumar et al., 2021; Yin et al., 2023a). However, the modeling of refined finite element vehicles is complex, and the accurate response results are computationally time-consuming (Ni et al., 2022). Hence, many scholars have used the surrogate model based on data-driven methods to predict the dynamic response of bridges to reduce the consumption of computational processing (Lei et al., 2023a; 2023b). Lei et al. (Lei et al., 2021) proposed a deep convolutional GAN to reconstruct lost structural health monitoring data, ensuring accurate modal identification and strain analysis. Li et al. (2021) adopted the feedforward neural network and long short-term memory (LSTM) network to model the VBI system. Aloisio et al. (2023) explored the use of artificial neural networks and genetic programming techniques to develop surrogate models capable of effectively predicting the VBI dynamic response under braking conditions. Once the surrogate model based on data-driven is established and well-trained, it allows low-cost simulations that can be used to conduct the subsequent dynamic analysis of the long-span bridge (Deng et al., 2023; Huang et al., 2025; Rachedi et al., 2021; Zhu et al., 2022). However, the surrogate model faces the challenge of insufficient labeling data, which limits the surrogate model from extracting features of the vibration response (Levine et al., 2022; Sun et al., 2023). Notably, the accurate physical information of the VBI system can be provided by the reasonable 3D vehicle model, which improves the reliability and prediction accuracy of the surrogate model.

This study introduces a novel approach to predicting the dynamic response of long-span cable-stayed bridges by integrating a 3D vehicle model with a bidirectional long short-term memory (Bi-LSTM) network. The 3D vehicle model used in this research accounts for the tire-pavement contact area, providing a more realistic representation of the vehicle-bridge interaction (VBI) system compared to traditional 3D vehicle models. This model also simplifies the complexity of vehicle modeling relative to refined finite element (FE) vehicle models. The enhanced 3D vehicle model significantly improves the accuracy of dynamic calculations for long-span cable-stayed bridges. Furthermore, the Bi-LSTM network reduces the computational cost of structural dynamic analysis by capturing the input-output relationship between vehicle loading and bridge response. Compared to the standard LSTM network, the Bi-LSTM network excels in processing time series data due to its bidirectional architecture. The effectiveness and accuracy of the proposed method were validated through field tests on large-span cable-stayed bridges. Key factors, including prediction accuracy, computational efficiency, and the physical model, are thoroughly evaluated and discussed.

Data-driven-based surrogate modeling

In this section, the surrogate model via 3D vehicle models and Bi-LSTM network is developed. Specifically, the 3D vehicle model is built entirely on general FE software ABAQUS without the need for complex co-simulation with other software applications such as MATLAB or SIMPACK (Borjigin et al., 2018; Nguyen et al., 2017; Zou et al., 2022). Furthermore, the road surface roughness can be easily considered with the built-in contact approach of ABAQUS (Yang et al., 2024). The training dataset given to the surrogate model to capture the dynamic characteristics of the large-span cable-stayed bridge can be obtained from the VBI system built in ABAQUS. The accuracy of the predictive capability of the resulting surrogate models is then corroborated by considering a validation dataset (i.e., new data that were not included in the training dataset).

Modeling method of the VBI system

3D vehicle model

The 3D vehicle model is illustrated in Figure 1. Vehicle body is a rigid body, and has 3 DOFs, which are vertical displacement ( $Z_{v}$ ), pitching rotation ( $θ_{v}$ ), and roll displacement ( $φ_{v}$ ), respectively. $m_{v}$ is the vehicle body’s mass. The suspension system of the vehicle is simulated by a series of the spring-damper. $m_{w}$ is the wheel mass, and $Z_{w}$ is the vertical displacement of wheels (Stoura et al., 2020). Therefore, the entire vehicle model has a total of 9 DOFs. To consider the tire-pavement contact area, a tire model based on the Mooney-Rivlin approach is simulated. Mooney-Rivlin is a mathematical model used to describe the material of hyper-elastic materials, such as rubber or certain polymers (Sidhu et al., 2023). Mooney-Rivlin can be written as:

W = C_{01} ({\bar{I}}_{2} - 3) + C_{10} ({\bar{I}}_{1} - 3) + \frac{1}{D_{1}} {(J - 1)}^{2}

(1)

where

W

is the strain energy per unit volume,

C_{01}

and

C_{10}

are material constants related to the distortional response,

{\bar{I}}_{1}

and

{\bar{I}}_{2}

are first and second invariants of the strain tensor, respectively;

D_{1}

is the material constant related to the volumetric response, and

J

is the Fick’s laws of diffusion, which is expressed as:

J = - D_{c} \frac{d φ}{d x}

(2)

where

D_{c}

is the diffusion coefficient,

φ

is the concentration, with a dimension of the amount of substance per unit volume, and

x

is position.

Figure 1.

The schematic diagram of the 3D vehicle model.

Furthermore, the 3D vehicle model can be established from existing elements in ABAQUS software. The S8R element is used to build the vehicle’s body and axle, the vehicle body and axle are connected by rigid rods, the SPRING element is simulated to the suspension system, and the C3D8R element is employed to generate the tire model (Yin et al., 2023b). The parameters of the 3D vehicle model are listed in Table 1.

Table 1.

The parameters of the 3D vehicle model in ABAQUS (Yin et al., 2023a).

Member	Element type	Symbol	Value
Vehicle body	S8R	$m_{v}$	29990 kg
		$θ_{v}$	172160 kg·m²
		$φ_{v}$	31496 kg·m²
Axle	S8R	$m_{a f}$	824 kg
Axle	S8R	$m_{a r}$	968 kg
Suspension system	SPRING	$C_{1} = C_{2}$	2190 N·s/m
		$K_{1} = K_{2}$	242604 N/m
		$C_{3} = C_{4} = C_{5} = C_{6}$	7882 N·s/m
		$K_{3} = K_{4} = K_{5} = K_{6}$	1903172 N/m
Wheel	/	$m_{w 1} = m_{w 2}$	65 kg
		$m_{w 3} = m_{w 4}$	82 kg
		$m_{w 5} = m_{w 6}$	88 kg
Tire	C3D8R	$C_{01}$	0.48 MPa
		$C_{10}$	0.16 MPa
		$D_{1}$	0.001 MPa^-1

The road surface roughness

Road surface roughness is an important factor in causing coupled vibration between vehicles and bridges. Normally, the road surface roughness is constructed from the power spectral density (PSD) function. However, the PSD is converted to a series of real numbers using a 2D inverse fast Fourier transform (IFFT) to simulate 3D road surface roughness since the bridges and vehicles are modeled in 3D, as shown in Figure 2. The 3D road surface roughness formulation can be written as:

r_{z} (m, n) = \frac{1}{M N} \sum_{p = 1}^{M - 1} \sum_{q = 1}^{N - 1} F (f_{p}, f_{q}) e^{j 2 π (\frac{m}{M} p + \frac{n}{N} q)}

(3)

where

r_{z}

is the 3D road surface elevation, as well as M and N are the number of sampling points on the lateral and longitudinal directions of the road, respectively;

f_{p}

and

f_{q}

are the discrete frequency components of the road in the lateral and longitudinal directions, respectively;

p

and q are the time series components in the lateral and longitudinal directions, respectively;

F (f_{p}, f_{q})

is the Fourier transform for both

f_{p}

and

f_{q}

. More details about the derivation of the 3D road surface roughness can be found in (Oliva et al., 2013; Yao et al., 2018).

Figure 2.

The 3D road surface roughness.

The VBI system in ABAQUS

The VBI system is established in the general FE software ABAQUS due to its powerful nonlinear analysis performance. The dynamic equation of the VBI system can be written as follows:

[\begin{array}{l} M_{b} & 0 \\ 0 & M_{v} \end{array}] {\begin{cases} {\ddot{d}}_{b} \\ {\ddot{d}}_{v} \end{cases}} + [\begin{array}{l} C_{b} & 0 \\ 0 & C_{v} \end{array}] {\begin{cases} {\dot{d}}_{b} \\ {\dot{d}}_{v} \end{cases}} + [\begin{array}{l} K_{b} & 0 \\ 0 & K_{v} \end{array}] {\begin{cases} d_{b} \\ d_{v} \end{cases}} = {\begin{cases} F_{b}^{c} + F_{v}^{G} \\ F_{v}^{c} \end{cases}}

(4)

where

C_{b}

and

C_{v}

are the damping matrices of the bridge and vehicle, respectively;

F_{b}^{c}

and

F_{v}^{c}

are contact forces between the bridge and the vehicle, respectively;

F_{v}^{G}

is the self-weight of the vehicle.

In the constrained optimization problems within the FE model, particularly those involving normal contact and lateral friction, the penalty method emerges as a vital algorithm for simulating the VBI model (Lu et al., 2020), and is delineated as follows:

{\begin{cases} \min \prod p (u) \\ δ_{i} (u) \neq 0, i = 1, 2, . . ., m \end{cases}

(5)

where

\prod p (u)

is the potential energy of the VBI system, and

δ_{i} (u)

is the gap function of the contact area that is a function of the distance

u

between the wheel and the bridge deck contact point. In the iterative computation of the FE model, the contact situation

δ_{i} (u)

is treated as a constraint, and the potential energy

\prod p (u)

is taken as an objective function. To minimize

\prod p (u)

, a virtual elastic penalty item is introduced in equation (5), which transforms constrained optimization into unconstrained optimization. Hence, the general principle of the potential energy function is rewritten as:

f_{\min} (u, r_{k}) = \min {\prod p (u) + r_{k} \sum_{i = 1}^{m} G [δ_{i} (u)]}

(6)

where

G [δ_{i} (u)]

is the penalty function, and

r_{k}

is the penalty parameter. The optimal solution of equation (6) is convergent to that of equation (5).

In ABAQUS, the dynamic response of the VBI system can be solved by using the Hilber-Hughes-Taylor-α (HHT-α) algorithm, in which the displacement, velocity, and acceleration of the VBI system are expressed as:

u_{t + Δ t} = u_{t} + Δ t \cdot {\dot{u}}_{t} + \frac{Δ t^{2}}{2} [(1 - 2 β) \cdot {\ddot{u}}_{t} + 2 β \cdot {\ddot{u}}_{t + Δ t}]

(7)

{\dot{u}}_{t + Δ t} = {\dot{u}}_{t} + Δ t \cdot [(1 - γ) \cdot {\ddot{u}}_{t} + γ \cdot {\ddot{u}}_{t + Δ t}]

(8)

{\ddot{u}}_{t + Δ t} = α \cdot {\ddot{u}}_{t + Δ t} + (1 - α) \cdot {\ddot{u}}_{t}

(9)

where

u_{t + ∆ t}

{\dot{u}}_{t + ∆ t}

{\ddot{u}}_{t + ∆ t}

are the displacement, velocity, and acceleration at time

t + Δ t

, respectively;

Δ t

is the time step,

β

and

γ

are parameters related to the Newmark-β method that affect stability and accuracy, and

α

is the HHT-α algorithm-specific parameter that introduces algorithmic damping.

Moreover, the Newton-Raphson method can be implemented to iteratively solve equation (4). Notably, the HHT-α and Newton-Raphson methods have been already integrated into ABAQUS, in which the detailed theoretical derivation and parameter settings can be found in (Lu et al., 2020; Melo et al., 2018). Figure 3 shows the computation process of the VBI system in ABAQUS. The main steps are as follows:

Step-1. Create a VBI system based on the mass, stiffness, and damping matrices of the vehicle and bridge. The road surface roughness is used as the dynamic input to the VBI system.

Step-2. Determine the initial time ( $t_{s t a r} = 0$ ) and step time ( $∆ t$ ) of the system. The penalty method is utilized to calculate the contact behavior between vehicles and bridges.

Step-3. Calculate the preliminary dynamic response of the system at time point $t = t_{0} + ∆ t$ by the HHT-α based time integration method and determine the initial contact condition for this time integration step.

Step-4. Check the results of the calculations from Step 3, including contact conditions, residual forces, and residual displacements. Ensure the vehicle-bridge contact is reasonable. The contact force residuals should be within a 1.0% error margin. Displacement convergence should meet a tolerance of is 1 × 10^-6 using the Newton-Raphson method.

Step-5. Iteratively solve equation (4) using the Newton-Raphson method. The same procedure calculates the system vibration response for the next time step $(t_{0} + ∆ t) + ∆ t$ until the end of the calculation.

Figure 3.

The computation process of the VBI system.

The Bi-LSTM network architecture

The LSTM network is a variant of the conventional recurrent neural networks designed to improve the dynamic capture performance in time series data processing (Lu et al., 2023). The dynamic response of the bridge is a typical time series signal with a strong connection between the data. Hence, the LSTM network can be implemented to focus on the collected temporal information for accurate vibration signal prediction (Chen et al., 2023). While convolutional neural networks (CNNs) excel at capturing local dependencies and spatial hierarchies, making them highly efficient for tasks such as image classification, they are less effective at modeling long-range dependencies compared to bidirectional long short-term memory networks (Bi-LSTMs) (Zhang et al., 2022). In contrast, the transformer models are particularly well-suited for capturing long-range dependencies and handling large-scale data, especially in natural language processing tasks, although they generally require more computational resources (Zrira et al., 2024). Consequently, Bi-LSTMs provide a balanced choice, offering a good trade-off between accuracy and computational efficiency, particularly for tasks that demand contextual understanding. The LSTM network consists of LSTM cells, and its architecture is shown in Figure 4. As shown in Figure 5, three gate units are designed in the LSTM cells, that is, the input gate, the forget gate, and the output gate, to reduce the network overfitting phenomenon. The input gate plays a role in managing the retention of incoming data. The forget gate is capable of adjusting the extent to which long-term dependencies from the prior timestep are considered. Updates to the cell’s information are influenced by the results from both the input and forget gates. The output gate is employed to alter the hidden state output of the LSTM cell, taking into account the refreshed state of the cell. The internal computations of the LSTM cell can be described as

{\hat{c}}_{i} = \tan h (W_{\hat{c}} h_{i - 1} + W_{\hat{c}} x_{i} + b_{\hat{c}})

(10)

e_{i} = σ (W_{e} h_{i - 1} + W_{e} x_{i} + b_{e})

(11)

f_{i} = σ (W_{f} h_{i - 1} + W_{f} x_{i} + b_{f})

(12)

o_{i} = \tan h (W_{o} h_{i - 1} + W_{o} x_{i} + b_{o})

(13)

c_{i} = c_{i - 1} ⊙ f_{i} + e_{i} ⊙ {\hat{c}}_{i}

(14)

h_{i} = o_{i} ⊙ \tan h (c_{i})

(15)

where

c_{i}

e_{i}

f_{i}

o_{i}

, and

h_{i}

are the cell state, the input gate, the forget gate, the output gate, and the hidden output, respectively;

W_{p} (p = c, e, f, o)

is the weight matrices corresponding to different gate units,

b_{p} (p = c, e, f, o)

is the corresponding bias vector,

σ (\cdot)

is the logistic sigmoid activation function,

\tanh (\cdot)

is the hyperbolic tangent activation function, and the operator

⊙

is the vector dot product.

Figure 4.

The architecture of the LSTM network.

Figure 5.

The LSTM cell.

The Bi-LSTM comprises a pair of unidirectional LSTM layers, each processing the input data in a distinct direction (forward and backward in time), subsequently integrating their respective outputs. This architecture is illustrated in Figure 6. The Bi-LSTM networks can capture information that may be overlooked by the unidirectional LSTM networks, suggesting the use of the Bi-LSTM instead of the unidirectional LSTM in time series analysis to solve the prediction problem (Roy and Chen, 2021).

Figure 6.

The Bi-LSTM structure.

Framework of the surrogate model based on data-driven

As shown in Figure 7, the innovative surrogate model framework that fuses the VBI system and the Bi-LSTM networks is developed to analyze the dynamic response of the long-span cable-stayed bridge. The proposed framework comprises three parts.

Figure 7.

Framework of the surrogate model based on data-driven.

In part one, the basic information of the cable-stayed bridge, including component size, modal frequency, and material parameters, is obtained by field measurement. Similarly, test vehicle information is obtained for gross vehicle weight, suspension stiffness, suspension damping, and tire parameters. More details will be described in Section 3.

In part two, the physical model of bridges and vehicles are built in ABAQUS according to the bridge and vehicle information, respectively. Furthermore, the vehicle and bridge models are combined by a penalty method to create the VBI system. A large amount of bridge vibration simulation data is acquired by the VBI system.

In the last part, the Bi-LSTM network architecture is built, and simulated data is fed into the network for training to find the physical characteristics of bridge vibration. The trained Bi-LSTM network is used for vibration prediction of the real bridge.

Case study

Field measurement data collection

Prototype bridge

The Jiayu Yangtze River Bridge is implemented as the test bridge to verify the accuracy and efficiency of the proposed method. Jiayu Yangtze River Bridge, located in Hubei Province, China, is a large-span double-tower cable-stayed bridge across the Yangtze River with a span arrangement of (70 + 85+72 + 73) m + 920 m+(330 + 100) m, as shown in Figure 8. There are 240 stay cables in the whole bridge. The stay cables are finished low-relaxation high-strength parallel steel wire cables. The standard strength of the steel wires is 1770 MPa. The main girder is composed of steel box girders, with a width of 38.5 m and a height of 3.8 m (Figure 8(b)). The bridge has eight lanes (four in each direction).

Figure 8.

Diagram of Jiayu Yangtze River Bridge (unit: cm): (a) Panoramic view of the bridge (b) schematic modeling of the bridge.

The full-scale FE model of the Jiayu Yangtze River Bridge can be established using ABAQUS software, and the main materials and mechanical properties of the bridge components are listed in Table 2. In the bridge FE model, the main girders are constructed using the C3D8R solid element, an eight-node brick element with reduced integration. The pylons and piers of the bridge are built using the B31 beam element, which is designed to optimize the computational efficiency while taking into account the deflection deformations. Moreover, the stay cable of the bridge is modeled by B31 the truss element. The whole bridge is discretized into 73,359 nodes and 78,616 elements. The connection between the stay cable and the main girder is simulated with a set of coupled DOFs. The support between the main girder and the pylon is simulated by coupling the vertical and transversal DOFs, and all DOFs at the bottom of the pylons and piers are restrained, as shown in Figure 9.

Table 2.

Main parameters of the bridge component.

Component	Material type	Density (kg·m^-3)	Young’s modulus (GPa)	Poisson ratio
Main girder	Steel	7850	206	0.30
Pylon	Concrete	2600	34.5	0.23
Stay cable	Steel	7830	195	0.30
Pier	Concrete	2600	34.5	0.23

Figure 9.

The full-scale FE model of Jiayu Yangtze River Bridge.

The wireless modal test and analysis system (DH5907 A) was used to conduct the field modal test of the bridge, with a frequency range of 0.06 to 40 Hz. The test results were used to update the FE bridge model, and the first six frequencies of the updated model, along with their differences from the measured frequencies, are listed in Table 3. The maximum difference between the modeled and measured frequencies is about 15%, suggesting some room for improvement in the FEM accuracy.

Table 3.

Compared the first six-mode frequencies of the FE model and measured data.

Mode type	Measured (Hz)	FE model (Hz)	Different (%)
1^st-order vibration mode	/	0.085	/
2^nd-order vibration mode	0.166	0.141	15.06
3^rd-order vibration mode	0.254	0.228	10.24
4^th-order vibration mode	0.298	0.291	2.35
5^th-order vibration mode	0.381	0.360	5.51
6^th-order vibration mode	0.396	0.384	3.03

Notably, the 1st-order mode corresponds to a rigid-body mode, which is the longitudinal translation of the girder. This mode involves extremely low-frequency vibrations, and its frequency is outside the operational range of the testing equipment, which spans from 0.06 Hz to 40 Hz. As a result, the frequency of the 1^st-order mode could not be captured during the field test due to the low-frequency limit of the equipment.

To reduce this discrepancy, potential refinements include refining material properties based on additional testing, improving boundary conditions to better reflect the actual constraints, refining the mesh in critical regions, including higher-order modes for a more complete dynamic response, and calibrating damping parameters with experimental data. These improvements could help align the model more closely with the measured frequencies.

Test vehicles

The 35,500 kg three-axle truck was utilized for loading in this field test, as shown in Figure 10. The static weighing system was used to control the axle weight and gross weight of each vehicle. The actual weight deviation of a single-loaded vehicle ranged from −0.94 % to 1.00 %, with an average weight of 35,510 kg and an axle weight distribution of 7110 kg + 14,200 kg + 14,200 kg. The stiffness and damping of the vehicle’s suspension were determined from the vehicle design information.

Figure 10.

Test vehicle: (a) vehicle weighing (b) axle weight distribution (c) vehicle size (cm).

In the ABAQUS software, the 3D vehicle model is constructed using predefined elements. The S8R element is utilized for the fabrication of the vehicle’s body and axle, which are interconnected via rigid rods. The SPRING element is employed to model the suspension system, whereas the C3D8R element is applied in the development of the tire model. Details of the vehicles are given in Table 1 of Section 2, and will not be repeated here.

Modeling of the VBI system

The road surface roughness simulation and verification

The critical factor in establishing a VBI system in ABAQUS is to simulate the road surface roughness to provide vibration excitation to the bridge. However, by generating the irregular road surface model directly in the FE model, the model mesh must be very refined to obtain a more realistic road surface state, which will lead to the consumption of huge computational resources by the VBI system. A new road model can be generated by considering the road surface roughness as an external force applied to the axle coupling system (Yao et al., 2018), as shown in Figure 11. This model has the same accuracy as other FE models while being easier to generate in the commercial FE software. The road surface roughness formula can be written as

F_{r o a d} = K_{v} r [\begin{array}{l} - {N} \\ 1 \end{array}] = [\begin{array}{l} M_{b} \\ M_{v} \end{array}] (\begin{array}{l} {\ddot{d}}_{b} \\ {\ddot{d}}_{v} \end{array}) + [\begin{array}{l} K_{b} + v^{2} M_{v} {N} \frac{\partial^{2} {N}^{T}}{\partial x^{2}} + K_{v} {N} {N}^{T} & - K_{v} {N} \\ - K_{v} {N}^{T} & K_{v} \end{array}] (\begin{array}{l} d_{b} \\ d_{v} \end{array}) - [\begin{array}{l} - M_{v} g {N} \\ 0 \end{array}]

(16)

where

M_{b}

and

M_{v}

are the mass matrices of the bridge and vehicle, respectively; m and n are the numbers of DOFs of the bridge and vehicle, respectively;

{\ddot{d}}_{b}

and

{\ddot{d}}_{v}

are the acceleration vector of the bridge and vehicle, respectively;

K_{b}

and

K_{v}

are the stiffness matrices of the bridge and vehicle, respectively; v is the vehicle speed,

{N}

is a vector containing the cubic Hermitian interpolation functions,

d_{b}

and

d_{v}

are the displacement vector of the bridge and vehicle, respectively;

r

is the road surface profile, which can be generated from the displacement spectral density, and

K_{v} r

is the external force caused by the road surface roughness.

Figure 11.

Schematic diagram of the equivalent method for road surface roughness.

Numerical simulations are carried out to study the accuracy of the simulated approach to the road surface roughness by comparing it with the author’s previous literature (Yin et al., 2017). The reference result was obtained using the Newmark-β integration approach. The first three intrinsic frequencies of the bridge were 2.81 Hz, 4.35 Hz, and 6.57 Hz. Two vehicle speeds (30 km/h and 40 km/h) and three road levels (very good, average, and measured) were considered. Since the initial state of the vehicle in the axle interaction procedure in MATLAB is a static equilibrium state, the dynamical equilibrium state of the vehicle in ABAQUS was also used as the initial state for comparison between the two methods.

As shown in Figure 12, the vertical displacements in the span of the bridge are considered for different road surfaces when the speed is 30 km/h and 40 km/h. It can be seen that the differences between the proposed method and the reference results are small. These small vibration differences indicate that the proposed method can accurately simulate the VBI system for various pavement classes.

Figure 12.

Bridge displacement at different road surface levels: (a) the road surface level is very good at vehicle speed = 30 km/h (b) the road surface level is average (c) the road surface level is measured (d) the road surface level is very good at vehicle speed = 40 km/h (e) the road surface level is average (f) the road surface level is measured.

Validation of VBI system using field-measured data

To determine the simulation accuracy of the VBI system, the measured main girder deflection data of the bridge was used to validate the complete system. The overall deflection value of the main girder was observed by a gas-liquid coupled differential pressure deflectometer, and the deflection measurement points of the bridge were arranged at the field, as shown in Figure 13. The measurement points are arranged as four equal points for each span of the north side span (70 m + 85 m + 72 m + 73 m), 16 equal points for the main span of 920 m, eight equal points for the south side span of 330m, and four equal points for the south side span of 100m. A total of 90 measurement points for the whole bridge.

Figure 13.

Bridge deflection measurement point arrangement and system calibration.

The 52 test vehicles were used to load the bridge slowly onto the bridge to obtain the deflection values of the bridge. Similarly, the bridge model was loaded using the same vehicle model in the VBI system, as shown in Figure 14. The measured and numerical data were compared to assess the accuracy of the VBI system simulation, as shown in Figure 15.

Figure 14.

Vehicle loading in the bridge.

Figure 15.

Comparison of measured and simulated deflection values of the bridge.

Furthermore, the differences were quantified using the Root Mean Square Error (RMSE), a metric typically used to measure the difference between modeled and measured values. The smaller the RMSE value, the less the numerical data deviates from the measured data. The RMSE values for the two sides of the bridge are 1.77 and 1.93, respectively, indicating that the difference between the simulated and measured deflections is small.

Data-driven framework for bridge dynamic prediction

The surrogate model construction and training

The architecture of the network and the hyperparameter settings are essential steps when designing a surrogate model based on the Bi-LSTM network. In this study, the architecture of the Bi-LSTM network is illustrated in Table 4. The input data is the vibration response of the VBI system at 2400 time points, and the output data is the predicted response of the bridge at 2000 time points. The network hyperparameters are set as follows: initial learning rate is 0.001, the Adam optimization algorithm is used, the batch size is 32, the dropout value is 0.5, and the number of iterations is 500. The surrogate model training is carried out on a standard personal computer equipped with an Inter(R) Core i7-10,700 CPU@2.9 GHz and an NVIDIA GEFORCE RTX 2060 12 GB graphics card.

Table 4.

Architecture of the Bi-LSTM network.

Layer	Kernel	Input dimension	Output dimension	Activation function
Input	/	1 × 2400	1 × 2400	/
Bi-LSTM	128	1 × 2400	1 × 2400 × 256	Tanh
Pooling	/	1 × 2400 × 256	1 × 2000 × 256	/
Full connectivity	128	1 × 2000 × 256	1 × 2000 × 128	ReLU
Output	/	1 × 2000 × 128	1 × 2000	/

The division of the dataset is critical to ensure that the surrogate model can learn bridge vibration features effectively and has the ability to generalize. The 300 vibration data collected from the VBI system were divided into a training set (200) and a validation set (100). The validation set was used for parameter tuning during model training and model generalization capability assessment. The data collected from the real bridge was used as a test set to evaluate the prediction performance of the trained surrogate model.

As shown in Figure 16, the training loss history obtained from the training process of the Bi-LSTM network is generally decreasing. The training loss value for the bridge displacement response is converged to 0.02, while the training loss value for the bridge acceleration response is converged to 0.10. In addition, the training loss for the displacement response is smoother than the training loss for the acceleration response. The results show that the training effect of the bridge displacement response is better than the acceleration response, as the training loss value of the former network is reduced more significantly. The reason may be that predicting the acceleration response requires the model to capture more complex nonlinear relationships than the displacement response.

Figure 16.

Training loss history: (a) the displacement response (b) the acceleration response.

Normally, displacement data is smoother than acceleration data because acceleration is the second-order derivative of displacement. Smooth data is easier for the model to learn and predict. Therefore, displacement data will be better trained than acceleration data training.

Analysis and discussion of results

The predicted results of the displacement response of the bridge are shown in Figure 17. It is observed that the predicted values are close to the measured data. Similarly, Figure 18 shows the predicted results of the acceleration response of the bridge. There is a significant difference between the predicted acceleration and the measured data. The displacement data is smoother than the acceleration data, resulting in smoother data that is easier for the surrogate model to learn features and predictions. Furthermore, to compare the effectiveness of the predictions of the displacements and accelerations, the difference values (RMSE) between the measured and predicted at different locations of the bridge are presented in Table 5. The maximum RMSE value predicted for displacement response is 2.25, while the maximum RMSE value predicted for acceleration reaches 7.44. Although the RMSE value for acceleration is greater than that for displacement, both RMSE values have not exceeded 10, indicating that the prediction results of displacement and acceleration reached effective accuracy. Therefore, the results demonstrated that the proposed method can accurately predict the bridge response and the bridge displacement response is predicted better than the acceleration response.

Figure 17.

The comparison of displacement responses between the measured data and prediction value: (a) mid-span (b) 1/4 span.

Figure 18.

The comparison of acceleration responses between the measured data and prediction value: (a) mid-span (b) 1/4 span.

Table 5.

The RMSE value from different locations of the bridge.

	Mid-span	1/4 span	1/8 span	1/16 span
Displacement (mm)	1.67	2.25	1.91	2.06
Acceleration (mm/s²)	7.44	6.12	5.56	5.22

To evaluate the prediction accuracy and computational efficiency of the proposed method, the numerical calculation results of the VBI system and the prediction results of the proposed method are applied for comparison. As shown in Figure 19(a), the numerically calculated RMSE values are smaller than the predicted RMSE values. The results show that the numerical model calculated results are closer to the actual bridge vibration response. However, the maximum difference between the numerical model calculated RMSE and the predicted RMSE is not more than 5%, indicating that the prediction accuracies of both methods are relatively close to each other. It is worth noting that the computation time of the proposed method is 82% lower than the numerical computation time, as shown in Figure 19(b). Although the accuracy of the numerical method is slightly higher than that of the proposed method, the computational efficiency is much lower than that of the proposed method.

Figure 19.

The comparison of the numerical model and proposed method: (a) the RMSE value (b) computational time.

As listed in Table 6, the Bi-LSTM model demonstrates better prediction accuracy compared to the other surrogate models, including Recurrent Neural Networks (RNN) and Long Short-Term Memory (LSTM) networks. Specifically, the Bi-LSTM model shows an improvement of approximately 10% in RMSE for acceleration response and 25% for displacement response compared to RNN and LSTM. Regarding computational efficiency, the Bi-LSTM model achieves an 82% reduction in computation time compared to FEM simulations and also exhibits improved efficiency relative to RNN and LSTM.

Table 6.

Comparison of Bi-LSTM with RNN and LSTM.

Model	RMSE (Displacement)	RMSE (Acceleration)	Training time (Hours)	Notes
Bi-LSTM	2.25	7.44	3.0	Best prediction accuracy with efficient time
RNN	2.80	8.45	3.0	Less accurate than Bi-LSTM, slower
LSTM	2.65	8.12	3.5	Slightly lower accuracy, more time-consuming

To assess the influence of accurate physical models on bridge vibration prediction, two physical models are compared, namely the vehicle model used in this study and the moving loads. The detailed modeling procedure for moving loads can be found in the literature (Yu and Cai, 2019) and will not be described in detail here. The prediction results of the two methods are shown in Figure 20. The predicted response of the surrogate model guided by the moving load deviates significantly from the measured data. The reason may be that the moving load model cannot effectively capture the physical behavior of vehicle-bridge coupled vibration, resulting in the surrogate model being unable to accurately predict the vibration response of the actual bridge under VBI. The RMSE values of the two methods are compared to quantify their vibration prediction accuracy. As shown in Table 7, the average RMSE value of the proposed method is 1.97, while the average RMSE value of the moving load-guided surrogate model reaches 13.14. The results show that the prediction accuracy of the proposed method is improved by 85% due to the reasonable physical model.

Figure 20.

The comparison of displacement responses of the measured data, proposed method, and moving load method: (a) mid-span (b) 1/4 span.

Table 7.

The RMSE value for two methods (mm).

	Mid-span	1/4 span	1/8 span	1/16 span	Average
Proposed	1.67	2.25	1.91	2.06	1.97
Moving load	13.75	13.54	12.35	12.98	13.14

Conclusions

This study presents a method for predicting the vibration response of long-span cable-stayed bridges by integrating a physical model with a neural network. The 3D vehicle model incorporates the contact area between the tires and the road surface, offering a more realistic representation of the vehicle-bridge interaction (VBI) physical model. The Bi-LSTM network leverages the input-output relationships between vehicle loads and bridge responses to accurately predict bridge vibration responses while reducing the computational cost of structural dynamics analysis. The primary conclusions are as follows:

(1) The proposed method accurately predicts bridge responses, with displacement responses being more precisely predicted than acceleration responses. The maximum RMSE value for displacement prediction is 2.25, and for acceleration prediction, it is 7.44. Both values are below 10, indicating effective accuracy in the predictions.

(2) The proposed method balances prediction accuracy and computational efficiency. The maximum difference between numerically calculated RMSE and predicted RMSE is within 5%, and the computation time of the proposed method is reduced by 82% compared to the numerical computation time.

(3) The prediction accuracy of the proposed method is significantly enhanced with a reasonable physical model. When compared with the traditional moving load model, the prediction accuracy is improved by 85%.

The key contribution of this paper is the development of a novel method that integrates a 3D vehicle model with a Bi-LSTM network to predict the dynamic response of long-span cable-stayed bridges. Field tests have validated the accuracy and robustness of this method. However, the study acknowledges that the method may have limitations when applied to other types of bridges due to differences in structural properties. Additionally, the method does not account for environmental variations, such as temperature changes, seismic activity, or wind loading effects. Future research will address these limitations by incorporating wind excitation and exploring the method’s applicability to various bridge types and conditions.

Footnotes

ORCID iDs

Xinfeng Yin

Ping Xiang

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China Grant Nos. (52078057), (52178207), and the Natural Science Foundation of Hunan Province Grant No. (2023JJ30044).

Declaration of conflicting interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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