Analysis of Vehicle-Induced Vibration Response and Impact Coefficient for Composite Truss Bridge with Partially Concrete-Filled Rectangular Steel Tube Members

Abstract

An innovative method is presented in this paper for the solution to the vehicle–bridge coupled vibration problem for highway bridges using the ANSYS software. By comparing the calculation results of this paper with those of the literature, the correctness and reliability of this method were verified. Taking a new structural bridge (composite truss bridge with partially concrete-filled rectangular steel tube members) as the engineering foundation, the vehicle-induced responses of the lower chords of the main truss were compared and analyzed in two states: partially concrete-filled and non-filled. Accordingly, the concept of the concrete filling coefficient was further developed, and the vehicle-induced responses of the new structural bridge and its impact coefficients were solved and analyzed for different concrete filling coefficients. The results showed that after partially filling the new structural bridge with concrete, the vertical dynamic stiffness of the bridge could be improved, the vehicle-induced response could be reduced, and the fatigue resistance could be increased. It is recommended that the concrete filling coefficient of the new structural bridge be within the range of 0.35 to 0.5. Under different concrete filling coefficients, the fitting results of the impact coefficient of the bridge submit to the extreme value distribution type-I, and the suggested value of the impact coefficient is 0.223 at a 95% assurance rate. For this new structural bridge, the impact coefficient values in Chinese specification are less than the suggested value of 0.223, which should be taken into account by bridge designers.

1. Introduction

A concrete-filled steel tube composite truss girder and a concrete-filled steel tube composite pier are suitable for heavy-load and long-span structures. They are widely used in engineering construction because of the high bending stiffness of their concrete-filled steel tubular members, the clear stress of their truss systems, and their high bearing efficiency [1,2,3,4,5]. Recent advances in structural dynamics, such as the geometrically nonlinear framework incorporating first-order shear deformation theory (FSDT first-order shear deformation theory) by Nazari et al. [6], have provided robust methodologies for analyzing complex bending behaviors in composite systems. These innovations have driven the widespread adoption of CFST (concrete filled steel tube) systems in modern engineering.

In light of their engineering design, Liu et al. [7] optimized the bottom slab and web of concrete box girders and proposed a new type of bridge structure, called a composite truss bridge with partially concrete-filled rectangular steel tubes. The first step of application of the new bridge in highway bridge construction in western China was taken. Liu et al. [8] carried out a real bridge test of the new bridge in practical engineering. It was found that the new bridge was still in an elastic working state under overload conditions, with static load coefficients of 1.90~3.05, and that the static load ratio between the concrete-filled steel tube chord and the hollow steel tube chord was consistent with its axial stiffness ratio. The researchers [9,10] conducted a bending test and analyzed the vertical deflection limit of the partially concrete-filled rectangular steel tube composite truss, examined the effect of joint deformation on the overall deformation of the truss, and then discussed the good service performance of the new bridge from the perspectives of durability and maintainability. In succession, the researchers [11,12,13] introduced the hot spot stress method to optimize the fatigue design of the new bridge joints.

There has been a lot of research on composite truss bridges with partially concrete-filled rectangular steel tube members so far. Most of these studies have focused on analyzing the static and fatigue performance of the whole bridge, its local components, and its joints [14]. However, there are few reports on the dynamic performance of the new bridge, including any vehicle-induced vibration response. Therefore, it is necessary to study the dynamic response of the new bridge.

The research methods for vehicle–bridge coupled vibration can be mainly divided into four categories: the field measurement method, classical theoretical analysis method, model testing method, and numerical analysis method [15,16,17,18,19,20,21,22,23,24,25]. The field measurement method is the main research method used to study vehicle–bridge coupled vibrations in the early stages of this research. It is time-consuming, and the results reflect a comprehensive view of all factors, which cannot be formulated as a rigorous theoretical framework. There are limitations to the classical theoretical analysis method due to its calculation conditions. Based on the simplified vehicle model and bridge model, approximate calculations are performed, and it is difficult to guarantee the accuracy of the analysis. Angelo et al. [26] compared three bridge response prediction methods under moving vehicles: a concentrated moving load, a single-degree-of-freedom model, and a two-degree-of-freedom model. This study evaluated scenarios requiring vehicle–bridge interaction modeling for accurate response estimation. Model testing is a complex, expensive method that does not take into account the randomness of actual traffic loads. With the advent and development of the computer and finite element method in the 1960s and 1970s, vehicle–bridge coupled vibration research made significant progress. The numerical analysis method has been widely applied to the study of the vehicle–bridge coupled vibration response of bridge structures. Yu P et al. [27] developed a refined vehicle–bridge interaction finite element model for long-span concrete-filled steel tubular arch bridges, systematically investigating the effects of vehicle speed, axle load, and deck roughness on their structural dynamic responses. Their study further quantified riding comfort indices through comprehensive vibration analysis under operational conditions. The majority of existing numerical methods require the derivation of the vibration equation of the vehicle–bridge system as well as the compilation of a complex solution program. The configuration of a coupled vibration analysis of vehicles and the bridge is a complex process that is difficult for engineers to master and apply.

In this paper, an innovative method for solving the vehicle–bridge coupled vibration problem was proposed based on the ANSYS software (ANSYS CFX 14.0). In this method, the vehicle finite element model and the bridge finite element model were built in the ANSYS classical environment using the APDL language. Through ANSYS constraint equations, the displacement coordination relationship between the wheel and the bridge deck contact point could be realized at any time. The vehicle–bridge coupled vibration problem was solved using the ANSYS transient dynamic solution function. It was determined that the method presented in this paper was correct and reliable by quantitatively comparing the calculated results with the original ones in the literature [28]. A composite truss bridge with partially concrete-filled rectangular steel tube members was taken as the research object, and a concrete filling coefficient was proposed. The vehicle-induced response and impact coefficient of the bridge were analyzed in relation to nine concrete filling coefficients, taking into account grade A bridge deck roughness [29]. In this field, researchers as well as engineering designers may find it useful to refer to the relevant conclusions and experiences within this study.

2. Method for Solving Vehicle–Bridge Coupled Vibration

2.1. Vehicle Model

Vehicles are complex vibration systems that should be simplified appropriately according to the problems analyzed. It was demonstrated in research on vehicle dynamics that the longitudinal vibration of the vehicle body, suspension, and wheel components along the vehicle operation direction has little effect on the vertical and transverse vibration of the bridge, and the coupling effect between the vertical and lateral vibration of the vehicle is weak [30,31]. Generally, the vertical vibration of a vehicle and the transverse vibration of a vehicle are calculated in their respective planes in order to make calculation easier. Consequently, when studying the vertical vibration of the vehicle–bridge coupled system, the vehicle model only needs to consider the three degrees of freedom of the vehicle body’s heave (vertical displacement), pitch, and roll, as well as the vertical displacement of each wheel. A simplified model of a two-axle vehicle, which consists of the upper structure (vehicle body) and the lower structure (wheels and suspensions), is presented as follows: (1) The vehicle body is considered a rigid body, and includes three degrees of freedom in terms of its heave, pitch, and roll, without taking into account the transverse movement of the vehicle body. (2) The wheels are also considered rigid bodies, and they only include one degree of freedom, namely vertical displacement. Due to the characteristic stiffness and damping of each wheel tire, they were fitted with a spring damper. (3) The suspensions are considered spring dampers, and provide vertical support for the vehicle body as well as vibration reduction. (4) There are rigid beams that connect the upper structure and lower structure of the vehicle, allowing displacement and force to be transferred from one to the other.

Figure 1 is a simplified three-dimensional model of a two-axle vehicle, where $M$ is the mass of the vehicle body; $I_x$ and $I_z$ are the rotational inertia of the vehicle body around the x-axis and z-axis; $m_{\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$ is the mass of each wheel; $k_{t\mathrm{i}},k_{s\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$ represent the stiffness of each wheel tire and suspension; $c_{t\mathrm{i}},c_{s\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$ represent the damping of each wheel tire and suspension; $Y$ is the vertical displacement of the vehicle body; $\theta$ and $\phi$ are the angular displacement of the vehicle body around the x-axis and z-axis; and $Y_{\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$ is the vertical displacement of each wheel.

The ANSYS software integrates multiple physical fields such as structure, fluid, and electromagnetic fields. There are a wide variety of element types available in the ANSYS software, so it is convenient to accurately model a simplified vehicle model with the software. The rigid body of the vehicle body as well as the rigid body of the wheel can be simulated using the MASS21 element, the spring dampers of the suspension and wheel tire can be simulated using the COMBIN14 element, and the rigid beam can be simulated using the MPC184 element.

Table 1. Element types and their options for the components of the simplified vehicle model in the ANSYSY software (ANSYS CFX 14.0).

Components of Vehicle	ANSYS Element Type	ANSYS Element Option
The vehicle body	MASS21	KEYOPT(3) = 0
The wheel	MASS21	KEYOPT(3) = 2
The suspension and wheel tire	COMBIN14	KEYOPT(2) = 2
Rigid beam	MPC184	KEYOPT(1) = 1

illustrates the types of elements and their setting options corresponding to all components of the simplified vehicle model in the ANSYS software (ANSYS CFX 14.0) [32].

Following this, a finite element model of the simplified vehicle can be built by compiling the APDL command flow using the ANSYS software. Figure 2 shows an ANSYS finite element model of a spatial two-axle simplified vehicle, where M0 represents the MASS21 element and K0 represents the COMBIN14 element.

Based on the above method, the simplified multi-rigid body finite element model of common automobile models (multi-axle or trailer) can be established using the ANSYS software. Considering the traffic flow through a bridge, the traffic flow model can be formed by establishing multiple independent vehicles in each lane.

2.2. Bridge Model

In the vehicle–bridge coupled vibration analysis, the bridge structure types are extensive, including girder bridges, arch bridges, rigid frame bridges, suspension bridges, cable-stayed bridges, and various composite systems bridges. With powerful functions and an extensive element library and material library, ANSYS can be used for the full-bridge simulation analysis of bridges with any structural system [32]. Using solid elements in the bridge finite element model for vehicle–bridge coupled vibration analysis results in a large number of elements and costs a lot of resources for the solution. In contrast, using beam and plate elements can reduce the number of elements and the amount of resources for the solution. In this paper, the bridge model is simulated by the ANSYS BEAM188 beam element or SHELL181 plate element.

2.3. Bridge Deck Roughness Model

Bridge deck roughness refers to the deviation of the actual bridge deck from its absolute ideal datum plane (a smooth horizontal plane) due to the unevenness of the bridge deck surface, which is one of the main factors affecting vehicle–bridge coupled vibration. As a steady gauss stochastic process with a zero mean and ergodicity, bridge deck roughness can be described statistically by power spectrum density (PSD). The general form of the power spectrum density of road roughness specified by ISO (International Organization for Standardization) and the Chinese national standard GB/T7031-2005 [29] is fitted by Equation (1):

$$G_x(n)=G_x(n_0){\left({\displaystyle \frac{n}{n_0}}\right)}^{-w}$$

(1)

where $n_0$ is the reference spatial frequency, $n_0=0.1$ m⁻¹, and $w$ is the exponent of the fitted PSD.

The inverse Fourier transform technique is a common simulation method for bridge deck roughness samples because it is a simple method with a clear concept and a simple operation, and its calculation results are highly consistent with the expected power spectrum. Equations (2)–(4) represent the discrete Fourier inverse transformation method for solving bridge deck roughness samples [33].

$$\left|X_k\right|=\sqrt{{\displaystyle \frac{N}{2\Delta l}}{G}_x(n_k)} , k=0, 1, 2,\cdots ,{\displaystyle \frac{N}{2}}$$

(2)

$$X_k=\left|X_k\right|e^{j{\phi }_k}, k=0, 1, 2,\cdots ,{\displaystyle \frac{N}{2}}$$

(3)

$$r_m={\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=0}^{N-1}{X}_k{e}^{\frac{j2\pi km}{N}}} , m=0, 1, 2\cdots N-1$$

(4)

where $n_k$ is the discrete spatial frequency; $\Delta l$ is the sampling interval; $N$ is the number of sampling points; $G_x(n_k)$ is the power spectral density of the bridge deck roughness at the discrete spatial frequency $n_k$; $X_k$ is a Fourier spectrum of time-domain samples of bridge deck roughness that represents a complex number; $\left|X_k\right|$ is a module of the complex number $X_k$; ${\phi }_k$ is a random phase that is subject to a uniform distribution on [0, 2π]; $r_m$ represents the time-domain samples of bridge deck roughness; and $j$ is an imaginary unit.

In order to simulate the time-domain samples of bridge deck roughness numerically, the M file is compiled via MATLAB software (MATLAB R2023b) using the Fourier inverse transformation method. Figure 3 shows the bridge deck roughness samples of class A and D in accordance with the Chinese standard GB/T7031-2005.

2.4. Displacement Compatibility Relation of Vehicle–Bridge Coupled System and Its Implementation

Figure 4 shows a schematic diagram of the vehicle–bridge coupled vibration model in which beam and plate (shell) elements are used to simulate the bridge, while the vehicle is modeled as a spatial two-axle vehicle, as discussed previously. In the vehicle model, $L_{\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$ represents the number of nodes at the contact position between the wheel tire and bridge deck, and $y_{v\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$ is the vertical displacement of the nodes, numbered $L_{\mathrm{i}}$ $(\mathrm{i}=1,2,3,4)$. Oxyz represents the global coordinate system. A bridge model was built based upon the Oxyz global coordinate system, with x as the longitudinal direction, y as the vertical direction, and z as the transverse direction for the bridge.

When the vehicle passes through the bridge, the trajectory of the vehicle is usually approximately parallel to the central axis of the bridge, and its driving state is usually in the state of uniform or uniform variable speed motion. Assuming that the coordinates of the nodes numbered $L_{\mathrm{i}}$ of the vehicle model are $x_{L_{i0}}$, $y_{L_{i0}}$, and $z_{L_{i0}}$ at the initial moment $t_0=0$, the vehicle’s initial speed is $v_0$ and its acceleration is $a$. The coordinates of the nodes numbered $L_{\mathrm{i}}$ of the vehicle model at any moment $t$ are calculated via Equation (5).

$$\left\{\begin{array}{l}{x}_{L_i}(t)=x_{L_{i0}}+v_{0}t+a{t}^2/2\\ y_{L_i}(t)=y_{L_{i0}}\\ z_{L_i}(t)=z_{L_{i0}}\end{array}\right.(i=1, 2, 3, 4)$$

(5)

At any time, the position of each node at the contact position between the wheel tire and bridge deck in the vehicle model can be determined according to Equation (5). In the case that these nodes are located exactly at the node positions of the bridge finite element model, the vertical displacement of the bridge deck can be determined directly by the node of the bridge element at the corresponding position. Alternatively, if these nodes are located outside the node positions of the bridge finite element model, the vertical displacement of the bridge deck at a given position can be calculated by the interpolation of the bridge element shape function. Figure 5 is the diagram of displacements at the beam element nodes, and $O\overline{x\mathrm{y}\mathrm{z}}$ represents the element’s local coordinate system. The vertical displacement at any position in the beam element can be obtained by Equation (6). A vertical displacement at any position in a plate element can also be expressed as a function of the degree of freedom at the relevant nodes of the plate element according to the element shape function.

$$\left\{\begin{array}{l}v(\overline{x})=(1-3{\xi }^2+2{\xi }^3)v_1+(3{\xi }^2-2{\xi }^3)v_2\\ +l(\xi -2{\xi }^2+{\xi }^3){\theta }_1+l({\xi }^3-{\xi }^2){\theta }_2\\ \xi =\overline{x}/l\end{array}\right.$$

(6)

where $l$ is the length of the beam element; $v_1 , v_2$ are the vertical displacements of the nodes at both ends of the beam element; and ${\theta }_1,{\theta }_2$ are the rotation angles of the nodes at both ends of the beam element around the element axis $\overline{z}$.

It is assumed that when a vehicle is driving on a bridge, the wheel tires are in close contact with the bridge deck without separation. Figure 6 illustrates a diagram of the displacement compatibility relationship for the contact point between the wheel tire and the bridge deck.

In Figure 6, the node $L_i$ represents the point of contact between the wheel tire and the bridge deck in the vehicle model. $y_{vi}$ represents the vertical displacement of the node $L_i$; $y_{bi}$ represents the vertical displacement of the bridge deck at the corresponding wheel tire node $L_i$ position, which can be calculated by the displacement of the relevant bridge element node according to the element shape function in Equation (6); and $r_i$ represents the bridge deck roughness value at the corresponding wheel tire node $L_i$ position. On the basis of the above assumptions, the geometric displacement compatibility relation between the vehicle and bridge, as shown in Equation (7), can be derived.

$$r_i=y_{vi}-y_{bi}$$

(7)

As can be seen from the literature [34], the ANSYS constraint equation is a linear equation (Equation (8)) that relates the degrees of freedom of nodes, which can replace the coupling of degrees of freedom of nodes, and it is more general.

$$\mathrm{Constant}={\displaystyle \sum _{k=1}^m\left(Coef(k)\times U(k)\right)}$$

(8)

where $U(k)$ represents the degrees of freedom of the nodes, including translational degrees of freedom and rotational degrees of freedom. $Coef(k)$ is a coefficient of the $U(k)$ degree of freedom; $m$ is the number of terms in the equation.

By using the constraint equation function of the ANSYS software, the displacement compatibility relationship between the vehicle and bridge can be established at any load substep (any time) to realize the close contact between the wheel tire and bridge deck. As shown in Equation (7), the constraint equation form in ANSYS is as follows:

$$\mathrm{CE}, r_i, L_i, UY, 1, N_{bi}, Lab, C_{bi}$$

where $UY$ represents the vertical translation degree of freedom of node $L_i$; $N_{bi}$ represents the number of nodes of the bridge finite element model; and $Lab$ and $C_{bi}$ represent the degree of freedom of node $N_{bi}$ and its corresponding coefficient, which can be determined by the element shape function interpolation for calculating the vertical displacement of the bridge deck at the node $L_i$ position.

2.5. Solution Steps of Vehicle–Bridge Coupled Vibration Problem in ANSYS

Based on the transient dynamic analysis function of the large general finite element analysis ANSYS platform, the vehicle–bridge coupled vibration analysis system for highway bridges was compiled using the APDL language. Figure 7 depicts this algorithm flow, which consists of the following steps:

Step 1:

In the ANSYS software, the bridge finite element model and vehicle finite element model are established.

Step 2:

Based on the inverse Fourier transform method, the time-domain sample data of bridge deck roughness are obtained using MATLAB M-files, which are then read into ANSYS tables. As a result, the surface roughness time-domain sample value at the position of a vehicle–bridge interaction at any time is determined using the index rule of “0 row, 0 column” of the ANSYS table array and its automatic interpolation function.

Step 3:

The modal solution of the finite element model of the bridge is conducted, and the fundamental frequency and natural vibration period $T$ of the bridge structure can be determined. The appropriate time integral step length $\Delta t$ for transient dynamic calculation can be selected, and the general time integral step length can be taken as $\Delta t\le {\displaystyle \frac{T}{10}}$.

Step 4:

By using the ANSYS constraint equation, the vertical displacement constraint conditions of the contact point between the wheel tire and the bridge deck at any given time can be determined according to Equation (7) (the geometric displacement compatibility relation between the vehicle and bridge). The dynamic time-history analysis of the vehicle crossing the bridge is realized based on the transient dynamic analysis function of the ANSYS software.

2.6. Verification of the Method

In order to verify the correctness and reliability of the proposed method and the self-compiled program, a numerical example from the literature [28] was simulated using the proposed method. Due to limited space, only a few comparative data and results are presented. As shown in Figure 8, the vertical displacements at the midspan of the simply supported girder bridge in this paper were compared to those calculated by the Runge–Kutta method in the literature [28]. It was evident that the calculation results of this paper were in excellent agreement with those of the literature [28], and the maximum relative error of both methods under different driving speeds was less than 5%.

3. Vehicle-Induced Vibration Response of Composite Truss Bridge with Partially Concrete-Filled Rectangular Steel Tube Members

3.1. Engineering Overview

A composite truss bridge with partially concrete-filled rectangular steel tube members (Huang Yan Bridge) was constructed in China, and the overall length of the bridge was 96.96 m. The main bridge was composed of two parallel rectangular steel tubular truss girders with a triangular web member truss system. The length of the main truss was 88 m (the span arrangement is 24 m + 40 m + 24 m), with a total of 22 internodes (the internode length is 4 m). The bridge deck width was 5.5 m, and the truss height was 2.5 m. The height–span ratio of the truss was 1/16. C50 micro expansive concrete was symmetrically grouted in the range of 2.5 m along the length of the lower chord segment near the edge fulcrum and 13 m along the length of the lower chord segment near the middle fulcrum. The bridge pier was a double-limb Y-shaped concrete-filled steel tubular pier stiffened with PBL. There was a rigid connection between the pier and the main truss, and the pier cap was octagonal. The bridge abutment was a lightweight bridge abutment with a direct connection between its piles and bent cap. The bent cap measured 5.5 m in transverse, 1.8 m in longitude, and 1.5 m in height, and the foundation was a bored pile foundation. All the steel plates of the whole bridge were made of Q345D steel. The cast-on-site concrete bridge deck was connected with the main truss through the PBL perforated steel plate connectors of the upper chord, which were designed according to crack control. The front view of the bridge is shown in Figure 9, and in Figure 10 is a structural diagram of the interface between the concrete-filled segment and the empty steel tube segment. The cross section of the main members of the bridge is shown in Figure 11.

3.2. Finite Element Model

The finite element model of the composite truss bridge with partially concrete-filled rectangular steel tube members was established using the ANSYS software through a bottom–up method. All members of the main truss and piers were simulated by the 3D linear finite strain beam element BEAM188, and the bridge deck was simulated by the 3D finite strain shell element Shell181. Fixed constraints were imposed on the bottom of the pier. The full-bridge ANSYS finite element model is shown in Figure 12.

The vehicle model adopted the simplified model of a two-axle vehicle (see Figure 1), and the corresponding ANSYS finite element model of a two-axle vehicle is shown in Figure 2. The technical parameters of the two-axle vehicle are shown in

Table 2. Technical parameters of the two-axle vehicle.

Technical Parameters	Value
Mass of vehicle body $M$/kg	38,500
Rotational inertia $I_z$/kg·m2	2,446,000
Rotational inertia $I_x$/kg·m2	1,223,000
Mass of wheel $m_1,m_2,m_3,m_4$/kg	2165
Tire stiffness $k_{ti}(i=1,2,3,4)$/N·m−1	2,140,000
Tire damping $c_{ti}(i=1,2,3,4)$/kg·s−1	49,000
Suspension stiffness $k_{si}(i=1,2,3,4)$/N·m−1	1,267,500
Suspension damping $c_{si}(i=1,2,3,4)$/kg·s−1	98,000
Wheel base $l$/m	8.4
wheel track $b$/m	3.0

.

3.3. Analysis of Vehicle-Induced Vibration Response

Due to the high level of maintenance and management of the highway bridge deck in China, it was difficult for the highway bridge deck to appear as class C and class D in the Chinese national standard GB/T7031-2005. Moreover, existing studies have shown that the influence of vehicle speed on vehicle–bridge coupled vibration is complex and has no obvious regularity. Accordingly, when analyzing the vehicle-induced vibration response in this paper, the roughness level of the bridge deck was set at class A and the vehicle speed was set at 60 km/h. In addition, because of the plate–truss combination effect for the composite truss bridge, the force of the upper chords of the main truss was complex, which was affected by many factors. Therefore, in this paper, the vehicle-induced response analysis and impact coefficient calculation of the composite truss bridge were only carried out with the lower chords of the composite truss bridge as an example.

When the vehicle passed through the bridge, the time history of vertical displacement and time history of axial force (negative value of axial force represents compression) of every lower chord of the main truss for the composite truss bridge under the two working conditions of partially concrete-filled and non-filled are shown in Figure 13 and Figure 14, respectively. It can be seen from Figure 13 and Figure 14 that the shape of the time-history curves of the response of lower chords for the bridge did not change after the bridge was partially filled with concrete. A partially concrete-filled structure had little effect on the position of the maximum value of the response of the bridge, which occurred at the midspan of the middle-span.

The maximum dynamic response of the lower chords of the composite truss bridge with rectangular steel tube members under two working conditions is shown in

Table 3. The maximum dynamic response of lower chords under two working conditions for composite truss bridge with rectangular steel tube members.

Dynamic Response	Partially Concrete-Filled	Non-Filled
Vertical displacement/mm	−29.29	−31.77
Axial force/kN	1317.59	1436.24

. According to Table 3, the maximum vertical displacement of the bridge under partially concrete-filled conditions decreased by 2.48 mm, with a decreasing amplitude of about 7.80%. At the same time, the maximum axial force of the lower chords decreased by 118.65 kN, and the decreasing amplitude was about 8.26%.

Under the two working conditions, partially concrete-filled and non-filled, the dynamic response increments of the midspan lower chords at the middle-span and side-span for the composite truss bridge with rectangular steel tube members are shown in

Table 4. Dynamic increments of lower chords for composite truss bridge with rectangular steel tube members under two working conditions.

Dynamic Parameters		Partially Concrete-Filled	Non-Filled
Midspan of midddle-span	Increment of vertical displacement/mm	−0.46	−0.52
	Increment of axial force/kN	41.93	46.80
Midspan of side-span	Increment of vertical displacement/mm	−0.41	−0.43
	Increment of axial force/kN	57.85	58.58

. Among them, the dynamic response increment refers to the maximum response value of the bridge considering the dynamic effect relative to the maximum response value without considering the dynamic effect when the vehicle crossed the bridge. According to Table 4, after being partially concrete-filled, the dynamic response increments of the midspan lower chords at the middle-span and side-span for the composite truss bridge showed a downward trend, and the dynamic response increment of the midspan lower chords at the middle-span decreased greatly. The dynamic displacement increment of the midspan lower chords at the middle-span decreased by 11.54% and the dynamic axial force increment decreased by 10.41%. It was demonstrated that pouring concrete into the hollow steel tubes of the lower chords of the composite truss bridge could effectively improve the vertical dynamic stiffness of the bridge and reduce its dynamic response, as well as improve the fatigue resistance of the composite truss bridge under a vehicle load.

4. Influence of Concrete-Filled Segment Length on Dynamic Response of Composite Truss Bridge

4.1. Concrete Filling Coefficient

In order to further explore the influence of the concrete-filled segment length on the dynamic response of the composite truss bridge with partially concrete-filled rectangular steel tube members, the concrete filling coefficient was defined as follows:

$$\lambda ={\displaystyle \frac{2(D_1+D_2)}{D}}$$

(9)

where $D_1$ is the concrete-filled segment length of the lower chord near the edge fulcrum, $D_2$ is the concrete-filled segment length of the lower chord near the middle fulcrum, and $D$ is the overall length of main truss, as indicated in Figure 9. The concrete filling coefficient $\lambda$ has a range of values between 0 and 1.

According to the internode arrangement of the lower chords of the main truss, the concrete filling coefficients $\lambda$ = 0, 0.125, 0.216, 0.352, 0.489, 0.625, 0.761, 0.852, and 1 were taken only by changing $D_2$. Under nine different concrete filling coefficients, the vehicle-induced vibration analysis of the composite truss bridge was carried out. The concrete-filled segment length under the nine different concrete filling coefficients is shown in

Table 5. Concrete-filled segment length under nine concrete filling coefficients.

$\mathit{\lambda }$	Concrete-Filled Segment Length/m
	${\mathit{D}}_1$	${\mathit{D}}_2$
0	0	0
0.125	2.5	3.0
0.216	2.5	7.0
0.352	2.5	13.0
0.489	2.5	19.0
0.625	2.5	25.0
0.761	2.5	31.0
0.852	2.5	35.0
1	2.5	41.5

.

4.2. Dynamic Response Analysis of the Composite Truss Bridge Under Nine Concrete Filling Coefficients

Based on a bridge deck roughness class A and a driving speed of 60 km/h, the dynamic responses of the composite truss bridge were calculated under nine concrete filling coefficients. The vertical displacements and axial forces of the midspan lower chords at both the middle-span and side-span were selected for analysis. As a result of the symmetry of the bridge structure, only the left half of the bridge was considered for analysis. The time histories of the vertical displacements and axial forces of the midspan lower chords at both the middle-span and side-span under nine different concrete filling coefficients are depicted in Figure 15 and Figure 16, respectively. The maximum vertical displacement and axial force of the midspan lower chords at both the middle-span and side-span under nine different concrete filling coefficients are shown in

Table 6. Maximum dynamic response of lower chords of main truss for composite truss bridge under nine different concrete filling coefficients.

$\mathit{\lambda }$	Midspan of the Middle-Span		Midspan of the Side-Span
	Vertical Displacement/ mm	Axial Force/kN	Vertical Displacement/mm	Axial Force/kN
0	−31.77	1436.24	−6.91	561.26
0.125	−31.27	1411.08	−6.86	547.60
0.216	−30.43	1374.70	−6.78	520.56
0.352	−29.29	1317.59	−6.76	505.30
0.489	−29.11	1311.89	−6.68	494.17
0.625	−28.99	1346.27	−6.63	518.68
0.761	−28.45	1413.48	−6.30	542.53
0.852	−27.79	1470.96	−6.25	555.55
1	−26.33	1588.02	−6.19	589.54

.

It can be seen from Figure 15 and Figure 16 and Table 6 that with the increase in the concrete filling coefficient, the maximum vertical displacements of the midspan lower chords at both the middle-span and side-span gradually decreased, indicating that partial concrete filling could effectively improve the vertical dynamic stiffness of the bridge. With the increase in the filling coefficient, the maximum axial forces of the midspan lower chords at both the middle-span and side-span decreased first and then increased, which was mainly due to the change in the relative stiffness of the lower chords. As soon as the filling coefficient was set to 0.4886, the axial forces of the lower chords reached their minimum value. The minimum axial forces of the midspan lower chords at both the middle-span and side-span were 1311.89 kN and 494.17 kN, respectively. The recommended concrete filling coefficient for the composite truss bridge with partially concrete-filled rectangular steel tube members was in the range of 0.35 to 0.5.

5. Impact Coefficient of the Composite Truss Bridge with Partially Concrete-Filled Rectangular Steel Tube Members

5.1. Impact Coefficient of Bridge

In general, the impact coefficient of a bridge is defined as the ratio between the dynamic response and the static response of a bridge structure to moving vehicles [35]. The calculation formula of the impact coefficient is as follows:

$$\mu ={\displaystyle \frac{\left(R_{dy}-R_{st}\right)}{R_{st}}}$$

(10)

where $R_{dy}$ and $R_{st}$ are the maximum dynamic response and maximum static response of a bridge caused by vehicle loads.

In this paper, the impact coefficient of a bridge is calculated with the vertical displacement (deflection) and axial force of the lower chord, since the specification does not specify which response should be used. In general, the impact coefficient of a bridge under multi-vehicle working conditions is smaller than that under single-vehicle working conditions [36]. Based on conservative considerations, this paper uses the impact coefficient under the action of a single vehicle as a representative for analysis.

5.2. Influence of Concrete-Filled Length on Impact Coefficient of Composite Truss Bridge

The calculated values of the impact coefficients of the composite truss bridge under nine concrete filling coefficients are shown in Figure 17. With increasing concrete filling coefficients, the impact coefficients of the composite truss bridge fluctuated within a certain range, with the exception of the position of the edge fulcrum and the middle fulcrum, as shown in Figure 17. There was no clear correlation between the impact coefficients of the bridge and the concrete filling coefficients. The deflection impact coefficients near the middle fulcrum were larger, with a maximum value of 1.41, and the axial force impact coefficients near the edge fulcrum were larger, with a maximum value of 3.91. As a result of the supporting effects of the fulcrum, the actual dynamic response values of the members were low (see Figure 13 and Figure 14), and the calculated values of the impact coefficients of the members could not reflect the overall impact effect of the vehicle on the bridge.

5.3. Suggested Value of Impact Coefficient

Considering the value of the impact coefficient of a bridge in the United States, Canada, China, and other national bridge specifications, the sample values of impact coefficients greater than 0.5 near the edge fulcrum and the middle fulcrum were removed. In this paper, a goodness-of-fit test was carried out for 867 impact coefficients, which are shown in Figure 18 to follow the extreme value distribution type-I.

Table 7. Statistical parameters of impact coefficient of composite truss bridge.

Statistical Parameters	Value
Mean value $\mu$	0.114
Standard deviation $\sigma$	0.058

provides the statistical parameters of the impact coefficients. Practically, the statistical value of the parameter with a guarantee rate of 95% was taken as the suggested value. In this paper, the probability density function was fitted using MATLAB software, and the impact coefficient corresponding to the 95% guarantee rate was 0.223.

A comparison of the impact coefficient values of five national specifications under nine concrete filling coefficients is shown in Figure 19. In the working conditions discussed in this paper, the impact coefficient values in the American, Canadian, British, and Australian specifications were all fixed values [35], namely 0.33, 0.3, 0.25, and 0.3, respectively, which were significantly higher than the suggested value of 0.223. The values of the impact coefficient in these national specifications were conservative for this new type of bridge. The value of this impact coefficient in the Chinese specification is related to the bridge’s fundamental frequency [37]. With the increase in the concrete filling coefficient, the value increases slightly, as shown in

Table 8. Impact coefficient values of Chinese specification under nine concrete filling coefficients.

Concrete Filling Coefficient	Fundamental Frequency of Bridge	Value of Impact Coefficient
0	3.21	0.19
0.13	3.23	0.19
0.22	3.26	0.19
0.35	3.31	0.20
0.49	3.35	0.20
0.63	3.39	0.20
0.76	3.45	0.20
0.85	3.50	0.21
1	3.58	0.21

. Based on the Chinese specifications, the impact coefficient values for the bridge under the nine concrete filling coefficients were less than the suggested value of 0.223, which should be taken into account by bridge designers.

6. Conclusions

This paper presents a new method to solve the vehicle–bridge coupled vibration problem based on an ANSYS constraint equation, and its correctness and reliability are verified. This method can be used to analyze vehicle-induced dynamic responses and impact coefficients for a new type of composite truss bridge with partially concrete-filled rectangular steel tube members. The conclusions are as follows:

• The method in this paper is used to calculate the simply supported beam in the literature. The calculation results of the midspan deflection of the bridge are highly consistent with the literature results, and the maximum relative error is less than 5% at different speeds. The accuracy and reliability of the method were verified. For vehicle–bridge coupling vibration analysis, this method eliminates the need for engineers to design complicated programming programs, resulting in greatly improved efficiency.
• The partial concrete filling of a bridge does not change the shape of the time history curve of the dynamic responses of the lower chords of the bridge and has little impact on the position at which the maximum dynamic responses occur. In composite truss bridges, the partial concrete filling can improve the vertical dynamic stiffness, reduce its dynamic response, and improve fatigue resistance.
• The concrete filling can change the relative stiffness of the chord and the bridge. With the increase in the concrete filling coefficient, the axial forces of the midspan lower chords at both the middle-span and side-span first decrease and then increase for the composite truss bridge. The recommended concrete filling coefficient for the composite truss bridge with partially concrete-filled rectangular steel tube members is in the range of 0.35 to 0.5.
• A goodness-of-fit test was conducted on 867 calculated values of the impact coefficient of the new bridge under nine concrete filling coefficients, and the fitting results were submitted to the extreme value distribution type-I. At a 95% guarantee rate, the impact coefficient of the composite truss bridge was 0.223, which was the suggested value of the impact coefficient. The suggested value of the impact coefficient was less than the value of the impact coefficient in the American specification, Australian specification, British specification, and Canadian specification, but greater than the value of the impact coefficient in the Chinese specification, and should be highly valued by bridge designers.