#### Abstract

In the current vehicle-bridge dynamics research studies, displacement impact coefficients are often used to replace the moment and shear force impact coefficients, and the vehicle model is also simplified as a moving-load model without considering the contribution of vehicle stiffness and damping to the system in some concerned research studies, which cannot really reflect the mechanical behavior of the structures under vehicle dynamic loads. This paper presents a vehicle-bridge coupling model for the prediction of dynamic responses and impact coefficient of the long-span curved bending beam bridge. The element stiffness matrix and mass matrix of a curved box girder bridge with 9 freedom degrees are directly deduced based on the principle of virtual work and dynamic finite element theory. The vibration equations of vehicle-bridge coupling are established by introducing vehicle mode with 7 freedom degrees. The Newmark-*β* method is adopted to solve vibration response of the system under vehicle dynamic loads, and the influences of flatness of bridge surface, vehicle speed, load weight, and primary beam stiffness on the impact coefficient are comprehensively discussed. The results indicate that the impact coefficient presents a nonlinear increment as the flatness of bridge surface changes from good to terrible. The vehicle-bridge coupling system resonates when the vehicle speeds reach 60 km/h and 100 km/h. The moment design value will maximally increase by 2.89%, and the shear force design value will maximally decrease by 34.9% when replacing moment and shear force impact coefficients with the displacement impact coefficient for the section internal force design. The load weight has a little influence on the impact coefficient; the displacement and moment impact coefficients are decreased with an increase in primary beam stiffness, while the shear force impact coefficient is increased with an increase in primary beam stiffness. The theoretical results presented in this paper agree well with the ANSYS results.

#### 1. Introduction

Bridge structures will bear dynamic loads except for constant loads during their service period, and vehicle load is the most common dynamic load. The research on the vehicle-bridge coupling mainly focuses on the dynamic responses of the system [1–5]. The bridge will vibrate when the vehicles pass over it, leading to an increase in the internal force and displacement. The impact coefficient, a dynamic amplification coefficient, reflects the magnitude of the impact action. The impact coefficient value is of great importance to the safety of bridge structures, and the impact effect is prominent especially when the bridge vibration frequency and the vehicle frequency are equal (resonance occurs).

A great deal of research studies on vehicle-bridge coupling vibration responses and dynamic impact coefficients have been carried out so far, in which the dynamic interactions of the bridge-vehicle system and wheel-rail contact are critical issues. Generally, there are several approaches to simulate the dynamic interaction of the bridge-vehicle system: wheel-rail coupling method, Hertzian spring method, penalty stiffness method, and equation coupling method. Due to the same degrees of freedom, there is no contact issue in the wheel-rail coupling method. Therefore, it cannot simulate track irregularity. The Hertzian spring method, possessing few degrees of freedom, is the most popular method to predict the dynamic response of the vehicle-bridge system. Zakeri et al. [6–8] adopted the Hertzian spring method to investigate dynamic responses of suspension bridges and curved railway bridges, considering the effects of curvature radius, track irregularity, high speed, and running modes. Neves et al. [9] presented a direct method for analyzing the nonlinear vehicle-structure interaction based on the Hertzian spring model. Mohammed et al. [10] researched the dynamic impact coefficient of highway bridges under overweight vehicles with equidistant axle. Although the Hertzian spring method considers the wheel-rail contact, it does not belong to real contact analysis. The wheel and rail are always in contact, and there is no separation of the two components. To deal with this problem, Fan et al. [11] investigated the vehicle jumping phenomenon caused by irregularity based on the penalty stiffness algorithm. The result indicated that for rail irregularity, the interfacial contact algorithm is more suitable for the simulation of interaction condition of vehicle-bridge dynamics than the Hertzian spring method. Besides, the equation coupling method based on the numerical model is also an efficient approach to predict the dynamic interaction of the vehicle-bridge system. Ji et al. [12, 13] studied the dynamic impact coefficient of PC simply supported box girder bridges with corrugated steel webs using the equation coupling method and compared the calculated results with AASHTO. Deng et al. [14–16] also established the vehicle-bridge coupling finite element mode, in which the cross section type, bridge deck roughness, vehicle stiffness, and damping were considered.

The aforementioned research studies have enriched and improved the vehicle-bridge coupling issue. However, in terms of research object, the current research studies mainly focus on the straight beam bridge. Few research studies have investigated the dynamic response of the curved beam bridge. It is not reasonable for some concerned research studies to simplify the vehicle model as moving-load model without considering the contribution of vehicle stiffness and damping to the system. Besides, replacing moment and shear force impact coefficients with the displacement impact coefficient cannot really reflect the mechanical behavior of the structures under vehicle dynamic loads. This paper presents a vehicle-bridge coupling model by directly deducing the element stiffness matrix and mass matrix and introducing the vehicle model. The Newmark-*β* method is used in the present work to solve the dynamic response of the system. The displacement, moment, and shear force impact coefficients are respectively calculated, and the influences of the flatness of bridge surface, vehicle speed, load weight, and stiffness on the impact coefficient are considered.

#### 2. Theoretical Model of Vehicle-Bridge Coupling

##### 2.1. Assumptions

Considering that the spatial mechanical model of a box girder is so complicated, some assumptions are made to simplify it:(1)The materials are in elastic phase(2)The distortion and distortional warping are neglected(3)The flowing cylindrical coordinate system is adopted when deducing the element matrix

Figure 1 shows the spatial coordinate system of curved bending bridge, where *x*, *y*, and *z* represent radial, vertical, and axial direction, respectively. The element node displacement matrix is given by:where *i* and *j* are node numbers; *u*, , and , respectively, represent longitudinal, vertical, and transverse displacement; *φ*, *v'*, and *w'*, respectively, represent torsion, vertical, and lateral angle; *w''* represents lateral curvature; *β* is constraint torsion displacement; and *ζ* is the largest shear displacement difference. A lot of previous findings demonstrated that the shear lag will contribute to an additional moment (called shear lag moment). This could consequently lead to an increment of the vertical displacement of the box girder bridge [17, 18]. Therefore, it is necessary to consider the shear lag effect for the box girder bridge to accurately calculate dynamic responses of the system.

##### 2.2. Stiffness Matrix

Based on the geometric equation of elastic mechanics, the generalized strain {*ε*} is

Equation (3) shows the corresponding matrix form:where the first matrix is differential operator vector denoted as {*P*}, the second matrix is denoted as {*δ*}, and *R* is the curvature radius of the box girder. Then, equation (3) can be written as {*ε*} = {*P*}{*δ*}.

It is assumed that the lateral displacement is interpolated by the fifth polynomial, the vertical displacement and the torsion angle are interpolated the cubic polynomial, and the axial displacement is interpolated by the linear polynomial. Therefore, the shape function matrix {*N*} can be obtained. The element displacement iswhere {*δ*}^{e} represents the nodal displacement. Then, according to equations (3) and (4), (5) could be given bywhere {*B*}is called the strain matrix, and the elastic matrix {*D*} is given bywhere *E* and *G* are the shear modulus and elastic modulus, *A* is the cross section area, *I*_{x} and *I*_{y} represent the vertical and lateral moment of inertia, and *I*_{d} and represent the free torsional and constraint torsional moment of inertia. The element virtual strain is given by

The stress does the virtual work for a virtual strain, and it can be given by

Meanwhile, the nodal force does the virtual work for a virtual displacement. It can be given by

Based on the virtual work principle [19], there is *δW*_{1} = *δW*_{2}. Therefore, the nodal force column matrix {*F*}^{e} is given by

The element stiffness matrix is shown as equations (11a) and equation (11b):

##### 2.3. Mass Matrix

For the box girder, the mass center is not coincident to the torsion center; therefore, the lateral internal force is given bywhere *ρ* is the box girder density and *e* is the distance from torsion center to the centroid. Obviously, the lateral internal force will also contribute to an additional torque, and it is given bywhere *I*_{ρ} is the polar moment of inertia. The internal force vectors of box girder are shown as equation (14):

Substitute into equation (14), and {*q*} can be written as

The equivalent node force is given by

Substitute equation (15) into equation (16), there is

The mass matrix {*M*} could be obtained from equation (18), and it is given by

##### 2.4. Vehicle Model

The 2-axes vehicle model is shown in Figure 2, where the tire and suspension system have independent degrees of freedom. Except for the vertical translation and torsion degrees of freedom, the transverse torsion degree of freedom is also included. There are totally 7 degrees of freedom in the model.

In Figure 2, *K*_{si} (*i* = 1, 2, 3, 4) and *C*_{si} (*i* = 1, 2, 3, 4) are the stiffness and damping of the suspension system, respectively, *K*_{ti} (*i* = 1, 2, 3, 4) and *C*_{ti} (*i* = 1, 2, 3, 4) are the stiffness and damping of the tire, respectively, and *M*_{ti} (*i* = 1, 2, 3, 4) is the mass of tires. *M*_{s}, *J*_{θ}, and *J*_{α} are the vehicle weight, vertical torsion inertia moment, and transverse torsion inertia moment, respectively. *a*_{1} and *a*_{2} are the distances from the vehicle gravity center to the front axis and back axis, respectively, *b*_{1} and *b*_{2} are the distances from the vehicle gravity center to the left axis and right axis, respectively. The details of vehicle parameters are shown in Table 1 [20].

The Rayleigh damping is adopted in the present work, and it is given bywhere *ξ* and *η* are the damping coefficients [21]. The vehicle-bridge coupling vibration equation is acquired by assembling the element stiffness matrix, mass matrix, damping matrix, and load matrix:where the subscripts and *b* denote the vehicle and the bridge, respectively, are acceleration, velocity, and displacement, respectively. The dynamic responses can be obtained through the Newmark-*β* method or Wilson-*θ* method [21].

##### 2.5. Basic Principles of Vehicle-Bridge Coupling

The schematic diagram of vehicle-bridge coupling is shown in Figure 3 [22]. The equation coupling method is adopted in the present work. The element node of the bridge surface is coupled with the spring element node of tire. The details are as follows:

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Figure 3(a) shows the coupling principle for the smooth bridge surface. The high-order displacement interpolation functions are utilized. The equations for vertical displacement are given aswhere *N*_{1} and *N*_{2} are vertical displacement interpolation functions, *G*_{1} and *G*_{2} are angle interpolation functions, *l* is the length of an element, and *x* is the distance from the tire to the element node. Assuming that the tires are in close contact with the bridge deck and there is no separation between them during the whole driving process, the vertical displacement of the tire is given by

However, the rough bridge surface as shown in Figure 3(b) will cause the separation between the tire and bridge surface and then contact again, leading to a great impact force on the bridge surface. Herein, the random irregularity function is introduced, and the vertical displacement of tire is given by

The random irregularity function is created by using the harmonic wave superposition method in the present work:where *a*_{i} is the amplitude of *i*th sine wave, *n*_{i} is the spatial sampling frequency point, *x* is the longitudinal location of bridge surface, and *φ*_{i} is the random phase angle within the range of (0, 2*π*). The power spectral density function is adopted to describe the random irregularity:where *n*_{0} is the standard spatial frequency, *n* is the spatial frequency, and *G*_{0} is the roughness coefficient of the bridge surface corresponding to *n*_{0}. The irregularity curves of the bridge surface are created by using triangular series on MATLAB, and they are shown in Figure 4. It is obvious that from A grade to D grade, the flatness of the bridge surface is excellent, good, average, and bad, respectively.

#### 3. Calculations and Discussions

##### 3.1. 3D Model Verification

The sectional dimension of a PC simply supported curved box girder is shown in Figure 5. The axial length is of 60 m, the radius of curvature *R* = 1000 m, Young's modulus *E* = 3.4 × 10^{4} MPa, and the density = 2500 kg/m^{3}.

The Newmark-*β* method is adopted to calculate dynamic responses of the bridge. The box girder is meshed into appropriate numbers of elements according to the trial calculation. Meanwhile, the nonlinear finite element model of the moving-load model is established by ANSYS, where the girder is simulated by BEAM 188 element. The vehicle body is simulated by the moving load.

The dynamic responses of the curved box girder are shown in Figures 6–8. As illustrated in Figures 6–8, the theoretical results of the vertical displacement in this study agree well with the numerical results of ANSYS. Although there is certain deviation at local areas between them, the variation trends are basically the same. This is mainly attributed to the fact that there is no moment of inertia in the moving-load model, leading to smooth response curves. This indicates that it is reasonable to consider the inertia force in the vehicle model. Besides, the contributions of stiffness and damping of tire and suspension to the vehicle-bridge coupling system are also ignored in the moving-load model, resulting in certain deviation between them. On the other hand, there is also no independent degree of freedom in the moving-load model, and its responses need to be obtained from the bridge response. The moving-load model cannot calculate the effect of irregularity of the bridge surface.

**(a)**

**(b)**

**(c)**

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**(b)**

**(c)**

**(a)**

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**(c)**

In Figure 6, the maximum dynamic displacements in the present work at 1/4 cross section, midspan, and 3/4 cross section are 6.46 mm, 8.79 mm, and 6.49 mm, respectively, whilst the results calculated by ANSYS are 6.02 mm, 8.79 mm, and 6.49 mm, respectively. The errors between them are 7.31%, 6.39%, and 9.44%. This indicates that the bridge structures obtain maximum vertical dynamic displacement when the vehicle is located at the midspan; this is consistent with the influence line principle. Besides, compared with the moving-load model, the existence of the vehicle body has a certain influence on the responses of the bridge structures. It is found that the existence of the vehicle body has a maximum 9.44% increase in vertical displacement.

Similarly, in Figure 7, the bridge structure obtains a maximum dynamic moment of 3872 kN m when the vehicle is located at the midspan. Therefore, the midspan cross section will be chosen for the following analysis.

##### 3.2. Comparisons between Curved and Straight Beam Bridges

The comparisons of vertical displacement and bending moment responses of straight and curved bridges with the same span are shown in Figure 9. It can be seen from the figure that the dynamic response of the curved bridge is slightly greater than that of the straight bridge. This is mainly attributed to the fact that the bending-torsion coupling effect existed in the curved bridge when considering the curvature. However, due to the small central angle of the curved bridge, the difference between them is not so remarkable. This is consistent with that of the research study conducted by Xiang [23]. These results indicates that when the curvature radius is greater than 1000 m, the dynamic responses of the curved bridge can be computed according to the straight bridge.

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##### 3.3. Effect of Irregularity of the Bridge Surface

The impact coefficient is defined aswhere *Y*_{d, max} and *Y*_{s, max} are, respectively, the peak dynamic response and peak static response. They can be displacement, strain, moment, or shear force. In this section, the vertical displacement is adopted to analyze the influence of parameters on the impact coefficient.

Figures 10(a)–10(d) show the effect of the irregularity of the bridge surface on the vibration of the vehicle-bridge coupling system. As illustrated in Figures 10(a)–10(c), as the flatness of the bridge surface becomes worse and worse, the vertical displacement amplitude of the bridge and the vertical acceleration amplitude of the vehicle body are gradually increased, and the vehicle vibrates more and more serious. In addition, the contact force between the tire and the bridge deck is zero at certain times. This indicates that the vehicle and the bridge will transitorily separate due to the irregularity of the bridge surface during driving and contact again subsequently, leading to the considerable impact force on the bridge. As expected, in Figure 10(d), the impact coefficient is gradually increased as the irregularity of the bridge deck changes from good to bad. The impact coefficient reaches the maximum value of 2.54 when the flatness grade is D, and it reaches the minimum value of 0.06 when the bridge surface is smooth. These results demonstrate that in practical engineering, the maintenance of the bridge surface should be regularly carried out to prevent the bad flatness of the bridge surface due to the natural ageing materials.

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##### 3.4. Effect of Vehicle Speeds

The displacement and the strain impact coefficient were used to replace the moment and shear force impact coefficient in the previous research studies [12–14, 16], which cannot really reflect the mechanic behavior of the primary beam under dynamic vehicle loads. As a matter of fact, the impact coefficient is essentially a parameter that reflects the amplification of the primary beam internal force (moment and shear force) as shown in equation (21). Besides, there is also a classification of the moment and shear force impact coefficient in CEN Euro code [24]. Herein, the vertical displacement, moment, and shear force impact coefficient are respectively calculated and compared in the following analysis, and the influences of the vehicle speed, vehicle weight, and the stiffness of the primary beam are considered.

Figure 11 shows the impact coefficients under different vehicle speeds (20 km/h∼120 km/h). As illustrated in Figure 11, the variation trend of the moment impact coefficient is the same as that of the displacement impact coefficient. The moment and the displacement impact coefficients both obtain extreme values at the speed of 60 km/h and 100 km/h, wherein the external excitation frequencies (1.744 Hz and 1.45 Hz) related to the vehicle axles are close to the natural frequency of the bridge (1.544 Hz). This indicates that the vehicle-bridge coupling system can resonate at both low and high speeds, leading to a large dynamic response. Since speed is a sensitive parameter to influence responses of vehicle-bridge, this phenomenon is also shown in the research studies conducted by Majka and Hartnett [25] and Bucinskas and Andersen [26]. Besides, since the envelope curve of moment impact coefficient is covered by that of the displacement impact coefficient, replacing the moment impact coefficient with the displacement impact coefficient will overestimate the moment design value. The moment design value is maximally overestimated by 2.89% as the vehicle speed reaches 100 km/h.

It can also be found from Figure 11 that the shear force impact coefficient under different vehicle speeds varies between 0.036 and 0.468, and the fluctuation range is relatively great. Since the speed greatly affects the dynamic shear force response at the left side and due to the leap of shear force at the midspan section (action of vehicle loads), the variation trend of shear force impact coefficients at the left side and the right side are completely opposite, and the envelope curve of the displacement impact coefficient is covered by that of the shear force impact coefficient. This means replacing the shear force impact coefficient with the displacement impact coefficient will underestimate the shear force design value. The shear force is maximally underestimated by 34.9% as the vehicle speed reaches 120 km/h, which is insecure for bridge structures.

The impact coefficient calculating formulas in the *General Specification for Design of Highway Bridges and Culverts* (JTG D60-2015) [27] arewhere *f* is the fundamental vibration frequency of the simply supported beam bridge. The impact coefficient of the box girder bridge calculated by equation (27) is 0.061 (*f* = 1.544 Hz), which is significantly smaller than that in this study. The difference between them is great, especially for the shear force impact coefficient. This indicates that it is not reasonable to only define the impact coefficient as the function of the fundamental vibration frequency of bridge structures, and the influence of the vehicle speed should be also considered.

##### 3.5. Effect of Load Weight

The impact coefficient under different load weights is shown as Figure 12. As illustrated in Figure 12, except for the differences of the shear force impact coefficient at the right side, the other impact coefficients under different load weights are almost the same, which indicates that load weights have little influence on the impact coefficient. Herein, the numbers of axle remain unchanged in the vehicle model, and only the load weight varies.

Ji et al. [12], Deng et al. [16], and Han et al. [28] also studied the influence of vehicle weight on impact coefficients. In their research studies, different vehicle models are considered including two-axle, three-axle, and five-axle vehicles. They concluded that the impact coefficient decreased with an increase in vehicle weights. As a matter of fact, it is the numbers of the vehicle axle that affect the impact coefficient. It is noteworthy that the more the axles, the heavier the vehicle, whilst the load weight has little influence on the impact coefficients.

##### 3.6. Effect of Primary Beam Stiffness

Deng et al. [16] studied the influence of cross section types of primary beams on the impact coefficient, whilst the influence mechanism of the impact coefficient was unrevealed. As a matter of fact, different cross section types will contribute to different inertia moments (*I*), and the primary beam stiffness (*EI*) was thus affected. In order to study the influence of the primary beam stiffness on the impact coefficient, the stiffness of the primary beam varies by changing the elastic modulus of concrete in the present work. The C40∼C80, C100, and C120 concretes are setup in the present work, and the detailed values of the elastic modulus are shown in Table 2. Figure 13 shows the impact coefficient under different primary beam stiffness. As illustrated in Figure 13, as a whole, the displacement and moment impact coefficient are decreased with an increase of the primary beam stiffness (elastic modulus increases), and they are only 0.053 and 0.061, respectively, when the concrete grade is C120. However, the shear force impact coefficient is generally increased with an increase of the primary beam stiffness, and it reaches 0.337 as the concrete grade is C100.

Similarly, replacing the moment and shear force impact coefficient with the displacement impact coefficient will have little influence on the moment design value but make the shear force design value conservative.

#### 4. Conclusions

A vehicle-bridge coupling model was proposed to predict the dynamic responses of the system in the present work, and the effect of flatness of bridge deck, vehicle speed, vehicle weight, and the primary beam stiffness on the impact coefficients are considered. Based on the results and discussions presented in this paper, the conclusions can be drawn as follows:(1)The vehicle-bridge coupling system can resonate at both low speeds and high speeds. When the curvature radius is greater than 1000 m, the dynamic responses of the curved bridge can be computed according to the straight bridge.(2)Replacing the moment and shear force impact coefficient with the displacement impact coefficient will have little influence on the moment design value but make shear force design value conservative, which is insecure for bridge structures.(3)The load weights have little influence on the impact coefficient. The moment and shear force impact coefficient decrease with the increase of the primary beam stiffness, whilst the shear force impact coefficient increases with the increase of the primary beam stiffness; a worse flatness of the bridge surface can lead to a larger impact force on the bridge structure.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare no conflicts of interest for this paper.

#### Acknowledgments

The present work was supported by the National Natural Science Foundation of China (51478397). The authors would like to express their gratitude to it.