Vehicle – Bridge Interaction Dynamic Analysis of Continuous Rigid Frame Composite Box Girder Bridge with Corrugated Steel Webs under Seismic Excitation

. To study the vehicle – bridge interaction (VBI) of highway bridges under seismic excitation, a vehicle – bridge couple analysis method based on Ansys is proposed. The 1/2 vehicle model and space beam element model were established to analyze the VBI response of


Introduction
The composite box girder bridges with corrugated steel webs, originated in France, have been widely used in bridge engineering in Japan and China due to its advantages such as lightweight and superior load-bearing performance compared to ordinary prestressed concrete bridges.Among the various composite box girder bridge with corrugated steel webs, the continuous rigid frame box girder bridge with corrugated steel web, which has the advantages of small deformation, high stiffness, comfortable driving experience, and ease of construction, has been constructed more in China.By the end of 2022, there were already more than 21 built and underconstruction continuous rigid frame box girder bridges with corrugated steel webs in China.
China is a country prone to frequent earthquakes.On one hand, as the span of these bridges continues to increase, it increases the possibility of seismic-vehicle-bridge coupled vibration.On the other hand, although scholars have conducted relatively systematic research on the vehicle-bridge coupled vibration of concrete box girder railway bridges under seismic excitation, compared to concrete box girder bridges, this type of bridge can significantly reduce seismic forces.When this type of bridge is subjected to vehicle-bridge coupled and seismic coupled interaction, the sensitivity of various factors to the dynamic response of the vehicle-bridge coupled interaction may differ from that of concrete bridges.Therefore, it is necessary to study the vehicle-bridge coupled vibration of highway bridges under seismic excitation and clarify the relevant laws of seismic-vehicle-bridge coupled vibration caused by various factors.However, few studies have been done on the vehicle-bridge coupled vibration of highway bridges under seismic excitation so far, especially regarding composite box girder bridge with corrugated steel webs [1,2].
At present, the seismic-vehicle-bridge coupling has been widely studied in the field of railway bridges, mainly focusing on the response of vehicles and bridges under seismic excitation, as well as issues related to vehicle safety.Zhongxian et al. [3] analyzed the light rail railroad and compared the vibration of the Tianjin wave and El Centro wave on the vehicle-bridge coupled system.It was concluded that the seismic-vehicle-bridge coupling interaction satisfies the superposition principle, and the coupling effect can be approximated as the sum of the bridge's vibration response under seismic loads and the vehicle-bridge coupled dynamic response.Zhihui et al. [4] and Hujun et al. [5] conducted a comparative analysis of the dynamic response of bridges and trains under different train speeds and seismic intensities and provided threshold values for the safe operation of trains under various levels of seismic motion.Du et al. [6] compared and analyzed the effects of seismic motion displacement and acceleration input modes on the seismic-vehicle-bridge coupling dynamic response.Hujun and Xiaozhen [7] further compared commonly used nonuniform analysis methods in seismicvehicle-bridge coupling analysis, which are direct solution method, relative motion method, large mass method, and large stiffness method.Zhi et al. [8] compared and analyzed the influence of seismic spectrum characteristics on the dynamic response of bridges.Ziming et al. [9] based on the principle of minimum potential energy and with the seismicvehicle-bridge coupling, compared the dynamic response of the horizontal and vertical seismic waves on the railroad bridge.In summary, the current research mainly focuses on vibration analysis of railway bridges under seismic excitation, while research on seismic-vehicle-bridge coupling vibration in highway bridges is scarce.
In terms of research methods for vehicle-bridge coupled vibration, the current methods mainly include development methods for vehicle-bridge coupled interaction based on Fortran language, combined simulation methods using multibody software and finite element software, and combined simulation methods based on MATLAB and Ansys finite element.The current Ansys vehicle-bridge coupled method based on secondary development.Chengzhao [10], Wang et al. [11], and Li et al. [12] established the bridge model by Ansys preprocessing function, introduced two conditions of geometric compatibility of vehicle and bridge and the opposite direction of the vehicle-bridge coupled forces by using Fortran language, and realized the vehicle-bridge coupling based on Newmark-β stepwise integration method.Jianrong [13] adopts the method of combining Ansys with the self-programmed VBDIP (Vehicle Bridge Dynamic Interaction Program); Jianrong [14] adopts the method of combining Ansys and UM software; Peiwen et al. [15] and Shizhong [16] propose the method of implementing vehicle-bridge coupling in a single Ansys environment; however, the proposed methods are subject to the secondary development of Ansys language, which is difficult to implement for the general researchers.To address this problem, this paper proposes a contact constraint-based vibration analysis method for axle coupling, which can better realize the simulation of road surface irregularity [17] and simplify the analysis method of vehicle-bridge coupled interaction.Jin et al. [18] based on a numerical model of a train-trackbridge system with 31°of freedom showed that vertical seismic excitation promotes wheel bulging and increases the chance of derailment.In addition, the risk of derailment generally increases with the ratio of lateral to vertical deck acceleration.Paraskeva et al. [19] investigated the seismic response of a vehicle-bridge interaction (VBI) system under vertical seismic excitation, modeling a truck vehicle as a rigid body assembly.A parametric study is carried out based on a realistic highway (straight R/C) bridge-truck case.The analysis presents two main sources of dynamic excitation which is an inherently unpredictable "vehicle-bridge-seismic time" problem requiring probabilistic treatment.Zhou et al. [20] established the equations of motion of the monorail train and solved them by self-programming on the MATLAB platform.A finite element model of the bridge was developed and solved using Ansys programing method.The bridge and vehicle subsystems are coupled by global iteration with the exchange of wheel-rail contact forces.The cosimulation method is validated by the dynamic response of the monorail train and the bridge reported in the related literature.Su et al. [21] proposes a method for evaluating the fatigue life of tied-arch bridge suspenders by considering the effects of random cyclic traffic loads and environmental erosion.Yu et al. [22] in order to cover the complexity of coding and extend the generality on the road vehicle-bridge iteration, a process to solve VBI considering varied vehicle speed based on a convenient combination of MATLAB Simulink and Ansys is presented.Zou et al. [23] cover the effects of different VBI models on the bridge responses are studied and the results from different models are compared in terms of their accuracy, efficiency, and suitability.
Compared to railway bridges, highway bridges have smaller live loads, and the vibration characteristics caused by vehicles may differ from those of trains.Moreover, the seismic performance of composite box girder bridge with corrugated steel webs is better than that of railway bridges.Therefore, whether the seismic-vehicle-bridge coupling laws of railway bridges can be directly applied to highway bridges requires further research.In this regard, this paper proposes a contact-constraint-based vehicle-bridge coupling vibration analysis method.Subsequently, taking the Xiaoshagou Bridge as a case study, a 1/2 vehicle model is established, and the contact-constraint-based vehicle-bridge coupling vibration analysis method is used to investigate the influence of seismic motion types, directions, and intensities on the vertical dynamic response of seismic-vehicle-bridge coupling.Then, the correlation between the vertical dynamic response of the bridge is summarized and the influence of factors such as vehicle speed, road surface irregularity level, and randomness of road surface irregularity is analyzed under seismic excitation coupling.Finally, a comparison is made between the vehicle-bridge coupling vibration and the dynamic response of the bridge under seismic action, to show the simplified analysis of seismic-vehicle-bridge coupling vibration in highway bridges.

2
Advances in Civil Engineering  1), is always used for the current analysis, because it can reflect the vibration characteristics of the vehicle and is also simpler and more efficient than the whole vehicle model.
Based on the vibration of various components of the vehicle model, the equivalent forms of various types of vehicles in Ansys finite element software are summarized, as shown in Table 1.The beam element can replace the Mpc184 element in the diagram.

Bridge
Model.The bridge model can be built by Ansys finite element software, which should be matched with the vehicle model in the modeling process.For the 1/2 vehicle model, the bridge is used as a beam element model.The more commonly used in vehicle-bridge coupling analysis is Rayleigh damping, calculated as shown in Equation (1): where α and β are the mass damping coefficient and the stiffness damping coefficient, respectively.

Road Surface Irregularity
Simulation.Road surface irregularity is an important excitation in vehicle-bridge coupling vibration and has strong randomness, which must be considered in vehicle-bridge coupling vibration analysis.Currently, in China, road surface irregularity is recorded using power spectral density functions based on measured values.According to the Chinese standard GB/T7031-2005, the expression for road surface power spectrum density can be given as shown in Equation ( 2): In the equation, n and n 0 are the spatial frequency and spatial reference frequency, respectively.Where n 0 is usually taken as 0.1 m −1 ; G q (n 0 ) and G q (n) are the pavement leveling coefficient and pavement power spectral density function in m 2 /m −1 , respectively.w is the frequency index, generally taken as 2.
In this study, road surface irregularity is simulated based on the inverse Fourier transform, and the main equations are shown in Equations ( 3)-( 5): In the above, jX k j: is the modal value of the discrete Fourier transform; n x , ΔL, N are the spatial frequency sampling points, sampling spacing and number of samples, respectively; G q ðn x Þ: is the value of the pavement power spectral density function corresponding to the spatial frequency sampling points; and φ K is the random phase angle on the interval (0, 2π).X m is the pavement road surface irregularity.

Ansys-Based Axle Coupling Method. Using Ansys, vehicle model and bridge model can be established, by linking the vehicle and the bridge through the displacement coordination relationship.
It is supposed that the wheels always maintain contact with the bridge when vehicle moves.This can guarantee the vehicle and the bridge satisfy the displacement coordination relationship in Equation ( 6): where Y vi is the vertical displacement of the vehicle wheelbase node; Y bi is the vertical displacement of the bridge at the contact position between the wheelbase node and the bridge; and Δ i is the sample value of the road surface irregularity at the bottom node of the wheel.The schematic diagram of the vehicle-bridge coupling calculation model used in this paper is shown in Figure 2.

Vehicle component categories
Figure 2 represents the implementation of the 1/2 vehicle model.The contact-based analysis method proposed in this paper is entirely based on modeling and solving within the Ansys environment.

Method Validation.
To validate the accuracy of the method proposed in this paper, the vehicle-bridge coupling dynamic model [24] was used for comparison.All parameters were consistent with the values reported in the literature.The calculation model included an 8 m approach bridge, resulting in a total travel length of 40 m.The results of the midspan deflection for the simply supported beam without considering road surface irregularity are shown in Figure 3.
From Figure 3, it shows that the variation pattern of the corresponding time curve in the paper is consistent with that of the literature [24].When the speed of the vehicle is 40, 60, 120, and 160 km/hr without considering the smoothness of the road surface, the peak vertical displacement response is 8.28, 8.37, 8.73, and 9.11 mm; respectively, calculated by the method of this paper, while the results of the literature [24] are 8.48, 8.48, and 9.11 mm; respectively, 8.51, 8.8, 8.69, and 9.22 mm, the error is less than 2.0%, and the method is more accurate.The modal analysis of the main structure of the Xiaoshaogou Bridge is performed by using the Ritz vector method.The free vibration frequencies of the first six modes are shown in Table 3, The control section 1-7 (middle span and pier top of each span) and finite element model are shown in Figure 6.

Vehicle-Bridge Interaction Dynamic Model
To verify the proposed bridge model, the free vibration frequencies and vibration modes were compared with those of the literature [26].The bridge-pier-pile model developed in the literature [26] has first-order transverse bending and first-order antisymmetric vertical bending frequencies of 0.452287 and 1.453605 Hz, respectively, which are slightly smaller than the results of this paper.The flexibility of the bridge will be slightly increased and the free vibration frequency will be reduced after considering the influence of piles.The correctness of the bridge model in this paper can be verified.

Vehicle-Bridge Coupled Response under Seismic Excitation
Based on the proposed vehicle-bridge coupling method, the direct input of seismic acceleration records was used.
According to the "Specifications for Seismic Design of Highway Bridges" (JTG/T 2231-01-2020) [27], assuming that the ground motion is consistent at all support locations of the bridge and considering that the continuous rigid frame bridge does not exceed 600 m, the effect of wave propagation can be neglected.The integration time step and seismic acceleration records were set to 0.02 s.Advances in Civil Engineering and the peak shear force changes by 20.4%.The peak of each response does not correspond to the same seismic wave, and the peak displacement, bending moment, and shear dynamic response of the section does not correspond to the same seismic wave.It is suggested that all the displacement, bending moment, and shear response under different seismic waves should be considered respectively.
Combining the results of the seven different control sections, the maximum difference of 31.4% for the bridge girder and 39.5% for the bridge pier after considering different types of seismic waves.The main reason is that the coupled seismic vehicle-bridge response is not only affected by the peak seismic.It should also be related to the spectral characteristics of the ground vibration, and the cross-section at the location of the bridge pier is also the same law.

Seismic Wave Direction
Effects.Keeping the vehicle speed at 60 km/hr and other factors constant, the EI Centro waves of 0.1 g were input in the transverse, vertical, and longitudinal directions as shown in Figure 7(a), and the coupled seismic-vehicle-bridge analysis was performed respectively.The maximum dynamic response of the bridge girder and the top of the bridge pier are shown in Figures 9 and 10.
From Figures 9 and 10, it shows that the vertical response of the bridge under transverse ground shaking is much less than that of vertical and longitudinal ground shaking in most cases, while the longitudinal ground shaking has the greatest effect on the moment response of the bridge piers and the vertical ground shaking has the greatest effect on the shear force of the bridge piers and the effect on the displacement of the bridge deck cannot be ignored.For the bridge girder, the ratio of the bridge response under vertical or longitudinal Seismic excitation to the response of transverse seismic excitation is 1.50-12.31;for the bridge pier, the ratio is 2.48-26.61.The result imply that the effect of vertical and longitudinal seismic excitation must be considered when analyzing the vertical dynamic response of the earthquakevehicle-bridge coupling.

Advances in Civil Engineering
From Figures 11 and 12, it can be seen that the dynamic response of the beam increases approximately in proportion to the increase in the seismic intensity level.Taking Section 1 as an example, the calculated displacements under seismic waves at 0.05, 0.1, 0.15, and 0.2 g are 2.702, 5.794, 8.889, and 11.985 mm, respectively, with a displacement ratio of 1 : 2.14 : 3.289 : 4.435; the calculated bending moment values are 5,437, 10,295, 15,154, and 20,016 kN•m with bending moment ratio of 1 : 1.894 : 2.787 : 3.681; the calculated shear values are: 210, 407, 628, and 850 kN, the ratio is 1 : 1.938 : 2.99 : 4.048.The response at crosssection of the bridge pier is also by this law.The displacement, bending moment, and shear force are not kept exactly same proportional because of other factors such as bridge self-weight, vehicle system, and damping of the coupled system.For general cases, it can be approximated that the ground vibration intensity is proportional to the response value.

Simplification of Seismic-Vehicle-Bridge Coupling Vibration
In this section, based on the research results from Section 4.2, only vertical and longitudinal seismic waves were input to analyze the seismic-vehicle-bridge coupling vibrations.

Effects of Vehicle Speed.
Keeping other parameters constant, the EI Centro seismic wave with an amplitude of 0.1 g was input in the longitudinal and vertical directions.The dynamic responses of the bridge were studied under vehicle speeds of 60, 90, and 120 km/hr, with the same duration of seismic motion of 18.84 s.The maximum dynamic responses on the bridge girder and pier tops are shown in Figures 13  and 14.
From Figures 13 and 14, it can be obtained that the displacement, bending moment, and shear force responses of the bridge are slightly different under different vehicle speeds, but the peak values of the dynamic responses are not significantly affected.The dynamic responses of the control sections are generally consistent.Take Section 1 as an example, when the vehicle is traveling at 60, 90, and 120 km/hr, the peak displacement of the bridge is 9.256, 9.145, and 9.246 mm, the peak bending moment is 20,075, 20,388, and 19,856 kN•m, and the peak shear force is 858 and 853 kN.The peak shear forces are: 858, 853, and 844 kN, the difference between the maximum and minimum value is about 1.2%.The effect of vehicle speed is less than 7.2%, and the effect of vehicle speed can be ignored to simplify the calculation.

Effects of Road Surface Irregularity
Randomness.Road surface irregularity exhibits strong randomness, and many domestic and foreign researchers have conducted extensive studies on the randomness of vehicle-bridge coupling vibration [28].To study the impact of road surface irregularity randomness on the seismic-vehicle-bridge coupling vibration, a vehicle speed of 60 km/hr was maintained, and three different road surface irregularity sequences of the same grade were selected.The maximum dynamic responses on the bridge deck and pier tops are shown in Figures 15 and 16.
From Figures 15 and 16, it can be obtained that for different pavements of the same road surface irregularity, the peak values of displacement, bending moment, and shear response of road surface irregularity sequence for the coupled seismic-vehicle-bridge vibration are almost the same, and the overall difference is within 3% without considering the effect of the randomness of pavement unevenness.Therefore, it is concluded that the randomness of road surface irregularity can be neglected in the seismic-vehiclebridge coupled vibration.

Effects of Road Surface Irregularity
Grade.With a seismic excitation speed of 60 km/hr and other parameters constant, three different grades of road surface irregularity sequences (A, B, and C) were selected to study the effects of different road surface irregularity grades on the seismic-vehicle-bridge coupling vibration.A comparison was also made with the seismic bridge response and the vehicle-bridge coupling response (considering road surface irregularity Grade D).The maximum dynamic responses on the bridge deck and pier tops are shown in Figures 17 and 18.In the figures, G1, G2, G3, G4, G5, and G6 represent the seismic-vehicle-bridge coupling responses with road roughness grades A, B, C, and D, the seismic bridge vibration response without considering   Advances in Civil Engineering response, while the seismic bridge response is consistent with the coupled seismic-vehicle-bridge response, mainly because the vehicle load is smaller compared to the self-weight of the bridge.Another reason is that because for each midspan section, the time of the peak vibration response of the coupled vehicle-bridge response is different from the time of the peak seismic bridge response.Therefore, the dynamic response problem of seismic-vehicle-bridge-coupling can be simplified to the seismic-bridge vibration problem.The difference in the dynamic response of the bridge under each level of road 12 Advances in Civil Engineering surface irregularity not exceed 1%, the effect of road surface irregularity level can be ignored when performing seismicvehicle-bridge coupling.

Conclusion
In this study, a contact-constrained vehicle-bridge coupling vibration analysis method was proposed, and a seismic-vehicle-bridge coupling vibration model was established to analyze the dynamic response of bridges under seismic loads.The effects of seismic wave categories, directions, and intensities on the bridge's dynamic response were investigated.Furthermore, the effects of vehicle speed, road surface irregularity grade, and randomness in the seismic-vehicle-bridge coupling were studied by inputting vertical and longitudinal seismic motions.A comparison was made between the seismic bridge vibration, seismicvehicle-bridge coupling vibration, and vehicle-bridge coupling vibration responses.The following conclusions can be drawn: Advances in Civil Engineering 13 (1) The vehicle model was simplified by introducing contact nodes and establishing contact equations with the bridge and constraint equations with the vehicle.Compared to existing methods, this approach is more efficient and ensures a higher level of accuracy in considering the effects of road surface irregularity on seismic-vehicle-bridge coupling vibration.(2) Seismic wave directions have a different impact on dynamic response of the bridge.When studying the vertical vibration of the bridge, it is necessary to consider vertical and longitudinal seismic excitation.Furthermore, a comprehensive analysis including displacement, bending moment, and shear force dynamic responses should be conducted.(3) The ground shaking intensity is basically approximately proportional to the dynamic response of the bridge.The dynamic response of the coupled seismic-vehicle bridge is not only related to the seismic intensity but also the seismic category.(4) After considering seismic loads, the influence of vehicle speed and road surface irregularity is generally less than 3% on the structure response and can be neglected.The randomness of road surface irregularity can also be neglected, too.In practical engineering, for simplicity, the analysis of bridge response under seismic-vehicle-bridge coupling can directly focus on the bridge's response under seismic action.(5) This study did not consider the motion of seismic waves at different supports and did not analyze the dynamic response of vehicle sequence, multilane vehicles under seismic conditions for continuous rigid frame box girder bridge with corrugated steel webs.Further research may be carried out on these aspects in the future.

FIGURE 1 :
FIGURE 1: Modern vehicle model and element equivalence: 1/2 vehicle model and equivalent.

FIGURE 3 :
FIGURE 3: Comparison of vertical displacement time history curves: (a) method of this paper and (b) results of literature [24].

3. 2 .
Vehicle Models.The 1/2 vehicle model provides more accurate results compared to the full vehicle model, and its computational speed is faster.Based on reference[25], the 1/2 two-axle vehicle model used in this paper was selected.Four degrees of freedom were considered in the analysis: vertical bounce, pitch, and vertical bounce between the vehicle body and the wheelset.The vehicle diagram and specific parameters are shown in Figure5and Table2 .

FIGURE 5 :
FIGURE 5: Diagram of vehicle model diagram.

FIGURE 8 :
FIGURE 8: Maximum dynamic response of bridge piers under different seismic wave: (a) bending moment and (b) shear force.

FIGURE 9 :
FIGURE 9: Maximum dynamic response of bridge deck in different seismic wave directions: (a) vertical displacement, (b) bending moment, and (c) shear force. .

FIGURE 10 :FIGURE 11 :
FIGURE 10: Maximum dynamic response of bridge piers in different seismic wave directions: (a) bending moment and (b) shear force.

FIGURE 14 :
FIGURE 14: Maximum dynamic response of bridge piers at different vehicle speeds: (a) vertical displacement and (b) bending moment.

ðcÞFIGURE 13 :
FIGURE 13: Maximum dynamic response of bridge girder at different vehicle speeds: (a) vertical displacement, (b) bending moment, and (c) shear force.

ðcÞFIGURE 15 :FIGURE 16 :
FIGURE 15: The maximum dynamic response of bridge girder with different road surface irregularity: (a) vertical displacement, (b) bending moment, and (c) shear force.

ðcÞFIGURE 17 :FIGURE 18 :
FIGURE 17: Maximum dynamic response of bridge decks with a different grad of road irregularity: (a) vertical displacement, (b) bending moment, and (c) shear force.

TABLE 1 :
Unit selection and setting of vehicle model components in Ansys.

TABLE 3 :
First six natural frequencies and modes of the Xiaoshagou bridge.Figures7 and 8show the maximum dynamic responses of the bridge girder and the bridge pier for the three types of seismic wave inputs.From Figures7 and 8, the peak displacement dynamic responses of EI Centro, Taft, and San Fernando waves are 5.794, 4.937, and 5.422 mm, respectively, at Section 1; the peak bending moment responses are 10295 × 10 6 , 10750 × 10 6 , and 11,652 × 10 6 kN•m, respectively.The peak shear response was 407, 463, and 490 kN.The peak displacement at Section 1 changes by 17.3%, the peak bending moment changes by 13.2%, at 60 km/hr, the EI Centro wave, Taft wave, and San Fernando wave were respectively modified to have an acceleration amplitude of 0.1 g and used to analyze the seismic-vehiclebridge coupling responses for different seismic wave types.
From Figures17 and 18, it can be obtained that the peak displacement of the bridge at Section 1 under the six conditions mentioned in the paper are: 9.255, 9.253, 9.266, 9.309, 9.256, and 1.452 mm, the peak bending moment are: 20,074, 20,004, 20,003, 19,883, and 3,571 kN•m, and the peak shear force are: 858, 857, 858, 857, 858, 344, 20,075, and 3,571 kN•m; the peak shear forces are: 858, 857, 858, 857, 858, and 344 kN, and similar results for other sections.Without seismic excitation, the peak dynamic response is significantly smaller compared to the coupled seismic-vehicle-bridge