Dynamic Characteristics of Metro Vehicle under Thermal Deformation of Long-Span Cable-Stayed Bridge

In order to study the influence of thermal deformation of long-span cable-stayed bridge (LSCSB) on the dynamic characteristics of metro vehicle on the bridge, based on the theory of vehicle-track coupled dynamics, the rigid-flexible coupled dynamic model of metro vehicle-track-LSCSB system is established by using finite element method and multi-rigid-body dynamics. Adopting this model, the deformation of LSCSB subject to temperature is analyzed, then the comprehensive effect of track random irregularity and rail deformation caused by temperature load is considered to study the dynamic characteristics of metro vehicle running through the bridge, and finally the influences of temperature increment and running speed on concerned dynamic indices of vehicle are studied. The results show that the LSCSB deforms obviously subject to temperature load, and the overall performance is that the cooling is arched, and the heating is bent, and the shape variable changes almost linearly with the temperature load. According to the parameters studied in this paper, the rail deformation caused by temperature load increases the wheel-rail vertical force, derailment coefficient and wheel load reduction rate by 1.5%, 3.1% and 5% respectively. The vertical acceleration of the vehicle body decreases by 2.4% under the cooling condition, while increases by 3.7% under the heating condition. The dynamic response of the bridge changes under temperature load. The maximum vertical and horizontal displacement in the middle of the main beam span are 6.24 mm and 2.19 mm respectively, and the maximum vertical and horizontal acceleration are 1.29 cm/s 2 and of vehicles on LSCSB.


Introduction
With the development of science and technology, rail transit has become an important mode of transportation. The development of rail transit affects the sustainable development of a city. Some cities in some areas of our country need to build long-span bridges to achieve the purpose of traffic because of geographical conditions. Cable-stayed bridge is the main type of super long-span bridge. With the increase of span, the structure becomes more complex [1]. However, the stability of LSCSB is greatly affected by external environmental factors, especially the effect of temperature load. Under the influence of temperature, the main beam will have a large vertical deformation, which will cause the deformation of track structure on the bridge, and then affect the dynamic characteristics of rail vehicles [2]. In serious cases, it may also lead to safety accidents [3]. Therefore, it is urgent to study the dynamic characteristics of railway vehicles under temperature deformation of LSCSB.
At present, some scholars have made some achievements on bridge deformation and train vibration caused by temperature load. Anna Saetta et al. [4] proposed a numerical method for stress-strain analysis of concrete structures under time-space varying thermal load based on the finite element method, and proved the effectiveness and reliability of the numerical method in practical structural design through the analysis of an example. Zhu et al. [5] studied the interface damage evolution of slab track under the joint action of temperature change and vehicle dynamic load and its impact on dynamic response. The analysis showed that the precast slab had lateral warpage, resulting in serious longitudinal interface damage on both sides of slab track during the temperature drop process.
Hambly et al. [6] studied how to calculate the temperature induced stress in simply supported and continuous structures according to the temperature distribution. Scanlan et al. [7] proposed a structural analysis method based on flutter derivative formula for the aeroelastic response of cable-stayed bridges to wind. Combined with examples, it is concluded that the analysis method can be successfully used to explain the phenomena observed in full-scale tests and practices. Xu et al. [8] proposed a dynamic analysis framework of train cable-stayed bridge coupling system under crosswind, and then applied the framework to a practical LSCSB. The final results show that the framework and the corresponding computer program can effectively predict the dynamic response of train cable-stayed bridge coupling system under crosswind. Hu et al. [9] reviewed the Q. M. Long et al. World Journal of Engineering and Technology latest design practices of some special bridges in China's high-speed railway projects, classified them according to the length of the main span, summarized the key technical characteristics of long-span high-speed railway bridges, and discussed the feasibility of long-span high-speed railway bridges. The bridge model established by previous scholars using finite element method is rarely verified by field tests, and the accuracy of its simulation needs to be discussed. This paper verifies that the model meets the simulation requirements through field tests. Previous studies have mainly studied the dynamic response of vehicles and bridges on the bridge separately. This paper comprehensively studies the dynamic response of the bridge under external excitation. For the research of vehicle dynamic response on LSCSB, predecessors have proposed a variety of different external excitation and excitation treatment methods. This paper mainly studies the influence of temperature load on the system. On the basis of previous studies, based on the dynamic theory of vehicle track coupling system [2], this paper establishes the finite element model of LSCSB with the actual engineering background, and then explores the action law of temperature load on LSCSB based on the thermodynamic principle [10]. By establishing the rigid-flexible coupling dynamic model of vehicle track bridge, study the influence of temperature load on vehicle dynamic response, explore the influence law of temperature load and speed on vehicle operation index, and draw relevant conclusions, which can provide some reference for subsequent research.

Vehicle Track Bridge Coupling Dynamic Model Considering Temperature Effect
Based on the vehicle-track coupling dynamics theory, a rigid-flexible coupling model of metro vehicle-track-LSCSB is established. The model includes vehicle subsystem model, track subsystem model and Bridge subsystem model, and the wheel-rail interaction and bridge rail interaction are used to connect the subsystems [11].

Vehicle Dynamics Model
The vehicle model is composed of car body, bogie, wheelset and suspension system. The suspension system is simulated by spring and damping. The vehicle body, bogie and wheelset have five degrees of freedom: lateral movement, heave, roll, nod and shake head. A vehicle subsystem model includes one vehicle body, two frames and four wheelsets. Therefore, the vehicle subsystem dynamics model has a total of 35 degrees of freedom [2]. Table 1 shows the degrees of freedom of the vehicle subsystem model. Figure 1 is the dynamic model of vehicle subsystem. The vibration differential equation of vehicle subsystem is: where,

, ,
v v v M C K are the total mass matrix, total damping matrix and total Q. M. Long et al.

Track Subsystem Model
The rail is simulated by Euler Bernoulli beam element [13]. The discrete elastic coefficient and viscous damping coefficient between rail and track plate are respectively K d and C d , from the vibration equation of Euler Bernoulli beam, the vibration differential equation of track subsystem is:

Bridge Subsystem Model
The bridge subsystem model is established based on the finite element theory, and its vibration differential equation is: M C K are the total mass matrix, total damping matrix and total stiffness matrix of the bridge beam subsystem, respectively. Where, , , , X X X F   are the acceleration, velocity, displacement of the bridge beam subsystem and the force of the track plate subsystem on the bridge beam subsystem [13].
The structural damping of the bridge is simulated by Rayleigh damping, and the Rayleigh damping formula is: where M and K are the mass matrix and stiffness matrix of the bridge structure respectively, and where, ω 1 , ω 2 , ξ 1 and ξ 2 are the first-order and second-order natural vibration frequencies of the structure and their corresponding damping ratios, respectively. For rigid bridges, ξ 1 and ξ 2 generally 2% ~ 3%. Therefore, as long as the stiffness matrix and mass matrix of the bridge structure are obtained, the damping matrix of the bridge structure can be obtained [12].

Rigid-Flexible Coupling Model of Vehicle Track Bridge
The vehicle, track and bridge are regarded as a coupled dynamic system, with wheel-rail contact as the interface, which is connected through the geometric compatibility conditions and interaction force balance conditions of wheel-rail [14].
Based on the above equations, the motion equation of vehicle track bridge coupling system is:

Track Excitation Model
It is very important to correctly describe and reasonably select the track excitation model for the study of vehicle-track-bridge dynamic interaction [3]. Due to the rail deformation caused by temperature load, the actual track excitation can be expressed as: where 0 r Z means random irregularity of the line, rT Z represents the rail deformation caused by temperature load.
1) According to the comparison between the maximum running speed of vehicles allowed by different levels of lines and the actual situation, this paper uses the American level 5 track irregularity spectrum to generate the random irregularity sample of track, and is 0 r Z .
2) The temperature load causing bridge deformation mainly considers the effects of uniform temperature difference, sunshine temperature difference and cooling temperature difference. This paper only analyzes the bridge deformation caused by sunshine temperature difference under temperature load. Liu once used exponential function to describe the temperature difference distribution of concrete structure along the direction of wall and slab [15]. In view of the fact that scholars at home and abroad have expressed the temperature distribution in the direction of pier thickness and beam height of box girder bridge in the form of index, and have been verified by a large number of experimental data. This paper intends to analyze and study the temperature distribution of box girder along the direction of beam height in the form of index [16]. Preliminarily fit the temperature distribution form in the height direction of the main beam and box girder according to the index curve, as shown in the following formula: where, T y and T 0 are the temperature difference between the position of the observation point and the beam height direction respectively, y represents the distance from the observation point to the upper surface, and a is the coefficient. T 0 and a can be selected according to the code for design of reinforced concrete and prestressed concrete structures of Railway Bridges and culverts tb1002.3-2005, taking 20˚C and 5 m −1 respectively [17].
The temperature difference along the thickness curve of the box girder can be calculated according to the following formula: where δ is the plate thickness, which a′ is selected according to Table 2.
Based on the principle of energy conservation and heat balance equation [18], the structure can be obtained by thermodynamic analysis of the above temperature difference load by using the finite element method, can be obtained rT Z .

System Parameters of Metro Vehicle-Track-LSCSB
In this paper, the Dongshuimen Yangtze River Bridge of Chongqing Rail Transit Line 6 is taken as the research object. The operating model is type B vehicle, and its vehicle dynamic parameters are shown in Table 3.   60 kg/m rail is selected for Metro Line 6, and its main parameters are shown in Table 4.
The spatial finite element model of the bridge is established based on the finite element theory [19], as shown in Figure 4. Among them, the main truss beam,

Experimental Study on Natural Vibration Characteristics of LSCSB and Model Verification
The natural vibration characteristic of the bridge not only represents the stiffness index of the bridge, but also is an important factor in the dynamic response of the vehicle bridge coupling system [20]. The natural frequency test of Dongshuimen Yangtze River Bridge, namely the bridge modal monitoring test, can verify the accuracy of the bridge finite element model. Through the comparison and analysis between the bridge natural frequency tested on site and the corresponding vibration mode frequency of the bridge finite element simulation model. The error is within a reasonable range and the accuracy of subsequent numerical calculation is ensured [21]. The natural vibration test site of the bridge in this paper is the upper deck of Dongshuimen Yangtze River Bridge. The test section is arranged at the middle section of the secondary side span and the middle span. And the test points are arranged by three survey lines, namely, the upper, middle and lower reaches. The test instrument is a capacitive acceleration sensor. The main principle is the driving excitation method, that is to identify the structural natural vibration characteristic parameters through the residual vibration signal of the bridge structure caused by the vehicle leaving the bridge deck. The field test is shown in Figure 5.
The subspace iteration method is used to calculate the modal analysis of the bridge model. The first 20 frequencies are calculated, and the first 4 frequencies are compared with the measured frequencies. Table 5

Deformation of LSCSB under Temperature Load
By consulting the relevant meteorological data to determine the temperature load, and then applying the temperature load to the bridge based on the prin- In order to study the influence of temperature on track deformation, the overall temperature load is applied to the bridge model [22].

Dynamic Behaviors of Metro Vehicle Running through LSCSB Subject to Temperature Load
Based on the rigid-flexible coupling model of metro vehicle-track-LSCSB [16], this section explores the influence of temperature load on vehicle dynamic response and the influence law of vehicle speed and temperature load on vehicle dynamic response.

Dynamic Response of Bridge under Temperature Load
For the bridge structure, considering that the dynamic response in the middle of the main beam span is larger than that in the side span and other parts. This pa-

Dynamic Response of Vehicle under Temperature Load
Since the effect of temperature load is mainly reflected in the vertical deformation of steel rail caused by the vertical deformation of the bridge, studying the influence of temperature load on the dynamic response of the train. The wheel/ rail vertical force, wheel load reduction rate, derailment coefficient and vertical acceleration index when the vehicle crosses the bridge are mainly extracted, and the speed is 70 km/h. See Table 7 for the maximum value of vehicle dynamic response under temperature load.
The vertical force of wheel/rail under temperature load is shown in Figure 9.
Under the initial irregularity condition, the maximum vertical wheel-rail force of that the track deformation caused by the overall temperature rise and fall is a smooth transition state, which has little influence on the wheel-rail vertical force. The derailment coefficient index of vehicle is shown in Figure 10. Under the initial irregularity condition, the derailment coefficient of vehicle is 0.122. It is 0.1258 under temperature rise and 0.1257 under temperature drop. Due to track deformation caused by temperature rise and fall, the positions of bridges with extreme values of derailment coefficient are inconsistent. Compared with the derailment coefficient without temperature load, the derailment coefficient under overall temperature rise and temperature drop increases by 3.1% and 3% respectively, with a small difference in growth. The reason is that the track deformation amplitude caused by the overall temperature rise and fall is basically the same. Based on the above results, it can be seen that the temperature load added in this paper has little impact on the derailment coefficient index during train operation, and all of them do not exceed the safety limit of 0.8, with a large safety margin.
Calculate the wheel load reduction rate under temperature load, as shown in Figure 11. Under the initial rough conditions, the wheel load reduction rate of the vehicle is 0.2547, 0.2672 under the overall temperature rise and 0.268 under the overall temperature drop. Compared with the wheel load reduction rate  without temperature load, the overall temperature drop increases by 5%, and the wheel load reduction rate increases by 4.9% under the overall temperature rise.
The influence of temperature load on wheel load reduction rate is greater than that of derailment coefficient. However, there is little difference in wheel load reduction rate under the condition of overall temperature rise and temperature drop. In general, under the temperature load added in this paper, the wheel load reduction rate under all working conditions is within the safety limit of 0.6, which is generally safe.
The vertical acceleration of vehicle body under temperature load is shown in

Comparison of Vehicle Dynamic Response under Different Vehicle Speed and Temperature Load
In  Figure 13. It can be seen from the figure that the bridge shape variable is almost linearly correlated with the temperature load.
Six different temperature loads plus no temperature load, a total of 7 loading modes and 3 different running speeds are combined into 21 different working conditions. The corresponding extreme values of vehicle running dynamic response are obtained.
As can be seen from Figure 14, the wheel/rail vertical force of the vehicle   As can be seen from Figure

Conclusions
Based on the vehicle-track coupling dynamics theory, this paper takes Dong- 1) The deformation of LSCSB under temperature load is significant. When extreme high temperature load is applied in this paper, the maximum deformation in the middle of the bridge span reaches 227 mm, and the shape variable is almost linearly related to the temperature load.
2) The dynamic response of the bridge changes under temperature load. The maximum vertical and horizontal displacement in the middle of the main beam span are 6.24 mm and 2.19 mm respectively. And the maximum vertical and horizontal acceleration are 1.29 cm/s 2 and 2.54 cm/s 2 respectively. The dynamic response index of the girder midspan is more significantly affected by the running speed of vehicles on the bridge.
3) The track deformation caused by temperature load has an impact on the dynamic response of vehicles. The main performance is that the wheel-rail vertical force increases under the load of rising and falling temperature, with an increase of 1.5% and 1% respectively. The derailment coefficient also increased, with an increase of 3.1% and 3% respectively. The wheel load reduction rate also increased, with an increase of 4.9% and 5% respectively. The vertical acceleration of the vehicle body decreases by 2.4% under the cooling load and increases by 3.7% under the heating load.
4) The influence of vehicle speed and temperature load on vehicle dynamic response has a certain law. Temperature load has a greater influence on the derailment coefficient than vehicle speed. Vehicle speed has a greater influence on wheel-rail vertical force and wheel load reduction rate than temperature load. At the same speed, the influence of temperature load on the vertical acceleration of vehicle body is almost linear.