Layout Optimization of Rail Expansion Joint on Long-Span Cable-Stayed Bridge for High-Speed Railway

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China Structural Health Monitoring and Control Institute, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Shanghai Key Laboratory of Rail Infrastructure Durability and System Safety, College of Transportation Engineering, Tongji University, Shanghai 201804, China China Railway Siyuan Survey and Design Group Co., Ltd., Wuhan 430000, China


Introduction
As a kind of cable system, cable-stayed bridge has been widely applied in high-speed railway construction due to its large spanning capability. e bridge-track interactionis aggravated under the effects of various loads including the temperature, train load, and seismic load [1][2][3][4][5][6][7][8]. e bridgetrack interaction and distribution of rail longitudinal force on the long-span bridges are more complex than those on simply supported bridges and concrete continuous bridges [9,10]. e track and cable-stayed bridge are investigated in different aspects, including the exploration of nonlinear finite element model (FEM) of cable-stayed bridges [11], nonlinear analysis program [12], nonlinear static analysis of cable structures [13], the influence of distinct parameters on the mechanical properties of track on long-span cable-stayed bridges by FEM [14], experimental tuning of the model [15], the distribution of rail longitudinal force including expansion force, and braking force on the cable-stayed bridges [16][17][18][19][20]. e simulation process and design of CWR on cable-stayed bridges are simplified [21]. As a result, the optimal layout of CWR is relatively scarce.
e Rail Expansion Joint (REJ) is a necessary device on long-span bridges, which can alleviate the bridge-track interaction decreasing the rail longitudinal force. Some researches about REJ have been carried out. Xie et al. studied the dynamic response of REJs located at continuous beam bridge under the earthquake [22]. Min et al. adjusted REJ by measuring the installation location to improve the maintenance efficiency and reduce the cost [23]. Bae et al. analyzed the longitudinal displacement of the sleepers in REJ zone by simulating the ballast track with beam and spring elements [24]. Chung and Lee analyzed the longitudinal force of REJs on bridge under temperature load [25]. In the previous studies, the influence of REJ installation was investigated, but the layout of REJ in influencing the rail and piers longitudinal force is ignored. A reasonable layout of REJ can reduce the rail and pier longitudinal force simultaneously and improve the operating state of railways. However, the research on REJ layout of cable-stayed bridges is still limited. e optimization methods are not unified [26].
is research will mainly take a cable-stayed bridge with laying CWR under construction in China as a case; the influence of REJ on the rail and piers longitudinal force is investigated. An optimization method for choosing the installation location of REJ on the cable-stayed bridge is put forward. e field monitoring data of the existing cable-stayed bridge are compared with the simulation results of the new bridge to verify the optimal scheme. e conclusions will be helpful for the layout optimization method of REJ on the cable-stayed bridges.

Calculation
eory. e calculation method of rail longitudinal force is established based on the bridge-track interaction theory. e derivation of bridge-track interaction theory is based on equilibrium condition of rail microelement as shown in Figure 1. u r(x) denotes the rail longitudinal displacement and u b(x) refers to the longitudinal displacement of the upper surface of the bridge; both u r(x) and u b(x) are positive to the right; the beam-rail relative displacement is u and is shown in the following formula: (1) Consider a part of rail with an infinitesimal length of dx in Figure 1. P is the rail axial force (take tension as positive); r(u) is the gradient of longitudinal resistance. According to the longitudinal balance principle of infinitesimal rail, the following formula can be deduced: By changing formulae (2), the following (3) can be derived: Considering rail material as linear elastic material, according to the geometric equation and physical equation of elasticity mechanics, the following formulae can be derived: where ε is the rail strain. σ is the rail stress; E is the elastic modulus of rail; F is the cross-sectional area of rail. Substituting formulae (5) into (4), the following formula can be derived: Substituting formulae (6) into (3), the following formula can be obtained: Substituting formula (1) into formulae (7), the following can be derived: Formula (9) is the differential equation of bridge-track interaction.
According to the rail geometric boundary condition that there is no displacement of rail in the rail fixed area. e following formula is Once the description form of r(u) is determined, the rail longitudinal displacement and rail longitudinal force can be solved by formulae (8)- (10). Due to the complicated interaction between the cable-stayed bridge and CWR system, formulae (8)- (10) are too difficult to solve, so the finite element method is used to analyze CWR on the cable-stayed bridge.

Railway Track on the Cable-Stayed Bridge.
A cable-stayed bridge under construction in China is modeled in this investigation, with four-line high-speed railway on it. e main beam is steel-concrete composite box beam with six spans of 50 + 274 + 672 + 224 + 50 + 50 m. e type of track on bridge is ballast track. e existing models for analyzing bridge-track interaction of CWR on cable-stayed bridges are two dimensional models, in which the factors include influence of multiline of rail, mechanical behavior of cable-stayed bridges, and geometric nonlinearity which are ignored. us, a trackbeam-cable-tower spatial model for analyzing bridge-track interaction of CWR on cable-stayed bridge is established. e approach spans at left and right sides of cable-stayed bridge are simplified as 5 × 48 m and 5 × 32 m, simply-supported concrete beam bridges, respectively, which are shown in Figure 2(a).

Ballast Track Model.
e track comprises 60 kg/m rail, type III concrete sleepers, elastic bar II-type fasteners, and small resistance fasteners. e rail is connected to the sleepers by fasteners; sleepers are connected to the main beam of bridge by ballast bed; the Timoshenko beam element is adopted for the simulation of the rail so as to take the shear deformation into account. In order to reduce the impact of boundary conditions on the calculated results, both sides of rail are extended for 100 m out of the bridges; two ends of rail are fixed. Sleepers are divided into beam elements by supporting point of rail; the fasteners and ballast bed elements are modeled by three-dimensional spring system. One end of a fastener spring connects with a rail node and the other end connects with a sleeper node. e 2 Advances in Civil Engineering two ends of the ballast bed spring connect to the sleeper node and the main beam node, respectively. e longitudinal and transverse resistance of fasteners and ballast bed are simulated as nonlinear spring elements. e vertical constraints of that are modeled by linear spring elements.

Cable-Stayed Bridge
Model. e cable-stayed bridge is a separate tower-beam and consolidated tower-pier form that is composed of beams, stay cables, and double towers. e H-typed main towers are constructed by reinforced concrete with heights of 252 m and 242 m, respectively. 304 stay cables are fabricated by 8 kinds of parallel wire bundles. e damping devices are used to connect towers with main beam. e geometric nonlinearity of cable-stayed bridge is considered. e stay cables subjecting to the axial force are simulated by the link elements; the impact of stay cable sag effect on bridge-track interaction is so little that can be ignored [27]. ree-dimensional beam elements are applied to simulate the main towers, main beam, and piers by taking into account the bearing axial force and bending moment. e damping devices are modeled by spring elements and only used when trains brake and accelerate. e fixed bearings of the simply supported beam approach bridges on each side are set apart from the cable-stayed bridge. e beams, piers, and abutments of approach bridges are simulated as beam elements. e beams and piers are tied on horizontal and vertical direction at fixed bearings, while the bridge has longitudinal freedom at expansion bearings. e bottom of the main towers and piers is piles and soil.

Pile-Soil Interaction
Model. Pile-soil interaction is a difficult problem in seismic research of bridges. Spring elements are used to simulate the interaction of pile-soil structure. e piles are modeled as beam elements laid on elastic foundation. According to the soil conditions and the mechanism of pile-soil interaction, the soil around piles is simplified as equivalent soil   Advances in Civil Engineering spring elements according to the equal stiffness principle when calculating the horizontal resistance of piles. e method assumes that the soil is linear elastic continuous medium, and the equivalent spring stiffness is calculated by "m-method." e calculation method is as shown as where σ zx is the lateral resistance that the soil acts to pile; m is foundation coefficient which can be got from the code [28]; z is the depth of soil layer; x z is the pile lateral displacement at the depth of z.
From the definition of stiffness of soil spring element, the following formula can be derived: where K S is the stiffness of soil spring element and A is the contact area between pile and soil. Substituting formulae (11) into (12), the following formula can be derived as where a is the thickness of the soil layer and b P is the width of the soil layer in the direction perpendicular to the plane where the calculation model is located.

Parameters and Loads.
All track parameters are obtained from the actual situations and code [29]. e longitudinal resistance-displacement diagrams of constant resistance fastener, small resistance fastener, and ballast bed are shown in Figure 3. e parameters of bridges are got from China Railway Siyuan Survey and Design Group C. Table 1 shows the parameters of the bridge, towers, stay cables, and rail. e height and vertical moment of inertia of 48 m simply supported beam section are 3.2 m and 11.81 m 4 , and those for 32 m simply supported beam are 2.6 m and 10.24 m 4 . e longitudinal stiffness of abutments and piers of approach bridges is 1.2 × 10 5 kN/m and 6 × 10 4 kN/m, respectively. e pile is drilling pile, the diameter of pile laid on the bottle of approach bridges is 1.5 m, and that for main span and towers is 2.5 m. e longitudinal stiffness of damping devices is 200 kN/m under braking condition [30].
When calculating rail expansion force, the temperature differential amplitudes of steel beam and stay cables on the ballast bed are 25°C, and those for concrete beams and main towers are 15°C. e highest and the lowest rail temperatures are 61.9°C and −27.4°C, respectively. e maximum rising and dropping temperature differentials of rail are 41.9°C and 53.4°C, respectively. e vehicle type is CRH3 train, the axle load of the train is 17 t, the length of the wheelbase is 2.5 m, and the dynamic bending stress caused by the train at the rail head (bottom) is 149.40 MPa (128.04 MPa). e ZK standard live load as shown in Figure 4 is applied to the track. e adhesion coefficient between wheel and rail is 0.164. e length of bending and braking loads is 400 m [29,31].

Designing of CWR
CWR on bridges should meet the requirements of rail strength and track stability to ensure the running safety of train.
ough REJ can coordinate the longitudinal deformation of bridge and CWR and release rail longitudinal force automatically, its structural characteristics make it the weak region of the track structure; REJ should be avoided as much as possible. When designing CWR on cable-stayed bridges, it is better to evaluate constant resistance fasteners without REJ according to the requirement. If it is not satisfied, the small resistance fasteners will be preferred. When that two schemes cannot satisfy the requirements, REJ needs to be used. e layout and quantity of REJ need to be determined reasonably according to the engineering practice.

Control and Contrast Index.
For the design and checkcalculation of CWR, the codes formulated by various countries and organizations are different. China has built the longest high-speed railway in the world; the design and calculation technology of CWR are mature. us, codes for designing CWR [29] made by China railway organization are used as structural calculation rules. CWR should definitely meet the requirements of the rail strength and the track stability.

Rail Strength.
To meet the requirement of rail strength, the compressive stress at the rail head and tensile stress at the rail bottom shall not be greater than the permissible stress of 351.50 MPa. According to literature research [29,32], the calculation result of rail longitudinal stress tends to be conservative through routine linear superposition method which has been adopted in code. e code formulated of rail longitudinal stress can be calculated with the following formula: where σ d is dynamic bending stress at the rail head and bottom; σ t is rail basic thermal stress; σ f is rail additional stress, which is larger one of rail expansion stress and rail bending stress; and σ z is rail braking stress; [σ] is permissible stress of rail; σ s is yield strength of rail (457 MPa); and K is the safety factor (1.3).

Track Stability.
Many research programs are conducted on the stability of CWR all over the world [33][34][35]. e stability of CWR on the ballast bed is calculated by a "unified formula" according to the code [29]. e permissible force is 1505.10 kN. In order to keep the stability of ballast bed, the beam-rail relative displacement under braking load is no more than 4 mm when constant resistance fasteners are used and that cannot be beyond 30 mm when small resistance fasteners are applied.

Initial Layout of REJ.
When designing CWR on longspan cable-stayed bridges, the schemes that constant resistance fasteners or small resistance fasteners are laid on the whole bridge should be adopted at first. e rail temperature dropping amplitude is greater than that of temperature rising. e temperature dropping condition is considered as the most unfavorable condition in this investigation. e temperature of steel beam and stay cables reduces by 25°C, that for the concrete beams and the main towers is 15°C, and the rail temperature reduces to 53.4°C. Figure 5 shows rail longitudinal force when the main beam and the concrete supported beams are paved with the constant or small resistance fasteners. Under temperature dropping condition ( Figure 5(a)), the amplitudes of rail longitudinal force are 2623.20 kN and 2202.70 kN corresponding to the constant and small resistance fasteners, respectively. It can be concluded that small resistance fastener is of great benefit for reducing rail expansion force. At the main beam joint, the additional force reaches maximum owing to the maximum expansion amounts of main beam.   It can be concluded from Figure 5 that the maximum rail longitudinal force appears at the main beam joints under temperature dropping, bending, and braking condition. e amplitude of rail longitudinal force under bending condition is much smaller than that under temperature dropping condition, so rail longitudinal force under temperature load is considered in the design of CWR.
Dynamic bending stress at the rail head and bottom, the basic thermal stress, expansion additional stress, and braking stress are selected to check rail strength. Table 2 shows that rail effective stress is greater than the requirement.   e main beams of cable-stayed bridges are longer; the force and deformation of CWR laid on the bridge are larger than common continuous bridges. e rail strength and track stability on the stayed-cable bridge are not able to satisfy the requirement stipulated in the code [29] without REJ according to the numerical simulation. So REJ should be used on the bridge. Two kinds of REJ laying schemes are proposed. Two sets of unidirectional REJ are arranged at the main beam ends as shown in Figure 6.
According to practical engineering experience, switch rails are fixed with constant fasteners while stock rails expand, and small resistance fasteners should at least cover the stock rail expansion zones of rails. For scheme 1, switch rails are located at the main beam; stock rails are set up on the simply supported beams and cross over beam joints. For scheme 2, switch rails are located at the simply supported beams; stock rails are set up on the main beam and cross over beam joints.

Optimization of REJ Layout
e feasibility of the two REJ laying schemes needs to be determined by rail strength and track stability.

Rail Longitudinal Force with REJs.
e temperature of the cable-stayed bridge components and rail drops and rises, respectively. Figure 7(a) shows the rail longitudinal force with variation of temperature (expansion condition). e amplitudes of rail longitudinal force are 1023.79 kN and 803.32 kN at both ends of rail, which is considered as rail basic thermal forces. e maximum rail longitudinal forces on the main bridge are 770.17 kN and 552.66 kN. It can be known from Figure 7 that rail longitudinal force of the section laying small resistance fasteners is smaller than that of section laying constant resistance fasteners. But the effect of small resistance fasteners on reducing the peak value of rail longitudinal force is not obvious. e most unfavorable bending condition can be determined by changing the position of train load. Figure 8 shows rail longitudinal force under rail temperature variation and bending loads (bending condition). e maximum rail tensile force is 1015.02 kN at the mid-span of the main beam when four-line ZK-train loads apply from left tower to 724 m zone and rail temperature reduces to 53.4°C. Four-line ZK-train loads apply from 564 m to 964 m zone and rail temperature increases by 41.9°C; the maximum compressive force is 964.70 kN, locating at the position that is 235 m away from the right main beam joint. e amplitudes of rail longitudinal force under bending condition are larger than those under expansion condition after laying REJs, so rail longitudinal force under bending condition is taken to deduce rail strength and track stability.
Four-line braking loads are applied from 660 m to 1060 m (case 1) and the section is from 1085 m to 1485 m (case 2); CWR on main beam is under the most unfavorable condition with superposition of thermal force, bending, and braking force. Because small resistance fasteners have little influence on rail braking force, only scheme 1 is considered under braking condition. Figure 9(a) shows rail longitudinal force under braking conditions. e maximum rail tensile force under case 1 is 56.24 kN on the left end of loading section. In case 2, the maximum compressive force is 63.34 kN on the left end of loading section. Figure 9(b) shows the beam-rail relative displacement. Under case 1, the peak value of relative displacement is 0.83 mm on the middle of loading section; the beam-rail relative displacement distributes about the loading of mid-span antisymmetric. For case 2, the maximum relative displacement is 2.63 mm on R1 pier. Under both cases, the peak values of beam-rail relative displacement are less than 4 mm.
It can be known that CWR on mid-span of main beam is the weakest location of tensile force under temperature dropping and bending loads. e rail head compressive stress is 282.14 MPa and rail bottom tensile stress is 266.36 MPa, both of which are smaller than the permissible stress of 351.50 MPa. In addition, rail resultant force of basic thermal compressive force and rail bending force is 964.7 kN which is smaller than the permissible force of 1505.10 kN.
Compared to the schemes without REJ, the rail longitudinal force gets reduced significantly. Even, in the cablestayed bridge, the strength and stability of CWR can confirm to the requirements. According to the calculations, it is found that the peak values of rail longitudinal force in two REJ paving schemes are almost the same.

Optimization of REJ Layout.
rough rail longitudinal force under complex loads, the feasibility of two proposed schemes is determined. REJ layouts are optimized by the stock rail longitudinal force at the main beam joint, adjacent pier longitudinal force, and beam-rail relative displacement. Figure 10(a) shows the stock rail longitudinal force of REJ at the beam joint when REJ paving locations are different Advances in Civil Engineering 7 distances from the beam joint. It can be found that the further REJ is from the beam joint, the larger stock rail longitudinal force is. ere is a linear relationship between the distance and stock rail longitudinal force. Stock rail longitudinal force is minimal with REJ located at the beam joint. In order to reduce stock rail longitudinal force, the stock rail should not be far away from the beam joint. e switch rail should be more than 2 m away from the beam joint to avoid the vibration.

Fixed Pier Longitudinal
Force. e rail longitudinal force produced by temperature load will lead to the fixed pier longitudinal force and the pier longitudinal force of L5 and R5 which are considered in Figure 10(b). Under bending condition, the fixed pier longitudinal force of L5 is 3071.49 kN and 5759.33 kN in schemes 1 and 2, respectively, and that for R5 is 2047.74 kN and 3839.61 kN. Compared with scheme 2, the pier longitudinal force of L5 and R5 in scheme 1 reduces by 46.69% and 46.67%. When small resistance fasteners are laid on the simply supported beams, the bridge-track interaction can be alleviated; the fixed pier longitudinal force near the main beam reduces.

Beam-Rail Relative Displacement.
In order to investigate the influence of the switch rail, stock rail, and small resistance fasteners on the beam-rail relative displacement, the braking loads are applied from −242.5 m to 157.5 m. As shown in Figure 11, the maximum beam-rail relative   Advances in Civil Engineering displacement in scheme 1 and scheme 2 are both 3.75 mm, which locates at the first bridge abutment. e beam-rail relative displacement of the section laying small resistance fasteners is larger than that of the section laying constant resistance fasteners. Both peak values are less than the threshold of 30 mm. e expansion of REJ under braking condition is 0.69 mm. For scheme 1, small resistance fasteners on approach bridges can reduce fixed pier longitudinal force near the main beam. e switch rails are laid on the main beam. e constant resistance fasteners are set up. It can increase the integrity of the structure and reduce the deflection of the steel box beam. It is also conducive to keep the ballast bed and track geometry in a good condition. e switch rails are located at the main beam; stock rails and small resistance fasteners are located at the simply supported beam approach bridges; scheme 1 is the better arrangement scheme of REJ.

Rail Gap Value under Braking Condition.
e probability of two or more rails braking at the same time is very little, so only one rail braking is considered when calculating rail gap value of scheme1. Considering the most unfavorable condition, rail temperature reduces by 53.4°C. Rail brakes on the middle of main beam (case 1) or the location of pier L1 (case

Expansion Range of REJ.
e expansion range of REJ is an important factor for choosing REJ. ere are so many loads acting on the bridge. Factors that influence expansion amounts of REJ on long-span bridges are worth determining.
e expansion amounts of REJs under different conditions of temperature, train, and seismic loads are shown in Figure 13. e historical temperature differential is 69.3°C at the region where the bridge locates. Considering the most unfavorable conditions, temperature differential amplitude of main towers, stay cables, and beams is 34.65°C. e rail dropping and rising temperature amplitudes are 53.4°C and 41.9°C, respectively. e expansion amounts (Figure 13 Advances in Civil Engineering 9 expansion amount is 21.74 mm. Due to the damping devices and the constraint of stay cables, the REJ expansion amount is even smaller under braking condition. Considering traveling wave effect, the wave velocity is 2000 m/s; Ninghe Seismic Wave in Tianjin is applied to pile foundation of bridge with large-mass method. e expansion amounts (Figure 13(b)) of REJ on two sides are 73.66 mm and 145.23 mm, respectively. It can be found that REJ expansion amounts induced by rail braking and seismic loads are much smaller than those under temperature differentials.
It can be concluded that temperature, rail braking, and earthquake condition have a big impact on the expansion amount of REJ laid on cable-stayed bridge; the bending and

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Advances in Civil Engineering braking loads have little effect on expansion range of REJ that can be ignored. e probability of simultaneous action of four kinds of loads is very small, but the superposition effect of the expansion and the surplus of expansion should be considered to ensure the safety of high-speed railway. e recommended range of REJ on the newly built railway cable-stayed bridge can be 330.39 + 93.75 + 145.23 � 569.37 mm under the most disadvantageous conditions. Taking the range of the existing REJ production and the safety of CWR into account, it is suggested that the range of REJ is ± 600 mm.
REJ with large expansion range should be used along with sleeper-supporting apparatus to dispose of uneven stiffness of track at the beam joints caused by the large displacement of main beam.

Field Validation
e bridge calculated previously is still under construction. In order to design the distribution of REJ along the bridge better, the monitoring data of REJ on Tongling Yangtze River cablestayed bridge which is similar to the new cable-stayed bridge is applied and analyzed thoroughly. e existing bridge is a steel truss cable-stayed bridge with twin towers and its main span is 645 m. e design speed of the bridge is 250 km/h and REJs are installed at the ends of the main beam. e predicted expansion displacements of REJ based on the monitoring data can be used as verification and reference for the design of REJ on the new cable-stayed bridge.

Monitoring Method.
e sensors for the long-term monitoring of high-speed railway should be equipped with high security, high stability, corrosion resistance, anti-interference, and sufficient accuracy. Fiber Bragg Grating (FBG) technology and Image Intelligent Identification technology are used.

FBG Technology.
FBG method has advantages of interference immunity and high precision, which can meet requirements of monitoring rail longitudinal force and temperature [36][37][38].
ere is 1 point to measure environment temperature, and there are 4 points to measure the rail temperature in the range of REJ. Due to the special position of REJ, it is difficult to measure expansion amounts of REJ with FBG technology. However, Image Intelligent Identification technology is an ideal choice [39][40][41].

Image Intelligent Identification Technology.
e images about the relative position of REJ structure and scale are obtained by cameras. Based on the Image Identification technology, the video camera is used to record the expansion displacement of REJ. Two monitoring points locate at the switch rail; one point locates at the stock rail for one set of REJ as shown in Figure 14. QR code is pasted on stock rail; the cloud corresponding to QR code can store monitoring information which can be obtained automatically by scanning QR code.

Monitoring Data Analysis and Discussion.
e relationship between the expansion displacement of REJ at Tongling Yangtze River cable-stayed bridge and the environment temperature is shown in Figure 15(a). e trend of REJ expansion displacement is similar to environment temperature variation. To further reveal the relationship between the daily expansion displacement (y/mm) of REJ and the daily variation of environment temperature (T/°C), the fitting and analysis of the monitoring data are carried out as shown in the red line in Figure 15(b). e relationship can be expressed as follows:

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where T is the environment temperature (°C) and y is REJ expansion displacement (mm). e goodness of the formula fitting (R-square) is 0.81. In view of the consistent bridge with similar span and surrounding environment of the Tongling bridge, the variation of REJ expansion displacement has a valuable reference for the cable-stayed bridge being designed. When temperature changes, the expansion of the main beam of cable-stayed bridge will lead to deformation of rail, which is linear with the expansion of main beam. It can be known from Δl � α × l × Δt that the expansion deformation of beam is linear with the length of main beam when temperature changes consistently. Since the structure type and the length of main beam of the bridge in this paper are similar to Tongling bridge, the variation of expansion displacement and daily environment temperature of the bridge in this paper can be deduced through the fitting curve by the monitoring data of REJ on Tongling bridge. e length of main beam of Tongling bridge and the bridge in this paper is 1290 m and 1320 m, respectively, so the formula of expansion displacement and daily environment temperature of bridge in this paper can be deduced: as shown in the blue line in Figure 15(b). e relationship between the variation of the expansion displacement and the environment temperature differential is calculated by the numerical simulation model: y � 9.14 × T, as shown in the black line in Figure 15(b).
Predicted by the field monitoring data of the existing bridge, the maximum displacement of REJ on the new bridge is 304.5 mm under extreme temperature. e maximum displacement deduced by the numerical model is 316.7 mm. e expansion displacement of REJ on the new cable-stayed bridge deduced from the fitted curve of the field monitoring data is basically consistent with the numerical simulation results. is means that the optimal scheme and expansion range of REJ proposed by the numerical simulation are feasible. Based on the field monitoring data, the established numerical model can predict the layout optimization of REJ on cable-stayed bridges.

Conclusions
e numerical coupling model of CWR on the long-span cable-stayed bridge is established by taking the influence of pile-soil interaction into account. Rail longitudinal force on cable-stayed bridge under different conditions is discussed. e layout optimization method of REJ on cable-stayed bridge is proposed. e numerical simulation is verified by comparing with the field monitoring data. e conclusions are drawn as follows: (1) e longitudinal thermal stress of CWR on the long-span cable-stayed bridges plays a controlling role in strength and stability of CWR. e application of REJ is essential for CWR on the long-span cable-stayed bridge to reduce the rail longitudinal force effectively. (2) From the perspective of technical and field conditions, the unidirectional REJ should be laid at both ends of the long-span cable-stayed bridge. Switch rail of REJ should be located at the main beam, stock rail should be set on the approach bridge and crossing over beam expansion joint, and several-meter small resistance fasteners should be laid on the side of stock rail to reduce fixed pier longitudinal force near the main beam. (3) e expansion range of REJ laid on cable-stayed bridge is mainly decided by the variation of rail and bridge temperature, rail breaking, and earthquake loads; the bending and braking load have little influence on it so that the loads can be ignored. (4) e monitoring data of REJ on Tongling Yangtze River can prove that the method to optimiazte the layout of REJ is effective. e method can be provided as reference for the layout of REJ on other cable-stayed bridges.

Data Availability
e monitoring data used to support the findings of this study are available from the corresponding author Yongjie Gao, who works at China Railway Siyuan Survey and Design Group Co., Ltd.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Advances in Civil Engineering 13