The analysis for fatigue caused by vibration of railway composite beam considering time-dependent effect

The time-dependent effects in steel-concrete composite beam bridges can intensify track irregularities, subsequently leading to amplified train-bridge coupling vibrations. This phenomenon may increase the stress amplitudes in the bridge steel, thereby impacting the fatigue performance of the composite structures. This paper employs multiple rigid body dynamics to construct a high-speed train model and utilizes the finite element method to develop a steel-concrete composite beam element model that accounts for time-dependent effects, interfacial slip, and shear hysteresis. This approach enables the computational analysis of the train-bridge coupling system, facilitating an investigation into the influence of concrete’s time-dependent effects on the fatigue performance of railway steel-concrete composite bridges. Focusing on a 40-m simply supported composite bridge, the train-bridge coupling dynamic responses were computed for each operational year within a decade of the completion of construction. Applying the P-M linear fatigue damage accumulation theory, statistical analysis of stress history data across various operational periods was conducted to quantify the fatigue damage induced by a single eight-car high-speed train on the lower flange of the mid-span steel beam and the beam-end studs. The findings reveal that the beam-end studs sustain greater damage than the mid-span steel beam. Moreover, the detrimental impact of time-dependent effects diminishes with the increase of operational years. Notably, compared to the initial year, the fatigue damage to the lower flange of the mid-span steel beam by an eight-car train in the tenth year has surged by 39.3%. Conversely, the damage to the beam-end studs has decreased by 47.5%.


Introduction
Steel-concrete composite beams connect concrete slabs and steel beams through shear-resistant connectors, which can make full use of the tensile properties of steel and the compressive properties of concrete, with good stress performance and cost efficiency.Therefore, they are widely used in building and road construction projects (Jiang et al. 1999;Shen et al. 2022;Lou et al. 2022;Fan et al. 2022), and are one of the ideal structural forms for high-speed railway bridges with the main span in the range of 40~100m.In order to meet the requirements of regularity and stability of highspeed railway lines and reduce the impact on the environment along the lines, a large number of railway bridges need to be constructed.
Due to the shear deformation of stud joints in composite beams under external forces, interfacial slip will arise between the concrete slab and steel beams, thus reducing the structural stiffness of the composite beam and increasing the structural deformation (Chisari and Amadio 2014;Colajanni et al. 2017).Under the action of vertical load, the junction of steel web and concrete slab will exhibit shear deformation of concrete deck slab in the process of transferring shear flow through the studs, resulting in non-uniform distribution of stress along the cross-section of concrete deck slab, and the decreases of interface connection in both directions, i.e., shear hysteresis, which does not give full play to the load carrying capacity of the concrete slab (Kristek and Studnicka 1988;Chang and Ding 1988).Therefore, the slip effect and shear hysteresis effect should be fully considered in the design of composite box beam bridges to avoid the occurrence of insufficient bearing capacity.For composite box beam bridges of high-speed railways, the effects of slip and shear hysteresis on the traveling performance should also be considered (Zhu et al. 2020(Zhu et al. , 2023)).During the operation phase, the shrinkage and creep of concrete will lead to structural deformation and additional internal forces in the composite beam, and this effect will be more manifest with the increase of the operation time, which is also a key concern in the research on composite beams (Cao et al. 2018;Huang et al. 2019;Zhao et al. 2023).At present, many studies are carried out by using the finite element model simulation (Gil and Kodur 2023;Naser et al. 2024;Zhang et al. 2017;Banerji and Kodur 2023;Kumar and Kodur 2023).For example, Kwak et al. ignored the interfacial slip effect, and established a one-dimensional theoretical model and a finite beam element model of composite beams considering shrinkage and creep effects based on Euler beam theory (Kwak and Seo 2000;Kwak et al. 2023Kwak et al. , 2000)).Mari et al. proposed a finite beam element model similar to Kwak, and analyzed the long-term changes of stress, strain, deflection and cracks of composite beams (Mari et al. 2003).Souici et a proposed a method to study the effect of creep in composite beams by neglecting the interfacial slip between concrete slabs and steel beamsl (Souici et al. 2015).Zhang et al. used user-defined subroutines to simulate temperature effects and concrete shrinkage and creep effects to study the long-term performance of composite beams (Zhang et al. 2023).Based on the generalized beam theory, Henriques et al. proposed a finite element method to analyze the time-dependent effects in steel-concrete composite beams by considering the effects of shear lag and concrete cracking (Henriques et al. 2020).Huang et al. investigated the time-dependent shear hysteresis induced by concrete shrinkage and creep by studying the composite beams with concrete age differences (Huang et al. 2022).Based on the prediction results of the long-term performance model, Kumar et al. fitted the calculation formula for predicting the long-term deflection of the composite beams under normal service load (Kumar et al. 2021).It can be seen from the research of relevant scholars that the long-term deformation of composite beams cannot be ignored only under the influence of concrete shrinkage and creep effect.However, under various loads, interfacial slip, shear hysteresis effect, and shrinkage and creep of composite box beams can occur simultaneously, which affects the dynamic response of the bridge structure as well as the train running on it (Gao et al. 2022).
Due to the large loads of railway trains, the fatigue problem of railway bridges during operation is more prominent than that of other structures.Compared to highway bridges, high-speed railway bridges have determinable active loads and predictable dynamic responses, and their stresses are less randomized.Many scholars have studied the train-bridge coupling system considering complex effects.Based on Euler beam, Wang and Zhu established a dynamic model of train-track composite beam coupling system considering interfacial slip and constrained torsion effect, and studied the influence of interfacial slip effect on the displacement amplitude of track composite beam (Wang and Zhu 2018).Podworna established a dynamic model of train-track-composite bridge coupling system, and analyzed the influence of random track irregularity on the dynamic response of composite bridge under train load (Podworna 2017).Matsuoka et al. established a dynamic model of train-track-composite bridge coupling system and evaluated the influence of local deck slab vibration on peak acceleration (Matsuoka et al. 2019).
Railway bridges can often have a more defined fatigue life expectancy.Compared to reinforced concrete bridges or steel beams, the shear connectors of composite beams are additional components that need to be focused on for fatigue damage.Oehlers and Foley (1985) investigated the fatigue of composite beams and found that there were damage modes such as buckling of the studs, transverse cracking of the concrete at the top of the studs, and crushing of the concrete around the studs.The fatigue performance of the structure was investigated by Gattesco et al. (1997), and the results showed that the local crushing of the concrete adjacent to the studs and the shear damage of the studs were the main damage forms, and the studs started to fracture in a symmetrical manner in the front and back.Albrecht et al. (1995) conducted fatigue test studies on the composite structure, and concluded that the fatigue strength limit of the composite structure was controlled by the quality of the weld of the steel plate in the tensile flange.Nie and Wang (2012) summarized that the main fatigue damage modes of mixed beam bridges are the tensile damage of steel beam tensile flange and the shear damage of studs.
At present, most of the studies on the fatigue of composite box beams focus on the static performance, and there is a lack of studies on the vibration fatigue of composite beams in the operation stage.In order to make up for the vacancy in this field, this paper studies the influence of time-dependent effect on the fatigue performance of railway composite box beams.In this paper, by simulating a railway train through the method of multiple rigid body dynamics, by simulating a bridge through the finite element model of composite box beam considering slip and shear hysteresis, and by considering the time-dependent effect based on the step-by-step calculation method of the Kelvin rheological model and superimposing it on the track irregularity, the prediction of stresses in key parts of a bridge within 10 years of its operation is realized.Based on the P-M linear fatigue damage accumulation theory, the data for the fatigue damage of a high-speed train on the bridge structure are obtained, and the influence of time-dependent effects on the fatigue performance of the railway bridge is investigated.

Train-bridge coupling model
The train-bridge coupled vibration system consists of vehicle subsystem, bridge subsystem and the interaction between the two subsystems.In this paper, the vehicle subsystem is simulated by multiple rigid body dynamics method, the bridge subsystem is simulated by numerical analysis method, and the time-dependent effect is numerically analyzed by the step-by-step method of the Kelvin rheological model, and the deflection of the bridge caused by time-dependent effect will be superimposed on the track unevenness.MATLAB software is used to write some codes to build the rigid model of the train and the FE model of the steel-concrete composite beams.The bridge subsystem and vehicle subsystem are coupled together using wheel-rail contact theory.

Train model
The main structure of each train is divided into multiple rigid bodies, including one car body, two bogies and four wheel pairs to form a 27-degree-of-freedom train computational model (Zhu et al. 2020), and the interconnection between each rigid body is shown in Fig. 1.Each vehicle body / bogie is considered to have 5 degrees of freedom for wobble, floating, side roll, rocking and nodding, and each wheel pair is considered to have 3 degrees of freedom for wobble, floating and side roll.Then the equations of motion for car number i are shown in Eqs. ( 1) and (2) below: (1) (2) q vi = q vci q vt j i q vw k i T

Fig. 1 Train model with 27 DOFs
where M vi , C vi and K vi denote the mass, damping and stiffness matrices, respectively; F vi is the load matrix; qvi and qvi denote the first-order and second-order derivatives of the time variable q vi with respect to time t, respectively; q vi denotes the displacement sub- matrix of car body number i, and q vci , q vt j i , q vw k i denote the displacement sub-matrices of the car body number i, the bogie number j, and the wheelset number k, respectively.
Disregarding the connecting effect between each car, a single car can be extended to a train with Nv-car formations, and the train power balance equation corresponding to it can be obtained by Eq. ( 3).
where M vv , C vv and K vv are the mass matrix, damping matrix, and stiffness matrix of the train subsystem, respectively, and the corresponding chunking forms are as follows: F vv is the load column vector of the train subsystem in the overall coordinate system, corresponding to the following chunking form: The displacement column vector for the qvv train subsystem corresponds to the following chunking form: The method of the multiple rigid body dynamics is given in Reference (Zhu et al. 2020(Zhu et al. , 2023;;Gao et al. 2022).

FEM for composite bridge
A 2-node 18-degree-of-freedom combined bridge finite element model is established based on the theory of elastic Euler beams (Zhu et al. 2020).The model is aimed at the elastic stress stage, ignoring the shear deformation caused by bending effect, and considering the vertical and the transverse bending curvature of the steel beam and the concrete slab as the same, with no vertical or transverse relative displacements between them.The stiffness of the connection between the steel beam and the concrete slab in the longitudinal direction is considered to be constant, and only the shear hysteresis effect of the structure under vertical bending is considered.
In the coordinate system shown in Fig. 2, O c represents the centroid of the concrete slab, O s represents the centroid of the steel beam, and C s represents the torsion center of the converted section of the composite beam.Then the transverse displacement u (x, y, z) and vertical displacement v (x, y, z) at any point on the composite box beam, the longitudinal displacement w c (x, y, z) of the concrete slab and the longitudinal displacement w s (x, y, z) of the steel beam are: (3) where y h represents the y-direction coordinate of the torsion center C s and x h represents the x-direction coordinate of the torsion center C s ; w c0 and w s0 are the longitudinal displacements of the concrete slab centroid O s and the steel beam centroid O s , respectively; u 0 is the transverse displacement of the concrete slab centroid or the steel beam centroid, and v 0 is the vertical displacement of the concrete slab centroid or the steel beam centroid; φ is the angle of torsion of the overall structure; f c and f s are the strength functions of shear hysteresis warpage of concrete slab and steel beam, respectively; ψ c (x) and ψ s (x) are the shape functions of shear hysteresis warpage of concrete slab and steel beam, respectively.
And then the one-dimensional theoretical model of the composite box beam is obtained based on the classical principle of virtual work.The virtual work of the beam is: where A s and A c represent the cross-sectional areas of the steel beam and concrete slab, respectively; L is the span length of the composite beam; δε T s σ s dadz represent the internal virtual work generated by the deformation of the concrete slab and the steel beam, respectively; L δd sl q sl dz represents the internal virtual work generated by the interfacial slip between the steel beam and the concrete slab; δW T Q represents the external virtual work generated by the centralized load applied to any position of the composite beam, and L δW T qdz represents the external virtual work generated by the distributed load applied to any position of the composite beam; ε c is the strain matrix of the concrete slab; σ c is the stress matrix of the concrete slab; ε s is the strain matrix of the steel beam; σ s is the stress matrix of the steel beam; d sl is the (7) shear force; q sl is the interface slip displacement; W is the displacement vector at any position of the steel beam or the concrete slab; Q is the concentrated load; q is the distributed load.
The creep behavior of concrete can be simulated using a linear creep model.When the effect of temperature on concrete is not considered, the total concrete strain is the superposition of instantaneous strain, shrinkage strain and creep strain.Among them, the shrinkage strain is independent of the structural forces, and the consideration of the shrinkage effect in the overall structural calculation model is equivalent to applying the initial strain to the structure (Bazant and L'Hermite 1988).Instantaneous and creep strains evolve with time and are stress dependent.
The instantaneous and creep strains are calculated by introducing the creep function J(t,t0) of Eq. ( 9) below: where E c (t) is the modulus of elasticity of concrete at moment t, and C(t,t 0 ) is the creep function of concrete at moment t when the initial loading age is t 0 .Then the instantaneous strain ε c,e (t) and the creep strain ε c,cr (t) can be calculated by Eqs. ( 10) and ( 11).The Dirichlet series is introduced to fit the creep degree function C(t,t 0 ), as shown in Eq. ( 12).Long-term creep is considered using a stepwise calculation method that does not store stress and strain histories (Zhao et al. 2023).The relationship X c is shown in Eq. ( 13); μ c is Poisson's ratio of concrete; α j (t 0 ) (i = 1, 2, …, m) is solved by the least squares method provided that C(t, t 0 ) is known; The delay time τ j = 10 j−1 (j = 1, 2, …, m).
By dividing the steel-concrete composite box beam into n b elements and n b + 1 nodes, the dynamic equation of the composite box beam subsystem is: where M bb is the overall mass matrix of the composite box beam; C bb is the overall damping matrix; K bb is the overall stiffness matrix; the specific expressions are as follows: q b is the column vector of displacements of the steel-concrete composite beam (9) subsystem; F b is the column vector of nodal loads of the steel-concrete composite box beam subsystem in the overall coordinate system.
The FE model and the proof of its validity is described in detail in Reference (Zhu et al. 2020(Zhu et al. , 2023;;Zhao et al. 2023;Gao et al. 2022).

Track irregularity and interaction
In this paper, the commonly used German high-speed line low interference spectrum is used as the track irregularity spectrum, and its longitudinal irregularity and crosslevel irregularity are shown in Fig. 3. Assuming that the train travels at a uniform speed on the composite box beam bridge, there is no wheel-rail relative displacement in the floating and rolling directions, i.e., it is assumed that the wheel and the rail closely contact each other in the vertical direction, and the creep force in the transverse direction is taken into account according to the Kalker linear creep theory (Kalker 1991).
The power balance equation of the train-bridge coupling system is as follows: If the elements on the right-hand side of Eq. ( 15) are moved so that all unknown elements are on the left-hand side, the remaining terms are known track irregularities.Thus, the equation of motion for the train-bridge coupling system are obtained as follows: The train-bridge coupling model is described in detail in Reference (Zhu et al. 2020(Zhu et al. , 2023;;Gao et al. 2022). (15) Fig. 3 Track irregularity in the coupling model 3 Stress history of the key parts

Parameters of the train-bridge system
A large number of studies have shown that in the fatigue damage of steel box beams, the damage mainly occurs at the lower flange of the mid-span section.For steel-concrete composite beams, the slip near the beam end is larger, and the deterioration of the studs due to fatigue is more serious, so this area is also a dangerous part for fatigue damage.In order to investigate the influence of time-dependent effect on the fatigue performance of high-speed railway composite bridges, this chapter adopts the commonly used CRH2 train and CRH3 train to establish the vehicle model (Gao et al. 2022), with the average speed taken as 300km/h, to simulate a single train with 8-car formation passing through the simply-supported composite beam bridge.In order to simulate the actual bridgecrossing situation, a bridge with simply supported composite box beams with a span of 3 × 40m is set up, with the main focus on the dynamic response of the middle span.This is mainly because the bridge is in the vibration state before the train passes the middle span.The main parameters of the bridge are shown in Tables 1 and 2 (Su 2022).For the meanings of the parameters in Table 1, refer to Fig. 2; in Table 2, f ck is the concrete cubic compressive strength at day 28; RH is the ambient relative humidity; t sh is the concrete curing age; ρ c is the density of reinforced concrete; ρ s is the density of steel; p y is the dead load of the bridge deck in the second stage; υ c and υ s are Poisson's ratios of concrete and steel beams, respectively; Es is the elastic modulus of steel.
The deflection values generated by the time-dependent model at each operational period were calculated, as shown in Fig. 4. If the deformation between the track and the bridge is ignored, it can be assumed that the longitudinal track irregularity at each  Fig. 4 The displacement of simply supported beam time period is generated by the original irregularity superimposed on the deflection produced by the time-dependent effect.However, this will produce a drastic change in the track irregularity at the beam end, whereas in reality the track at the beam end does not produce the above situation due to the stiffness of the track itself and the connection between the track and the bridge being non-rigid.Therefore, the time-dependent deflections at the beam ends are fitted to transform them into smoother curves as shown in Fig. 5.The fitting result is superimposed on the track irregularity at the corresponding position to generate a new track irregularity, and the newly generated track irregularity is utilized as the self-excitation of the coupling system of vehicle and bridge to investigate the fatigue performance of steel-concrete composite beam bridges under the action of time-dependent effect.The influence of time-dependent effect of train-bridge coupling system is described in detail in Reference (Gao et al. 2022).

Lower flange Stress at mid-span of steel beam
The longitudinal displacement w s of the steel beam at any point can be obtained from Eq. ( 7), and the positive strain ε s of the steel beam can be obtained by deriving the longitudinal displacement of the steel beam in the composite beam, as shown in Eq. ( 17): The displacement of any point of the model can be obtained through the node displacement and the corresponding shape function matrix, through the train-bridge coupled dynamic analysis program, it is possible to obtain the nodal displacements of (17 Fig. 5 Comparison between the calculated and the fitted displacement the bridge elements when the train crosses the bridge, and then get the time history of the longitudinal displacements at the lower flange of the steel girder in the middle of the span.In this paper, a 2-node 18-degree-of-freedom finite beam element model considering slip effect and shear hysteresis effect is used, so the longitudinal displacement of the lower flange of the steel beam can be expressed as: where q se denotes the local nodal displacement vector of the steel beam element; N se denotes the shape function matrix corresponding to the longitudinal displacement field function w s in this beam element.The local nodal displacement of the steel beam element can be expressed as: where the subscripts i and j refer to the node numbers at both ends of the beam element, respectively; φ is the rotation angle around the y-axis; θ is the rotation angle around the z-axis.Since the structure is in the elastic phase, the stress-strain relationship of the steel beam can be expressed as: Substituting Eqs. ( 17)~( 19) into Eq.( 20), the positive stresses at the corresponding positions of the steel beams can be obtained.In the calculation of CRH2 trains and CRH3 trains in the operating time of 0 days and 10 years, through the combination of the dynamic response caused by the beam bridge, the results of the timedependent calculation of the positive stresses in the lower flange of the steel girder at mid-span are shown in Figs. 6 and 7 below.As can be seen from the figures, for a CRH2 train crossing the bridge, the time-dependent effect on the maximum stress in the lower flange of the steel beam is small, but the rest of the stresses which originally had smaller amplitude are enhanced.For a CRH3 train crossing the bridge, the timedependent effect has an even smaller impact. (18)

Stress of studs at beam end
In this paper, the studs for the bridge model used are of φ22mm-300mm type, and six are arranged in each row along the bridge transverse direction.Assuming that the interface shear connection stiffness ρ sh between the steel beam and the concrete slab is a constant value along the longitudinal direction of the bridge, then the relationship between the bond-slip force q sl (x) and the slip Δ(x) at the interface is shown in Eq. ( 21): where ρ sh = R/s, R is the shear stiffness of the studs and s is the longitudinal spacing of the studs.Oehlers and Coughlan (1986) carried out a large number of roll-out tests and proposed an approximate formula for the shear stiffness of the studs on the basis of the test results: where: d s denotes the stud cross-sectional diameter; N u denotes the stud shear bearing capacity; f c denotes the concrete compressive strength, and for ordinary concrete, α is taken as 0.08.
The slip Δ(x) can be obtained from the node displacement by Eq. ( 23): where q he denotes the unit local node displacement vector associated with the slip displacement field function; N he denotes the shape function matrix corresponding to the slip displacement field function Δ(x) in this beam element.The local nodal displacement of the steel beam element can be expressed as: where: i and j refer to the node numbers at each end of the beam element respectively.The shear force of a single row of studs can be expressed as: (21) where: q(x) is the bond-slip force.
Based on the previous assumptions, it is considered that the longitudinal shear force at the interface between the steel beam and the concrete slab in a steel-concrete composite bridge is entirely borne by the studs, and the shear stress τ x of a single stud at the beam end of a steel-concrete composite bridge beam can be found out by adopting Eq. ( 26).
where: n denotes the number of studs in the same cross-section of the composite bridge; d s denotes the diameter of the studs' cross-section.
The concrete used for this paper is of C50 grade.Referring to the research results of Wang et al. (2020), the shear stiffness R is taken to be 411kN/mm, which can get the shear stiffness ρ sh =8kN/mm 2 .
Through the train-bridge coupling dynamic analysis program, the node displacement of the bridge element when the train passes the bridge can be obtained, and then the slip time history of the bridge end of the composite beam can be obtained, and finally the time history of the stress produced by the beam-end studs under the train load can be obtained through Eq. ( 26).The time histories of the shear stress of the studs are shown in Figs. 8 and 9 when each train passes the bridge.As can be seen from the figures, for a CRH2 train passing the bridge, the time-dependent effect makes the maximum stress of the beam-end studs decrease significantly, and the stress amplitude is more concentrated.For a CRH3 train crossing the bridge, the conclusion is similar to that of the CRH2 train.
4 The evaluation for fatigue damage

Statistics of stress history
In order to obtain the fatigue damage of the bridge, it is necessary to convert the stress time history when a train crosses the bridge into a stress spectrum, and the rainfall counting method, which is widely used in engineering, is used to count the stress history.The contribution of the stress amplitude less than 1 Mpa to fatigue damage is small (Li et al. 2019).When the stress amplitude equal to 1MPa, according to Eq. ( 28) N is equal to 2.8184 × 10 13 and very huge.And the frequencies of occurrence of stress amplitude n less than 1 MPa is very small.Therefore, n/N is tiny and can be neglected in Eq. ( 30), and the stress amplitude less than 1 Mpa is not counted in the statistical process.The fatigue stress amplitude spectra of the lower flange of the steel beam at the mid-span and the beam-end studs are shown in the form of histograms, as shown in Figs. 10,11,12 and 13, when a single train of each formation passes the simply supported composite box beam bridge with the operational time of 0 days and 10 years.From Figs. 10 and 11, it can be seen that: for a CRH2 train crossing the bridge, the time-dependent effect makes the stress amplitude of the lower flange of the steel beam at the mid-span increase in the number of times in the range of 5 MPa-6 MPa, while the number of times in the range of 2 MPa-5 MPa decreases, and the number of times in the range of stress amplitude above 1 MPa decreases; The stress amplitude of the beam-end studs is significantly larger compared to the lower flange of the steel girder at the mid-span, and the number of stress amplitude occurrences worth counting is more than double that of the steel girder's lower flange; the time-dependent effect results in an increase in the number of stress amplitude occurrences for the studs above 1 MPa and a greater concentration below a stress amplitude of 5 MPa.From Figs. 12 and 13, it can be seen that: for a CRH3 train crossing the bridge, the time-dependent effect makes the number of stress amplitude occurrences above 1 MPa in the lower flange of the steel beam at the mid-span decrease significantly, with the stress amplitude of 1 MPa-2 MPa being the most obvious; the time-dependent effect increases the number of times worth counting for the stress amplitude of the beam-end studs, and the overall level of the stress amplitude and the maximum stress amplitude both increase significantly.

Calculation model for fatigue life
After obtaining the stress spectrum of the critical area, the fatigue damage should be calculated according to the corresponding S-N curve, which can be expressed by the following equation: where: Δσ is the actual stress amplitude; N is the total number of cycles when the structure fails by fatigue; m and C are the coefficients related to material and construction.
According to China's Code for Design on Steel Structure of Railway Bridge (TB10091-2017) (National Railway Administration of PRC 2017), the category for allowable amplitude of fatigue stress of the lower flange at mid-span of steel-concrete composite box beam treated in this paper is Class V, and its S-N fatigue curve equation is shown in Eq. ( 28): The S-N fatigue curve of the studs is chosen from the stud fatigue life calculation formula of Nie (2005) with a 95% guarantee on a test basis, as shown in Eq. ( 29):

Influence of time-dependent effect
Among the theories for calculating the structural fatigue damage accumulation, the linear accumulation theory has been recognized by most scholars and widely used.Palmgren-Miner theory is the most classical linear accumulation theory, which counts each damage caused by stress amplitude to the structural nodes as 1/Ni, and the total damage is obtained by the linear accumulation of each damage, as shown in Eq. ( 30): where D is the total fatigue damage sustained by the critical nodes in the operation stage; n i is the number of cycles of stress amplitude Δσ i ; N i is the maximum number of cycles corresponding to stress amplitude Δσ i ; and k is the total number of types of stress amplitude.
Thus, the damage caused to the lower flange of the steel beam at mid-span and the beam-end studs by a 8-car CRH2 train and CRH3 train crossing the bridge under different operation stages can be obtained, and the calculation results are shown in Table 3.
From the above results, it can be seen that the damage of beam-end studs is greater than that of the steel beams at mid-span, and the damage of beam-end studs is the controlling factor for the fatigue damage of the composite beams at this time.The influence of time-dependent effect is obvious in the early stage of operation with the increase of service time, but it tends to stabilize in the later stage, and the influence of time-dependent effect in the 10th year is already very small.On the one hand, the timedependent effect (concrete shrinkage and creep) may release and decrease the stud shear stress, which is beneficial for the mechanical behaviour of the studs.On the other hand, the time-dependent effect will enlarge the dynamic responses and furtherly enlarge the interface damage, which is not beneficial for the mechanical behaviour of the studs.Therefore, it is hard to say that the time-dependent effect increase or decrease the damage of the beam-end studs.

Conclusion
In this paper, a precise 2-node 18-degree-of-freedom beam element is used to simulate the steel-concrete composite beam.Considering the longitudinal interface slip, shear hysteresis and time-dependent effect, a train-bridge coupling system is established.Through the coupling system, the dynamic response of the composite beam is analyzed, and the quantitative fatigue stress and the degree of fatigue damage caused by the train passing through the bridge in different years are obtained.The influence of time-dependent effect on the fatigue damage of the composite bridge during operation is explored.However, the effects of geometric, residual stresses, complex mechanical effects and concrete shrinkage and creep effects on the dynamic and fatigue properties of railway composite bridges still need to be further studied.The main conclusions of this paper are as follows: (1) The time-dependent effect has less influence on the maximum stress of the lower flange of the steel beam, but enhances the originally smaller stress; it makes the number of occurrences of the stress amplitude worth counting in the lower flange of the steel beam decrease, but the amplitude increases.
(2) The time-dependent effect makes the maximum stress of the beam-end studs decrease significantly, but makes the stress amplitude more concentrated; it makes the number of times worth counting for the stress amplitude of the beam-end studs increase, with the majority of values being less than 5 MPa.(3) The stress level and fatigue stress amplitude of the beam-end studs are obviously larger compared with the lower flange of the steel beam, with the number of stress amplitude occurrences worth counting being more than twice that of the lower flange of the steel beam, and the maximum stress amplitude being more than 8 times that of the lower flange of the steel beam.(4) The time-dependent effect of concrete shrinkage and creep will not only reduce the shear stress of stud, but also enlarge the dynamic response and the interface damage.So it's difficult to determine the influence of time-dependent effect on the damage of the beam-end studs.

Fig. 2
Fig. 2 Geometric parameters of the steel-concrete composite box beam

Fig. 6
Fig. 6 Lower flange stress of steel beam when a CRH2 train passes the bridge

Fig. 7
Fig. 7 Lower flange stress of steel beam when a CRH3 train passes the bridge

Fig. 8 Fig. 9
Fig. 8 Studs stress history when a CRH2 train passes the bridge

Fig. 10
Fig. 10 Lower flange stress at mid-span for CRH2 train

Table 1
Section geometric dimensions of the composite box beam bridge

Table 2
Load cases of the composite box beam bridge

Table 3
Calculation results of fatigue damage when train passes by