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Rail. Eng. Science (2021) 29(1):59–73 https://doi.org/10.1007/s40534-020-00226-7 Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 1,2 1,2 1,2 1,2 1,2 • • • • Yuan Gao Ping Wang Kai Wang Jingmang Xu Zhiguo Dong Received: 6 August 2020 / Revised: 1 November 2020 / Accepted: 4 November 2020 / Published online: 14 December 2020 The Author(s) 2020 Abstract Broken gap is an extremely dangerous state in 1 Introduction the service of high-speed rails, and the violent wheel–rail impact forces will be intensified when a vehicle passes the As an infrastructure for vehicles running on high-speed gap at high speeds, which may cause a secondary fracture rails, track is a critical component that directly bears the to rail and threaten the running safety of the vehicle. To multi-field coupling effect of vehicle and temperature load. recognize the damage tolerance of rail fracture length, the During the long-term service of rails, under the effect of implicit–explicit sequential approach is adopted to simulate cyclic dynamic wheel load, construction deficiency and the wheel–rail high-frequency impact, which considers the temperature effect, the rails are subject to various damages factors such as the coupling effect between frictional (see Fig. 1). In the section of seamless rails, the changes in contact and structural vibration, nonlinear material and real temperature can cause enormous temperature force inside geometric profile. The results demonstrate that the plastic rails, thereby causing them to break eventually. To name a deformation and stress are distributed in crescent shape few, the rail of the upper line at K489 ? 140 m on Hen- during the impact at the back rail end, increasing with the gyang–Liuzhou railway was broken, the rail of the upper rail fracture length. The axle box acceleration in the fre- line at K2322 ? 490 m on Guangzhou–Shenzhen–Hong quency domain displays two characteristic modes with Kong high-speed railway was broken vertically, and the frequencies around 1,637 and 404 Hz. The limit of the rail base metal of the rail of the upper line at K359 ? 790 m on fracture length is 60 mm for high-speed railway at a speed Qingdao-Jinan railway line was broken (see Fig. 1). It has of 250 km/h. been admitted that the rail fracture is an unpredictable and extremely dangerous damage that seriously threatens the Keywords Rail broken gap Explicit FE method Damage running safety of vehicles (Fig. 2). High-frequency impact Stress mechanism The rail fracture has a great significant influence on wheel–rail interaction and running safety of vehicles. In terms of geometric properties, after rails are broken the geometry and continuity of tracks are destroyed, which is mainly represented by three factors: broken gap, step and bending angle. Moreover in terms of mechanical proper- ties, the geometrical discontinuity of rails caused by the broken gap results in the loss of the vertical bending stiffness of rails. For this reason, during the wheel passing & Jingmang Xu over the broken gap, a violent dynamic force of wheel–rail mang080887@163.com impact will be excited, aggravating the wheel–rail inter- action and vibration of rail. In addition, most of the rail MOE Key Laboratory of High-Speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China fracture can be detected by a safety system; however, due to the complex layout of track circuit wire in turnout, there School of Civil Engineering, Southwest Jiaotong University, still remains some rail broken gap in turnout that cannot be Chengdu 610031, China 123 60 Y. Gao et al. detected by the safety system (see Fig. 3). The track circuit can work normally when a rail is broken. In addition, there exists transverse track circuit wire at the heel of the frog; therefore the broken rail cannot be detected by the track circuit wire. The real-time monitoring of the rail broken gap through track circuit wire in turnout has not been realized. As the occurrence frequency of rail fractures in high-speed railways is uncertain, if it cannot be detected in time, it will aggravate the damage to the rails or even threaten the traffic safety. China Academy of Railway Sciences (CARS) carried out a safety test on a vehicle passing the rail with broken gap, in which the speeds of the vehicle were 20 km/h and 85 km/h, respectively. CHN50 rails and ballast track bed as well as wooden sleepers were used in the test line. The test measured the lateral displacement of the front rail, and the step value (height difference between the front rail and back rail under wheel load), and evaluated the running safety of the vehicle based on the degree of crescent- shaped damage at the free broken gap end [1]. However, the test method held certain limitations: the velocity of the train was relatively low compared to the high-speed rail- Fig. 1 Formation reasons of broken gap way, and the wheel–rail contact stress could not be directly and accurately measured by instrument. For this reason, aiming at the high-frequency wheel–rail impact at broken gap and its influence on running safety, many scholars have used the numerical simulation method for safety analysis. Suare et al. [2, 3] proposed a numerical simulation method by considering the train–track interaction to investigate the dynamic derailment behavior at the broken rail and then the running safety during the passage of an underground vehicle over the highly resilient straight and curved slab tracks. Schafer et al. [4] analyzed the factors of derailments caused by a mainline broken rail and proposed a feasible tool which is capable of identifying locations with a high probability of rail fracture by considering the rail mecha- nism characteristics, infrastructure information, mainte- nance and operational conditions. Allan [5] focused on the Fig. 2 Steel rail fracture with secondary fracture risk management techniques in controlling derailment of vehicles on broken rails and put forward the guidelines to Fig. 3 Schematic diagram of track circuit wire in turnout Rail. Eng. Science (2021) 29(1):59–73 Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 61 reduce the risk of derailments of vehicles on broken rails. there is no experiment data of ballastless rail gap in China. Matsuda [6] conducted a survey about the relationship In the Temporary Standard for New Railway Passenger between the occurrence of flaking leading to the rail Dedicated Line Design at the Speed of 250–350 km, the breakage and rail wear by on-site investigations and found gap value is set as 70 mm, which is determined by engi- that the balance between the wear and fatigue is the key neering experience [7]. Recently, the explicit finite element factor for flaking damage. Cai et al. [7] proposed a vehicle- method has become an efficient and feasible method, which track-bridge coupling model with commercial software is capable to simulate the dynamic response of wheel–rail ABAQUS and investigated the influence of factors high-frequency impact and determine the length of broken including the rail gap position, gap width and velocities on gap. the dynamic response of wheel–rail interaction. Ekberg In this paper, to recognize the secondary fracture of rail et al. [8] compared experiences of broken rails in different and determine the tolerance of rail fracture length, a three- countries, which is beneficial for networks to assess if a dimensional explicit finite element model of rail fracture is good control on stress free temperatures can be guaranteed. established to simulate the process where wheels impact Magel et al. [9] investigated the reasons and conditions of the back rail at the broken gap, and the wheel–rail dynamic derailment at broken rails and proposed the methods to high-frequency impact and vehicle running safety as well minimize the occurrence probability of broken rails, which as the stress mechanism of the rails at the broken gap are can provide a better understanding and mitigating strate- investigated; then the possibility of secondary fracture to gies to reduce the risk of failures. Tai et al. [10] investi- the back rail under the effect of high-frequency impact can gated the influence of rail service age, annual traffic density be evaluated, which can provide a certain reference for the and inspection frequency on the derailment risk of a service and maintenance of steel rails. vehicle at the broken rail and established the preliminary risk analysis model for the prediction of broken rail risks, which can potentially aid to mitigate derailment risk. 2 Finite element model of broken gap Sheperd et al. [11] investigated the effect of rolling contact fatigue on multiple fractures of rail, found that the rail Figure 4 illustrates the explicit finite element model for the fracture occurred in the vicinity of obstacles with the BNSF simulation of wheel–rail interaction at broken gap for high- railway broken rail data, and revealed the occurrence fre- speed railways. The model is of 25 m length and mainly quency of rail breaks. An et al. [12] proposed a grid-based consists of the wheel, rail, fasteners and other components analysis model for the effects of casual factors on rail break closely related to wheel–rail dynamic response. The profile risk based on the Cox’s proportional hazards regression of the rail model is selected according to the standard analysis method and compared the simulation results with CHN60 used on the high-speed railway, while the geo- the data of broken rails and casual factors, which can metric profile of the wheelset is LMA. In the explicit finite provide accurate understanding of the mechanism of rail element model, the broken gap is located in the middle of break. Most research mainly concerns the effect of broken the rail to simulate the most unfavorable condition, which gap on the running safety of vehicles [13–19], whereas a can be modeled by deleting the solid elements of the rail. few consider the wheel–rail contact behavior or even the As shown in Fig. 4, the wheelsets and rails are simulated secondary fracture caused by the wheel–rail impact if the by 8-node hexahedral elements SOLID165 with the one- rail fracture is not detected timely [20–23]. In addition, point integration method. In order to reduce the calculation (a) Running direction Back rail M (d) (b) Broken gap Torque X Front rail Surface-Surface contact (penalty function) (e) (c) Rail with broken gap Rail with no damage Fig. 4 Wheel–rail interaction model with a close-up of mesh Rail. Eng. Science (2021) 29(1):59–73 62 Y. Gao et al. amount and guarantee the accuracy of the contact solution, master wheel node N , the slave wheel node N which is r w the ALE (arbitrary Lagrange–Euler) self-adaptive mesh is closed to the master wheel node N can be searched first. If adopted in the model to ensure that the quality of element the slave wheel node N and master rail node N do not w r under violent wheel–rail impact force remains high. The coincide, the slave wheel node N can be shown whether it ALE self-adaptive method combines the features of lies in one of the segments S when it crosses the main Lagrange algorithm and Euler algorithm, which is mainly surface via the following equation: used to keep the high quality of elements during the sim- ðÞ s s ulation process; therefore, no huge distortions and defor- ðÞ s s [ 0 ðÞ s sðÞ s s [ 0 i ¼ 1; 2; 3; 4 ; i iþ1 i iþ1 mation will occur. The main principle for the ALE self- ð1Þ adaptive method is to detach the element and material, and then, the element will flow independently. In addition, the where s and s represent the vectors along the edges of i i?1 ALE self-adaptive method does not change the topology of segment and point outwards from N ; s represents the the element. The element size in the potential contact projection of the vector from N to N . Based on the two m w regions around the broken gap is 1 mm, while the maxi- equations and Newton–Raphson iterations, the normal mum element size in the non-contact regions can reach up contact force vector f can be obtained, which is cN to 20 mm. The fasteners can be simulated by linear springs proportional to the penetration as follows: and viscous dampers; thus, only the stiffness and damping l ¼ n ½ t rðÞ g ; h \0; ð2Þ i c of the fasteners which are closely associated with wheel– rail interaction are considered. After many trials, the single f ¼lkn ; ð3Þ cN rubber pad is modeled by a 7 9 7 discretely distributed where rðÞ g ; h is the master segment; t is the vector to the spring–damper pairs, which can effectively reduce the salve wheel node N ; n is a unit outwardly normal to a influence of the concentrated load. The total stiffness and . or or or or boundary element on C, n ¼ ; k is the damping of a fastener are 22 MN/m and 200 kNs/m, i og oh og oh respectively; the stiffness and damping for the each spring– 2 penalty contact stiffness k ¼ f KA =V , in which K is the SI damper pair are 0.44898 MN/m and 4.0816 kNs/m, bulk modulus, V is the element volume, A is the area face respectively. As the vibration wavelength of the sprung of the segment and f is the scale factor for the penalty SI mass is always expressed in meters [16], which is much contact stiffness. The equivalent contact force distributed longer than the size of contact patch (10 mm), the wheel– on the four main nodes of the master segment can be rail contact solutions will not be influenced by the sprung obtained as follows: mass vibration. For this reason, the body, bogie and other f ¼/ðÞ g ; h f ; ð4Þ jm j c cN components over the primary suspension can be simplified as a rectangular rigid body supported on the wheelset axle where /ðÞ g ; h is the value of the two-dimensional j c c through the primary suspensions. The primary suspension shape function at the contact point, and is also simulated by linear springs and viscous dampers, its P /ðÞ g ; h ¼ 1; The Coulomb friction law is applied j c c j¼1 total stiffness and damping being 0.88 MN/m and 4 kNs/m, to solve the tangential contact between wheel and rail: respectively. The stiffness and damping for each spring– t t damper pair are 0.01796 MN/m and 0.0816 kNs/m, f ¼ l f ; ð5Þ bound cN respectively. Moreover, the axle load is 16 t. The material t t tþ1 f kDef kDe\l f of U71Mn rail steel can be simulated by the bilinear cT cT cN tþ1 tþ1 f ¼ f f ; ð6Þ elastoplastic material. The wheel–rail elastic modulus is cT bound t tþ1 > f kDe [ l f : t cT cN 210 Mpa, and the yield strength is 650 MPa; the strain f kDe cT hardening modulus is 14.879 GPa; the material density is 3 m 7,800 kg/m ; and the Poisson’s ratio is 0.3. The friction t T t t f ¼ N (f þ f )dC ; ð7Þ con cN cT between wheelsets and rails is defined by Coulomb’s M¼1 c;e friction law, and neither climate nor geographical condi- where f is the tangential contact force of the node at time tions are considered in the model. The friction coefficient is t t step t ? 1, and f ¼ f kDe; f is the tangential contact set to 0.5 to simulate dry and clear friction conditions. Both cT cT ends of the broken gap and the wheelsets are of free force at time t; De is the increment along the slave node on boundaries, and the corresponding distal rail end is applied rail surface; k is the penalty contact stiffness; f is the bound t T with a symmetric constraint. traction bound at time t; f is the contact force vector; N con The interaction and impact between wheelsets and rails is the transpose of the shape matrix constructed by the can be solved by the surface-to-surface contact algorithm shape functions; l is the friction coefficient. based on the penalty function [24, 25]. For an arbitrary Rail. Eng. Science (2021) 29(1):59–73 Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 63 After the establishment of the explicit finite element unreal dynamic disturbance will be introduced. In order to model, the typical complex calculation process mainly avoid the effect of the initial disturbance on the accuracy of consists of two parts: the implicit static calculation and the simulation results, a 1.2 m long section of dynamic relax- explicit dynamic calculation. First, the wheelset is set to ation region shall be introduced to ensure that the energy of stand still on the rail, and then, the gravity acceleration is the initial disturbance is consumed when the wheel–rail applied to calculate the displacement field of the wheel–rail contact patch is located at the broken gap (Fig. 5). system (static implicit solution) under static wheel load. Before wheel–rail impact at the gap, the vertical wheel– Afterward, the obtained nodal displacement of the model is rail contact force equals to the wheel load, and the wheel taken as the initial state of the wheel–rail system for and rail will be in a steady contact status. Thus, the fric- the explicit calculation, then the initial forward speed tional rolling contact solutions in contact patch can be (250 km/P) are prescribed on the nodes of the wheel and validated by comparing the results from CONTACT and vehicle to simulate the forward motion, and the angular Hertz, as shown in Table 1, where the simulation results of speed is set on the nodes of the wheel to rotate around the the explicit finite element method are consistent with those wheel axle, which equals to the quotient of forward obtained from CONTACT and Hertz. velocity and wheel rolling radius. The frictional rolling contact solutions and dynamic response of wheel–rail impact (transient explicit solution) are derived based on the 3 Simulation results explicit time integration algorithm. Due to the differences of integration methods and motion states in the above two- In this section, the three-dimensional explicit finite element step calculation process, the initial state of the dynamic model is used to simulate the wheel–rail interaction during analysis should be processed by stress initialization the passage of wheel over the gap under actual operation according to the static implicit solution; for this purpose, an conditions; then, the stress mechanism of the rails and the Fig. 5 Sketch of rail subjected to broken gap Table 1 Comparison of the simulated results Approach Contact patch Maximum stress Major semi-axis (mm) Minor semi-axis (mm) Area (mm ) Vertical (MPa) Shear (MPa) Explicit FE 7.31 5.62 129.06 834.5 224.3 CONTACT 7.35 5.51 127.23 848.1 234.1 Difference w.r.t. CONTACT 0.04 - 0.11 - 1.83 13.6 9.8 Hertz 7.40 5.8 134.83 857.1 – Difference w.r.t. Hertz 0.09 0.18 5.77 22.6 – Rail. Eng. Science (2021) 29(1):59–73 64 Y. Gao et al. Rail. Eng. Science (2021) 29(1):59–73 (a) Vertical force 60 Vertical force 30 Vertical force 20 Vertical force Front rail Front rail Front rail 400 Front rail 40 20 separation 10 Back rail Back rail Back rail Back rail region 20 10 separation region 10 S5 S6 Running direction Running direction 0 0 Running direction 0 Running direction 1.255 1.260 1.265 1.270 1.275 1.280 200 1.255 1.260 1.265 1.270 1.275 1.280 1.260 1.265 1.270 1.275 1.280 1.285 1.2651.2701.2751.2801.2851.290 S7 S4 S8 S1 S3 S3 S2 separation point separation point separation point v v separation point Broken gap (50 mm) Broken gap (70 mm) v Broken gap (60 mm) Broken gap (80 mm) Broken gap Broken gap Broken gap Broken gap -200 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 Longitudinal coordinate (m) Longitudinal coordinate (m) Longitudinal coordinate (m) Longitudinal coordinate (m) (b) 120 15 Longitudinal force Longitudinal force Longitudinal force Longitudinal force Front rail 20 Front rail 10 Front rail 10 Front rail Separation Separation region Back rail Back rail region Back rail Back rail 0 Running direction 60 0 -10 Running direction 1.260 1.265 1.270 1.275 1.280 1.285 1.255 1.260 1.265 1.270 1.275 1.280 1.255 1.260 1.265 1.270 1.275 1.280 1.270 1.275 1.280 1.285 1.290 Running direction Running direction v v v Broken gap (60 mm) Broken gap (70 mm) v Broken gap (80 mm) Broken gap (50 mm) Broken gap Broken gap Broken gap Broken gap -60 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 Longitudinal coordinate (m) Longitudinal coordinate (m) Longitudinal coordinate (m) Longitudinal coordinate (m) (c) 6 4 Lateral force Lateral force Lateral force Lateral force 3 6 Front rail Front rail 6 Front rail Front rail 2 Back rail Back rail Back rail Back rail 2 1 Running direction 2 0 0 Running direction 1.260 1.265 1.270 1.275 1.280 1.285 1.255 1.260 1.265 1.270 1.275 1.280 1.255 1.260 1.265 1.270 1.275 1.280 Running direction Running direction 1.270 1.275 1.280 1.285 1.290 T1 separation point separation point separation point separation point v v v Broken gap (70 mm) v Broken gap (50 mm) Broken gap (60 mm) Broken gap (80 mm) Broken gap Broken gap Broken gap Broken gap -50 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 Longitudinal coordinate (m) Longitudinal coordinate (m) Longitudinal coordinate (m) Longitudinal coordinate (m) Fig. 6 History curve of contact force: a vertical contact force; b longitudinal contact force; c lateral contact force Vertical contact force (kN) Longitudinal contact force (kN) Lateral contact force (kN) Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 65 Fig. 7 Contact behavior between the wheel and rail with different gap lengths contact force evolution are analyzed, and the possibility of displacement of front rail end will increase under the action secondary fracture to the rails in case of the steel rail of the lateral impact force, which causes a certain degree of fracture is explored. deviation between the center line of the rail and the running direction of the wheelset. Therefore, the misalignment 3.1 Wheel–rail dynamic response between longitudinal axle of the front rail and back rail will increase the lateral creepage. Since the wheel–rail contact Figure 6 shows the time-history curves of the wheel–rail patch is not located at the center of the rail, the lateral force contact forces when wheels roll through the rail with dif- decays slower than the vertical force, and the lateral con- ferent broken gap lengths (50, 60, 70 and 80 mm), and the tact force will decrease to zero in a longer time, resulting in simulation results include vertical, lateral and longitudinal a relatively longer time of detachment between the wheel contact force. and rail. As indicated in Fig. 6, the front rail and back rail can be As shown Fig. 7 shows, when the length of gap is less regarded as the cantilever beam, when the wheel–rail than 60 mm, the wheel–rail contact patch is located in front contact patch is located on front rail, the rail end will sink of the front rail end. During the violent impact between the under the effect of wheel load, thus forming a height dif- wheel and back rails, the dynamic response of wheel–rail ference between the top surface of the back rail and front impact is relatively smaller due to the supporting effect of rail; when the wheelset impacts the back rail, due to the the front rail compared with the gap exceeding 60 mm. loss of vertical bending stiffness, the vertical wheel–rail When the gap length equals to 60 mm, the wheel will contact force will gradually decrease along the front rail. In continue rolling forward, and the contact patch between addition, the wheel–rail interaction during impact between wheelset and front rail will move forward to the end of the solids inherently consists of high-frequency shock and rail, where the wheel is in contact with back rail and front vibration; hence, the back rail will be subjected to enor- rail simultaneously, and the bending of front rail will be mous high-frequency impact force. The maximum vertical intensified under the influence of impact, leading to bigger wheel–rail force during impact can reach up to 308.3 kN, step values. Therefore, the dynamic response of wheel–rail which is about 4 times that of the static wheel load and impact will be intensified. When the gap length exceeds occurs about 0.2 ms after the impact, namely about 13 mm 60 mm, the wheel is in contact with the back rail, the wheel away from the back rail end at the broken gap. When the will detach from the front rail, and then, the wheel will wheels continue to roll forward, due to the release of track impact the back rail under the effect of gravity. As there is elastic energy, the wheels will continue to impact the back no supporting effect of the front rail during the impact, the rail. The wheel–rail contact force will fluctuate with time dynamic response of wheel–rail impact will be aggravated, and gradually become stable under the influence of fastener and the increase velocity is far greater than the one at the damping and structural damping. moment when the length of gap is less than 60 mm. In The evolution law of the lateral and longitudinal contact addition, when the gap length exceeds 60 mm, the sharp forces is similar to that of vertical force. Taking the 60 mm corner of the back rail end will be subjected to the entirely length of gap as example, the maximum values of the wheel dynamic loading independently, which is very easy longitudinal force and lateral force during the impact are to cause the spall of sharp corner and the unexpected 58.4 and 57.1 kN, respectively. Due to the freely con- expansion of the length of gap; further the derailment or strained end of front rail at the broken gap, the lateral overturning of the vehicle is easily to happen. Therefore, Rail. Eng. Science (2021) 29(1):59–73 66 Y. Gao et al. the 60 mm length of gap can be regarded as a special influencing the dynamic wheel–rail impact and running condition that distinguishes the contact behavior during the safety of vehicle. The front rail is sunk by the action of passage of wheel over the gap. dynamic vertical wheel–rail contact force, forming a height As the rail fracture seriously threatens the running safety difference with the top surface of the back rail. The height of vehicles, it is necessary to analyze the derailment difference is the first major affected factor, also known as coefficient, wheel unloading rate, step value and lateral step. If the step value is relatively large, the wheel will displacement difference between the back rail and front impact the back rail, leading to violent dynamic response rail, which are closely associated with the running safety of of the wheel–rail interaction. According to Fig. 10a, the vehicle. value of step height is up to 2.9 mm. Moreover, due to the As seen from Fig. 8, when the gap length exceeds flexural rigidity of rails, the vertical displacement of the 60 mm, the wheelset will detach from the rail during the back rail has not yet reached the maximum when the ver- impact, the derailment coefficient is considered to be tical contact force reaches the maximum. The right rail is exceeding the limit because the vertical contact force at broken while the left rail is non-damaged, which means this instant is 0; therefore, the derailment coefficient equals that the difference of vertical disturbance degree between to zero at the moment of wheel–rail separation. When the the left and right rails will cause the tilting of car body, gap length is less than 60 mm, the wheel will keep in leading to violent lateral wheel–rail contact force. More- contact with the back rail and front rail, and thus, the over, the lateral force will cause the lateral displacement of derailment coefficient is greater than zero. The derailment the wheel and rail. If it is too large, the flange may con- coefficient increases as the gap length, especially when the sequently climb on rails when the contact patch is trans- gap length exceeds 60 mm, the growth rate of maximum mitted from the front rail to back rail, thereby causing the derailment coefficient increases significantly due to the train to derail and threatening the driving safety. Thus, the change of wheel–rail contact behavior at broken gap. In second major affected factor is the lateral displacement addition, the maximum value of derailment coefficient difference between the front and back rails. As shown in corresponding to the 80 mm gap length is 0.268, which is Fig. 10b, due to the freely restrained end of the front rail at less than the threshold specified in [7]. However, due to the the broken gap, the maximum lateral displacement differ- violent impact induced by gap, the maximum wheel ence of 2.13 mm between the front and back rails occurs at unloading rate corresponding to the 80 mm gap length can the front rail end, which is less than the measured limit [7]. reach up to 0.754, which exceeds the threshold of the safety The step and lateral displacement difference under dif- index [7] and threatens the running safety of the vehicle. ferent gap lengths is shown in Fig. 11, where the step and When the gap length exceeds 60 mm, due to the change of lateral displacement difference can be regarded as irregu- larities to some extent, which can intensify the wheel–rail wheel–rail contact behavior at broken gap, the dynamic response of the wheel–rail impact will be intensified, and impact and threaten the running safety of vehicle. Espe- the wheel unloading rate will exceed the threshold [7] cially when the gap length exceeds 60 mm, the growth rate (Fig. 9). of the step, vertical displacement and lateral displacement The step height and lateral displacement difference is greater than that in the case of the gap length being less between the front rail and back rail are two main factors than 60 mm. The dynamic response of wheel–rail 0. 0.06 06 50 m 50 mm m 50 mm 60 mm 70 mm 80 mm 0. 0.4 4 60 m 60 mm m 0. 0.04 04 70 m 70 mm m 0.9 0. 0.3 3 0. 0.02 02 Limit: 0.65 80 m 80 mm m 0. 0.00 00 R Running unning dir dire ec ction tion 0. 0.2 2 -0 -0.0 .02 2 1. 1.22 22 1. 1.24 24 1. 1.26 26 1. 1.28 28 0.0 0. 0.1 1 Running direction 0. 0.0 0 Separation point separation point separation point vv v Br Bro ok ke en n g ga ap p Broken gap -0.9 -0 -0.1 .1 Broken gap Broken gap Broken gap 1. 1.0 0 1 1. .2 2 1. 1.4 4 1 1. .6 6 1.0 1.2 1.4 1.6 L Lo ong ngitudinal coordinate (m itudinal coordinate (m) ) Longitudinal coordinate (m) Fig. 8 Distribution of derailment coefficient Fig. 9 Distribution of wheel unloading rate Rail. Eng. Science (2021) 29(1):59–73 Derailmen Derailment co t coefficien efficient t Wheel unloading rate Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 67 (a) (b) 0.4 Front rail Front rail 0.2 Back rail Back rail Step value 0.0 Running direction -0.2 -3 Running direction -3 -0.4 separation point separation point -6 -0.6 v Broken gap -6 Broken gap Broken gap Broken gap -0.8 1.0 1.2 1.4 1.6 1.01.2 1.41.6 Longitudinal coordinate (m) Longitudinal coordinate (m) Fig. 10 History curves of wheel displacement (60 mm): a vertical displacement; b lateral displacement (a) (b) 5.0 Maximum vertical displacement of front rail Maximum lateral displacement of front rail Maximum vertical displacement of back rail Maximum lateral displacement of back rail 4.5 Step height 3 4.0 3.5 3.0 2.5 0 50 60 70 80 50 60 70 80 Length of broken gap (mm) Length of broken gap (mm) Fig. 11 Maximum step and rail displacements under different gap lengths: a vertical displacement and step height; b lateral displacement MPa Irregular contact patch region S1=0.99 m S2=1.25 m S3=1.26m S4=1.28 m Un-contact between wheel and rail Front rail Back rail S5=1.29 m S6=1.33m S7=1.38 m Broken gap Running direction S8=1.58 m v=250 km/h 60 mm Longitudinal Fig. 12 Surface normal pressure distribution (60 mm) Rail. Eng. Science (2021) 29(1):59–73 Lateral (m) Vertical displacement (mm) Displacement (mm) Step value (mm) Lateral displacement (mm) Lateral displacement (mm) 68 Y. Gao et al. interaction will increase with the gap length, leading to the wheelset just leaves the end of the back rail, the vertical violent dynamic response of wheel–rail interaction, and the free section of the back rail at the broken gap is bent significantly increased risk of vehicle derailment. downward under the dynamic impact loading. Because the flexural rigidity of the broken rail is much smaller than that 3.2 Evolution of frictional rolling contact behavior of the ordinary rail, the bending deformation of the back rail is relatively large, leading to a decrease in the contact An accurate stress analysis is the other key factor to rec- stiffness and a reduction size of front contact patch (S6– ognize the rail secondary fracture and the wheel/rail con- S7). When the wheel–rail contact patch is far away from tact behavior at broken gap. Eight typical instants (S1–S8) the rail end, the contact patch will return to its elliptical are selected for analysis, including the instant with the shape (S8). maximum vertical wheel–rail force (Fig. 6). In order to Figure 13 demonstrates the magnitude and distribution determine whether the rail will suffer from the secondary of the normal stress along the lateral–vertical section of the fracture, the split Hopkinson pressure bar (SHPB) and the rails at eight typical moments. Specifically, (a)–(c) show split Hopkinson tensile bar (SHB) [18] are used to obtain the contact patch which are located at the front rail, evaluation indexes: the ultimate tensile strength of U71Mn because the vertical wheel–rail force continues to decrease steel is 1188Mpa and the yield strength is 650 Mpa. For the along the longitudinal direction of the front rail, and the ease of description, the following figures all take the normal contact stress is continuously reduced during this 60 mm length of gap as a display objective, and finally the period; (d)–(h) show the contact patch which is located at maximum objective values corresponding to different gap the back rail. During the impact, the normal contact stress lengths are summarized. reaches the maximum in a short time, and then, the contact Figure 12 illustrates the magnitude and distribution of stress will slowly decrease after the impact. The maximum contact stress along the longitudinal axle of the rail. The normal contact stress is located at the rail surface with a contact stress before impact is relatively small and slowly value up to 1,953 MPa, and it decreases sharply with the decreases due to the deflexibility of front rail (S1–S3). increase in the depth. After the impact, the wheel–rail Then, the contact stress increases rapidly after the wheelset normal stress is of crescent-shaped distribution; when the impacts the back rail, reaches the maximum value in about wheel continues rolling forward, the crescent-shaped stress 0.2 ms after the occurrence of impact. After the impact, the will gradually return to the size as Fig. 13a. Compared with contact stress will rapidly decrease under the effect of the case where the wheel–rail contact patch is located at the structural damping and fasteners damping. Furthermore, front rail, the stress is substantially increased. when the contact patch is located near the end of the front Figure 14 indicates the distribution of the surface shear stress along the longitudinal axle of the rail. When the rail at the broken gap, the shape of the contact patch changes from ellipse to semi-ellipse (S4) due to the influ- contact patch is located at the front rail, the maximum ence of broken gap, and the geometric area of contact patch shear stress occurs in the rear of the wheel–rail contact on front rail gradually decreases to zero. When the patch due to the influence of longitudinal contact force wheelset impacts the back rail, the end of the back rail will (S1–S3). After the impact, the sharp corner of the end of withstand severe contact stress (S5), and its sharp corners the back rail is used as the pivot point during the passage of will be subjected to the maximum stress. Besides, when the wheel over broken rail; therefore, the maximum shear Fig. 13 Normal stress distribution at lateral–vertical Sect. (60 mm) for S1–S8 (a–h) Rail. Eng. Science (2021) 29(1):59–73 Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 69 MPa Irregular contact S1=0.99 m patch region S2=1.25 m S3=1.26 m S4=1.28 m Un-contact between Back rail Front rail S5=1.29 m wheel and rail S6=1.33 m Running direction S7=1.38 m Broken gap S8=1.58 m v=250 km/h 60 mm Longitudinal Fig. 14 Surface shear pressure distribution (60 mm) Fig. 15 Subsurface shear stress distribution (60 mm) for S1–S8 (a–h) Fig. 16 Subsurface tensile stress distribution (60 mm) for S1–S8 (a–h) stress is concentrated on the sharp corner of rail end (S4– rail under the dynamic impact load will undergo large S5). Similar to the distribution rules of normal contact bending deflection, thereby resulting in the non-elliptical stress, because the vertical flexural rigidity at the broken distribution of shear stress (S6–S7), and the location of the rail is much smaller than that of the ordinary rail, the back maximum shear stress will gradually move to the rear of Rail. Eng. Science (2021) 29(1):59–73 Lateral (m) 70 Y. Gao et al. the contact patch. When the wheel–rail contact patch is Broken gap away from the rail end, the distribution of shear stress No damage returns to an elliptical shape, and it will remain stable when the shear stress gradually drops to a certain value (S8). Figure 15 shows the magnitude and distribution of the maximum shear stress along the lateral–vertical section of 0 the rails at eight typical moments. Similarly, (a)–(c) show the contact patch which are located at the front rail; the rail -200 Front rail Back rail bends along the longitudinal direction and activate an effect like unloading, the shear stress continues to decrease, Broken gap -400 Broken gap and the maximum shear stress before impact is located at 1.01.2 1.41.6 2.2 mm below the rail surface. (d)–(h) show the contact Longitudinal coordinate (m) patch which are located at the back rail, the value of the maximum shear stress is significantly increased due to the Fig. 17 Time history curve of axle acceleration (60 mm) wheel–rail impact, which can reach up to 1060 MPa, and the impact causes the maximum shear stress to move to the narrow crescent. Therefore if the rail fracture cannot be rail surface. Under the coupled effect of the normal found in time, the wheels will repeatedly hit the sharp dynamic impact load, the crescent-shaped distribution can corners at the rail end, which may cause rail secondary be observed on the lateral–vertical section of rails. When fracture and threaten the running safety of the vehicle. the wheelset is away from the broken gap, the value of the As shown in Table 2, the contact stress and shear stress maximum shear stress gradually decreases and returns back as well as the tensile stress are all positively correlated with below the rail surface. the length of broken gap. As the gap length exceeds the Based on the first strength theory, the first principal 60 mm, the growth rate of the stress increase much faster, stress can be equivalent to the maximum tensile stress, and the tensile stress corresponding to 70 mm length of broken the rail secondary fracture can be predicted by comparison gap is slightly higher than the ultimate tensile strength of of the threshold of first principal stress and the ultimate the rail material. If the length of broken gap is less than tensile strength. Figure 16 shows the distribution of the 60 mm, the value of stress will be slightly lower than the first principal stress along the lateral–vertical plane of steel ultimate failure strength, and the rail secondary fracture rails at eight typical moments. will not occur immediately. If the rail fracture is not Specifically, (a)–(c) show the contact patch which are detected timely, the occurrence of secondary fracture will located at the back rail, and the tensile stress continues to be further intensified. Therefore, it is suggested that the decrease along the longitudinal direction of the front rail. length of gap should be controlled within 60 mm from the (d)–(h) show the contact patch which is located at the back angle of the stress and rail secondary fracture. rail. Due to the violent wheel–rail impact, the maximum tensile stress of the rail surface is increased with a crescent 3.3 Axle acceleration analysis shape. The value of the maximum tensile stress can reach up tp 1,207 MPa (the gap length equals to 60 mm), slightly Based on the analysis above, broken gaps are a type of higher than the ultimate tensile strength of the rail material; unpredictable damage to high-speed railway track. Some if the gap length exceeds 60 mm, the violent force induced parts of the rail fracture cannot be detected by real-time by impact may cause rail secondary fracture. In addition, safety system, for instance the turnout. If maintenance is when the gap length exceeds 60 mm, the maximum contact not conducted timely, the enormous impact load will be stress, shear stress and maximum tensile stress are all excited when the wheelset passes through the broken gap, located at the sharp corners of the back rail end during which may cause rail secondary fracture. As the dynamic impact, and the distribution of tensile stress is a long and response of wheel–rail impact between solids inherently Table 2 Maximum stresses at different gap lengths Length of broken gap (mm) Contact stress (MPa) Shear stress (MPa) Tensile stress (MPa) 50 1,721 891 1,040 60 1,953 1,060 1,207 70 2,201 1,232 1,401 80 2,445 1,454 1,673 Rail. Eng. Science (2021) 29(1):59–73 Axle box accleration (m/s ) Damage tolerance of fractured rails on continuous welded rail track for high-speed railways 71 acceleration is 42 m/s , and the signal mainly fluctuates 729 Hz between the range of - 42 and 30 m/s , which is consistent Broken gap rail No damage rail with the experimental results in Ref [26].. The broken gap will evoke more intense amplitude fluctuations in the axle acceleration, the range of axle acceleration fluctuations is between - 150 and 400 m/s , and its maximum value can reach up to 400 m/s . In Fig. 17, the maximum axle 2850 Hz acceleration and maximum wheel–rail vertical force are located at 1.315 and 1.293 m, respectively. Therefore, the maximum energy of the axle acceleration mainly occurs after the impact. Figure 18 is the time–frequency distribution of axle 03 69 12 Frequency (kHz) acceleration processed by wavelet transform. In case of non-damaged rails, the energy distribution curve of the axle Fig. 18 Frequency domain response (S-transformation) at 60 mm acceleration is relatively smooth. In the case of a broken gap, the energy of axle acceleration curve is mainly dis- consists of high-frequency vibration, to develop an tributed between 404 Hz and 1,637 Hz (see Fig. 19), which insightful understanding of the wheel–rail contact behavior corresponds to the high-frequency impact load caused by at broken gap, the dynamic response of the wheel–rail the first impact and low frequency excited by the subse- impact in frequency domain is investigated. quent vibration, respectively. The frequency of the impact Figure 17 compares the axle acceleration in frequency excited at the broken gap is relatively high, which is caused domain between the broken gap and undamaged rail. The by unconstrained end at the broken rails and wheel–rail result is processed by the low-pass filtered method to high-speed impact. The high-frequency load decays eliminate the interference of high-frequency noise. The quickly under the effect of the inertia force of the rail, and above figure indicates a slight fluctuation in the axle there is no enough time to transmit it to the vehicle and the acceleration of wheels in case of no damage, which is rail substructure. caused by the transient contact between wheels and rails. In As shown in Table 3, it is evident that the axle accel- this segment of signal, the maximum value of the axle erations caused by the broken gap and non-damaged rails (a) (b) 7500 200 0.015 0.017 0.019 0.021 0.023 0.015 0.017 0.019 0.021 0.023 Time (s) Time (s) Fig. 19 Time–frequency distribution map of axle acceleration with a non-damage rail and b broken rail at 60 mm gap Table 3 Maximum axle acceleration and frequency for different gap lengths Length of broken gap (mm) Axle box acceleration (m/s ) Frequency First shock (Hz) Subsequent vibration (Hz) 50 356.5 1,604 398 60 401.1 1,637 404 70 460.5 1,710 432 80 522.2 1,736 440 Rail. Eng. Science (2021) 29(1):59–73 Amplitude Frequency (Hz) Power spectral density (km/s ) Frequency (Hz) Power spectral density (km/s ) 72 Y. Gao et al. are remarkably different in domains of time and frequency, the impact. When the gap length exceeds 60 mm, the where the frequency of axle accelerations are concentrated maximum shear stress and the maximum tensile stress between 1,500 Hz and 1,800 Hz. Besides, given the influ- borne by the rail are slightly more than the ultimate ence of high-frequency load, the surface of rail end near the breaking strength, and the rail secondary fracture may occur. broken gap is a relatively dangerous region. The wheel will vibrate more violently with the gap 4. The frequency domain energy excited by the broken gap is mainly concentrated between 1,500 and length increasing. The different lengths correspond to the different values of the frequency of first impact and sub- 1800 Hz, and the difference of frequency value between the broken gap and non-damage rail is sequent vibrations, and the amplitude of axle box accel- eration is positively correlated with the length of broken relative large. gap. However, the frequency characteristic has no con- 5. The limited broken gap for high-speed railways should nection with the length of broken gap. During the wheel be less than 60 mm, if the length exceeds the 60 mm passage over gaps. The frequencies corresponding to the (LMA profile and at 250 km/h), the rail and wheel may peak value of the amplitude under different gap lengths are suffer a secondary fracture. The damage tolerance of consistent, which are mainly concentrated near the 404 Hz rail fracture length is 60 mm for high-speed railways. and 1,637 Hz as shown in Table 3. In addition, it is evident that the axle accelerations caused by the broken gap and Acknowledgements The work is supported by the National Natural non-damaged rails are remarkably different in domains of Science Foundation of China (Nos.51608459, 51778542 and U1734207) and Fundamental Research Funds for the Central time and frequency, which is probably related to the Universities (No.2682018CX01) and Cultivation Program for the reduction of the vertical structural stiffness at the gap. This Excellent Doctoral Dissertation of Southwest Jiaotong University. is similar to the free end of a cantilever beam. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as 4 Conclusions long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate Aiming at the effect of rail fracture on the wheel–rail if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless contact behavior and running safety of trains on high-speed indicated otherwise in a credit line to the material. If material is not railways, we simulate the process when the wheels impact included in the article’s Creative Commons licence and your intended the broken rail on the basis of the three-dimensional use is not permitted by statutory regulation or exceeds the permitted explicit finite element model, investigate the rail stress use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. mechanism and vibration rule of the wheel at the broken org/licenses/by/4.0/. gap and reach the following conclusions from the above analysis: 1. The maximum wheel–rail contact force caused by the References broken gap can be 4–5 times that of the static wheel load. When the gap length exceeds 60 mm, the 1. Tong D (1986) Railroad track. China Railway Publishing House, derailment coefficient is smaller than the standard Beijing value, but the wheel unloading rate is higher than the 2. 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Railway Engineering Science – Springer Journals
Published: Dec 14, 2020
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