Non-uniform variation characteristics of temperature field and corresponding thermal effect of longitudinal continuous slab ballastless track structure

Contents lists available at ScienceDirect Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite Non-uniform variation characteristics of temperature field and corresponding thermal effect of longitudinal continuous slab ballastless track structure Q.Q. Xub, Y. Liua,b,*, X.D. Sunb,c, D.Y. Jiangb, Z.Y. Jia,b, Y. Xud, R. Arcose,f, J. Romeuf aMOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu, 610031, China bSchool of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China cNational Engineering Research Center of Geological Disaster Prevention Technology in Land Transportation, Chengdu, 610031, China dRailway Engineering Research Institute, China Academy of Railway Sciences Corporation Limited, Beijing, 100081, China eSerra Húnter Fellow, Universitat Politècnica de Catalunya, Spain fAcoustical and Mechanical Engineering Laboratory, Universitat Politècnica de Catalunya, Terrassa, (Barcelona), Spain HIGHLIGHTS •Non-uniform temperature field of longitudinal continuous slab track structure was established. •Interface damage between track slab and CA mortar layer under a non-uniform temperature load is obtained. •Strong non-uniformity is found in both temporal and spatial variation for temperature as well as for thermal deformation. •Solar radiation and heat convection are the dominant factors for the temperature field in the daytime and night, respectively. •Non-uniform temperature load produces remarkably different thermal effects to those given by the design temperature load. ARTICLE INFO Handling Editor: Huihe Qiu Keywords: Longitudinal continuous slab ballastless track structure Temporal-spatial non-uniformity Temperature field Design temperature load Thermal effect ABSTRACT The longitudinal continuous slab ballastless track structure (LCSBTS) easily suffers from interface and joint damage when subjected to extremely high-temperature loads, significantly affecting the running smoothness and safety of high-speed trains. This paper establishes a numerical model of the LCSBTS using the heat transfer principle and real-time shadow technology to replicate the time-varying evolution and spatial distribution of the temperature field. The corresponding thermal effect is obtained based on the sequential thermal-mechanical coupling method. The results show that the temperature variation and the corresponding thermal effect exhibit strong temporal-spatial non-uniformity on the horizontal and vertical planes of the LCSBTS. Solar radiation and heat convection are the dominant factors for the temperature field of the LCSBTS in the daytime and night, respectively, resulting in a higher temperature and greater thermal deformation on the sunny side than on the shady one. The temperature and vertical thermal deformation along the longitudinal direction vary periodically with a specific wavelength between the two adjacent rail support concrete blocks. The thermal deformation increments under the studied nonuniform temperature loads cause an interface damage increment. Finally, the consideration of a * Corresponding author. MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu, 610031, China. E-mail address: [email protected]u.cn (Y. Liu). https://doi.org/10.1016/j.csite.2024.104992 Received 21 June 2024; Received in revised form 7 August 2024; Accepted 16 August 2024 Case Studies in Thermal Engineering 61 (2024) 104992 Available online 17 August 2024 2214-157X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Q.Q. Xu et al. non-uniform temperature load produces remarkably different thermal effects from those given by the design temperature load. 1. Introduction China Railway Track System Type-II slab ballastless track structure is a longitudinal continuous slab ballastless track structure (LCSBTS) constructed by connecting adjacent unit track slabs with six tension locks. Due to the track slabs being longitudinally connected, thermal deformation of LCSBTS under temperature load is restrained, leading to high smoothness and stability of high-speed railways [1]. The temperature field of the slab ballastless track structures (SBTS) is directly influenced by environmental factors such as wind, rain, snow, temperature dropping, solar radiation, and shading. The different thermal properties among the track slab, cement asphalt (CA) mortar layer, and base plate cause the temporal-spatial non-uniform variation of the temperature load of the LCSBTS under complex environmental conditions, resulting in significant and complex thermal effects [2]. Temporal-spatial non-uniform temperature loads result in incompatible thermal deformation between different layers, leading to thermal damage in the interface and joints [3–5]. These thermal damages deteriorate the stability and service life of the LCSBTS, threatening the safety of high-speed railway operations. Therefore, the distribution and evolution of temporal-spatial temperature loads and the thermal damage mechanism of the LCSBTS arise as a topic that deserves attention from the scientific community. Four methods have mainly been used to obtain the temperature load of SBTS: theoretical analysis, field measurements, scale test, and numerical simulation. Based on the heat transfer theory and a large amount of meteorological data gathered, the nonlinear varying evolution of the temperature load along the vertical direction within the SBTS is obtained using theoretical analysis in various works [6–9]. However, the theoretical equations are difficult to solve due to the complexity of the thermal boundary conditions. To obtain the analytical solution, 3D theoretical models are usually simplified to one or two-dimensional models, and the suggested values of temperature loads for different regions are obtained, including overall temperature rise and temperature gradient of track slabs [8,9]. Field measurements allow for temperature data to be gathered from the construction period to the operation period by prelaying sensors on the SBTS. Temperature data from field measurements is more realistic and reliable than those from the theoretical analysis [10,11]. Typically, temperature data obtained from a limited number of sensors makes the process of reproducing on-site temperature loads quite difficult. Although more sensors can be used in the model test to enhance the completeness of the gathered temperature data, the complex thermal boundary conditions of the on-site temperature field in high-speed railways can hardly be duplicated, resulting in the temperature field obtained from the scale tests are hardly verified by the field measurements [12–16]. With the development of finite-element analysis (FEA) software, reproducing the actual 3D temperature field of SBTS using numerical simulation technologies become mainstream [17–19]. With FEM models, the complex coupling relationships between parameters, such as air temperature, solar radiation, wind speed, etc., can be considered for solving the theoretical equations of temperature fields in different climate areas, and the variation evolution and strong temporal-spatial non-uniformity of the temperature load can be deeply investigated [20]. Research conducted during the last decade concluded that the interface damage of SBTS is mainly related to the incompatible thermal deformation between different layers [4]. Traditionally, the temperature of the SBTS is often simplified to be uniform, such as overall temperature rise/decline, or considers the linear or nonlinear gradient along the vertical direction, like the maximum positive and negative temperature gradients of the SBTS are 90 °C/m and −45 °C/m, respectively, provided by the "High-speed Railway Design Code" [21]. Under the overall temperature rise, the LCSBTS will expand in the longitudinal direction, causing damage in the interface and joints or buckling of the track slab [22–24]. Once the vertical warping deformation caused by the temperature gradient is too large, the interface delamination between the layers will be generated [25]. Zhou et al. (2022) [26] further compared the displacement and stress of the CRTS II slab ballastless track under six cases of overall temperature rises and temperature gradients, respectively, showing that the combination of the cohesive zone model and concrete damaged plasticity model could provide the worst damage scenarios. Currently, the research on the temperature effect of the SBTS using scale tests is gradually increasing. Based on 1:4 the scale tests with constraint and freedom boundary conditions, Zhou et al. (2022, 2023) systematically investigated the thermal effect of the LCSBTS under cyclic temperature loads [15] and compared the internal temperature distribution, stress, and displacement under three environmental temperatures rising and dropping cases [16], indicating the boundary condition of the slab track have a significant influence on thermal effects of the slab track. In FEM, using the sequential thermal-mechanical coupling method with the simulated temperature field, the evolution of thermal deformation [27–29], joint damage [30], and buckling of the track slab [31] can be reproduced. Besides, Zhou et al. (2024) [32] further investigated the coupling effect of environmental temperature and moving train loads on the mechanical performance of the SBTS, indicating that the coupling effect notably increases the dynamic stress with a more pronounced effect on the dynamic displacement of the longitudinal double-block ballastless tracks than the unitized. The temperature load used in Zhou et al. (2022) [26] and Zhou et al. (2024) [32] follows the traditional way in terms of the combination of a single value of overall temperature rise, which is the same everywhere within LCSBTS, and a temperature gradient only describing the temperature difference within LCSBTS along depth but is the same everywhere on the horizontal plane. This definition of temperature load makes it difficult to exhibit the temperature variation along the horizontal direction, which is particularly important for LCSBTS in explaining the local stress concentration and damage near the edge and within the joint. According to the site investigation of operating high-speed rail lines, in addition to causing periodic track irregularities and local support effects between layers [3–4], the thermal effects will also cause local stress concentration, especially in the T-shape joints [30], resulting in non-uniformity and randomness of structural damage. When designing SBTS in certain regions with large values of traditional temperature loads, a fully safe and non-redundant solution is commonly impossible to reach. Additionally, the thermal deCase Studies in Thermal Engineering 61 (2024) 104992 2 Q.Q. Xu et al. formation and damage under the traditional temperature load show a centrosymmetric distribution, spatially speaking [22–26], which makes it difficult to explain the causes of actual non-uniform damage of the SBTS. In the scale test of Zhou et al. (2022,2023) [15,16], the sample was put into the temperature chamber, which changes the temperature by heating or cooling the air within the chamber. In this way, the variation of temperature load is the same for everywhere on the scale model. The real-time variation of the solar radiation cannot be simulated, making it difficult to reproduce the non-uniformity of the temperature related to the solar radiation. In the actual operating environment, the change of the shadow induced by factors such as solar radiation and incident angle has strong temporal and spatial variation characteristics, which is one of the main reasons for the non-uniform temperature of the SBTS. However, real-time shadow technology is rarely considered in the temperature field of the SBTS. Although some studies have analyzed the lateral non-uniformity of temperature and temperature deformation of the SBTS in FEA [29–31], the temperature effect of the LCSBTS under the non-uniform temperature field showing strong spatial non-uniformity and time-varying characteristics are still rarely involved in existing studies. In the present work, a numerical model of the LCSBTS is established based on the heat conduction theory. The main thermal boundary conditions considered in the model are solar radiation, thermal convection, heat conduction, and real-time shadow. Using the model, the temporal evolution and spatial distribution characteristics of the non-uniform temperature load are reproduced based on the heat transfer principle, verified by the field test on-site. The corresponding thermal deformation and thermal damage of the LCSBTS under a three-day temporal-spatial non-uniform temperature load are calculated by the sequential thermal-mechanical coupling method. The action of the non-uniform temperature load and the design temperature load have been compared in terms of thermal deformation and thermal damage of the LCSBTS. The results obtained in this paper can provide theoretical support for further research in high-speed railway maintenance. 2. Methodology 2.1. Calculation theory A numerical model of the temperature field of LCSBTS can be established using the finite element method (FEM). In this work, the thermal boundary conditions for LCSBTS are mainly concerned on solar radiation, convective heat transfer, and radiation heat transfer, represented by the orange arrow, green arrow, and light blue arrow, respectively, as shown in Fig. 1(a). The total heat flux qtransferred from the environment into the track structure under these three types of heat transfer can be expressed as q=ql+qc+qs, (1) where qlis the heat flux generated by radiation heat transfer on the structure surface; qcis the heat flux generated by convection heat transfer on the structure surface and qsis the heat flux generated by solar radiation on the structure surface. Fig. 1. Thermal boundary conditions adopted for the LCSBTS. Case Studies in Thermal Engineering 61 (2024) 104992 3 Q.Q. Xu et al. Radiation heat transfer is defined as the phenomenon of electromagnetic wave radiation from objects with temperature. During the radiation heat transfer process of LCSBTS, the heat transfer qlfollows the Stefan-Boltzmann law, which can be calculated by [33] ql=1 2𝛼lC0(273 +Ta)4[(Aa+1)+(Aa−1)sin 𝛽n]−AlC0(273 +T)4, (2) where αlis the longwave radiation absorptivity, C0is the Stefan-Boltzmann constant, εais the atmospheric radiation coefficient, Tais the air temperature, εlis the thermal radiation emissivity and βnis the angle of surfaces of the track structure. The value of βnis 90°for the horizontal surfaces, while it takes a value of 0°for the vertical ones. Assuming the variation law of air temperature follows the double sine function [34]. The air temperature Tais given by Ta=Ta+ Ta[0.96 sin 𝜔(𝜏−𝜏0)+0.146 sin 2𝜔(𝜏−𝜏0)], (3) Where Taand Taare the average daily air temperature and the amplitude of air temperature, they can be obtained from the local historical meteorological data. τis the time in hours which is set to be 0 h at 6:00 a.m. for the beginning of time-history. τ0is the time difference between the peak solar radiation moment and the maximum air temperature moment. According to Newton's law of cooling, the heat flux generated by convection heat transfer qccan be given by qc=hc(Ta−T), (4) where hcis the convective heat transfer coefficient of the track structure surfaces. According to Jürges-Nusselt formulation [35], for wind speed v≤5.0 m/s, hccan be determined by hc=2.6(4 √||T−Ta||+1.54v), (5) The heat flux generated by solar radiation can be written as qs=𝛼s(ID+Id𝛽+Ir𝛽), (6) where αsis the radiation absorption rate; IDis the direct solar radiation intensity; Idβis the sky scattering intensity, which describes the solar radiation reflected by atmospheric molecules, water vapor, dust, and other components; and Irβis the ground reflection intensity, which describes the direct radiation as well as the scattered radiation reflected after reaching the ground surface. The direct solar radiation intensity IDcan be given by ID=I0P 1 sin Hcos 𝛾, (7) where I0is the solar radiation constant, His the solar altitude angle (i.e., the angle between the solar radiation and the ground level), Pis the atmospheric transparency coefficient, and γis the solar incident angle. Furthermore, Idβand Irβcan be derived as      Id𝛽= 1+ sin 𝛽n 20.271 I0−0.294 IDsin H, Ir𝛽= 1− sin 𝛽n 2 reIDsin H+ 2Id𝛽 1+ sin 𝛽n, (8) where reis the ground surface shortwave reflectivity. At any given moment, to calculate the direct solar radiation intensity on the external surface of the track structure, it is necessary to first determine the real-time shadow zone. The real-time shadow mainly includes the self shadow, which mainly appears on the vertical surface of the track structure that is back facing the sun, and the cast shadow, which usually exists on the surface of the base plate generated by the shelter from the track slab, as shown in Fig. 1(b). The self shadow zone is formed when the angle between the normal vector of the external surface of the track structure and0. the sun rays is greater than 90°, leading to the external surface of the track structure not being directly exposed to the sun rays. The real-time shadow zone is directly related to the real-time position of the sun. Based on the real-time position of the sun and the geographical location of the track structure, the real-time relative relationship between the sun ray and the surface of the track structure is determined. The real-time position of the sun can be determined by its altitude angle Hand azimuth angle V. Among them, the solar azimuth angle refers to the angle between the projection of the incident sunlight on the ground plane and the due south direction. The calculation formula for both [36] is as follows: sinH =sin 𝛿sin 𝜑+cos 𝛿cos 𝜑cos 𝜔, (9) cos V=sin Hsin 𝜑− sin 𝛿 cos Hcos 𝜑, (10) Case Studies in Thermal Engineering 61 (2024) 104992 4 Q.Q. Xu et al. In the formula, φis the latitude; ωis the hour angle, which refers to the angle between the hour circle where the sun is located and the local meridian; δis the declination angle, which is the angle between the line connecting the center of the Earth and the center of the sun and the equatorial plane of the Earth. 2.2. Simulation model Based on the calculation theory, the numerical model of the LCSBTS on the bridge is established using ABAQUS, as illustrated in Fig. 2. The model consists of five track slabs, a CA mortar layer, a base plate, and four joints between two adjacent track slabs referred to the wide and narrow joints in the figure. The total longitudinal length of the model is 32.45 m, and the system is oriented to the north-south direction. Each track slab is made of C60 concrete with a geometry of 0.2 m thick, 6.45 m long, and 2.55 m wide. The width of the wide joint is 210 mm, while the narrow joint is 50 mm wide, and both are 100 mm thick. A base plate is made of C30 concrete with a thickness of 0.2 m and a width of 2.95 m. Track slabs and base plates are bonded by a 0.03-m-thick CA mortar layer. In the model, the track slab, the CA mortar layer, and the base plate are established using 151,736 three-dimensional solid elements, of which the smallest size is 74 mm × 81 mm × 20 mm, and the corresponding elastic modulus of the three layers is 36 GPa, 10 GPa, and 22 GPa, respectively. The thermal expansion coefficients of solid elements are set to be 10−51/°C. In ABAQUS, the DFLUX subroutine is employed to realize the input of the heat flux, and the subroutine FILM is utilized to realize the variation of environmental temperature and to set the heat exchange coefficient. The self shadow area and cast shadow areas produced by changing the angle of the sun are simulated by real-time shadow technology, referred to Ref. [20]. Since the delamination is most likely to occur on the surface between the track slab and the CA mortar layer, a bilinear cohesive zone model (CZM) is employed to simulate the interface, characterizing the force-displacement relationship of the interface during the whole damage process from damage initiation to delamination. The contact between them is defined as a hard contact in the normal direction and a frictional one in the tangential direction, with a penalty friction coefficient of 0.3. The nodes on contact surfaces in the simulation model are considered to be tied, except for the interface between the track slab and the CA mortar layer. According to the bilinear CZM, the parameters such as the interface stiffness and the fracture toughness are calculated [25]. Besides, the degree of interface damage is represented by SDEG, when SDEG takes the value 0, no damage occurs; when SDEG takes the value 1, delamination occurs; when SDEG takes the value between 0 and 1, interface damage occurs. At the bottom surface of the model, all the degrees of freedom of nodes are constrained. To eliminate the influence of thermal boundary conditions, the base plate is 1 m longer than the track slab at both ends of the model. The base plate, the CA mortar layer, and the track slabs are discretized by using 3D thermal solid elements. Thermal boundary conditions and contacts between different layers follow the basic principles of heat transfer. Thermal conduction between the base plate and the bridge deck is neglected, considering the thermal barrier between them and the temperature changing slightly in the base plate. The thermal parameters of the model are listed in Table 1. To calculate the temperature field of LCSBTS, the initiate temperature field is obtained from observation at the Beijing-Shanghai high-speed railway, Suzhou, Anhui Province. The average daily air temperature Taand the amplitude of air temperature Taare 28 °C and 4 °C, respectively, and then, the air temperature is calculated by equation (3). The hour angle, declination angle, and sun angle are referred to Ref. [36]. The values of the shortwave radiation absorption rate αs, the longwave radiation absorptivity αl, the thermal radiation emissivity εland the atmospheric radiation coefficient εaare referred to Ref. [20]. The time difference between the maximum solar radiation and the maximum air temperature τ0is 3 [34], the atmospheric transparency coefficient is 0.6, and the average wind speed is 2 m/s, obtained from the ECMWF Re-Analysis 5 database which is analyzed by the European Centre for Medium-Range Weather Forecasts. All the mentioned calculation parameters are shown in Table 2. Fig. 2. Finite element model of the LCSBTS system considered in this study. Table 1 Thermal parameters of model. Layers Thermal Conductivity W/(m·°C) Density kg/m3Specific heat capacity J/(kg·°C) Track slab 1.67 2500 900 CA mortar layer 0.93 2450 1200 Base plate 1.67 2500 900 Case Studies in Thermal Engineering 61 (2024) 104992 5 Q.Q. Xu et al. Table 2 Parameters of temperature field. Parameters Values Ta28 °C Ta4 °C C05.67×10−8W∙m−2∙K−4 Atmospheric transparency coefficient 0.6 αl0.9 αs0.6 εl0.9 εa1 − 0.26 exp (−7.776 × 10−4Ta2) Solar constant I01367 [1 + 0.033 cos (365 360N)] Hour angle ω15(tbj +(λ−120)/15 −12) Declination angle δ0.3723 + 23.2567 sin θ+ 0.1149 sin 2θ− 0.1712 sin 3θ− 0.758 cos θ+ 0.3656 cos 2θ+ 0.0201 cos 3θ Day angle θ365.2422 2π× (N− 79.6764 − 0.2422 × (Y− 1985) + INT [(Y− 1985) /4]) tbj is Beijing time; λis longitude; Nis the number of the day in the current year; Yis the year. 3. Non-uniform temperature field and corresponding thermal effects 3.1. Non-uniform temperature field The measured data on September 12th, 2010, in Suzhou, is selected as verification with simulated temperature of the track structure, as shown in Figs. 3 and 4. Before the field test, a total of 7 temperature sensors were arranged along the vertical direction of the track structure at the depths of 0, 50, 100, 200, 215, 230, and 280 mm from the top surface of track slab, respectively. Among them, the first four sensors correspond to the top surface, interior, and bottom of the track slab, and the last three correspond to the interior and bottom of the CA mortar and the interior of the base plate, respectively. Fig. 3 shows the top-surface temperature and the temperature gradient of the track slab in 24 h. In Fig. 3, the measured data are collected each hour, and the simulated data are calculated at equal intervals of 5 min, both of which are obtained at the central point of the track slab. As shown in Fig. 3, the simulation results of the top-surface temperature and temperature gradient of the track slab agree well with the observation data at most times of the day, and the maximum difference during the daytime is about 1.5 °C at 15:00. Fig. 4 shows the temperature variation along the depth direction of the track slab every hour in a day. The simulation results also agree well with the observation temperature data [11], indicating that the FEM model can accurately reproduce the temperature field of the LCSBTS under an actual service environment. Taking the measured data at 0:00 on September 12th, 2010, as the initial temperature field, the following three-day hightemperature field of the LCSBTS is calculated. Fig. 5 presents temperatures at the top and bottom surfaces and the temperature gradient of the track slab in a three-day hot weather. The temperatures of the top surface and bottom surface, as well as the temperature gradient of the track slab, change periodically every day. The variation of the temperature field mainly includes two stages, such as the temperature rise stage and temperature drop stage, with the corresponding time from 6:00 a.m. to 14:00 p.m. and 14:00 p.m. to 6:00 a.m. on the following day. The changing law of the temperature gradient is almost synchronized with that of the temperature of the top surface of the track slab, which reaches the maximum negative and positive values at around 6:00 and 14:00 p.m., respectively, as observed in Ref. [11]. In the three-day high-temperature environment, the maximum temperature at the top and bottom surfaces gradually increases, while the maximum negative and positive temperature gradient gradually decreases. Three middle track slabs of the LCSBTS are selected for the demonstration of the non-uniform temperature field and the corresponding thermal effect, as shown in Fig. 6. In Fig. 6, two north-to-south longitudinal sections are marked with A-A′and B–B′, respectively. Ten rail support concrete blocks on the east side of the middle track slab are numbered 1–10 from south to north, whereas ten Fig. 3. Top-surface temperature and temperature gradient of the track slab in 24 h. Case Studies in Thermal Engineering 61 (2024) 104992 6 Q.Q. Xu et al. Fig. 4. Temperature variation along the depth of the track slab at selected times. Fig. 5. Three-day temperature field of the LCSBTS in hot weather. Fig. 6. Longitudinal and cross sections selected in the LCSBTS. rail support concrete blocks on the west side are numbered 1′-10′from south to north. It is worth noticing that all longitudinal or cross sections pass through the mid-point of the rail support concrete blocks. Fig. 7 shows the temperature distribution on the cross-section 5-5′and longitudinal section A-A′of the LCSBTS at eight selected moments. As shown in Fig. 7(a), the temperature of the track slab surface shows the opposite trend in two periods of time. During the temperature rise stage, the temperature shows an increasing trend, and the temperature is higher on the east side than on the west side of the track slab. The temperature differences on horizontal planes in the cross-section 5-5′can reach the peak value of Case Studies in Thermal Engineering 61 (2024) 104992 7 Q.Q. Xu et al. Fig. 7. Temperature distribution on cross and longitudinal sections at selected moments. 5.70 °C at 12:00 a.m. On the top surface of the base plate, the temperature of the center is 17.93 °C higher than the temperature of the edges at 14:00 p.m. However, the temperature differences between the east side and the west side of the track slab gradually decrease in the temperature drop stage. Especially after 18:00, the temperature almost symmetrically distributes along the longitudinal center line of the track slab. The noticeable change in the temperature distribution characteristic of the track slab surface during the day and night indicates that solar radiation and the incident angle of the sun have a significant impact on the asymmetric distribution of temperature in the same cross-section during the daytime, from 6:00 a.m. to 18:00 p.m. As shown in Fig. 7(b), within the range of 10 cm below the top surface of the track slab, the temperature fluctuates at a fixed wavelength along the longitudinal direction of the track slab. The wavelength is the distance between two adjacent rail support concrete blocks, and each wave crest is located at the center of the concrete blocks. The maximum amplitude of temperature fluctuation is 3.1 °C, obtained at 14:00 p.m. This is mainly due to the thickness of the rail support concrete blocks being greater than the surrounding area, which increases the thermal conduction distance from the top to the bottom surface of the track slab. The temperature distribution of the fifth (counting from south to north) right rail support concrete block is shown in Fig. 8. As seen from Fig. 8, two types of temperature difference can be observed: the temperature difference between the center of the rail support concrete block and the surrounding area, and the temperature difference between the center and the edge of the track slab. The former temperature difference is due to the convective heat transfer at the four side surfaces of the rail support concrete block, which reduces the temperature near the edge of the rail support concrete block. The maximum temperature difference between the rail support concrete block and the surrounding areas reaches the peak at 18:00 p.m. and then gradually decreases. This temperature difference becomes negligible from 00:00 a.m. to 6:00 a.m. The temperature difference between the center and the edge of the track slab is the combined effect of the solar radiation and the convective heat transfer at the side surface of the track slab. In order to clarify the depth range where the temperature dramatically fluctuations, the temperature distribution of the track slab center along the vertical direction in 24 h has been presented in Fig. 9. As shown in Fig. 9, the temperature changes drastically within the depth range of 0∼10 cm below the top surface of the track slab, especially from 10:00 a.m. to 18:00 p.m. The maximum amplitude of temperature fluctuation reaches 15 °C at 14:00 p.m. As observed in Ref. [11], the deeper the distance below the top surface of the track slab, the more slightly the temperature fluctuates. The temperature within the base plate even barely fluctuates. The maximum vertical temperature difference of the CA mortar layer is 2 °C at 16:00, which is almost two times the temperature difference at the base plate. Moreover, there is an obvious phase temperature difference in the internal structure of LCSBTS, where the temperature change lags behind the top surface, and the lag phase temperature difference becomes larger with the increase in depth. Since the risFig. 8. Temperature distribution at the fifth right rail support concrete block (counting from south to north). Case Studies in Thermal Engineering 61 (2024) 104992 8 Q.Q. Xu et al. Fig. 9. Temperature distribution along the vertical direction in 24 h. ing moment of the bottom temperature lags behind the surface temperature, the positive temperature gradient would gradually increase to the summit, while, conversely, the negative temperature gradient gradually increases to the trough once the dropping moment of the bottom temperature lags behind the surface temperature. Furthermore, Fig. 10 shows the spatial temperature distribution of the LCSBTS at five typical depths for the maximum positive temperature gradient (occurring at 13:00 of the first day) and the maximum negative temperature gradient (occurring at 4:30 of the third day). The distances of 0 mm, 100 mm, 200 mm, 230 mm, and 280 mm from the top surface correspond to the top, middle, and bottom surfaces of the slab track, the bottom surface of the CA mortar layer, and the middle surface of the base plate, respectively. As seen in Fig. 10(a), at the time of maximum positive temperature gradient, the temperature of the ballastless track gradually decreases along the vertical direction. For the top surface of the track slab, the temperature of the rail support concrete block is higher than the surrounding area, with a maximum difference of 1.8 °C. Conversely, the temperature in the internal corresponding location is lower than the surrounding area, and the difference between them gradually decreases as the depth increases. For example, the maximum temperature difference is −5.56 °C at a depth of 100 mm, reducing to 0.3 °C once the depth exceeds 230 mm. Regarding the base plate, the edge temperature is significantly higher than the internal, with a maximum difference of 12.04 °C. The main reason is that the transverse width of the base plate is 200 mm wider than the track slab, resulting in this area being directly exposed to the environment, with temperatures nearly equal to the top surface (0 mm). As seen in Fig. 10(b), at the moment of the maximum negative temperature gradient, the temperature of the ballastless track gradually increases along the vertical direction. The temperature of the rail support concrete block is lower than the surrounding area, with a maximum difference of 1.25 °C. The greater the depth from the top surface, the smaller the difference between them in internal structure. The maximum difference for the middle surface of the track slab reduces to 0.42 °C, while the difference becomes negligible once the depth exceeds 200 mm. In contrast to the maximum positive temperature gradient, the edge temperature of the base plate is significantly lower than the internal temperature, with a maximum difference of 10.50 °C. From the comparison of the results presented, it can be seen that the temperature changes across the vertical direction from 33.37 °C to 53.37 °C at the time of the maximum positive temperature gradient, while it changes from 27.56 °C to 38.78 °C at the time of the maximum negative temperature gradient. Rail support concrete blocks significantly affect wavelengths of temperatures fluctuating at different depths. 3.2. Thermal deformation To investigate the temporal-spatial characteristics of temperature deformation under the non-uniform temperature field, six typical moments along one day are selected as listed in Table .3, while the corresponding vertical deformation distribution characteristics on the top horizontal plane of the LCSBTS are shown in Fig. 11. Among the six selected moments, the moment of 04:30 in the morning and 13:00 in the afternoon correspond to the maximum positive and negative temperature gradient moments at the center of the slab plate, respectively, which can exhibit the thermal effect characteristics at the peak of the temperature field. Before reaching the maximum positive temperature gradient, two critical moments of the initiation of interface damage (11:20) and the maximum solar altitude angle (12:00) are considered. The moment of 11:20 can help understand the relationship between interface damage and temperature deformation, and the moment of 12:00 is a critical moment for the change of lateral non-uniform temperature effect from the east to west. After reaching the maximum positive temperature gradient, two critical moments of the maximum vertical displacement of the track slab (14:00) and the moment of 16:00 when interface damage stops growing are selected. The time difference between the maximum vertical displacement moment and the maximum positive temperature gradient moment helps to understand the hysteresis Case Studies in Thermal Engineering 61 (2024) 104992 9 Q.Q. Xu et al. Fig. 20. Differences of deformation and interface damage on cross and longitudinal sections under design and non-uniform temperature loads. results under two types of temperature load. Taking the vertical displacement as an example, the design temperature load gives almost the same vertical displacement for all support concrete blocks, with only a slightly higher value at the support concrete blocks near the center of the track slab. On the contrary, the non-uniform temperature load gives a relatively higher value at the support concrete blocks near the longitudinal edge of the track slab. These make the difference of the vertical displacement vary along the longitudinal direction and produce a peak difference at the support concrete block at the longitudinal center of the track slab. 5. Conclusions In this paper, a calculation model of the non-uniform temperature field of the LCSBTS is established using real-time shadowing technology and the heat transfer principle. The model has been verified by on-site field testing. Subsequently, the sequential thermalmechanical coupling calculation method is applied to analyze the thermal effects of the LCSBTS under the previously simulated nonuniform temperature load. A comparison between the non-uniform temperature load and the design temperature load has been further discussed. The conclusions can be drawn as follows. 1) The temperature field of the LCSBTS exhibits periodicity and strong temporal-spatial non-uniformity. Both the temperature and temperature gradient of the LCSBTS vary periodically in days. In terms of spatial distribution, the temperature is symmetrically distributed in the transverse direction during night-time as mainly affected by convective heat transfer, while the temperature on the sunny side is significantly higher than on the shady one during daytime. Longitudinally speaking, the temperature fluctuates with a certain wavelength which is the distance between two adjacent rail support blocks, while the longitudinal fluctuations gradually weaken with the increase of depth. In the vertical direction, the temperature change in the internal structure often lags behind the top surface temperature, and the lag phase difference increases with the increase of depth. 2) The temperature deformation of the LCSBTS exhibits strong temporal-spatial non-uniformity along the horizontal plane under the non-uniform temperature field. Under null or weak solar irradiance, the deformation of the structure is transversely symmetrically distributed, while the temperature deformation on the sunny side is significantly higher than on the shady side under strong solar irradiance. Besides, the longitudinal deformation also fluctuates with a certain wavelength, which is the distance between two adjacent rail support concrete blocks, and the deformation of the rail support concrete block increases from the center area of the track slab to the joints. 3) Similar to the temporal-spatial non-uniformity of temperature deformation, the interface damage on the sunny side is significantly higher than on the shady side, and the value also fluctuates along the longitudinal direction with increases from the center of the track slab to the joints. Overall, the interface damage increment is caused by the temperature deformation increment under the non-uniform temperature load. Due to its irreversibility, the non-uniform distribution of interface damage will continue to be maintained after one day and then enter the next temperature effects calculation cycle as the initial interface damage. Case Studies in Thermal Engineering 61 (2024) 104992 16 Q.Q. Xu et al. 4) Significant differences exist in deformation and interface damage under design or non-uniform temperature loads. Typically, the design temperature is higher than the non-uniform temperature due to the strong spatial non-uniformity of the actual temperature field. As a result, except for the edge areas, the deformation and interface damage under the design temperature load are higher than those under the non-uniform temperature load. When compared to the design temperature load, the nonuniform temperature load is more effective in replicating the actual temperature field effect. Other meteorological factors besides solar radiation, environment temperature, and wind speed, are not included in the current thermal boundary in this work since the effects of these meteorological factors on the temperature of the track structure are slight compared to the effects of the solar radiation, environment temperature, and wind speed. Similarly, the shadow zone resulting from the shelter by other nearby structures is not considered. The sensitivity of meteorological factors on the temperature field of the LCSBTS has been reported in several references [6,37]. The authors also conducted a one-year on-site observation of the temperature field of the LCSBTS and analyzed the sensitivity of meteorological factors [38], which stands on the same side with references [6,18,37], proving the significance of the solar radiation, environment temperature, and wind speed on the temperature field of the LCSBTS. Considering the length limitation of this article and the research scope of exploring the non-uniformity of the temperature within LCSBTS different from the design temperature load and the corresponding displacement and damage of the LCSBTS by considering a nonuniform temperature load, the sensitivity analysis of meteorological factors is not addressed. Data availability The data used to support the findings of this study are available from the corresponding author upon request. CRediT authorship contribution statement Q.Q. Xu: Writing –original draft, Validation, Software. Y. Liu: Writing –review & editing, Writing –original draft, Supervision, Resources, Project administration, Methodology, Conceptualization. X.D. Sun: Writing –review & editing, Writing –original draft, Validation, Methodology, Conceptualization. D.Y. Jiang: Validation, Software, Methodology. Z.Y. Ji: Validation, Software, Methodology. Y. Xu: Supervision, Project administration. R. Arcos: Writing –review & editing. 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