A method for calculating deformation and damage of ballastless track base plate in seasonally frozen areas

Abstract

The present invention relates to the technical field of ballastless track, and specifically, to a method for calculating deformation and damage of a ballastless track base plate in a seasonally frozen area, which comprises the following steps: step 1, introducing damage evolution parameters, considering the tensile and compressive anisotropy of concrete, and building a concrete rate constitutive equation of the ballastless track base plate considering double scalar damage from the perspective of damage evolution; step 2, based on the concrete rate constitutive equation, establishing a ballastless track damage model considering the uneven frost heave effect of the roadbed; step 3, analyzing the deformation and damage of the ballastless track base plate under the action of frost heave position, frost heave wavelength, and frost heave amplitude. The present invention can more accurately and truly explore the deformation and damage of the ballastless track base plate components under the action of frost heave of the roadbed in the seasonally frozen area.

Description

Technical Field

[0001] The invention relates to the technical field of ballastless track, and in particular to a method for calculating deformation and damage of a base plate of a ballastless track in a seasonally frozen area. Background Art

[0002] With the development of high-speed railways, the operating mileage of railways in seasonally frozen areas continues to increase, and the problem of winter roadbed frost heave has attracted more and more attention. Under the action of uneven frost heave, the various components of ballastless track will deform to varying degrees, causing interlayer voids, structural damage and cracking, and other defects, affecting the durability of the structure and the smoothness and safety of driving. At present, the research on the deformation of ballastless track under the roadbed frost heave effect mainly focuses on revealing the deformation mapping mechanism and analyzing the mechanical state of the structure after deformation.

[0003] If we want to explore the coordinated mapping mechanism of deformation between subgrade and ballastless track, we first need to understand the frost heave law of subgrade in seasonally frozen areas. To this end, scholars have collected and analyzed subgrade frost heave data using artificial leveling observation, automatic monitoring, track dynamic monitoring and other means. The study found that the uneven frost heave curve of the subgrade conforms to the characteristics of the single-wave cosine function, which is also widely used in the characterization of track unevenness and subgrade settlement. Some scholars used the above-mentioned single-wave cosine function as the frost heave curve, established a subgrade-ballastless track finite element model, and used the frost heave curve action position, frost heave amplitude and frost heave wavelength as the subgrade frost heave characteristic parameters to analyze the deformation law of the track structure under different frost heave characteristics. On this basis, some scholars obtained the mapping relationship between the frost heave characteristics and the deformation of the track surface, track plate surface and base plate surface, revealing the coordination mechanism of subgrade frost heave and ballastless track deformation.

[0004] Regarding the study of the mechanical properties of ballastless track under the frost heave effect of roadbed, some scholars used the tensile strength of concrete as the criterion, calculated the tensile stress distribution of ballastless track components under the action of frost heave, and proposed the most unfavorable frost heave condition and corresponding maintenance standards. However, most of the above studies adopted the linear elastic constitutive model and did not consider the nonlinearity of concrete materials. Therefore, some scholars introduced the concrete damage constitutive model, established a ballastless track damage analysis model, and explored the stress deformation and damage law of the track structure under roadbed frost heave.

[0005] In summary, so far, scholars have made relatively complete research on the structural deformation and stress state of ballastless track under the action of roadbed frost heave. However, there are still some shortcomings: (1) Most models only consider the longitudinal heterogeneity when setting the roadbed frost heave curve. In actual engineering, the sunny and shady slopes of the roadbed will often experience different degrees of frost heave due to differences in environmental factors such as temperature and solar radiation. However, there are few studies in the existing literature on the deformation of track structures and the voiding under the slab under the effect of sunny and shady slopes; (2) When conducting deformation and damage analysis of ballastless track, the structural components are mostly based on the empirical constitutive model of concrete, which cannot well explain the generation and evolution of concrete damage and is not theoretically strong. Summary of the invention

[0006] The present invention provides a method for calculating deformation and damage of a ballastless track base plate in seasonally frozen areas, which can overcome certain or some defects of the prior art.

[0007] According to the present invention, a method for calculating deformation and damage of a base plate of a ballastless track in a seasonally frozen area comprises the following steps:

[0008] Step 1: Introduce damage evolution parameters, consider the tensile and compressive anisotropy of concrete, and build the concrete rate constitutive equation of ballastless track base plate considering double scalar damage from the perspective of damage evolution;

[0009] Step 2: Based on the concrete rate constitutive equation, a ballastless track damage model considering the uneven frost heave effect of the roadbed was established;

[0010] Step 3: Analyze the deformation and damage of the ballastless track base plate under the action of frost heave position, frost heave wavelength, and frost heave amplitude.

[0011] Preferably, in step 1, specifically:

[0012] Step 1.1. For isotropic elastic damaged materials, the following relationship exists:

[0013]

[0014] In the formula is the fourth-order elastic stiffness tensor, σ represents concrete stress, d represents the damage evolution parameter, ε represents concrete strain, and for elastic materials, the effective stress satisfy

[0015] Considering the strength difference of concrete in tension and compression process, the tensile damage evolution parameter is adopted. d + and compression damage evolution parameters d - To describe the evolution of concrete damage during service; similarly, by introducing the positive and negative stress tensors σ± Expresses the stress state of concrete under tension and compression, and the tensile damage evolution parameter d + Only by σ + The effect is generated, and the pressure damage evolution parameter d - Only by σ - The effect occurs; therefore, there is a relationship:

[0016]

[0017] In the formula, is the fourth-order damage tensor, is the fourth-order projection operator of the effective stress tensor under tension and compression; is a fourth-order unit tensor;

[0018] Step 1.2: Construct concrete damage evolution law

[0019] Introducing the damage energy release rate Y ± , there is a relationship:

[0020]

[0021] In the formula, is the Helmholtz free energy, and the stress-strain relationship satisfies Green elastic material, Helmholtz free energy is equal to the strain energy W, that is, the following relationship exists:

[0022]

[0023] use The Helmholtz free energy of concrete under different behaviors is expressed as follows:

[0024]

[0025] is the initial Helmholtz free energy of the material;

[0026] Substituting formula (4) into formula (3), we get:

[0027]

[0028] The damage energy release rate Y ± The function G constructed is used to represent the damage surface of the material, and the relationship is established:

[0029]

[0030] In the formula, Y, which controls the damage evolution, n± Equivalent to Y n ± The increasing function g ± (Y n ± ), because the function is arbitrary, take the function but E0 is the initial elastic modulus of concrete, is the fourth-order elastic flexibility tensor, which is related to the fourth-order elastic stiffness tensor Existence relation r n ± is the historical maximum damage energy release rate, which is calculated by formula (7):

[0031]

[0032] Among them, r0 ± is the threshold of the damage energy release rate when concrete is under tension or compression, and r0 ± =f0 ± =E0ε0 ± , f0 ± , ε0 ± are the linear elastic stress and strain of concrete in uniaxial tension and compression respectively; τ is the time interval;

[0033] The concrete damage criterion is determined by equations (6) and (7). On this basis, the damage parameter evolution criterion is introduced, as shown in equation (8):

[0034]

[0035] Concrete damage will only occur or develop when the damage function G = 0. At this time, Substituting into formula (8), we get:

[0036]

[0037] In formula (9), the concrete damage evolution parameter d ± and damage energy release rate r n ± The relationship between is as shown in formula (10):

[0038]

[0039] In the formula, A + A is the control parameter of concrete tensile stress-strain curve; - , B - is the control parameter of concrete compressive stress-strain curve;

[0040] Taking the derivative of both ends of equation (10) with respect to time, we can obtain the damage parameter evolution law: Among them, h ± is the damage evolution function, see formula (11):

[0041]

[0042] Step 1.3: Derivative both ends of equation (2) to obtain the concrete rate constitutive equation, as shown in equation (12):

[0043]

[0044] The rate of change of effective stress with time when under tension and compression Satisfy the relationship Among them, the fourth-order symmetric tensor Calculated by formula (13):

[0045]

[0046] In the formula, is the effective principal stress; P (ij) is a second-order symmetric tensor, and its calculation formula is shown in (14):

[0047]

[0048] is the direction vector corresponding to the effective principal stress;

[0049] For formula (12) Items are transformed, taking into account and when damage occurs or develops Therefore, the damage energy release rate that varies with time is obtained As shown in formula (15):

[0050]

[0051] The damage parameter evolution law Substitute into formula (15) In the equation, we get the following relationship:

[0052]

[0053] is a fourth-order tensor, and the relationship and Substituting into equation (12), we can obtain the concrete rate constitutive equation and the fourth-order tangent stiffness matrix: See formula (17):

[0054]

[0055] Preferably, in step 2, based on the concrete rate constitutive equation, a concrete damage constitutive subroutine is compiled, and then embedded in the CRTSⅢ ballastless track base plate in a material self-programming manner to establish a ballastless track damage model that takes into account the uneven frost heave effect of the roadbed.

[0056] As a preferred embodiment, in step 2, when compiling the concrete damage constitutive subroutine, the following requirements must be met: ① define the Jacobian matrix; ② update the stress and state variables used at the end of the incremental step;

[0057] The elastic Jacobian matrix is ​​constructed as shown in formula (18):

[0058]

[0059] In the formula, E0 is the elastic modulus and ν0 is the Poisson's ratio.

[0060] Preferably, in step 3, a yin-yang slope effect ratio parameter is introduced to describe the non-uniformity of the lateral frost heave of the roadbed, and the yin-yang slope effect ratio parameter η has the following relationship:

[0061]

[0062] Where A is the frost heave amplitude at the center of the yin-yang slope, b and c are the lateral non-uniform frost heave control parameters, and the yin-yang slope effect ratio η∈[0,1];

[0063] By adjusting the parameter η, the simulation of different yin-yang slope effects can be achieved. When η = 0, the frost heave amplitude of the yang slope A(bc) = the amplitude A at the center of the frost heave curve = the frost heave amplitude Ab of the yin slope. At this time, there is no lateral uneven frost heave on the frost heave curve; when η = 1, the frost heave amplitude of the yang slope A(bc) = 0, and the frost heave amplitude of the yin slope Ab = 2A. At this time, the yin-yang slope effect of the roadbed is the most significant.

[0064] Based on the thermodynamic constant under isothermal pure mechanics, the present invention constructs the concrete damage evolution law to determine the occurrence and development of concrete damage, establishes the concrete rate constitutive equation considering double scalar damage, and inputs it into the ballastless track base plate in a subroutine manner, which can accurately and truly explore the deformation and damage of the ballastless track base plate components under the frost heave of the roadbed in the seasonal freezing area. At the same time, the introduction of the positive and negative slope effect ratio parameter can better describe the unevenness of the lateral frost heave of the roadbed. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 This is a flow chart of the deformation and damage calculation method of the ballastless track base plate in seasonally frozen areas;

[0066] Figure 2 Schematic diagram of concrete damage evolution;

[0067] Figure 3The logic diagram for running the concrete damage constitutive subroutine;

[0068] Figure 4 This is the finite element model diagram of CRTSⅢ type slab ballastless track on the roadbed;

[0069] Figure 5 Set up schematic diagrams for model contact properties;

[0070] Figure 6 This is a schematic diagram of uneven frost heave of roadbed;

[0071] Figure 7 This is the distribution map of uneven frost heave locations on the roadbed;

[0072] Figure 8(a) is the stress cloud diagram of the base plate under the yin-yang slope effect when frost heave acts on position 1;

[0073] Figure 8(b) is the stress cloud diagram of the base plate under the yin-yang slope effect when frost heave acts on position 2;

[0074] Figure 9(a) is a schematic diagram of the frost heave when the base plate is damaged;

[0075] Figure 9(b) is a schematic diagram of the frost heave when macro cracks appear on the base plate;

[0076] Fig.10 Schematic diagram of the gap between the base plate and the roadbed. DETAILED DESCRIPTION

[0077] In order to further understand the content of the present invention, the present invention is described in detail in conjunction with the accompanying drawings and embodiments. It should be understood that the embodiments are only for explaining the present invention and are not intended to limit it.

[0078] Example

[0079] like Figure 1 As shown, this embodiment provides a method for calculating deformation and damage of a ballastless track base plate in a seasonally frozen area, which includes the following steps:

[0080] Step 1: Introduce damage evolution parameters, consider the tensile and compressive anisotropy of concrete, and build the concrete rate constitutive equation of ballastless track base plate considering double scalar damage from the perspective of damage evolution;

[0081] Step 2: Based on the concrete rate constitutive equation, a ballastless track damage model considering the uneven frost heave effect of the roadbed was established;

[0082] Step 3: Analyze the deformation and damage of the ballastless track base plate under the action of frost heave position, frost heave wavelength, and frost heave amplitude.

[0083] In step 1, specifically:

[0084] Step 1.1. For isotropic elastic damaged materials, the following relationship exists:

[0085]

[0086] In the formula is the fourth-order elastic stiffness tensor, σ represents concrete stress, d represents the damage evolution parameter, ε represents concrete strain, and for elastic materials, the effective stress satisfy

[0087] Considering the strength difference of concrete in the tensile and compressive processes, the tensile damage evolution parameter d + and the compression damage evolution parameter d - To describe the evolution of concrete damage during service; similarly, by introducing the positive and negative stress tensors σ ± Expresses the stress state of concrete under tension and compression, and the tensile damage evolution parameter d + Only by σ + The effect is generated, and the pressure damage evolution parameter d - Only by σ - The effect occurs; therefore, there is a relationship:

[0088]

[0089] In the formula, is the fourth-order damage tensor, is the fourth-order projection operator of the effective stress tensor under tension and compression; is a fourth-order unit tensor;

[0090] is the direction vector of the principal stress in the effective stress space, H(x) is the step function, and Step 1.2: Construct concrete damage evolution law

[0091] Introducing the damage energy release rate Y ± , there is a relationship:

[0092]

[0093] In the formula, is the Helmholtz free energy, and the stress-strain relationship satisfies Green elastic material, Helmholtz free energy is equal to the strain energy W, that is, the following relationship exists:

[0094]

[0095] Similar to the positive and negative decomposition of the stress tensor, The Helmholtz free energy of concrete under different behaviors is expressed as follows:

[0096]

[0097] is the initial Helmholtz free energy of the material;

[0098] Substituting formula (4) into formula (3), we get:

[0099]

[0100] The damage energy release rate Y ± The function G constructed is used to represent the damage surface of the material, and the relationship is established:

[0101]

[0102] In the formula, Y, which controls the damage evolution, n ± Equivalent to Y n ± The increasing function g ± (Y n ± ), because the function is arbitrary, take the function but E0 is the initial elastic modulus of concrete, is the fourth-order elastic flexibility tensor, which is related to the fourth-order elastic stiffness tensor Existence relation r n ± is the historical maximum damage energy release rate, which is calculated by formula (7):

[0103]

[0104] Among them, r0 ± is the threshold of the damage energy release rate when concrete is under tension or compression, and r0 ± =f0 ± =E0ε0 ± , f0 ± , ε0 ± are the linear elastic stress and strain of concrete in uniaxial tension and compression respectively; τ is the time interval;

[0105] Equations (6) and (7) describe the occurrence and development of concrete damage. Figure 2 The process is clearly expressed in the form of a schematic diagram. Figure 2 It can be seen that when the concrete is in process ①, the material damage energy release rate Y1 ±Does not exceed the threshold r0 ± (The historical maximum damage energy release rate is r0 ± ), the concrete is still in a linear state, the damage function G < 0, and there is no damage; in process ②, the material damage energy release rate Y2 ± Exceeding the threshold r0 ± , concrete begins to be damaged, at this time the damage function G=0, Y2 ± Replace r0 ± The largest damage energy release rate in history, r2 ± ; When concrete is in process ③, the material damage energy release rate is Y3 ± Does not exceed the historical maximum damage energy release rate r2 ± At this time, the damage function G is less than 0, and the concrete damage does not develop further; in the process ④, the material damage energy release rate Y4 ± Exceeds the historical maximum damage energy release rate r2 ± , concrete damage begins to develop, Y4 ± Replace r2 ± The highest damage energy release rate in history, r4 ± , at this time the damage function G=0.

[0106] The concrete damage criterion is determined by equations (6) and (7). On this basis, the damage parameter evolution criterion is introduced, as shown in equation (8):

[0107]

[0108] Concrete damage will only occur or develop when the damage function G = 0. At this time, Substituting into formula (8), we get:

[0109]

[0110] In formula (9), the concrete damage evolution parameter d ± and damage energy release rate r n ± The relationship between is as shown in formula (10):

[0111]

[0112] In the formula, A + A is the control parameter of concrete tensile stress-strain curve; - , B - is the control parameter of concrete compressive stress-strain curve;

[0113] Taking the derivative of both ends of equation (10) with respect to time, we can obtain the damage parameter evolution law: Among them, h ± is the damage evolution function, see formula (11):

[0114]

[0115] Step 1.3: When performing implicit analysis, ABAQUS uses the Newton iteration method to obtain the tangent stiffness matrix corresponding to the incremental step in the form of step-by-step approximation, and performs linear operations on this basis to obtain mechanical states such as unit integration point stress. Therefore, if you want to use the UMAT subroutine to perform ABAQUS implicit operations, you must obtain the tangent stiffness matrix. Based on the concrete constitutive equation proposed in formula (2) that considers double scalar damage, the concrete rate constitutive equation is constructed and the tangent stiffness matrix is ​​obtained.

[0116] Taking the derivative of both ends of formula (2), we get the concrete rate constitutive equation, as shown in formula (12):

[0117]

[0118] Formula (12) cannot reflect the tangent stiffness, so it is necessary to transform the above formula to satisfy In the form of, the fourth-order tensor It is the concrete tangent stiffness matrix.

[0119] The rate of change of effective stress with time when under tension and compression Satisfy the relationship Among them, the fourth-order symmetric tensor Calculated by formula (13):

[0120]

[0121] In the formula, is the effective principal stress; P (ij) is a second-order symmetric tensor, and its calculation formula is shown in (14):

[0122]

[0123] is the direction vector corresponding to the effective principal stress;

[0124] For formula (12) Items are transformed, taking into account and when damage occurs / develops Therefore, the damage energy release rate that varies with time is obtained As shown in formula (15):

[0125]

[0126] The damage parameter evolution law Substitute into formula (15) In the equation, we get the following relationship:

[0127]

[0128] is a fourth-order tensor, and the relationship and Substituting into equation (12), we can obtain the concrete rate constitutive equation and the fourth-order tangent stiffness matrix: See formula (17):

[0129]

[0130] In step 2, based on the concrete rate constitutive equation, a concrete damage constitutive subroutine is compiled, and then embedded in the CRTSⅢ ballastless track base plate in a material self-programming manner to establish a ballastless track damage model that considers the uneven frost heave effect of the roadbed.

[0131] Compilation of concrete damage constitutive subprogram and determination of parameters

[0132] When compiling the concrete damage constitutive subroutine, the following requirements must be met: ① define the Jacobian matrix; ② update the stress at the end of the increment step and the state variables used; on the basis of meeting the above requirements, compile the concrete constitutive subroutine, the operation logic of which is as follows: Figure 3 shown.

[0133] from Figure 3 It can be seen that the subroutine uses custom matrices and variables during the compilation process. Among them, EF_STRESS(K1) is the total effective stress DDS(K1,K2) is the elastic Jacobian matrix, which is constructed as shown in formula (18):

[0134]

[0135] In the formula, E0 is the elastic modulus and ν0 is the Poisson's ratio.

[0136] EF_STRESS_POSI(K1) and EF_STRESS_NEIG(K1) are the total effective stresses respectively. The effective tensile stress obtained by spectrum decomposition and the effective shear stress of concrete under compression Y_damage_POSI and Y_damage_NEIG are the positive and negative damage energy release rates of concrete Y ± ; DAMAGEOLD_POSI and DAMAGEOLD_NEIG represent the positive and negative damage evolution parameters d after the last cycle old ± rnold_POSI and rnold_NEIG are the historical maximum damage energy release rates r after the last cyclen_old ± ; r0_POSI, r0_NEIG are concrete damage energy thresholds r0 ± ; DAMAGE_POSI and DAMAGE_NEIG are the positive and negative damage evolution parameters d updated in this cycle ± rn_POSI and rn_NEIG are the historical maximum damage energy release rates r updated in this cycle n ± .

[0137] The concrete damage constitutive subroutine has 4 characteristic parameters and 3 curve control parameters for program operation, and 22 state variables for data transfer in the subroutine, as shown in Table 1.

[0138] Table 1 Subroutine parameters and state variables table

[0139]

[0140] Among the state variables in Table 1, STATEV(1) is used to store the tensile damage evolution parameters, STATEV(2) stores the compressive damage evolution parameters, STATEV(3) stores the historical maximum tensile damage energy release rate, and STATEV(4) stores the historical maximum compressive damage energy release rate. Figure 3 In the "Damage Generation and Development Determination" module, the value is first assigned to the damage evolution parameter d obtained in the previous cycle. old ± The historical maximum damage energy release rate r n_old ± , and read the damage energy release rate Y obtained by the "Effective Stress Calculation and Decomposition" module in this cycle ± The following two types of judgments are made:

[0141] The first type of judgment is used to update the damage evolution parameters. ± ≥r0 ± And Y ± ≥r n_old ± , then the damage occurs / continues to develop, and the damage evolution parameter d is recalculated ± And update the state variables STATEV(1) and STATEV(2); if the above inequality is not satisfied, it means that the damage has not occurred / developed, the damage evolution parameter value has not changed, d ± =d old ± .

[0142] The second type of judgment is used to update the historical maximum damage energy release rate. If Y ± ≥r0 ± , take the historical maximum damage energy release rate r after the last cyclen_old ± The damage energy release rate Y obtained in this cycle ± The larger one is taken as the historical maximum damage energy release rate r at the end of this cycle. n ± , and update the state variables STATEV(3) and STATEV(4); if the inequality is not satisfied, that is, Y ± <r0 ± , then the damage energy release rate Y calculated by this cycle is ± Update status variables STATEV(3) and STATEV(4).

[0143] The state variables STATEV(4+K1) are used to store the total effective stress components In the "Effective Stress Calculation and Decomposition" module, the program reads the state variable STATEV(4+K1) updated by the previous cycle and assigns it to the total effective stress The total effective stress of this cycle is calculated through the elastic Jacobian matrix DDS and the strain increment DSTRAN and the state variable STATEV(4+K1) is updated. On this basis, the effective stress is spectrally decomposed to obtain the positive and negative effective stresses. And update the status variables STATEV(10+K1) and STATEV(16+K1).

[0144] Study on deformation and damage of base plate under frost heave of roadbed

[0145] 1) Ballastless track damage model considering frost heave effect of roadbed

[0146] The above-established concrete deterministic damage constitutive model was adopted and embedded into the CRTSⅢ ballastless track base plate in a material self-programming manner, and a ballastless track damage model considering the uneven frost heave effect of the roadbed was established. On this basis, the damage of the base plate when frost heave acts at different positions was analyzed, and the calculation results were compared with those of relevant literature to verify the reasonable reliability of the model.

[0147] The CRTSⅢ slab track is composed of rails, fastener system, track plate, self-compacting concrete and base plate from top to bottom. Among them, the rails are selected from the 60kg / m type of high-speed rail, which are simulated with solid units. According to the actual engineering situation, the fasteners are selected as WJ-8 type with a spacing of 630mm. Considering that this model only studies the vertical deformation and damage of the track structure, in order to simplify the modeling and enhance the convergence of the model, the lateral and longitudinal stiffness of the fasteners are ignored, and the axial spring (springa) is used to simulate the fasteners, and the stiffness is set to 35kN / mm. The track plate adopts P5600 plate with dimensions of 5600mm (length) × 2500mm (width) × 210mm (height). The concrete strength grade of the plate is C60, and there is a 70mm plate gap between the two track plates. The strength grade of the self-compacting concrete layer is C40. Its length and width are the same as those of the track plate, and its height is 100mm. Two 1000mm (length) × 700mm (width) × 100mm (height) bosses are provided at the bottom to contact the base plate. In the model, both the track plate and the self-compacting concrete layer are simulated by solid elements. The results show that when the length of the base plate is between 2 and 5 track plates, the lateral stability of the structure is good and meets the concrete strength requirements. For this reason, this paper selects the length of three track plates as the longitudinal connection size of the base plate, and uses solid elements for simulation. The concrete strength grade is C40.

[0148] In actual engineering, in order to prevent the structure from cracking due to temperature effects, expansion joints with a width of 20 mm are set at both ends of the base plate, and the expansion joints are filled with caulking plates, and the top and sides are sealed with sealing caulking glue. At the same time, for base plates in sections where the roadbed filler is graded gravel, dowel bars are generally required to be installed at the expansion joints to maintain the longitudinal stability of the structure. Since the expansion joint is a weak part of the structure, and the longitudinal effect of the dowel bar will weaken the deformation of the track structure. Therefore, considering the most unfavorable conditions, it is believed that the filling material in the expansion joint is completely destroyed and the dowel bar between the base plates is broken.

[0149] In the concrete components of ballastless track, the presence of steel bars can not only improve the tensile strength and crack resistance of the structural components, but also enhance their stiffness and stability. Considering that the base plate is the main research object of this paper, in order to reduce the amount of model calculation and improve the calculation speed, only the internal steel bars of the base plate concrete are established here. The base plate steel mesh consists of upper steel bars, groove anti-cracking bars, groove reinforcement bars and lower steel bars. The steel bar type is CRB550 grade cold-rolled ribbed steel bars.

[0150] In order to eliminate the boundary effect, three base plates were built, and the middle plate was used as the research object. The lower part of the base plate is supported by the roadbed surface, and its longitudinal length is the same as the length of the three base plates, with dimensions of 51110mm (length) × 3100mm (width) × 400mm (height), and solid units are also used for simulation.

[0151] Assemble the components from bottom to top in sequence to obtain the finite element model of CRTSⅢ slab track on the roadbed, as shown in Figure 4 shown.

[0152] The parameters of each component of the track structure adopt the data in Table 3 below. Among them, the base plate is a user-defined material parameter.

[0153] Table 3 Parameters of various components of CRTSⅢ slab track

[0154]

[0155]

[0156] 2) Interaction and boundary condition setting

[0157] When the track slab is cast, gate-shaped steel bars are reserved to enhance the connectivity between the track slab and the self-compacting concrete layer, ensuring that there is no relative displacement between the two. Therefore, tie binding is used in the model to constrain the track slab and the self-compacting concrete layer. The steel mesh uses embedded constraints to interact with the base plate to simulate the reinforced base plate concrete.

[0158] In terms of contact property settings, geotextiles are laid between the bottom of the self-compacting concrete layer and the base plate surface in the project, so the normal direction of the two is considered to be hard contact, there is friction between the layers, and the penalty contact property is used, and the friction coefficient is set to 0.7; the self-compacting concrete bottom boss and the base plate groove are surrounded by polyurethane elastic cushions. The model uses soft contact properties to simulate its normal behavior, the contact stiffness is set to 100GPa, and it is considered that there is no friction between the two in the tangential direction; the base plate is directly cast on the surface of the roadbed. When setting the contact properties, the normal behavior is considered to be hard contact, the tangential direction is penalty contact, and the friction coefficient is set to 0.5. The contact property settings are as follows: Figure 5 shown.

[0159] Through trial calculation, the interaction between the self-compacting concrete layer and the base plate, the base plate and the roadbed surface, and the self-compacting concrete boss and the base plate groove is established by adopting a general contact method and assigning contact properties respectively. This method has low modeling difficulty and good convergence.

[0160] In terms of boundary condition setting, the longitudinal displacement of both ends of the rail is constrained to simulate the seamless line; since the axial spring springa has no rotational freedom and this embodiment only studies the vertical effect of the model, the longitudinal and lateral displacements of the fasteners are constrained; in order to simulate the supporting effect of the roadbed subgrade, the bottom surface of the roadbed surface layer is fully constrained.

[0161] The frost heave curve is input into the roadbed surface with continuous displacement boundary conditions using analytical fields to simulate the frost heave effect of the roadbed. Analysis shows that the track structure is most unfavorable under the classic cosine frost heave curve. Therefore, this embodiment selects the cosine curve as the longitudinal uneven frost heave of the roadbed, and the curve formula is as shown in formula (19).

[0162]

[0163] Where A is the frost heave amplitude, L1 is the frost heave wavelength, z0 is the longitudinal initial position of the frost heave curve, and z is the longitudinal position along the frost heave curve.

[0164] Since the roadbed of the high-speed railway is a strip structure, the two sides of its cross section will produce a more significant yin-yang slope effect under the influence of factors such as temperature and solar radiation, and the frost heave amplitude of the yang slope is smaller than that of the yin slope. It can be seen that the frost heave curve also needs to consider the lateral uneven frost heave of the roadbed. This embodiment assumes that the lateral frost heave of the roadbed changes uniformly, and the formula for the roadbed frost heave curve is obtained as shown in formula (20), and the schematic diagram is shown in Figure 6 shown.

[0165]

[0166] Where L2 is the width of the roadbed surface, b and c are the lateral uneven frost heave control parameters, x0 is the lateral initial position of the frost heave curve, and x is the lateral position along the frost heave curve.

[0167] From formula (20) and Figure 6 It can be seen that when the longitudinal position of the frost heave is at the midpoint of the frost heave wavelength, that is, When , the frost heave curve has a maximum frost heave. At this time, substituting x-x0=0 into formula (20), we can get the frost heave of the roadbed on the shady slope as Abmm; Substituting into the formula, we can get the frost heave of the roadbed at the center of the yin and yang slopes as Amm; substituting x-x0=L2 into the formula, we can get the frost heave of the roadbed at the yang slope as A(bc)mm.

[0168] 3) Analysis of base plate deformation and damage under the yin-yang slope effect

[0169] a. The influence of frost heave position on the base plate

[0170] According to the structural characteristics of CRTSⅢ slab ballastless track, the subgrade frost heave can be mainly divided into four types: frost heave in the middle of the track plate, frost heave at the limit groove of the base plate, frost heave at the seam of the track plate, and frost heave at the expansion joint of the base plate. On this basis, it can be further divided into 7 working conditions according to the specific action position of the frost heave curve. The uneven frost heave position distribution of the subgrade under different working conditions is as follows: Figure 7 shown.

[0171] Considering the non-uniformity of the transverse frost heave of the roadbed, the frost heave curve with frost heave wavelength L1 = 10m, frost heave amount A = 10mm at the center of the yin-yang slope, frost heave amplitude A(bc) = 5mm on the yang slope, and frost heave amplitude Ab = 15mm on the yin slope was used as an example to explore the influence of the frost heave position of the roadbed on the deformation and damage of the base plate under the yin-yang slope effect. The damage cloud map of the base plate under 7 working conditions was obtained.

[0172] The frost heave of the roadbed under each working condition when the base plate concrete begins to be damaged and the first macro crack appears is extracted to obtain the following Table 4.

[0173] Table 4 Damage development of base plate under frost heave of roadbed

[0174]

[0175] It can be seen from Table 5 and the damage cloud map that the damage to the base plate mainly occurs at the peak and foot of the frost heave slope. Among them, when the frost heave acts on position 1, the base plate is subjected to a relatively uniform force, and the four cracks are symmetrically distributed along the center line of the yin and yang slopes. When the frost heave acts on positions 2 to 7, the base plate presents an "S" shape under the action of embankment frost heave and gravity. The cracks on the upper surface are radial, developing from the peak of the yin slope to the foot of the yang slope, and an oblique crack develops from the foot of the yin slope to the peak of the yang slope on the lower surface.

[0176] According to the above analysis, when the lateral uneven frost heave of the roadbed is not considered, the base plate in the limit groove area will be damaged earlier when the frost heave acts on the track plate. Under the yin-yang slope effect, when the frost heave acts on the middle position of the track plate (position 1, position 5), the base plate will be damaged first. To explore the reason, the maximum principal stress cloud diagram of the base plate when the frost heave amount A = 1.7mm at the center of the yin-yang slope is extracted for the working conditions of position 1 and position 2, as shown in Figure 8(a) and Figure 8(b).

[0177] As shown in Figure 8(a) and Figure 8(b), under the Yin-Yang slope effect, the stress of the base plate develops from the peak of the Yin slope to the foot of the Yang slope. According to the above analysis, the four corners of the base plate limit groove are sharp in shape, which is prone to stress concentration. When frost heave acts on position 1, the corners of the limit groove are in the direction of stress development, so the stress concentration phenomenon at this position is more serious, and the maximum principal stress is 2.13MPa; when frost heave acts on position 2, the corners of the limit groove are not in the direction of stress development, and the stress concentration is not obvious. The maximum principal stress at this position is slightly smaller than that of working condition 1, which is 2.06MPa. According to the above analysis, when the Yin-Yang slope acts on the middle position of the track plate, the damage of the base plate will appear earlier.

[0178] In addition, a comprehensive comparison of the number of cracks in the base plate, the amount of frost heave when damage occurs, and the amount of frost heave when macro cracks occur shows that, compared with other working conditions, when the yin-yang slope acts on position 2, the base plate is in the most unfavorable damage state. Therefore, when analyzing the damage of the base plate, the working condition at position 2 can be selected as the main research object. b) Influence of frost heave amplitude and frost heave wavelength on the base plate

[0179] Assuming that the frost heave amplitude at the center of the yin-yang slope is A=10 mm, and introducing the yin-yang slope effect ratio parameter η, η has the following relationship:

[0180]

[0181] Where A is the frost heave amplitude at the center of the yin-yang slope, b and c are the lateral uneven frost heave control parameters, and the yin-yang slope effect ratio η∈[0,1].

[0182] According to formula (21), by adjusting the parameter η, the simulation of different yin-yang slope effects can be achieved. When η = 0, the frost heave amplitude of the yang slope A(bc) = the amplitude at the center of the frost heave curve A = the frost heave amplitude of the yin slope Ab, and at this time, there is no lateral uneven frost heave on the frost heave curve; when η = 1, the frost heave amplitude of the yang slope A(bc) = 0, and the frost heave amplitude of the yin slope Ab = 2A, and at this time, the yin-yang slope effect of the roadbed is the most significant. In this embodiment, η is set to 0, 1, and its parameter settings are shown in Table 5 below.

[0183] Table 5 Yin Yang slope parameter settings

[0184]

[0185] Combined with engineering practice and reference to relevant literature, four frost heave wavelengths L1 of 10m, 15m, 20m and 25m were selected, and together with the above-mentioned four yin-yang slope effect ratios η, a total of 16 working conditions were formed to explore the influence of frost heave amplitude and frost heave wavelength on the deformation and damage of the base plate.

[0186] According to the above, when analyzing the damage of the base plate, the condition where frost heave occurs at position 2 is selected as the research object. The frost heave amount at the center of the yin-yang slope when the base plate begins to be damaged and the first macro crack appears under different yin-yang slope effect ratios η and different frost heave wavelengths L1 is shown in Figure 9(a) and Figure 9(b).

[0187] According to Figure 9(a) and Figure 9(b), as the yin-yang slope effect ratio η increases and the frost heave wavelength L1 decreases, the frost heave amount at the center of the yin-yang slope gradually decreases when the concrete of the base plate begins to be damaged and the first macro crack appears. It can be seen that a larger yin-yang slope effect ratio and a shorter frost heave wavelength are more unfavorable to the track structure. When the frost heave wavelength is 10m, the yin-yang slope effect ratio η gradually increases from 0 to 1, and the frost heave amount at the center decreases from 2.2mm to 1.6mm when the base plate begins to be damaged, a decrease of 27.3%; when the frost heave wavelength is 15m, η gradually increases, and the frost heave amount decreases from 4.9mm to 2.8mm when damage occurs, a decrease of 42.9%; when the frost heave wavelength is 20m, η increases, and the frost heave amount at the center of the damaged area decreases from 8.7mm to 4.1mm, a decrease of 52.9%. It can be seen that the yin-yang slope effect ratio has a greater impact on the damage of the base plate under a longer frost heave wavelength.

[0188] Similarly, when the yin-yang slope effect ratio η is 0, 1. When the frost heave wavelength L1 gradually decreases from 20m to 10m, the frost heave amount at the center of the yin and yang slopes decreases by 74.7%, 74.4%, 67.8% and 61.0% respectively when the base plate is damaged. It can be seen that the influence of the frost heave wavelength on the base plate damage will increase when the yin and yang slope effect ratio is smaller.

[0189] Apply the frost heave curve to position 5 to explore the bottom void of the base plate under different yin-yang slope effect ratios η and different frost heave wavelengths L1. Fig.10 shown.

[0190] from Fig.10 It can be seen that as the frost heave wavelength L1 increases and the yin-yang slope effect ratio η decreases, the amount of void space between the base plate and the roadbed gradually decreases. When the frost heave wavelength increases from 10m to 20m (the wavelength increases by 100%), the void space under different yin-yang slope effect ratios η decreases by 88.7%, 62.5%, 60.0%, and 52.3%, respectively, with an average decrease of 65.9%; and when the yin-yang slope effect ratio is Increase to (The effect ratio increases by 100%), the bottom void volume of the plate under different frost heave wavelengths L1 decreases by 16.3%, 27.0%, 35.3%, and 44.4%, respectively, with an average decrease of 30.8%. It can be seen that the effect of frost heave wavelength on the bottom void volume of the plate is more significant.

[0191] The yin-yang slope effect ratio η=0, frost heave wavelength L1=10m( Fig.10 (a) of the Regulations); L1=10m( Fig.10 Part (b)) and L1=20m( Fig.10(c) The bottom degassing curves of the base plate under three working conditions. It can be seen that the maximum degassing of the base plate under the three working conditions is located at the corner of the base plate, and changing the yin-yang slope effect ratio and the frost heave wavelength has little effect on the position of the maximum degassing value. Horizontal comparison Fig.10 From parts (a) and (b), we can see that the effect ratio η only changes the amplitude and waveform of the air gap curve, but the air gap range does not change. Fig.10 In parts (b) and (c), the first peak of the frost heave curve almost disappears after increasing the frost heave wavelength, which shows that the frost heave wavelength has a greater impact on the voiding range.

[0192] Therefore, when the wavelength L1 of the roadbed frost heave curve is less than 25m and the yin-yang slope effect ratio η exceeds At the same time, the damage state of the base plate components and the air gap under the plate are relatively unfavorable, which will affect the service performance of the track structure to a certain extent.

[0193] The present invention and its embodiments are described schematically above, and the description is not restrictive. The drawings show only one embodiment of the present invention, and the actual structure is not limited thereto. Therefore, if a person skilled in the art is inspired by it and designs a structural method and an embodiment similar to the technical solution without creativity without departing from the purpose of the invention, they shall all fall within the protection scope of the present invention.

Claims

1. A method for calculating deformation and damage of a ballastless track base plate in seasonally frozen areas, characterized in that: The following steps are involved: Step 1: Introduce damage evolution parameters, consider the tensile and compressive anisotropy of concrete, and build the concrete rate constitutive equation of ballastless track base plate considering double scalar damage from the perspective of damage evolution; In step 1, specifically: Step 1.

1. For isotropic elastic damaged materials, the following relationship exists: In the formula is the fourth-order elastic stiffness tensor, σ represents concrete stress, d represents the damage evolution parameter, ε represents concrete strain, and for elastic materials, the effective stress satisfy Considering the strength difference of concrete in the tensile and compressive processes, the tensile damage evolution parameter d + and the compression damage evolution parameter d - To describe the evolution of concrete damage during service; similarly, by introducing the positive and negative stress tensors σ ± Expresses the stress state of concrete under tension and compression, and the tensile damage evolution parameter d + Only by σ + The effect is generated, and the pressure damage evolution parameter d - Only by σ - The effect occurs; therefore, there is a relationship: In the formula, is the fourth-order damage tensor, is the fourth-order projection operator of the effective stress tensor under tension and compression; is a fourth-order unit tensor; Step 1.2: Construct concrete damage evolution law Introducing the damage energy release rate Y ± , there is a relationship: In the formula, is the Helmholtz free energy, and the stress-strain relationship satisfies Green elastic material, Helmholtz free energy is equal to the strain energy W, that is, the following relationship exists: use The Helmholtz free energy of concrete under different behaviors is expressed as follows: is the initial Helmholtz free energy of the material; Substituting formula (4) into formula (3), we get: The damage energy release rate Y ± The function G constructed is used to represent the damage surface of the material, and the relationship is established: In the formula, the control damage evolution Equivalent to The increasing function Since the function is arbitrary, take the function but E0 is the initial elastic modulus of concrete, is the fourth-order elastic flexibility tensor, which is related to the fourth-order elastic stiffness tensor Existence relation is the historical maximum damage energy release rate, which is calculated by formula (7): in, is the threshold of the damage energy release rate when concrete is subjected to tension and compression, are the linear elastic stress and strain of concrete in uniaxial tension and compression respectively; τ is the time interval; The concrete damage criterion is determined by equations (6) and (7). On this basis, the damage parameter evolution criterion is introduced, as shown in equation (8): Concrete damage will only occur or develop when the damage function G = 0. At this time, Substituting into formula (8), we get: In formula (9), the concrete damage evolution parameter d ± Damage Energy Release Rate The relationship between is as shown in formula (10): In the formula, A + A is the control parameter of concrete tensile stress-strain curve; - , B - is the control parameter of concrete compressive stress-strain curve; Taking the derivative of both ends of equation (10) with respect to time, we can obtain the damage parameter evolution law: in, h± is the damage evolution function, see formula (11): Step 1.3: Derivative both ends of equation (2) to obtain the concrete rate constitutive equation, as shown in equation (12): The rate of change of effective stress with time when under tension and compression Satisfy the relationship Among them, the fourth-order symmetric tensor Calculated by formula (13): In the formula, is the effective principal stress; P(ij) is a second-order symmetric tensor, and its calculation formula is shown in (14): is the direction vector corresponding to the effective principal stress; For formula (12) Items are transformed, taking into account and when damage occurs or develops Therefore, the damage energy release rate that varies with time is obtained As shown in formula (15): The damage parameter evolution law Substitute into equation (15) In the equation, we get the following relationship: is a fourth-order tensor, and the relationship and Substituting into equation (12), we can obtain the concrete rate constitutive equation and the fourth-order tangent stiffness matrix: See formula (17): Step 2: Based on the concrete rate constitutive equation, a ballastless track damage model considering the uneven frost heave effect of the roadbed was established; Step 3: Analyze the deformation and damage of the ballastless track base plate under the action of frost heave position, frost heave wavelength, and frost heave amplitude.

2. The method for calculating deformation and damage of the base plate of ballastless track in seasonally frozen areas according to claim 1 is characterized in that: In step 2, based on the concrete rate constitutive equation, a concrete damage constitutive subroutine is compiled, and then embedded in the CRTSⅢ ballastless track base plate in a material self-programming manner to establish a ballastless track damage model that considers the uneven frost heave effect of the roadbed.

3. The method for calculating deformation and damage of the base plate of ballastless track in seasonally frozen areas according to claim 2 is characterized in that: In step 2, when compiling the concrete damage constitutive subroutine, the following requirements must be met: ① define the Jacobian matrix; ② update the stress and state variables used at the end of the increment step; The elastic Jacobian matrix is ​​constructed as shown in formula (18): In the formula, E0 is the elastic modulus, ν0 is Poisson's ratio.

4. The method for calculating deformation and damage of the base plate of ballastless track in seasonally frozen areas according to claim 3 is characterized by: In step 3, the yin-yang slope effect ratio parameter is introduced to describe the non-uniformity of the lateral frost heave of the roadbed. The yin-yang slope effect ratio parameter η has the following relationship: Where A is the frost heave amplitude at the center of the yin-yang slope, b and c are the lateral non-uniform frost heave control parameters, and the yin-yang slope effect ratio η∈[0,1]; By adjusting the parameter η, the simulation of different yin-yang slope effects can be achieved. When η = 0, the frost heave amplitude of the yang slope A(bc) = the amplitude A at the center of the frost heave curve = the frost heave amplitude Ab of the yin slope. At this time, there is no lateral uneven frost heave on the frost heave curve; when η = 1, the frost heave amplitude of the yang slope A(bc) = 0, and the frost heave amplitude of the yin slope Ab = 2A. At this time, the yin-yang slope effect of the roadbed is the most significant.

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