Calculation Method for Track Constraint Reduction Effect in Multi-Span Simply Supported Beam Bridges with Ballastless Track

Abstract

This invention discloses a method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track. The method includes acquiring parameter information of the multi-span simply supported beam bridge with ballastless track; constructing a calculation model of the reduction effect in the bridge-track system; analyzing the track constraint reduction effect under different span numbers; analyzing the law of action of the track constraint reduction effect between different bridge spans; and completing the calculation of the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track. This method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track provided by this invention can quickly determine the changing trend of the track constraint reduction effect, analyze the influence of the bridge substructure stiffness on the structural seismic response, and analyze the influence of the bridge subgrade track structure stiffness on the structural seismic response. Furthermore, this invention has high reliability and accuracy.

Description

Technical Field

[0001] This invention belongs to the field of civil engineering, specifically relating to a method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track. Background Technology

[0002] With the development of economy and technology and the improvement of people's living standards, high-speed railway systems have played an irreplaceable role in people's production and life.

[0003] Due to its advantages such as high smoothness and low maintenance, ballastless track has become the main structural form of high-speed railway tracks. To ensure the safety of high-speed railway bridges under seismic loads, the track constraint effect must be considered for bridges with seamless tracks. The presence of the track structure makes the bridge a weakly coupled continuous structure. Under longitudinal seismic loads, the continuously laid track slabs and rails cause relative displacement between the beam and rail. Because the longitudinal displacement of the main beam is constrained by the longitudinal resistance of the track, equal and opposite longitudinal forces are generated between the beam and rail. Through the interaction forces between the beam and rail, the track structure affects the seismic response of the bridge structure. This is the basic principle of the track system constraint effect under seismic loads. Considering that the longitudinal stiffness of the track system is usually large, it is essential to study the role of the track structure in the bridge structure during earthquakes.

[0004] Most existing studies have shown that track constraint effects have a significant impact on the seismic response of simply supported beam bridges for high-speed railways. The track structure is a longitudinally continuous structure on both the roadbed and the bridge. Under seismic loading, the longitudinal forces in the track structure are transmitted to the roadbed through the abutment anchorage mechanism. As the total number of bridge spans increases, the longitudinal forces in the track structure also increase. Since the longitudinal stiffness provided by the roadbed is limited, this leads to a gradual weakening of the influence of track constraint effects on the seismic response of multi-span simply supported beam bridges as the number of spans increases to a certain extent. This phenomenon is referred to as the reduction effect of track structure on bridge structure response.

[0005] Due to the track constraint reduction effect, it is impossible to determine whether the track structure can reduce the seismic response of a multi-span simply supported beam bridge when the number of spans is large. Therefore, it is crucial to explore the variation of the track structure's reduction effect on the bridge structure with the total number of spans. Existing studies often only select a few specific numbers of simply supported beam bridges when analyzing the influence of track constraint effects, which leads to several problems: 1) Existing methods cannot quickly reflect the changing trend of the reduction effect in the bridge-track system with the increase of the total number of spans; 2) The track constraint reduction effect in multi-span simply supported beam bridges with ballastless track under seismic loading is difficult to calculate quickly; 3) It is impossible to quickly determine the degree of influence of the total number of spans on the differences in the reduction effect between the spans in the bridge-track system. Summary of the Invention

[0006] The purpose of this invention is to provide a reliable and accurate method for calculating the track constraint reduction effect in multi-span simply supported beam bridges with ballastless track.

[0007] The method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track provided by this invention includes the following steps:

[0008] S1. Obtain parameter information for multi-span simply supported beam bridges with ballastless track;

[0009] S2. Based on the obtained parameter information, construct a calculation model for the reduction effect in the bridge track system;

[0010] S3. Based on the calculation model constructed in step S2, analyze the track constraint reduction effect under different span numbers;

[0011] S4. Based on the calculation results obtained in step S3, analyze the law of track constraint reduction effect between different bridge spans, and complete the calculation of track constraint reduction effect in multi-span simply supported beam bridges with ballastless track.

[0012] Step S2, which involves constructing a calculation model for the reduction effect in the bridge-rail system based on the obtained parameter information, specifically includes the following steps:

[0013] A model is created for a single-span railway simply supported beam bridge and rail system. The system is assumed to be in a linear elastic state, and springs and equivalent points are used to represent the structure of the bridge and rail system.

[0014] Let the magnitude of the seismic inertial force generated at a certain moment act on the center of mass of the main beam.

[0015] If the track structure is neglected, the force balance equation of the main beam is -F + k. d u bw +F b =0, where k d For the stiffness of the substructure, u bw The displacement of the left end of the main beam, F b This refers to the longitudinal force of the sliding support;

[0016] If the track structure is considered, the force balance equation of the main beam is:

[0017] -F+k d u b +F b +pL b +k c (u b -u1)=0

[0018] Where u b Let L be the displacement at the left end of the main beam, p be the beam-rail interaction force, and L be the displacement at the left end of the main beam. b Main beam length, k cdenoted as shear tooth groove, u1 is the track structure displacement at the shear tooth groove;

[0019] The internal forces in the track structure are at their maximum at the junction of the bridge and the left subgrade. At this point, the force balance equation of the bridge-track system is:

[0020] -F+k j u1+k d u b +F b +k j k b / (k j +k b u1=0

[0021] In the formula k b For the track structure stiffness of the bridge span; k j To obtain the equivalent stiffness of the roadbed track according to the equivalence principle and Where k0 is the end spike stiffness in the rear anchorage structure, k t For the track structure stiffness of the roadbed section and E t For the elastic modulus of the track structure, A t L is the area of ​​the track structure. j k is the length of the track structure in the roadbed section. m denoted as friction plate stiffness; j represents the variation in anchorage length behind the platform. L m The length of the friction plate;

[0022] According to the principle of virtual work, the displacement u at the left end of the main beam can be obtained. b The reaction force s1 at the fixed support is:

[0023]

[0024]

[0025] In the formula F e For the equivalent of the bridge-rail system and F e =FF b ;

[0026] The displacement difference Δu of the single-span main beam, considering and not considering the track structure, is calculated as follows:

[0027] Since both Δu and Δs are less than 0, it indicates that the track structure will reduce the displacement of the main beam and the support reaction force.

[0028] Considering that the stiffness of the shear tooth groove is much greater than that of other components, take Then u b The formulas for calculating s1 and Δu are simplified to: and

[0029]

[0030] When the number of bridge spans n is greater than 1, assuming that each span is subjected to the same external force F, then the displacement u of the main beam of any span of the bridge without considering the track structure is... bw (n) is

[0031] When considering bridges with track structures, the displacement u of the left end of the main beam under the action of a single external force F in any i-th span can be obtained based on the principle of virtual work. i (i,n) is Where k L (n) is an intermediate variable and k R (n) is an intermediate variable and According to the characteristic root method, the general term can be obtained. and Where b1 and b2 are the equivalent stiffness coefficients of the right-side bridge rail system and c1 and c2 are the equivalent stiffness coefficients of the left-side bridge rail system and B(n) is the equivalent stiffness of the bridge rail system on the right side of the i-th main beam across n spans, and C(n) is the equivalent stiffness of the bridge track system on the left side of the i-th main beam across n spans, and a1 is the stiffness coefficient of the substructure and a2 is the stiffness coefficient of the roadbed track structure and

[0032] Taking the total number of spans as 2n-1, and the displacement of the left end of the main beam in the first span under the action of a single external force F as u1(1,2n-1), then the displacement u of the left end of the main beam in the i-th span under the action of a single external force F in the first span can be calculated. i (1, 2n-1) is According to the characteristic root method, u can be obtained i The general term of (1, 2n-1) is: Where n1 and n2 are the displacements of the main beam in the first span under a single external force F in a bridge-track system with a total span of 2n-1. and X1(i) and X2(i) are the displacement coefficients of the main beam of the i-th span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

[0033]

[0034] According to the reciprocal displacement theorem, the displacement u(1,2n-1) at the left end of the main beam of the first span caused by the external forces in all spans is:

[0035]

[0036] In the formula, S1(n) and S2(n) are the total displacement coefficients of the first n spans of the main beam under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

[0037] Assuming that each span is subjected to the same external force F, the displacement u(i,2n-1) of the main beam of any span of the bridge considering the track structure is calculated as follows:

[0038]

[0039] According to the characteristic root method, the general term of u(i,2n-1) can be obtained as:

[0040]

[0041] In the formula, n3 and n4 represent the displacements of the main beam of the first span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

[0042] According to u bw The formulas for calculating (n) and u(i,2n-1) are used to calculate the displacement difference r(i,2n-1) of the main beam at different locations of the trackless bridge and the tracked bridge when each span is subjected to the same external force F.

[0043]

[0044] Step S3, which involves calculating the effect of track constraint reduction under different span numbers based on the computational model constructed in step S2, specifically includes the following steps:

[0045] Taking the k-th order partial derivative of r(i,2n-1) with respect to n, we have

[0046] When a1 and a2 satisfy When the first span is subjected to a single external force F, the displacement u1(1,2n-1) of the left end of the main beam increases with the increase of the total number of spans;

[0047] Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for calculating the k-th order partial derivative of r(i,2n-1) with respect to n, we obtain... and Therefore, in practical engineering, the reduction effect in the bridge-track system weakens as the total number of spans increases, and the rate of reduction decreases as the total number of spans increases.

[0048] As the number of spans approaches positive infinity, the bridge displacement has a limit value:

[0049]

[0050]

[0051] According to u max The formula for calculating (n) and u b The calculation formula is used to calculate the maximum displacement difference r between the main beam of the mid-span of the trackless bridge and the tracked bridge. max for

[0052] Therefore, we can conclude that: when the number of spans is 1, the reduction effect of the track structure on the bridge structure is the strongest; as the number of spans increases, the reduction effect of the track structure on the bridge structure weakens; the larger the number of spans, the smaller the reduction effect of the track structure on the main beam at the mid-span; when the number of spans approaches positive infinity, the maximum response of the tracked bridge will surpass that of the trackless bridge.

[0053] Step S4, which involves calculating the effect of track constraint reduction between different bridge spans based on the calculation results obtained in step S3, specifically includes the following steps:

[0054] Taking the partial derivative of r(i,2n-1) with respect to i, we have

[0055] Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for the partial derivative of r(i,2n-1) with respect to i, and letting n approach positive infinity, we obtain... and

[0056] According to u max (1) calculation formula and u max The formula for calculating (n) is used to calculate the maximum difference J in the span displacement of the bridge-rail system. max for

[0057] This leads to the conclusion that the shearing effect of the track structure on different bridge spans within the bridge-track system varies in intensity; when all bridge spans are subjected to external forces, the seismic response of the main beam in the middle span is the greatest, while that of the main beam in the side span is the smallest.

[0058] Taking the second mixed partial derivative of r(i,2n-1) with respect to i and u, we have

[0059]

[0060] Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for calculating the second-order mixed partial derivatives of r(i,2n-1) with respect to i and u, we obtain: and

[0061]

[0062] Therefore, we can conclude that as the total number of spans increases, the rate at which the reduction effect of the track structure on different bridge spans within the bridge-track system decreases varies; the closer the bridge span is to the roadbed track structure, the smaller the rate at which the reduction effect of track constraints decreases.

[0063] The method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track provided by this invention can quickly determine the changing trend of the track constraint reduction effect, analyze the influence of the stiffness of the bridge substructure on the seismic response of the structure, and analyze the influence of the stiffness of the bridge subgrade track structure on the seismic response of the structure. Moreover, this invention has high reliability and good accuracy. Attached Figure Description

[0064] Figure 1 This is a schematic diagram of the method flow of the present invention.

[0065] Figure 2 This is a schematic diagram of the track system structure of a single-span railway simply supported beam bridge according to the present invention.

[0066] Figure 3 This is a schematic diagram of the mechanical model of a single-span railway simply supported beam bridge track system according to the present invention.

[0067] Figure 4 Let u be the displacement of the left end of the main beam of the i-th span of the present invention under the action of external force F. i A schematic diagram of the mechanical model of (i,n).

[0068] Figure 5 This is a schematic diagram of the mechanical model of the displacement u1(i,2n-1) of the left end of the main beam of the i-th span under the action of external force F in the first span of the present invention.

[0069] Figure 6 This is a schematic diagram of the mechanical model of the displacement u(i,2n-1) of the left end of the main beam of the i-th span of all bridge spans under the action of external force F. Detailed Implementation

[0070] like Figure 1 The diagram shown is a schematic representation of the method flow of the present invention: The method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track provided by the present invention includes the following steps:

[0071] S1. Obtain parameter information for multi-span simply supported beam bridges with ballastless track;

[0072] S2. Based on the obtained parameter information, construct a calculation model for the reduction effect in the bridge-rail system; specifically including the following steps:

[0073] For a single-span railway simply supported beam bridge track system (structure such as...) Figure 2 Modeling is performed as shown, assuming the system is in a linear elastic state. The structure of the bridge-rail system is represented by springs and equivalent points, as shown. Figure 3 As shown;

[0074] Let the magnitude of the seismic inertial force generated at a certain moment act on the center of mass of the main beam.

[0075] If the track structure is neglected, the force balance equation of the main beam is -F + k. d u bw +F b =0, where k d For the stiffness of the substructure, u bw The displacement of the left end of the main beam, F b This refers to the longitudinal force of the sliding support;

[0076] If the track structure is considered, the force balance equation of the main beam is:

[0077] -F+k d u b +F b +pL b +k c (u b -u1)=0

[0078] Where u b Let L be the displacement at the left end of the main beam, p be the beam-rail interaction force, and L be the displacement at the left end of the main beam. b Main beam length, k c denoted as shear tooth groove, u1 is the track structure displacement at the shear tooth groove;

[0079] The internal forces in the track structure are at their maximum at the junction of the bridge and the left subgrade. At this point, the force balance equation of the bridge-track system is:

[0080] -F+k j u1+k d u b +F b +k j k b / (k j +k b u1=0

[0081] In the formula k b For the track structure stiffness of the bridge span; k j To obtain the equivalent stiffness of the roadbed track according to the equivalence principle and Where k0 is the end spike stiffness in the rear anchorage structure, k t For the track structure stiffness of the roadbed section and E t Let A be the elastic modulus of the track structure. t L is the area of ​​the track structure. jk is the length of the track structure in the roadbed section. m denoted as friction plate stiffness; j represents the variation in anchorage length behind the platform. L m The length of the friction plate;

[0082] According to the principle of virtual work, the displacement u at the left end of the main beam can be obtained. b The reaction force s1 at the fixed support is:

[0083]

[0084]

[0085] In the formula F e For the equivalent of the bridge-rail system and F e =FF b ;

[0086] The displacement difference Δu of the single-span main beam, considering and not considering the track structure, is calculated as follows:

[0087] Since both Δu and Δs are less than 0, it indicates that the track structure will reduce the displacement of the main beam and decrease the support reaction force; take Then u b The formulas for calculating s1 and Δu are simplified to: and

[0088] When the number of bridge spans n is greater than 1, assuming that each span is subjected to the same external force F, then the displacement u of the main beam of any span of the bridge without considering the track structure is... bw (n) is

[0089] like Figure 4 As shown, when considering a bridge with a track structure, the displacement u of the left end of the main beam under the action of a single external force F in any i-th span can be obtained based on the principle of virtual work. i (i,n) is Where k L (n) is an intermediate variable and k R (n) is an intermediate variable and According to the characteristic root method, the general term can be obtained. and Where b1 and b2 are the equivalent stiffness coefficients of the right-side bridge rail system and c1 and c2 are the equivalent stiffness coefficients of the left-side bridge rail system and B(n) is the equivalent stiffness of the bridge rail system on the right side of the i-th main beam across n spans, and C(n) is the equivalent stiffness of the bridge track system on the left side of the i-th main beam across n spans, and a1 is the stiffness coefficient of the substructure and a2 is the stiffness coefficient of the roadbed track structure and

[0090] like Figure 5 As shown, taking the total number of spans as 2n-1, the displacement of the left end of the main beam in the first span under the action of a single external force F is u1(1,2n-1). Then, the displacement u of the left end of the main beam in the i-th span under the action of a single external force F in the first span can be calculated. i (1, 2n-1) is According to the characteristic root method, u can be obtained i The general term of (1, 2n-1) is: Where n1 and n2 are the displacements of the main beam in the first span under a single external force F in a bridge-track system with a total span of 2n-1. and X1(i) and X2(i) are the displacement coefficients of the main beam of the i-th span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

[0091]

[0092] According to the reciprocal displacement theorem, the displacement u(1,2n-1) at the left end of the main beam of the first span caused by the external forces in all spans is:

[0093]

[0094] In the formula, S1(n) and S2(n) are the total displacement coefficients of the first n spans of the main beam under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

[0095] like Figure 6 As shown, assuming that each span is subjected to the same external force F, the displacement u(i,2n-1) of the main beam of any span of the bridge considering the track structure is calculated as follows:

[0096]

[0097] According to the characteristic root method, the general term of u(i,2n-1) can be obtained as:

[0098]

[0099] In the formula, n3 and n4 represent the displacements of the main beam of the first span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

[0100] According to u bwThe formulas for calculating (n) and u(i,2n-1) are used to calculate the displacement difference r(i,2n-1) of the main beam at different locations of the trackless bridge and the tracked bridge when each span is subjected to the same external force F.

[0101]

[0102] S3. Based on the computational model constructed in step S2, calculate the effect of track constraint reduction under different span numbers; specifically including the following steps:

[0103] Taking the k-th order partial derivative of r(i,2n-1) with respect to n, we have

[0104] When a1 and a2 satisfy When the first span is subjected to a single external force F, the displacement u1(1,2n-1) of the left end of the main beam increases with the increase of the total number of spans;

[0105] Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for calculating the k-th order partial derivative of r(i,2n-1) with respect to n, we obtain... and Therefore, in practical engineering, the reduction effect in the bridge-track system weakens as the total number of spans increases, and the rate of reduction decreases as the total number of spans increases.

[0106] As the number of spans approaches positive infinity, the bridge displacement has a limit value:

[0107]

[0108]

[0109] According to u max The formula for calculating (n) and u b The calculation formula is used to calculate the maximum displacement difference r between the main beam of the mid-span of the trackless bridge and the tracked bridge. max for

[0110] Therefore, we can conclude that: when the number of spans is 1, the reduction effect of the track structure on the bridge structure is the strongest; as the number of spans increases, the reduction effect of the track structure on the bridge structure weakens; the larger the number of spans, the smaller the reduction effect of the track structure on the main beam at the mid-span; when the number of spans approaches positive infinity, the maximum response of the tracked bridge will surpass that of the trackless bridge.

[0111] S4. Based on the calculation results obtained in step S3, calculate the law of track constraint reduction effect between different bridge spans, and complete the calculation of track constraint reduction effect in multi-span simply supported beam bridges with ballastless track; specifically including the following steps:

[0112] Taking the partial derivative of r(i,2n-1) with respect to i, we have

[0113] Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for the partial derivative of r(i,2n-1) with respect to i, and letting n approach positive infinity, we obtain... and

[0114] According to u max (1) calculation formula and u max The formula for calculating (n) is used to calculate the maximum difference J in the span displacement of the bridge-rail system. max for

[0115] This leads to the conclusion that the shearing effect of the track structure on different bridge spans within the bridge-track system varies in intensity; when all bridge spans are subjected to external forces, the seismic response of the main beam in the middle span is the greatest, while that of the main beam in the side span is the smallest.

[0116] Taking the second mixed partial derivative of r(i,2n-1) with respect to i and u, we have

[0117]

[0118] Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for calculating the second-order mixed partial derivatives of r(i,2n-1) with respect to i and u, we obtain: and

[0119]

[0120] Therefore, we can conclude that as the total number of spans increases, the rate at which the reduction effect of the track structure on different bridge spans within the bridge-track system decreases varies; the closer the bridge span is to the roadbed track structure, the smaller the rate at which the reduction effect of track constraints decreases.

[0121] The method of this invention can quickly determine the changing trend of the track constraint reduction effect. By analyzing the characteristics of the reduction effect function expression, the effect law of the reduction effect in the bridge-track system under different span numbers and between different bridge spans can be directly obtained, avoiding the need to repeatedly perform seismic calculations for bridges with different span numbers in previous studies.

[0122] The method of this invention can be used to analyze the influence of bridge substructure stiffness on the seismic response of the structure. By treating the reduction effect parameter expression as a functional analytical expression with the bridge substructure stiffness coefficient a1 as the independent variable, the influence of bridge substructure stiffness on the track constraint reduction effect in multi-span simply supported beam bridges is investigated.

[0123] The method of this invention can be used to analyze the influence of the stiffness of the bridge subgrade track structure on the seismic response of the structure. The reduction effect parameter expression is treated as a functional analytical expression with the equivalent stiffness coefficient a2 of the subgrade track as the independent variable, to explore the influence of the abutment anchorage length, friction plate stiffness, and end spike stiffness on the track constraint reduction effect in multi-span simply supported beam bridges.

Claims

1. A method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track, comprising the following steps: S1. Obtain parameter information for multi-span simply supported beam bridges with ballastless track; S2. Based on the obtained parameter information, construct a calculation model for the reduction effect in the bridge-rail system; including the following steps: A model is created for a single-span railway simply supported beam bridge and rail system. The system is assumed to be in a linear elastic state, and springs and equivalent points are used to represent the structure of the bridge and rail system. Let the magnitude of the seismic inertial force generated at a certain moment act on the center of mass of the main beam. Considering the track structure, the force balance equation for the main beam is -F + k. d u b +F b +pL b +k c (u b -u1)=0 Where u b Let L be the displacement at the left end of the main beam, p be the beam-rail interaction force, and L be the displacement at the left end of the main beam. b Main beam length, k c For shear tooth grooves, u1 is the track structure displacement at the shear tooth groove; k b For the track structure stiffness of the bridge span; F b Longitudinal force of sliding support Taking the total number of spans as 2n-1, and the displacement of the left end of the main beam in the first span under the action of a single external force F as u1(1,2n-1), then the displacement u of the left end of the main beam in the i-th span under the action of a single external force F in the first span can be calculated. i (1, 2n-1) is a1 is the stiffness coefficient of the substructure, a2 is the stiffness coefficient of the roadbed and track structure, and F e For the equivalent efficiency of the bridge-rail system, k b For the track structure stiffness of the bridge span, k R (1) is an intermediate variable; Assuming that each span is subjected to the same external force F, the displacement u(i,2n-1) of the main beam of any span of the bridge considering the track structure is calculated as follows: The displacement difference r(i,2n-1) of the main beam at different locations of the trackless bridge and the tracked bridge when each span is subjected to the same external force F is calculated as follows: n3 and n4 are the main beam displacements of the first span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1; X1(i) and X2(i) are the main beam displacement coefficients of the i-th span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1. S3. Based on the calculation model constructed in step S2, analyze the track constraint reduction effect under different span numbers; S4. Based on the calculation results obtained in step S3, analyze the law of track constraint reduction effect between different bridge spans, and complete the calculation of track constraint reduction effect in multi-span simply supported beam bridges with ballastless track.

2. The method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track according to claim 1, characterized in that... Step S2, which involves constructing a calculation model for the reduction effect in the bridge-rail system based on the obtained parameter information, specifically includes the following steps: If the track structure is neglected, the force balance equation of the main beam is -F + k. d u bw +F b =0, where k d For the stiffness of the substructure, u bw The displacement of the left end of the main beam, F b This refers to the longitudinal force of the sliding support; The internal forces in the track structure are at their maximum at the junction of the bridge and the left subgrade. At this point, the force balance equation of the bridge-track system is: -F+k j u1+k d u b +F b +k j k b / (k j +k b )u1=0 In the formula k b For the track structure stiffness of the bridge span; k j To obtain the equivalent stiffness of the roadbed track according to the equivalence principle and Where k0 is the end spike stiffness in the rear anchorage structure, k t For the track structure stiffness of the roadbed section and E t Let A be the elastic modulus of the track structure. t L is the area of ​​the track structure. j k is the length of the track structure in the roadbed section. m denoted as friction plate stiffness; j represents the variation in anchorage length behind the platform. L m The length of the friction plate; According to the principle of virtual work, the displacement u at the left end of the main beam can be obtained. b The reaction force s1 at the fixed support is: In the formula F e For the equivalent of the bridge-rail system and F e =FF b ; The displacement difference Δu of the single-span main beam, considering and not considering the track structure, is calculated as follows: Since both Δu and Δs are less than 0, it indicates that the track structure will reduce the displacement of the main beam and the support reaction force. Considering that the stiffness of the shear tooth groove is much greater than that of other components, take Then u b The formulas for calculating s1 and Δu are simplified to: and When the number of bridge spans n is greater than 1, assuming that each span is subjected to the same external force F, then the displacement u of the main beam of any span of the bridge without considering the track structure is... bw (n) is When considering bridges with track structures, the displacement u of the left end of the main beam under the action of a single external force F in any i-th span can be obtained based on the principle of virtual work. i (i,n) is where k L (n) is an intermediate variable and k R (n) is an intermediate variable and According to the characteristic root method, the general term can be obtained. and Where b1 and b2 are the equivalent stiffness coefficients of the right-side bridge rail system and c1 and c2 are the equivalent stiffness coefficients of the left-side bridge rail system and B(n) is the equivalent stiffness of the bridge rail system on the right side of the i-th main beam across n spans, and C(n) is the equivalent stiffness of the bridge track system on the left side of the i-th main beam across n spans, and a1 is the stiffness coefficient of the substructure and a2 is the stiffness coefficient of the roadbed track structure and According to the characteristic root method, u can be obtained i The general term of (1, 2n-1) is: Where n1 and n2 are the displacements of the main beam in the first span under a single external force F in a bridge-track system with a total span of 2n-1. and X1(i) and X2(i) are the displacement coefficients of the main beam of the i-th span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1. According to the reciprocal displacement theorem, the displacement u(1,2n-1) at the left end of the main beam of the first span caused by the external forces in all spans is: In the formula, S1(n) and S2(n) are the total displacement coefficients of the first n spans of the main beam under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1. According to the characteristic root method, the general term of u(i,2n-1) can be obtained as: In the formula, n3 and n4 represent the displacements of the main beam of the first span under the action of a single external force F in a bridge-track system with a total number of spans of 2n-1.

3. The method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track according to claim 2, characterized in that... Step S3, which involves calculating the effect of track constraint reduction under different span numbers based on the computational model constructed in step S2, specifically includes the following steps: Taking the k-th order partial derivative of r(i,2n-1) with respect to n, we have When a1 and a2 satisfy When the first span is subjected to a single external force F, the displacement u1(1,2n-1) of the left end of the main beam increases with the increase of the total number of spans; Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for calculating the k-th order partial derivative of r(i,2n-1) with respect to n, we obtain... and Therefore, in practical engineering, the reduction effect in the bridge-track system weakens as the total number of spans increases, and the rate of reduction decreases as the total number of spans increases. As the number of spans approaches positive infinity, the bridge displacement has a limit value: According to u max The formula for calculating (n) and u b The calculation formula is used to calculate the maximum displacement difference r between the main beam of the mid-span of the trackless bridge and the tracked bridge. max for Therefore, we can conclude that: when the number of spans is 1, the reduction effect of the track structure on the bridge structure is the strongest; as the number of spans increases, the reduction effect of the track structure on the bridge structure weakens; the larger the number of spans, the smaller the reduction effect of the track structure on the main beam at the mid-span; when the number of spans approaches positive infinity, the maximum response of the tracked bridge will surpass that of the trackless bridge.

4. The method for calculating the track constraint reduction effect in a multi-span simply supported beam bridge with ballastless track according to claim 3, characterized in that... Step S4, which involves calculating the effect of track constraint reduction between different bridge spans based on the calculation results obtained in step S3, specifically includes the following steps: Taking the partial derivative of r(i,2n-1) with respect to i, we have Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for the partial derivative of r(i,2n-1) with respect to i, and letting n approach positive infinity, we obtain... and According to u max (1) calculation formula and u max The formula for calculating (n) yields the maximum difference J in the span displacement of the bridge-rail system. max for This leads to the conclusion that the shearing effect of the track structure on different bridge spans within the bridge-track system varies in intensity; when all bridge spans are subjected to external forces, the seismic response of the main beam in the middle span is the greatest, while that of the main beam in the side span is the smallest. Taking the second-order mixed partial derivative of r(i,2n-1) with respect to i and u, we have Substituting the formulas for X1(i), X2(i), n3, and n4 into the formula for calculating the second-order mixed partial derivatives of r(i,2n-1) with respect to i and u, we obtain: and Therefore, we can conclude that as the total number of spans increases, the rate at which the reduction effect of the track structure on different bridge spans within the bridge-track system decreases varies; the closer the bridge span is to the roadbed track structure, the smaller the rate at which the reduction effect of track constraints decreases.

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