A method for mapping the lateral deformation of bridge-track structures considering the second-order gravity effects of piles, soil and piers

Abstract

The present invention discloses a method for mapping the lateral deformation of a bridge-track structure considering the influence of the second-order effects of pile-soil and pier gravity. The CRTS-Ⅱ type longitudinally connected ballastless track-bridge system is taken as the research object. Based on the elastic rod unit deflection equation, the pier top deformation formula considering the influence of pile-soil is derived. By iterating the influence of the vertical force on the pier top on the lateral deformation of the pier top, the influence of the second-order effect of pier gravity is incorporated into the above formula to form a main beam lateral deformation matrix considering the influence of pile-soil and gravity second-order effects. The main beam lateral deformation matrix is ​​substituted into the general mapping model of bridge-track system deformation, and a bridge-track integrated model lateral deformation mapping analysis method considering the influence of pile-soil and pier gravity second-order effects is established. The finite element model verifies that the method has good accuracy, and the calculation steps and parameter iteration logic are clear and the physical meaning is clear, and it has good practical and popularization value.

Description

Technical Field

[0001] The invention relates to a bridge track structure lateral deformation mapping method taking into account the influence of pile soil and bridge pier gravity second-order effect. Background Art

[0002] The mapping deformation of high-speed railway track-bridge system refers to how the deformation of the bridge structure affects the geometry of the railway track. This mapping relationship is crucial to ensure the safety of train operation and the stability of the track. The bridge structure may deform due to a variety of factors, such as temperature changes, earthquakes, material shrinkage creep, etc. If these deformations are not controlled, they may be transmitted to the track, causing the track to be uneven and affecting the smooth operation of the train.

[0003] At present, there are many methods for studying the mapping deformation of high-speed railway track-bridge systems. For example, Chen Zhaowei et al. derived the mapping relationship between the settlement of bridge piers and the vertical deformation of rails under a simply supported bridge system based on the deflection curve equation, and found that the accumulation of track deformation caused by the settlement of a single pier is equivalent to the track deformation caused by the settlement of multiple piers. Gou Hongye et al. established an analytical model for mapping the vertical and horizontal deformation of bridge structures and track surface deformation, and analyzed the influence of different bridge structure deformation modes on track deformation. Jiang Lizhong et al. derived the analytical relationship between the bridge deformation and track geometry of high-speed railway CRTS I, CRTS II and CRTS III slab tracks, respectively, and found that the rail deformation is approximately proportional to the settlement of bridge piers, and as the stiffness of the fasteners decreases, the rail deformation curve becomes smoother.

[0004] However, existing deformation mapping of high-speed railway bridge-track systems focuses on the research of mapping known main beam deformation to rail deformation, and has not yet included the bridge substructure including piers, piles and foundations in the deformation analysis of bridge-track system mapping. In fact, affected by adverse factors such as floods, earthquakes, pile side loading and concrete shrinkage creep in the structure itself, the substructure of high-speed railway bridges will suffer from pier settlement, soft foundation, beam misalignment and other defects during the service life. These defects are mapped to the rails layer by layer, which will also affect the smoothness of the track and endanger driving safety. In addition, after a period of service, the piers of high-speed railway bridges with a height of more than 60m will produce certain post-construction foundation deformation, causing the piers to settle or even tilt, and there is a second-order gravity effect. The impact of this on the deformation of the bridge-track system mapping is currently unclear.

[0005] Therefore, how to deeply understand the impact of high-speed railway bridge substructure deformation on the track-bridge system mapping deformation and provide scientific basis and technical support for the safe operation of high-speed railways is still a problem worthy of further exploration. Summary of the invention

[0006] In view of the above-mentioned existing high-speed railway bridge-track system deformation mapping, the substructure of the bridge including piers, piles and foundations has not yet been included in the deformation analysis of the bridge-track system mapping. It is still unclear whether the deformation caused by the service of high piers over 60m and the second-order effect of pier gravity have an impact on the deformation of the bridge-track system mapping. The present invention proposes a bridge-track structure lateral deformation mapping method that considers the second-order effects of pile soil and pier gravity. Based on the elastic rod unit deflection equation, the influence of the vertical force on the pier top on the lateral deformation of the pier top is iterated. Through the Cartesian theorem and the principle of linear superposition, a bridge-track integrated model lateral deformation mapping analysis method that considers the second-order effects of pile soil and pier gravity is established, in order to promote the further development of the high-speed railway bridge-track deformation mapping theoretical model and help improve the high-speed railway ballastless track smoothness maintenance technology. The specific technical scheme is as follows:

[0007] A method for mapping the lateral deformation of a bridge track structure considering the second-order effects of pile soil and bridge pier gravity includes the following steps:

[0008] 1) Taking the CRTS-Ⅱ longitudinally connected ballastless track-bridge system as the research object, based on the elastic rod unit deflection equation, the pier top deformation formula considering the influence of pile-soil is derived;

[0009] 2) By iterating the influence of the vertical force on the pier top on the lateral deformation of the pier top, the influence of the second-order effect of the pier gravity is incorporated into the pier top deformation formula considering the influence of piles and soil, so as to form a main beam lateral deformation matrix considering the influence of piles, soil and the second-order effect of gravity;

[0010] 3) Substitute the main beam lateral deformation matrix into the general deformation mapping model of the bridge-track system, and establish a lateral deformation mapping analysis method for the bridge-track integrated model that takes into account the influence of the second-order gravity effect of piles, soil and piers.

[0011] In the aforementioned bridge track structure lateral deformation mapping method considering the influence of pile soil and bridge pier gravity second-order effect, in step 1), the derivation process of the pier top deformation formula considering the influence of pile soil is as follows:

[0012] 1-1) Assume that the mechanical model of the bridge pier is a cantilever beam, and the horizontal load P acting on the pier top generates a horizontal deflection of Δ p and the rotation angle θ p , then according to the elastic rod unit deflection equation, the horizontal deflection of the pier top Δ p and the pier top rotation angle θ p They are expressed as follows:

[0013]

[0014] Substituting equation (2) into equation (1), we can obtain that when θ p When a corner is turned, the top of the pier will experience a Δ p The displacement is expressed as follows:

[0015]

[0016] Where: Δ p is the horizontal deflection of the cantilever beam pier top; θ p is the rotation angle of the pier top; P is the horizontal load acting on the pier top; L is the height of the pier; E is the elastic modulus of the pier; I is the moment of inertia of the pier section;

[0017] 1-2) Since there is a non-displacement but rotation constraint between the bridge pier and the foundation, the mechanical model of the bridge pier is modified to a primary statically indeterminate cantilever beam. According to the deflection curve equation of the elastic rod unit, it can be obtained:

[0018]

[0019] Where: Δ p' is the horizontal deflection of the pier top of the statically indeterminate cantilever beam; L1 is the pier height; L2 is the pile foundation height; θ B is the pier top rotation angle; θ A is the pier bottom corner;

[0020] 1-3) Combine (4) to (6), and we get that when the top and bottom of the pier occur θ B and θ A When the rotation angle is , the final deformation formula of the pier top considering the influence of pile and soil is:

[0021]

[0022] Where: Δ p' is the horizontal deflection of the statically indeterminate cantilever beam pier top; θ B is the pier top rotation angle; θ A is the pier bottom rotation angle; L1 is the pier height; L2 is the pile foundation height.

[0023] In the aforementioned method of mapping the lateral deformation of the bridge track structure considering the second-order effects of pile soil and pier gravity, in step 1-2), after the mechanical model of the pier is modified to a first-order statically indeterminate cantilever beam, the deflection curve equation of the elastic rod unit is substituted according to the displacement condition that the deflection at the statically indeterminate support is 0, and the deflection caused by the horizontal force P on point A can be obtained:

[0024]

[0025] Remove the statically indeterminate support and replace it with force F. Substituting it into the elastic rod unit deflection curve equation, the deflection caused by the support reaction force F on point A can be obtained:

[0026]

[0027] Since the displacement condition of point A is 0, the support reaction force is:

[0028]

[0029] Then the deflection caused by the horizontal force P on point B is:

[0030]

[0031] The deflection of point B caused by the support reaction force F is:

[0032]

[0033] The final displacement of point B on the pier top is:

[0034]

[0035] Similarly, substituting the rotation angle equation of the elastic rod unit deflection curve, the final rotation angle of point B on the pier top is:

[0036]

[0037] The final turning angle of point A at the bottom of the pier is:

[0038]

[0039] After sorting, we get equations (4) to (6).

[0040] In the aforementioned method for mapping the lateral deformation of the bridge track structure considering the second-order effects of pile soil and pier gravity, in steps 1-3), the process of linking equations (4) to (6) and calculating equation (48) is as follows:

[0041] Comparing formula (5) with formula (6), we can get:

[0042]

[0043] Substituting the comparison result into formula (4), we get:

[0044]

[0045] After rearrangement, we get formula (49).

[0046] In the aforementioned bridge track structure lateral deformation mapping method considering the influence of pile soil and bridge pier gravity second-order effect, in step 2), the derivation process of the main beam deformation formula considering the influence of pile soil and gravity second-order effect is as follows:

[0047] 2-1) Calculate the initial conditions by iteration:

[0048] Assume that the rotation angle θ from the pier top B and the pier bottom rotation angle θ A The initial pier top displacement caused by the axial displacement is ω1. According to formula (50), the initial pier top displacement ω1 is expressed as:

[0049]

[0050] According to Hooke's law and formula (4), the initial pier equivalent stiffness K1 is calculated as:

[0051]

[0052]

[0053] According to Hooke's law, the equivalent pier top horizontal force H is calculated by equations (11) and (12):

[0054] H = K1ω1 (53);

[0055] When the initial rotation angle is small enough, the initial rotation angle θ1 of the pier top can be calculated based on trigonometric functions and equivalent infinitesimal substitution formulas:

[0056]

[0057] Assume that when displacement Δ p' When , the geometric configuration of the pier changes. At this time, the normal component Vsinθcosθ of the vertical force V will cause the structure to produce an additional horizontal deformation Δω. The initial additional horizontal deformation Δω1 is:

[0058]

[0059] That is, the initial condition of the iterative expression is:

[0060]

[0061] 2-2) The generation of additional horizontal deformation will further change the structural geometric configuration of the pier. That is, considering the second-order gravity effect of the pier, the horizontal displacement and rotation angle of the pier top increase, the pier stiffness decreases and a new additional horizontal displacement is generated; the increased pier top horizontal displacement ω2, pier top rotation angle θ2, reduced pier equivalent stiffness K2 and pier top additional horizontal displacement Δω2 are respectively:

[0062] ω2=ω1+Δω1 (56);

[0063]

[0064] 2-3) Extending equation (60) to equation (61), we can obtain the pier top rotation angle θ B 、 Pier bottom rotation angle θ A The general expression of the displacement value of the pier top after the second-order effect of gravity is:

[0065] ω i+1 =ω i +Δω i (62);

[0066]

[0067] Where: i is the number of iterations, ranging from 10 to 20; ω i is the pier top displacement at the i-th iteration; ω i+1 is the displacement of the pier top at the i+1th iteration; Δω i is the additional displacement of the pier top at the i-th iteration; θ i+1 is the pier top rotation angle at the i+1th iteration; K i+1 is the equivalent stiffness of the bridge pier at the i+1th iteration; Δω i+1 is the additional displacement of the pier top at the i+1th iteration;

[0068] 2-4) Substituting the initial conditions of the bridge pier calculated in step 2-1) into the general expression of the pier top displacement value obtained in step 2-3), the pier top displacement considering the influence of the second-order gravity effect of the bridge pier can be calculated iteratively;

[0069] when When ω is calculated, it is considered that i+1 is the final calculation result. At this time, the final displacement value of the pier top is ω end ; Assume that the lateral deformation caused by the displacement of the pier top at any position x of the beam structure is d i , then the transverse deformation matrix Z of the main beam at this location is b1 It is expressed as:

[0070]

[0071] In the aforementioned bridge-track structure lateral deformation mapping method considering the influence of the second-order gravity effect of piles, soil and bridge piers, in step 3), the process of establishing the bridge-track integrated model lateral deformation mapping analysis method considering the influence of the second-order gravity effect of piles, soil and bridge piers is as follows:

[0072] 3-1) The transverse deformation matrix of the main beam considering the second-order effects of pile-soil and gravity obtained in step 2) is incorporated into the general mapping theoretical model of bridge-track system deformation. According to the second theorem of Karlsruhe, the deflection of a simply supported beam at any position under the action of a concentrated force is expressed as:

[0073]

[0074] Where: ω is the deflection of a simply supported beam at any position; F n is the magnitude of the concentrated force; E n is the elastic modulus of the simply supported beam; I n is the moment of inertia of the simply supported beam; l is the total length of the simply supported beam; a is the distance from the left end of the beam to the concentrated force; c is the distance from the deflection solution position to the left end of the beam;

[0075] 3-2)) According to Hooke's law, the spring force between each layer of the track structure and the displacement of each layer of the track structure are expressed as a general representation model as shown in the following formula:

[0076] A r F r =Z r (67);

[0077] -B r F r +C s F s =Z s (68);

[0078] -D s F s +G p F p =Z p (69);

[0079] -O p F p +Z b1 =Z b2 (70);

[0080] Where: r is the rail; s is the mortar layer; p is the contact layer; b1 is the input bridge lateral displacement; b2 is the output bridge lateral displacement (this item does not need to be derived, so it is omitted later); Z r represents the deformation matrix of the rail at the fastener position; Z s represents the deformation matrix of the track slab at the mortar spring position; Z p represents the deformation matrix of the base plate at the contact spring position; Z b2 represents the deformation matrix of the bridge at the contact spring location; Z b1 is the initial deformation matrix of the bridge at all contact spring locations;

[0081] A r , B r , C s , D s , G p , O p is the deformation influence matrix of each interlayer structure of the track under interlayer interaction, where: r , B r is the deformation influence matrix under the action of the fastener force between the rail and the track plate, B r The same as A in value r ; C s , D s is the deformation influence matrix under the action of mortar spring force between track plate and base plate, D s Same as C in value s ; G p , Op is the deformation influence matrix under the contact spring force between the base plate and the bridge, O p The same as G p ;

[0082] F r is the fastener force; F s is the mortar spring force; F p are the contact spring forces respectively; the specific expressions are:

[0083] F r =K r (Z s -Z r ) (71);

[0084] F s =K s (Z p -Z s ) (72);

[0085] F p =K p (Z b2 -Z p ) (73);

[0086] Where:

[0087] K r is the transverse stiffness matrix of rail fastener;

[0088] K s is the combined stiffness matrix that simultaneously considers the lateral spring stiffness of CA mortar, the spring stiffness of shear reinforcement and the spring stiffness of lateral stopper;

[0089] K p The composite stiffness matrix is ​​a matrix that takes into account the transverse spring stiffness of the extruded plate, the spring stiffness of the sliding layer in the area without the extruded plate, the shear tooth spring stiffness and the lateral stop spring stiffness.

[0090] 3-3) Solving equations (74) to (75) and equations (76) to (77) simultaneously, the deformation of each layer of the track structure is expressed as follows:

[0091]

[0092] 3-4) According to formula (32), when the main beam transverse deformation matrix Z is introduced, b1 After that, the rail mapping deformation of the ballastless track-main beam-pier-pile-soil integrated model can be obtained.

[0093] The above-mentioned bridge-track structure lateral deformation mapping method considering the influence of the second-order gravity effect of pile soil and bridge piers, the deformation influence matrix of each interlayer structure of the track under interlayer interaction is composed of: Assume that the total number of fasteners of the CRTS-Ⅱ type longitudinal ballastless track-bridge system is M, and the fastener spacing is l c , the total number of mortar springs and contact springs is N, and the spacing between mortar springs and contact springs is l k At the same time, a virtual fastener force of 0 is introduced; the deformation influence matrix of each interlayer structure of the track under interlayer interaction is as follows:

[0094]

[0095]

[0096]

[0097] The aforementioned method for mapping the lateral deformation of a bridge-track structure taking into account the influence of the second-order gravity effects of piles, soil and bridge piers also includes a mapping theoretical model verification step, and uses an existing finite element model to simulate and verify the calculation results of the method for mapping the lateral deformation of a bridge-track structure taking into account the second-order gravity effects of piles, soil and bridge piers.

[0098] The beneficial effects of the present invention are as follows:

[0099] 1) The present invention incorporates the bridge substructure including piers, piles and foundations into the bridge-track system mapping deformation analysis, improves the high-speed railway bridge-track deformation mapping theoretical model, and provides scientific basis and technical support for the safe operation of high-speed railways.

[0100] 2) The lateral deformation mapping analysis method of the bridge-track integrated model considering the second-order gravity effect of piles, soil and piers of the present invention has been verified to be in good agreement with the calculation results of the finite element model, with a maximum error of no more than 2.5%, and the calculation steps and parameter iteration logic are clear, have clear physical meaning, and have good accuracy.

[0101] 3) The method of the present invention analyzes the influence of considering and not considering the second-order effect of pile-soil and pier gravity on the deformation of bridge-track mapping, and finds that the pile-soil has a great influence on the deformation of rail mapping. Even a small pier bottom rotation angle has a significant impact on the deformation of bridge-track mapping, and the influence of pile-soil cannot be ignored. For high-speed railway piers, the influence of considering and not considering the second-order effect of pier gravity on the deformation of rail mapping is not significant. Even for 70m high piers, the influence does not exceed 0.5%, which provides a more comprehensive reference for the later construction and design of high-speed railway bridges.

[0102] 4) When the mapping model established by the present invention calculates the rail mapping deformation, the influence of the second-order gravity effect of the pier can be considered after only a few iterations, and the calculation result meets the engineering accuracy requirements, which has good practical and promotion value. BRIEF DESCRIPTION OF THE DRAWINGS

[0103] Figure 1 This is a schematic diagram of a mapping theoretical model structure of a bridge track structure lateral deformation mapping method considering the influence of the second-order gravity effect of pile soil and piers of the present invention;

[0104] Figure 2 This is a schematic diagram of the structure of the high-speed railway multi-span longitudinal ballastless track-main beam-bridge pier-pile-soil integrated model adopted by the present invention (in the figure: L1 is the bridge pier height, L2 is the pile foundation height);

[0105] Figure 3 The calculation principle of the integrated model of multi-span longitudinal track-main beam-bridge pier-pile foundation of high-speed railway of the present invention (in the figure: (a) is the calculation principle without considering the bridge pier pile foundation, (b) is the calculation principle considering the bridge pier pile foundation, and (c) is the calculation principle considering the second-order effect of bridge pier gravity);

[0106] Figure 4 It is a schematic diagram of solving the deflection of a simply supported beam at any position under the action of a concentrated force according to the present invention;

[0107] Figure 5 It is a schematic diagram of the forces on each layer of the track-bridge system of the present invention;

[0108] Figure 6 This is a cross-sectional view of the elliptical hollow bridge pier of the present invention;

[0109] Figure 7 To compare and verify the calculation results of the mapping model of the present invention with those of the finite element model;

[0110] Figure 8 The results of the analysis of the present invention with and without considering the influence of pile-soil on rail deformation are compared. DETAILED DESCRIPTION

[0111] The following will be combined with the embodiments and drawings to clearly and completely describe the technical solution of the present invention. Obviously, the described embodiments are only preferred embodiments of the present invention, not all embodiments, and are not intended to limit the present invention in other forms. Any technician familiar with the profession may use the disclosed technical content to make changes or modifications. However, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention still belongs to the protection scope of the technical solution of the present invention.

[0112] Example 1

[0113] This embodiment is aimed at the research on deformation mapping of existing high-speed railway bridge-track systems, which focuses on mapping known main beam deformation to rail deformation. The bridge substructure including piers, piles and foundations has not yet been included in the deformation analysis of bridge-track system mapping. It is still unclear whether the second-order gravity effect of piers has an impact on the deformation of bridge-track system mapping. To this end, the model of this embodiment considers the influence of piers, piles and foundations on the basis of the track-foundation structure multi-layer beam mapping model that considers the influence of damage to interlayer limiting components and interlayer connectors, and establishes a lateral deformation mapping analysis method for the bridge-track integrated model that considers the influence of the second-order gravity effects of piles, soil and piers. Figure 1 shown.

[0114] This embodiment takes the CRTS-Ⅱ type longitudinal ballastless track-bridge system as the research object, and adopts the high-speed railway multi-span longitudinal ballastless track-main beam-bridge pier-pile-soil integrated model, such as Figure 2 Compared with the existing model, the model adopted in this embodiment takes the bridge substructure (bridge piers, pile foundation) into consideration, and can upload the bridge pier deformation caused by the pile-soil effect layer by layer through the mapping relationship, and finally transform it into rail deformation, which improves the defect that other mapping models do not consider the bridge substructure.

[0115] First, based on the elastic rod unit deflection equation, the pier top deformation formula considering the influence of pile soil is derived. The process is as follows:

[0116] Since the bridge piers not only bear the vertical force of the bridge superstructure, but also bear various loads such as soil pressure and water flow impact, the mechanical calculation model of the bridge pier is assumed to be a cantilever beam, such as Figure 3 As shown in (a), it is assumed that the horizontal load P acting on the pier top produces a horizontal deflection of Δ p and the rotation angle θ p , then according to the elastic rod unit deflection equation, the horizontal deflection of the pier top Δ p and the pier top rotation angle θ p Can be expressed as the following formulas:

[0117]

[0118] Where: L is the height of the pier; E is the elastic modulus of the pier; I is the moment of inertia of the pier section;

[0119] Substituting formula (2) into formula (1) yields:

[0120]

[0121] From the above formula, we can see that when θ p When a corner is turned, the top of the pier will experience a Δ p displacement.

[0122] However, due to the constraint condition of no displacement but rotation between the pier and the foundation, according to the paper "A new method for calibrating the lateral displacement of light-pier railway bridges based on acceleration sensors and verification by shaking table tests", the mechanical model of the pier was modified to a primary hyperstatic cantilever beam, such as Figure 3 As shown in (b), according to the displacement condition that the deflection at the statically indeterminate support is 0, substituting it into the elastic rod unit deflection curve equation, the deflection caused by the horizontal force P on point A can be obtained:

[0123]

[0124] Remove the statically indeterminate support and replace it with force F. Substituting it into the elastic rod unit deflection curve equation, the deflection caused by the support reaction force F on point A can be obtained:

[0125]

[0126] Since the displacement condition of point A is 0, the support reaction force is:

[0127]

[0128] Then the deflection caused by the horizontal force P on point B is:

[0129]

[0130] The deflection of point B caused by the support reaction force F is:

[0131]

[0132] Then the final displacement of pier top B is:

[0133]

[0134] Similarly, substituting the elastic rod unit deflection curve rotation angle equation, the final rotation angle of pier top B is:

[0135]

[0136] The final turning angle of point A at the bottom of the pier is:

[0137]

[0138] The results after sorting are as follows:

[0139]

[0140] Where: L1 is the height of the statically indeterminate cantilever beam pier; L2 is the height of the pile foundation; θ B is the pier top rotation angle; θ A It is the corner of the pier bottom.

[0141] Combining (4) to (6), and comparing (5) with (6), we can obtain:

[0142]

[0143] Substituting formula (7) into formula (4), we can obtain:

[0144]

[0145] Arranged:

[0146]

[0147] Where: Δ p' is the horizontal deflection of the statically indeterminate cantilever beam pier top; θ B is the pier top rotation angle; θ A is the pier bottom rotation angle; L1 is the pier height; L2 is the pile foundation height.

[0148] From the above formula (88), we can see that when the top and bottom of the pier occur θ B and θ A When the rotation angle is Δ, the pier top occurrence Δ p' However, in reality, high pier bridges, especially those with piers over 60m, are often designed as flexible high piers, so the pier top displacement Δ p' In addition to the initial pier top rotation angle θ B and the pier bottom rotation angle θ A In addition to the influence of gravity on the bridge structure above the pier, the second-order gravity effect (P-Δ effect) must also be considered. Figure 3 (c) When the pier top is displaced, the geometric configuration of the structure changes. At this time, the normal component of the vertical force V sinθcosθ will cause the structure to produce an additional horizontal deformation Δω, and the additional horizontal deformation will further cause the geometric configuration of the structure to change, and so on, eventually reaching a state of equilibrium.

[0149] It is generally believed that in the above process, the structure is deformed continuously due to the gradual decrease in stiffness. Therefore, the initial pier top rotation angle θ can be solved iteratively according to this idea. B 、 Pier bottom rotation angle θ A and the displacement value of the pier top after the second-order gravity effect. The specific derivation process is as follows:

[0150] By θ B and θ A The initial pier top displacement ω1 caused by this method is expressed as follows.

[0151]

[0152] The initial stiffness K1 is calculated by Hooke's law and (4):

[0153]

[0154] According to Hooke's law, the equivalent pier top horizontal force H is calculated by equations (11) and (12):

[0155] H = K1ω1 (91);

[0156] When the initial rotation angle is small enough, the initial rotation angle θ1 of the pier top can be calculated based on trigonometric functions and equivalent infinitesimal substitution formulas:

[0157]

[0158] Assume that when displacement Δ p' When , the geometric configuration of the pier changes. At this time, the normal component Vsinθcosθ of the vertical force V will cause the structure to produce an additional horizontal deformation Δω. The initial additional horizontal deformation Δω1 is:

[0159]

[0160] That is, the initial condition of the iterative expression is:

[0161]

[0162] The additional horizontal deformation will further change the structural geometry of the pier. Considering the second-order gravity effect of the pier, the horizontal displacement and rotation angle of the pier top increase, the pier stiffness decreases and a new additional horizontal displacement is generated. The increased horizontal displacement of the pier top ω2, the rotation angle of the pier top θ2, the reduced equivalent stiffness of the pier K2 and the additional horizontal displacement of the pier top Δω2 are:

[0163] ω2=ω1+Δω1 (94);

[0164]

[0165]

[0166] By generalizing equation (98) to equation (99), we can obtain the equation considering the pier top rotation angle θ: B 、 Pier bottom rotation angle θ A The general expression of the displacement value of the pier top after the second-order effect of gravity is:

[0167] ω i+1 =ω i +Δω i (100);

[0168]

[0169] Where: i is the number of iterations, preferably in the range of 10 to 20; ωi is the pier top displacement at the i-th iteration; ω i+1 is the displacement of the pier top at the i+1th iteration; Δω i is the additional displacement of the pier top at the i-th iteration; θ i+1 is the pier top rotation angle at the i+1th iteration; K i+1 is the equivalent stiffness of the bridge pier at the i+1th iteration; Δω i+1 is the additional displacement of the pier top at the i+1th iteration;

[0170] Substituting the obtained initial conditions of the bridge pier into the general expression of the pier top displacement value, the pier top displacement considering the influence of the second-order effect of the bridge pier gravity can be calculated iteratively.

[0171] when When ω is calculated, it is considered that i+1 is the final calculation result. At this time, the final displacement value of the pier top is ω end ; Assume that the lateral deformation caused by the displacement of the pier top at any position x of the beam structure is d i , then the transverse deformation matrix Z of the main beam at this location is b1 It is expressed as:

[0172]

[0173] Then, the main beam deformation formula is substituted into the general mapping model of bridge-track system deformation, and a lateral deformation mapping analysis method of the bridge-track integrated model is established that considers the influence of the second-order effects of pile-soil and pier gravity. The general mapping theoretical model of bridge-track system deformation developed in the early stage is optimized, and the main beam deformation formula considering the second-order effects of pile-soil and gravity is incorporated into it. In the general mapping theoretical model of bridge-track system deformation, the bridge deformation is taken as input and the rail deformation is taken as output. The previous steps are all for obtaining the bridge deformation. In this step, the bridge deformation is taken as input, and the output is obtained after constructing the mapping theoretical model (the output is the rail deformation). The following formula is derived based on Castigliano's second theorem:

[0174] like Figure 4 As shown in Figure 2, according to the second theorem of Karlsruhe, the deflection of a simply supported beam at any position under the action of a concentrated force is expressed as:

[0175]

[0176] Where: ω is the deflection of a simply supported beam at any position; F n is the magnitude of the concentrated force; E n is the elastic modulus of the simply supported beam; I n is the moment of inertia of the simply supported beam; l is the total length of the simply supported beam; a is the distance from the left end of the beam to the concentrated force; c is the distance from the deflection solution position to the left end of the beam;

[0177] According to Hooke's law, the spring force between each layer of the track structure and the displacement of each layer of the track structure can be expressed as a general representation model as shown in the following formula:

[0178] AF r =Z r (105);

[0179] -BF r +CF s =Z s (106);

[0180] -DF s +GF p =Z p (107);

[0181] -OF p +Z b1 =Z b2 (108);

[0182] Where: r is the rail; s is the mortar layer; p is the contact layer; b1 is the input bridge lateral displacement; b2 is the output bridge lateral displacement (this item does not need to be derived, so it is omitted later);

[0183] Z r represents the deformation matrix of the rail at the fastener position; Z s represents the deformation matrix of the track slab at the mortar spring position; Z p represents the deformation matrix of the base plate at the contact spring position; Z b2 represents the deformation matrix of the bridge at the contact spring location; Z b1 is the initial deformation matrix of the bridge at all contact spring locations.

[0184] A r , B r , C s , D s , G p , O p is the deformation influence matrix of each interlayer structure of the track under interlayer interaction, where:

[0185] A r , B r is the deformation influence matrix under the action of the fastener force between the rail and the track plate, B r The same as A in value r (Note: Although B r and A r The significance of using different letters for the same numerical value is to distinguish the forces on each layer of the structure. Therefore, although the numerical value is the same, it needs to be expressed differently. The same below);

[0186] C s , Ds is the deformation influence matrix under the action of mortar spring force between track plate and base plate, D s Same as C in value s ;

[0187] G p , O p is the deformation influence matrix under the contact spring force between the base plate and the bridge, O p The same as G p .

[0188] The composition of the deformation influence matrix of the interlayer structure of the track under interlayer interaction is as follows: the total number of fasteners of the CRTS-Ⅱ longitudinal ballastless track-bridge system is M, the fastener spacing is lc, the total number of mortar springs and contact springs is N, the mortar spring and contact spring spacing is lk, and at the same time, a virtual fastener force of 0 is introduced; the deformation influence matrix of the interlayer structure of the track under interlayer interaction is as follows:

[0189]

[0190]

[0191]

[0192] F in formula (111) to formula (112) r is the fastener force; F s is the mortar spring force; F p are the contact spring forces;

[0193] The specific expression is:

[0194] F r =K r (Z s -Z r ) (113);

[0195] F s =K s (Z p -Z s ) (114);

[0196] F p =K p (Z b2 -Z p ) (115);

[0197] Where:

[0198] K r is the transverse stiffness matrix of rail fastener;

[0199] K sis the combined stiffness matrix that simultaneously considers the lateral spring stiffness of CA mortar, the spring stiffness of shear reinforcement and the spring stiffness of lateral stopper;

[0200] K p The composite stiffness matrix is ​​a matrix that takes into account the transverse spring stiffness of the extruded plate, the spring stiffness of the sliding layer in the area without the extruded plate, the shear tooth spring stiffness and the lateral stop spring stiffness.

[0201] By solving equations (116) to (117) and equations (118) to (119) simultaneously, the deformation of each layer of the track structure can be expressed as follows:

[0202]

[0203] According to formula (32), when the main beam transverse deformation matrix Z is introduced, b1 After that, the rail mapping deformation of the ballastless track-main beam-pier-pile-soil integrated model can be obtained.

[0204] Example 2

[0205] This embodiment uses ANSYS finite element to verify the lateral deformation mapping analysis method of the bridge-track integrated model considering the second-order effects of pile soil and bridge pier gravity described in Example 1. The details are as follows:

[0206] like Figure 6 As shown, in this embodiment, a five-span 32.5m standard span CRTS II ballastless track is used as the research object for verification. The CRTS II ballastless track has a pier height of 70m, a pile length of 20m, and an elliptical hollow pier, where B is 4.2m, R1 is 3.95m, and R2 is 2.56m. The verification process of this embodiment is as follows:

[0207] When 5×10 -5 rad, the rail mapping deformation obtained by the mapping theoretical model and the finite element model of the present invention is shown in the following figure. Figure 7 shown.

[0208] When 5×10 -5 rad、10×10 -5 rad、20×10 -5 rad、40×10 -5 rad、80×10 - 5 rad, the maximum rail mapping deformation values ​​obtained by the mapping model of the present invention and the finite element model are compared, as shown in Table 1.

[0209] Table 1. Comparison results of the maximum rail deformation calculated by the mapping theoretical model of the present invention and the finite element model

[0210]

[0211] Depend on Figure 7 As shown in Table 1, the mapping model of the present invention is consistent with the finite element model results, and the maximum difference does not exceed 2.5%. It can be seen that the analysis method of the present invention has good accuracy.

[0212] Example 3

[0213] This embodiment uses the lateral deformation mapping analysis method of the bridge-track integrated model considering the second-order effects of pile-soil and bridge pier gravity described in Embodiment 1 to perform case analysis. This embodiment takes the research object in Embodiment 2 as an example, and the specific analysis is as follows:

[0214] 1) Analyze the influence of pier pile soil on rail mapping deformation with and without considering it

[0215] When the 3# and 4# piers have a 5mm lateral misalignment, the analysis considers the pile soil (i.e. the 3# and 4# piers have a 1×10 -5 rad angle) and without considering the pile-soil condition, the influence of the rail mapping deformation is shown in the following figure. Figure 8 shown.

[0216] Depend on Figure 8 It can be seen that the maximum rail mapping deformation caused by the 5mm misalignment of the beam body on 3# and 4# piers is 5.07mm, and after considering the influence of pile soil (i.e., the 1×10 -5 rad rotation angle), the maximum rail mapping deformation increases to 9.10 mm, and the influence of the small rotation angle at the pier bottom on the rail mapping deformation is 79.5%. It can be seen that the influence of the small rotation angle at the pier bottom on the mapping deformation of the bridge-rail system is very significant, so the influence of the pile soil cannot be ignored.

[0217] 2) Analysis of the influence of the second-order gravity effect of bridge piers on rail mapping deformation

[0218] Assuming the initial pier bottom rotation angle is 5×10 -5 rad, the mapping model in Example 1 is used for iterative calculation, and the ANSYS finite element model in Example 2 is used for calculation (considering the second-order effect of the pier gravity), and the results of the two are compared and analyzed. The results are shown in Table 2.

[0219] Table 2. Calculation results of pier top displacement with and without considering the second-order effect of pier gravity

[0220]

[0221] It can be seen from the results in Table 2 that the error between the first-order solution calculated by the mapping model using the method described in Example 1 (not considering the second-order effect of pier gravity) and the finite element model (considering the second-order effect of pier gravity) is 0.24%. After three iterations, the difference between the calculation result using the method in Example 1 and the finite element model result is only 0.03%. It can be seen that the method of the present invention only needs a few iterations to consider the second-order effect of pier gravity, and the calculation result meets the engineering accuracy requirements.

[0222] It is obvious to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, the embodiments should be regarded as exemplary and non-restrictive from any point of view. In addition, it should be understood that although this specification is described in accordance with the embodiments, it does not contain only one technical solution. This narrative of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in the embodiments can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for mapping the lateral deformation of a bridge track structure considering the second-order gravity effect of piles, soil and piers, characterized by: The steps include: 1) Taking the CRTS-Ⅱ longitudinally connected ballastless track-bridge system as the research object, based on the elastic rod unit deflection equation, the pier top deformation formula considering the influence of pile-soil is derived; 2) By iterating the influence of the vertical force on the pier top on the lateral deformation of the pier top, the influence of the second-order effect of the pier gravity is incorporated into the pier top deformation formula considering the influence of piles and soil, so as to form a main beam lateral deformation matrix considering the influence of piles, soil and the second-order effect of gravity; 3) Substituting the main beam transverse deformation matrix into the general mapping model of bridge-rail system deformation, and establishing a transverse deformation mapping analysis method for the bridge-rail integrated model that takes into account the influence of the second-order gravity effect of piles, soil and piers; Among them, in step 1), the derivation process of the pier top deformation formula considering the influence of pile soil is as follows: 1-1) Assume that the mechanical model of the bridge pier is a cantilever beam, and the horizontal load P acting on the pier top generates a horizontal deflection of Δ p and the rotation angle θ p , then according to the elastic rod unit deflection equation, the horizontal deflection of the pier top △ p and the pier top rotation angle θ p They are expressed as follows: Substituting equation (2) into equation (1), we can obtain that when θ p When a corner is turned, the top of the pier will experience a Δ p The displacement is expressed as follows: Where: △ p is the horizontal deflection of the cantilever beam pier top; θ p is the corner of the pier top; P is the horizontal load acting on the pier top; L is the height of the pier; E is the elastic modulus of the pier; I is the moment of inertia of the pier section; 1-2) Since there is a non-displacement but rotation constraint between the bridge pier and the foundation, the mechanical model of the bridge pier is modified to a first-order statically indeterminate cantilever beam. According to the elastic rod unit deflection equation, we can get: Where: △ p' is the horizontal deflection of the pier top of the statically indeterminate cantilever beam; L1 is the height of the bridge pier; L2 is the height of the pile foundation; θ B It is the corner of the pier top; θ A is the pier bottom corner; 1-3) Combine (4) to (6), and we get that when the top and bottom of the pier occur θ B and θ A When the rotation angle is , the final deformation formula of the pier top considering the influence of pile and soil is: Where: △ p' is the horizontal deflection of the pier top of the statically indeterminate cantilever beam; θ B It is the corner of the pier top; θ A is the pier bottom corner; L1 is the height of the bridge pier; L2 is the height of the pile foundation.

2. The method for mapping the lateral deformation of a bridge track structure considering the second-order gravity effect of pile soil and piers according to claim 1 is characterized in that: In step 1-2), after the mechanical model of the pier is modified to a primary hyperstatic cantilever beam, the deflection of point A caused by the horizontal load P can be obtained by substituting the displacement condition that the deflection at the hyperstatic support is 0 into the elastic rod unit deflection equation: Where: F is the support reaction force; Remove the hyperstatic support and replace it with the support reaction F, and then substitute it into the elastic rod unit deflection equation to obtain the deflection caused by the support reaction F on point A: Since the displacement condition of point A is 0, the support reaction force F is: The deflection caused by the horizontal load P on point B is: The deflection of point B caused by the support reaction force F is: The final displacement of point B is: Similarly, substituting into the elastic rod unit rotation angle equation, the final rotation angle of point B is: The final turning angle of point A at the bottom of the pier is: After sorting, we get equations (4) to (6).

3. The method for mapping the lateral deformation of a bridge track structure considering the second-order gravity effect of pile soil and piers according to claim 1 is characterized in that: In step 1-3), the process of connecting equations (4) to (6) and calculating equation (7) is as follows: Comparing formula (5) with formula (6), we can get: Substituting the comparison result into formula (4), we get: Then we get formula (7).

4. The method for mapping the lateral deformation of a bridge track structure considering the second-order gravity effect of pile soil and piers according to claim 1 is characterized in that: In step 2), the derivation process of the main beam transverse deformation matrix considering the influence of pile soil and gravity second-order effect is as follows: 2-1) Calculate the initial conditions by iteration: Assume that the rotation angle θ from the pier top B and the pier bottom rotation angle θ A The initial pier top displacement induced is ω1, and the initial pier top displacement ω1 is expressed as: L1 is the height of the pier. According to Hooke's law and formula (4), the initial pier equivalent stiffness K1 is calculated as: Where: P is the horizontal load; L2 is the height of the pile foundation; E is the elastic modulus of the pier; I is the moment of inertia of the pier section. According to Hooke's law, the equivalent pier top horizontal force H is calculated by equations (8) and (9): H = K1ω1 (10); When the initial rotation angle is small enough, the initial rotation angle θ1 of the pier top can be calculated based on trigonometric functions and equivalent infinitesimal substitution formulas: Assume that when displacement △ p' When , the geometric configuration of the pier changes. At this time, the normal component Vsinθcosθ of the vertical force V will cause the structure to produce an additional horizontal deformation △ω. The initial additional horizontal deformation △ω1 is: That is, the initial condition of the iterative expression is: 2-2) The generation of additional horizontal deformation will further change the structural geometric configuration of the pier. That is, considering the second-order gravity effect of the pier, the horizontal displacement and rotation angle of the pier top increase, the pier stiffness decreases and a new additional horizontal displacement is generated; the increased pier top horizontal displacement ω2, pier top rotation angle θ2, reduced pier equivalent stiffness K2 and pier top additional horizontal displacement △ω2 are respectively: ω2=ω1+△ω1 (13); 2-3) Consider the pier top rotation angle θ B 、 Pier bottom rotation angle θ A The general expression of the displacement value of the pier top after the second-order effect of gravity is: oh i+1 =ω i +△ω i (17); Where: i is the number of iterations, ranging from 10 to 20; ω i is the pier top displacement at the i-th iteration; ω i+1 is the displacement of the pier top at the i+1th iteration; △ω i is the additional displacement of the pier top at the i-th iteration; θ i+1 is the pier top rotation angle at the i+1th iteration; K i+1 is the equivalent stiffness of the bridge pier at the i+1th iteration; △ω i+1 is the additional displacement of the pier top at the i+1th iteration; 2-4) Substituting the initial conditions of the bridge pier calculated in step 2-1) into the general expression of the pier top displacement value obtained in step 2-3), the pier top displacement considering the influence of the second-order gravity effect of the bridge pier can be calculated iteratively; when When ω is calculated, it is considered that i+1 is the final calculation result. At this time, the final displacement value of the pier top is ω end ; Assume that the lateral deformation caused by the displacement of the pier top at any position x of the beam structure is d i , then the transverse deformation matrix Z of the main beam at this location is b1 It is expressed as:

5. The method for mapping the lateral deformation of a bridge track structure considering the second-order gravity effect of piles, soil and piers according to claim 1 is characterized in that: In step 3), the process of establishing the lateral deformation mapping analysis method of the bridge-track integrated model considering the influence of the second-order effect of pile soil and bridge pier gravity is as follows: 3-1) The transverse deformation matrix of the main beam considering the second-order effects of pile-soil and gravity obtained in step 2) is incorporated into the general mapping theoretical model of bridge-track system deformation. According to the second theorem of Karlsruhe, the deflection of a simply supported beam at any position under the action of a concentrated force is expressed as: Where: ω is the deflection of the simply supported beam at any position; F n is the concentration force; E n is the elastic modulus of the simply supported beam; I n is the moment of inertia of the simply supported beam; l is the total length of the simply supported beam; a is the distance between the concentrated force and the left end of the beam; c is the distance from the deflection solution position to the left end of the beam; 3-2) According to Hooke's law, the spring force between each layer of the track structure and the displacement of each layer of the track structure are expressed as a general representation model as shown in the following formula: A r F r =Z r (22); -B r F r +C s F s =Z s (23); -D s F s +G p F p =Z p (24); -ABOUT p F p +Z b1 =Z b2 (25); Where: r is the rail; s is the mortar layer; p is the contact layer; b1 is the input bridge lateral displacement; b2 is the output bridge lateral displacement; Z r represents the deformation matrix of the rail at the fastener location; Z s represents the deformation matrix of the track slab at the location of the mortar spring; Z p represents the deformation matrix of the base plate at the location of the contact spring; Z b2 represents the deformation matrix of the bridge at the contact spring location; Z b1 is the initial deformation matrix of the bridge at all contact spring locations; A r , B r , C s , D s , G p , O p is the deformation influence matrix of each interlayer structure of the track under interlayer interaction, in: A r , B r is the deformation influence matrix under the action of the fastener force between the rail and the track plate, B r The same as A in value r ; C s , D s is the deformation influence matrix under the action of mortar spring force between track plate and base plate, D s The same as C s ; G p , O p is the deformation influence matrix under the contact spring force between the base plate and the bridge, O p The same as G p ; F r is the fastener force; F s is the mortar spring force; F p are the contact spring forces; According to Hooke's law, the specific expression is: F r =K r (WITH s -WITH r ) (26); F s =K s (WITH p -WITH s ) (27); F p =K p (WITH b2 -WITH p ) (28); Where: K r is the transverse stiffness matrix of rail fastener; K s is the combined stiffness matrix that simultaneously considers the lateral spring stiffness of CA mortar, the spring stiffness of shear reinforcement and the spring stiffness of lateral stopper; K p The composite stiffness matrix is ​​a matrix that takes into account the transverse spring stiffness of the extruded plate, the spring stiffness of the sliding layer in the area without the extruded plate, the shear tooth spring stiffness and the lateral stop spring stiffness. 3-3) The deformation of each layer of the track structure is expressed as follows: 3-4) According to formula (29), when the main beam transverse deformation matrix Z is introduced, b1 After that, the rail mapping deformation of the ballastless track-main beam-pier-pile-soil integrated model can be obtained.

6. The method for mapping the lateral deformation of a bridge track structure considering the second-order gravity effect of piles, soil and piers according to claim 5 is characterized by: The deformation influence matrix of each interlayer structure of the track under interlayer interaction is composed of: assuming that the total number of fasteners of the CRTS-Ⅱ longitudinal ballastless track-bridge system is M, the fastener spacing is lc, the total number of mortar springs and contact springs is N, and the spacing between mortar springs and contact springs is lk. At the same time, a virtual fastener force of 0 is introduced; the deformation influence matrix of each interlayer structure of the track under interlayer interaction is as follows:

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