A method for constructing a mapping model between temperature-induced deformation of arch bridges and track surface geometry

Abstract

The present invention proposes a method for constructing a mapping model between the temperature-induced deformation of an arch bridge and the track surface geometry. The mapping model between the vertical temperature-induced deformation of an arch bridge and the track surface geometry of a CRTSⅠ type double-block ballastless track established by this method is very fast and convenient for analyzing the track surface geometry characteristics under the action of temperature. It only requires inputting the overall temperature rise and fall amplitude of the arch bridge and the temperature gradient pattern of the continuous beam on the arch to calculate the track surface geometry of a long-span arch bridge when vertical temperature-induced deformation occurs under the action of temperature load. There is no need to perform complex finite element model calculations, which can greatly shorten the modeling time, improve the model solution efficiency, and better illustrate the influence of key parameters on the track surface geometry, thus having good promotion value.

Description

Technical Field

[0001] The present invention belongs to the technical field of track detection and evaluation, and in particular relates to a method for constructing a mapping model between the temperature-induced deformation of an arch bridge and the geometric form of a track surface. Background Art

[0002] The measured results of track irregularities show that there is a relatively clear mapping relationship between the deformation of the foundation structure and the geometry of the track surface. This foundation structure deformation, which changes with time and space, will lead to the deterioration of the track surface geometry, and then affect the safe and comfortable operation of the train through the wheel-rail dynamic effect.

[0003] Most of the existing research on the mapping relationship between offline structural deformation and track surface geometry focuses only on unidirectional, quasi-static permanent deformation modes such as pier settlement and creep arching of beam bridges. It is only applicable to small and medium-span bridges such as simply supported beams and continuous beams, and is difficult to generalize to special structures such as long-span arch bridges. For long-span bridges, their temperature-induced deformation is often greater than the structural deformation caused by live loads, and its impact on track smoothness and safe railway operation is greater. How to reveal the deformation mapping relationship between the temperature-induced deformation of long-span arch bridges and the track surface geometry is a key scientific issue for the rational control and prediction of sensitive elements, characteristic lengths and changing trends of track irregularities. It is also a "bottleneck" problem that needs to be solved urgently to restrict the laying of ballastless tracks on long-span arch bridges. Summary of the Invention

[0004] In response to the above problems existing in the prior art, the present invention provides a method for constructing a mapping model between the temperature-induced deformation of an arch bridge and the geometry of the rail surface, with the aim of realizing rapid calculation of the geometry of the rail surface when a large-span arch bridge undergoes temperature-induced deformation.

[0005] To achieve the above objectives, the present invention adopts the following technical solutions:

[0006] A method for constructing a mapping model between temperature-induced deformation and track surface geometry of an arch bridge, wherein the arch bridge comprises an arch ring, an arch upper beam, a track bed plate, and rails arranged in order from bottom to top, the rails being connected to the track bed plate via fasteners, and the track surface being a CRTS I type twin-block ballastless track surface. The construction method specifically comprises the following steps:

[0007] Step S101, obtaining the overall temperature rise and fall amplitude of the arch ring and the vertical temperature gradient pattern of the continuous steel-concrete composite beam on the arch according to on-site measurement or through formulation;

[0008] Step S102, establishing the overall rectangular coordinate system of the arch ring, the arch beam, the track bed plate and the rail respectively;

[0009] Step S103: Establish the rail deformation matrix W r , the rail deformation at all fastener positions within the calculation range is organized into a matrix form: W r =Cr P f , where W r is the rail deformation matrix at all fastener positions within the calculation range, C r is the influence matrix of fastener force on rail deformation, P f is the fastener force matrix;

[0010] Step S104: Establish the local roadbed plate deformation matrix W sm , the deformation of the roadbed slab at all fastener positions of the mth roadbed slab is organized into a matrix form: W sm =B sm +C sm P fm , where W sm is the deformation matrix of the roadbed slab at all fastener positions on the mth roadbed slab; B sm is the influence matrix of the deformation of the arch beam on the deformation of the mth trackbed plate; C sm is the influence matrix of the fastener force on the deformation of the mth trackbed plate; P fm is the fastener force matrix on the mth track slab;

[0011] Step S105: transform the local roadbed plate deformation matrix W sm Expand to all fasteners on the MN block roadbed within the calculation range to obtain the expanded roadbed deformation matrix

[0012] W s =B s +C s P f ,

[0013] Among them, W s is the deformation matrix of the MN slab at all fastener positions; B s is the influence matrix of the deformation of the arch beam on the deformation of the MN block roadbed plate; C s is the influence matrix of the fastener force on the deformation of the MN block roadbed plate; P f is the fastener force matrix;

[0014] Step S106: Analyze the temperature-induced deformation of the continuous steel-concrete composite beam on the arch under any vertical temperature gradient load to obtain the vertical deformation equation w of the continuous steel-concrete composite beam on the arch under nonlinear temperature gradient. T ;

[0015] Step S107: Establish an analytical model of the hingeless arch under the action of overall temperature rise and fall, and use elasticity theory to perform classical analysis on the arch ring to determine the bending moment M on any section of the arch ring. g and axial force N g ;

[0016] Step S108: The bending moment M on any section of the arch ring is g and axial force N g Substitute the differential equation of the arch ring's deflection curve under the action of overall temperature increase and decrease and integrate it to obtain the vertical deformation equation of the arch ring under the action of overall temperature increase and decrease w g ;

[0017] Step S109: Based on the bridge structure deformation coordination analysis method, the temperature-induced deformation of the arch ring is transferred to the beam body on the arch in the form of forced displacement at the continuous beam support position, and the redundant support reaction force is solved to determine the vertical deformation equation w of the continuous steel-concrete composite beam on the arch when forced displacement occurs. v ;

[0018] Step S110: The vertical deformation equation w of the continuous steel-concrete composite beam on the arch under the action of nonlinear temperature gradient in step S106 is converted to T The vertical deformation equation w when the continuous steel-concrete composite beam on the arch is forced to displace in step S109 is v The total temperature-induced deformation of the arch bridge under the action of temperature is determined by superposition, and the temperature-induced deformation function of the arch bridge w is used y Describe the bridge deck line shape of a long-span arch bridge when temperature-induced deformation occurs, and get w y =w T +w v ;

[0019] Step S111: Based on the fastener force and the deformation between the rail and the trackbed plate, the rail deformation matrix is ​​expressed as:

[0020] W r =k f C r (E+C s -C r ) -1 B s ,

[0021] Where E is the unit matrix; C s is the influence matrix of fastener force on trackbed plate deformation; C r is the influence matrix of fastener force on rail deformation; B s k is the influence matrix of the deformation of the arch beam on the roadbed plate deformation; f is the vertical stiffness of the fastener;

[0022] Step S112: The temperature-induced deformation function w of the arch bridge is y Input the influence matrix B of the deformation of the arch beam on the roadbed deformation s , and replace the vertical deformation of the arch beam to realize the mapping of the temperature-induced deformation of the arch bridge to the track surface.

[0023] Preferably, in step S102, when establishing the overall rectangular coordinate system, the origin of the horizontal coordinate axis of the overall rectangular coordinate system of the arch beam, the roadbed plate and the rail is set at the neutral axis position of the calculation starting point of the interlayer structure, the origin of the horizontal coordinate axis of the overall rectangular coordinate system of the arch ring is set at the arch top, and the origin of the vertical coordinate axis of the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail is set at the gravity balance position before the deformation of the corresponding structure. The coordinate axes are all rightward and downward as positive, satisfying the right-hand spiral rule, and the coordinate axes of the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail are obtained as (X g ,Y g ,Z g )、(X b ,Y b ,Z b )、(X s ,Y s ,Z s )、(X r ,Y r ,Z r ).

[0024] Preferably, in step S103, the rail deformation matrix W is established r Specifically, the rail deformation at the t-th fastener position is expressed as rt Expressed, we get:

[0025]

[0026] Where, L ri is the horizontal coordinate of the rail at the position of the ith fastener in the overall rectangular coordinate system; L r(total+1) is the total length of the rail; P f_i is the fastener force of the i-th fastener on the rail; E r I r is the vertical bending stiffness of the rail; total is the total number of fasteners on the MN block roadbed within the calculation range;

[0027] And get the following solution expression:

[0028] W r (t,1)=Y rt

[0029]

[0030] P f (i,1)=P f_i .

[0031] Preferably, in step S104, the local trackbed plate deformation matrix W is established. sm Specifically, establish the trackbed plate coordinate system x b Oy s, the deformation of the roadbed plate at the position of the ith fastener on the mth roadbed plate is expressed as y si Expressed, we get:

[0032]

[0033] Where x b is the horizontal coordinate of the local coordinate system of the trackbed plate; l si is the horizontal coordinate of the ith fastener on the mth trackbed slab in the local coordinate system of the trackbed slab; s(n+1) is the length of the mth track slab; l su k is the horizontal coordinate of the u-th fastener on the m-th trackbed slab in the local coordinate system of the trackbed slab; c is the support line stiffness of the isolation layer composed of geotextile and elastic cushion; E s I s is the vertical bending stiffness of the roadbed slab; ξ is a parameter related to the stiffness of the isolation layer and the roadbed slab, P fm_i is the i-th fastener force on the m-th trackbed slab; y b (x b ) is the vertical deformation of the arch beam; α, β, γ, δ are the horizontal coordinates of the roadbed plate x s The relevant function is specifically expressed by the following formula: α(x s =)cξx s h(, The overall rectangular coordinate system of the track bed plate (X s ,Y s ) and the local coordinate system of the trackbed plate (x b ,y s ) is: b =X s -L sm , y=Y s ;L sm is the horizontal coordinate of the roadbed slab at the neutral axis position of the starting point of the local coordinate system of the mth roadbed slab in the overall rectangular coordinate system of the roadbed slab;

[0034] And get the following solution expression:

[0035] W sm (i,1)=y si

[0036]

[0037]

[0038] Preferably, step S106 specifically includes the following sub-steps:

[0039] (1) Remove the middle support of the continuous steel-concrete composite beam, take the simply supported beam as the basic statically determinate system, establish the local coordinate system xOy of the basic statically determinate system of the simply supported beam, and determine the vertical deformation expression w of the basic statically determinate system of the composite beam under the action of nonlinear temperature gradient. T0 :

[0040]

[0041] Where, α c is the linear expansion coefficient of concrete, α s is the linear expansion coefficient of steel, E c is the elastic modulus of concrete, E s is the elastic modulus of steel, I c is the moment of inertia of the concrete bridge deck, I s is the moment of inertia of the steel beam, y a is the distance from the upper edge of the steel-concrete composite beam section to the centroid of the composite beam, b is the distance from the lower edge of the steel-concrete composite beam section to the centroid of the composite beam, d is the distance from the interface between the concrete bridge deck and the steel beam to the centroid of the composite beam, T(y) is the arbitrary vertical nonlinear temperature gradient acting on the composite beam, b(y) is the distribution of the cross-sectional width of the steel-concrete composite beam along the beam height, l l0 is the full length of the continuous steel-concrete composite beam on the arch; the overall rectangular coordinate system of the beam on the arch (X b ,Y b ) and the local coordinate system (x, y) of the arch beam are as follows: y=-Y b ;L M is the horizontal coordinate of the arch beam at the neutral axis position of the basic statically determinate system in the overall rectangular coordinate system of the arch beam;

[0042] (2) According to the selected basic static system, add the corresponding redundant support reaction at the original intermediate support position to establish the local coordinate system x of the continuous beam on the arch F Oy F , the coordinate origin is set at the basic statically determinate system edge support. Then, according to the geometric compatibility equation of continuous beam deformation and the physical relationship between force and displacement, the supplementary equation is determined, the redundant support reaction is solved, and the vertical deformation expression w of the basic statically determinate system of the composite beam under the action of the redundant support reaction is determined. Fj :

[0043]

[0044] Where, F j is the jth redundant support reaction force; E s is the elastic modulus of steel; I l Convert the section moment of inertia for the steel-concrete composite beam; l l0is the full length of the continuous steel-concrete composite beam on the arch; a j is the jth unknown redundant support reaction F j The horizontal axis of b j is the horizontal coordinate a j The relevant value, b j =l l0 -a j ;

[0045] (3) Superimpose the vertical deformation of the basic statically determinate system of the composite beam under the action of the redundant support reaction force obtained in step (2) and the vertical deformation of the basic statically determinate system of the composite beam under the action of the nonlinear temperature gradient obtained in step (1) to determine the vertical deformation equation w of the continuous steel-concrete composite beam on the arch under the action of the nonlinear temperature gradient T :

[0046]

[0047] Where w T0 is the vertical deformation equation of the basic statically determinate system of the composite beam under the action of nonlinear temperature gradient; w Fj is the vertical deformation equation of the basic statically determinate system of the composite beam under the reaction of the jth unknown redundant support; c is the linear expansion coefficient of concrete; α s is the linear expansion coefficient of steel; E c is the elastic modulus of concrete; E s is the elastic modulus of steel; I c is the moment of inertia of the concrete bridge deck; I s is the moment of inertia of the steel beam; y a y is the distance from the upper edge of the steel-concrete composite beam section to the centroid of the composite beam; b y is the distance from the lower edge of the steel-concrete composite beam section to the centroid of the composite beam; d is the distance from the interface between the concrete bridge deck and the steel beam to the centroid of the composite beam; T(y) is the arbitrary vertical nonlinear temperature gradient acting on the composite beam; b(y) is the distribution of the cross-sectional width of the steel-concrete composite beam along the beam height; l l0 is the total length of the continuous steel-concrete composite beam on the arch; the local coordinate system (x, y) of the beam on the arch and the local coordinate system (x F ,y F ) is as follows: y=y F .

[0048] Preferably, the bending moment M on any cross section of the arch ring in step S107 is g and axial force N g The specific calculation is as follows:

[0049]

[0050]

[0051] Where H x is the axial force acting on the elastic center; Y g is the vertical coordinate of the arch axis corresponding to any section of the arch ring in the overall rectangular coordinate system of the arch ring; gs is the vertical coordinate of the elastic center of the hingeless arch in the overall rectangular coordinate system of the arch ring; is the horizontal inclination angle of the arch axis; α c is the linear expansion coefficient of concrete; E c is the elastic modulus of concrete; I g is the moment of inertia of the arch ring section; t0 is the amplitude of the arch ring temperature change; l g is the calculated span of the arch ring; μ is the compression coefficient, A g is the area of ​​the arch cross section.

[0052] Preferably, in step S108, the vertical deformation equation of the arch ring under the action of overall temperature increase and decrease is w g The calculation process is as follows:

[0053]

[0054] Where, l g Calculate the span for the arch ring, is the horizontal inclination angle of the arch axis, E c is the elastic modulus of concrete, I g is the moment of inertia of the arch ring section, M g is the bending moment on any section of the arch ring, N g is the axial force on any section of the arch ring, A g is the area of ​​the arch cross section, ε t is the strain generated in the micro segment of the arch ring under the action of temperature.

[0055] Preferably, in step S109, the vertical deformation equation w of the continuous steel-concrete composite beam on the arch when forced displacement occurs is v The calculation process is as follows:

[0056]

[0057] Where, Δ A and Δ B are the forced displacements of the supports on both sides, is the vertical deformation equation of the continuous steel-concrete composite beam under the reaction of the jth redundant support, and where,

[0058]

[0059] Fvj is the support reaction force of the jth intermediate support when forced displacement occurs, l l0 is the full length of the continuous steel-concrete composite beam on the arch, E s is the elastic modulus of steel, I l is the converted section moment of inertia of the steel-concrete composite beam, a j is the jth unknown redundant support reaction F j The horizontal axis, b j is the horizontal coordinate a j The relevant value, b j =l l0 -a j .

[0060] Compared with the prior art, the present invention has at least the following beneficial effects:

[0061] The present invention establishes a mapping model between the vertical temperature-induced deformation of an arch bridge and the geometric morphology of the track surface of a CRTSⅠ type twin-block ballastless track. This model is used to quickly and conveniently analyze the geometric morphology of the track surface under the action of temperature. By simply inputting the overall temperature rise and fall amplitude of the arch bridge and the temperature gradient pattern of the continuous beam on the arch, the geometric morphology of the ballastless track surface when the long-span arch bridge undergoes vertical temperature-induced deformation under the action of temperature load can be calculated. Complex finite element model calculations are not required, which can significantly shorten modeling time, improve model solution efficiency, and better illustrate the influence of key parameters on the track surface geometry. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, constitute a part of this application, and do not constitute a limitation of the embodiments of the present invention. In the drawings:

[0063] Figure 1 It is a schematic diagram of the program solution process of the mapping model of the present invention;

[0064] Figure 2 It is a schematic diagram of the coordinate system arrangement of the present invention;

[0065] Figure 3 Schematic diagram of the track bed plate model established by the present invention;

[0066] Figure 4 It is a basic statically determinate system diagram of a simply supported steel-concrete composite beam established by the present invention;

[0067] Figure 5 Schematic diagram of the coordinate system for calculating the temperature effect of the composite beam of the present invention;

[0068] Figure 6 Schematic diagram of redundant support reaction force acting on a basic statically determinate system according to the present invention;

[0069] Figure 7Schematic diagram of the hingeless arch analysis model under the action of overall temperature rise and fall of the present invention;

[0070] Figure 8 It is a schematic diagram of the continuous beam under the action of forced displacement of the present invention. DETAILED DESCRIPTION

[0071] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with examples and drawings. The exemplary embodiments of the present invention and their descriptions are only used to explain the present invention and are not intended to limit the present invention.

[0072] like Figures 1 to 8 As shown, the present invention provides a method for constructing a mapping model between the temperature-induced deformation of an arch bridge and the geometry of the rail surface, see Figure 2 The arch bridge targeted by the method includes an arch ring, an arch beam, a roadbed plate and rails arranged in sequence from bottom to top, and the rails are connected to the roadbed plate by fasteners; preferably, a sleeper is provided on the roadbed plate, and a base plate and an isolation layer are further provided in sequence from bottom to top between the arch beam and the roadbed plate, and the track surface is a CRTS (i.e., slab ballastless track) type I double-block track surface; the construction method specifically includes the following steps:

[0073] Step S101, obtaining the overall temperature rise and fall amplitude of the arch ring and the vertical temperature gradient pattern of the continuous steel-concrete composite beam on the arch according to on-site measurement (or through formulation);

[0074] Step S102, respectively establish the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail, wherein the origin of the horizontal coordinate axis of the overall rectangular coordinate system of the arch beam, the roadbed plate and the rail is set at the neutral axis position of the calculation starting point of the interlayer structure, the origin of the horizontal coordinate axis of the overall rectangular coordinate system of the arch ring is set at the arch top, and the origin of the vertical coordinate axis of the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail is set at the gravity balance position of the corresponding structures (arch ring, arch beam, roadbed plate and rail) before deformation. The coordinate axes are all positive to the right and downward, satisfying the right-hand spiral rule, such as Figure 2 As shown in the figure, the coordinate axes of the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail are (X g ,Y g ,Z g )、(X b ,Y b ,Z b )、(X s ,Y s ,Z s )、(X r ,Y r ,Z r );

[0075] Step S103: Establish the rail deformation matrix W r , the rail deformation at all fastener positions within the calculation range is organized into a matrix form: W r =C r P f , where W r is the rail deformation matrix at all fastener positions within the calculation range, C r is the influence matrix of fastener force on rail deformation, P f is the fastener force matrix; specifically,

[0076] The rail deformation at the t-th fastener position is expressed as Y rt Expressed, we get:

[0077]

[0078] Where, L ri is the horizontal coordinate of the rail at the position of the ith fastener in the overall rectangular coordinate system; L r(total+1) is the total length of the rail; P f_i is the fastener force of the i-th fastener on the rail; E r I r is the vertical bending stiffness of the rail;

[0079] And get the following solution expression:

[0080] W r (t,1)=Y rt

[0081]

[0082] P f (i,1)=P f_i ;

[0083] Step S104: Establish the trackbed plate coordinate system x b Oy s ( Figure 3 ), establish the local roadbed plate deformation matrix W sm , the deformation of the roadbed slab at all fastener positions of the mth roadbed slab is organized into a matrix form: W sm =B sm +C sm P fm , where W sm is the deformation matrix of the roadbed slab at all fastener positions on the mth roadbed slab; B sm is the influence matrix of the deformation of the arch beam on the deformation of the mth trackbed plate; C sm is the influence matrix of the fastener force on the deformation of the mth trackbed plate; P fm is the fastener force matrix on the mth trackbed plate; specifically,

[0084] The deformation of the roadbed slab at the position of the i-th fastener on the m-th roadbed slab is expressed as y si Expressed, we get:

[0085]

[0086] Where x b is the horizontal coordinate of the local coordinate system of the trackbed plate; l si is the horizontal coordinate of the ith fastener on the mth trackbed slab in the local coordinate system of the trackbed slab; s(n+1) is the length of the mth track slab; l su k is the horizontal coordinate of the u-th fastener on the m-th trackbed slab in the local coordinate system of the trackbed slab; c is the support line stiffness of the isolation layer composed of geotextile and elastic cushion; E s I s is the vertical bending stiffness of the roadbed slab; ξ is a parameter related to the stiffness of the isolation layer and the roadbed slab, P fm_i is the i-th fastener force on the m-th trackbed slab; y b (x b ) is the vertical deformation of the arch beam; α, β, γ, δ are the horizontal coordinates of the roadbed plate x s The relevant function is specifically expressed by the following formula: α(x s =)cξx s h(, The overall rectangular coordinate system of the track bed plate (X s ,Y s ) and the local coordinate system of the trackbed plate (x b ,y s ) is: b =X s -L sm , y=Y s ;L sm is the horizontal coordinate of the roadbed slab at the neutral axis position of the starting point of the local coordinate system of the mth roadbed slab in the overall rectangular coordinate system of the roadbed slab;

[0087] And get the following solution expression:

[0088] W sm (i,1)=y si

[0089]

[0090]

[0091] P fm (i,1)=P fm_i

[0092] Step S105: Expand the local roadbed plate deformation matrix to all fasteners on the MN roadbed plates within the calculation range. Then the roadbed plate deformation can be expressed as: W s =B s +C s P f , the expression for solving each element is as follows:

[0093]

[0094]

[0095]

[0096]

[0097] Where W s is the deformation matrix of the MN slab at all fastener positions; B s is the influence matrix of the deformation of the arch beam on the deformation of the MN block roadbed plate; C s is the influence matrix of the fastener force on the deformation of the MN block roadbed plate; P f is the fastener force matrix;

[0098] Step S106, analyzing the temperature-induced deformation of the continuous steel-concrete composite beam on the arch under arbitrary vertical temperature gradient load, specifically includes the following sub-steps:

[0099] (1) Remove the intermediate support of the continuous steel-concrete composite beam, take the simply supported beam as the basic statically determinate system, and establish the local coordinate system xOy( Figure 4 、 Figure 5 ), determine the vertical deformation expression w of the basic statically determinate system of the composite beam under the action of nonlinear temperature gradient T0 :

[0100]

[0101] Where, α c is the linear expansion coefficient of concrete, α s is the linear expansion coefficient of steel, E c is the elastic modulus of concrete, E s is the elastic modulus of steel, I c is the moment of inertia of the concrete bridge deck, I s is the moment of inertia of the steel beam, y a is the distance from the upper edge of the steel-concrete composite beam section to the centroid of the composite beam, b is the distance from the lower edge of the steel-concrete composite beam section to the centroid of the composite beam, dis the distance from the interface between the concrete bridge deck and the steel beam to the centroid of the composite beam, T(y) is the arbitrary vertical nonlinear temperature gradient acting on the composite beam, b(y) is the distribution of the cross-sectional width of the steel-concrete composite beam along the beam height, l l0 is the full length of the continuous steel-concrete composite beam on the arch; the overall rectangular coordinate system of the beam on the arch (X b ,Y b ) and the local coordinate system (x, y) of the arch beam are as follows: y=-Y b ;L M is the horizontal coordinate of the arch beam at the neutral axis position of the basic statically determinate system in the overall rectangular coordinate system of the arch beam;

[0102] (2) According to the selected basic static system, add the corresponding redundant support reaction at the original intermediate support position to establish the local coordinate system x of the continuous beam on the arch F Oy F , the coordinate origin is set at the basic statically determinate system side support ( Figure 6 ), then, based on the geometric compatibility equation of continuous beam deformation and the physical relationship between force and displacement, the supplementary equation is determined, the redundant support reaction is solved, and the vertical deformation expression w of the basic statically determinate system of the composite beam under the action of the redundant support reaction is determined. Fj :

[0103]

[0104] Where, F j is the jth redundant support reaction force; E s is the elastic modulus of steel; I l Convert the section moment of inertia for the steel-concrete composite beam; l l0 is the full length of the continuous steel-concrete composite beam on the arch; a j is the jth unknown redundant support reaction F j The horizontal axis of b j is the horizontal coordinate a j The relevant value, b j =l l0 -a j ;

[0105] (3) Superimpose the vertical deformation of the basic statically determinate system of the composite beam under the action of the redundant support reaction force obtained in step (2) and the vertical deformation of the basic statically determinate system of the composite beam under the action of the nonlinear temperature gradient obtained in step (1) to determine the vertical deformation equation w of the continuous steel-concrete composite beam on the arch under the action of the nonlinear temperature gradient T :

[0106]

[0107] Where w T0is the vertical deformation equation of the basic statically determinate system of the composite beam under the action of nonlinear temperature gradient; w Fj is the vertical deformation equation of the basic statically determinate system of the composite beam under the reaction of the jth unknown redundant support; c is the linear expansion coefficient of concrete; α s is the linear expansion coefficient of steel; E c is the elastic modulus of concrete; E s is the elastic modulus of steel; I c is the moment of inertia of the concrete bridge deck; I s is the moment of inertia of the steel beam; y a y is the distance from the upper edge of the steel-concrete composite beam section to the centroid of the composite beam; b y is the distance from the lower edge of the steel-concrete composite beam section to the centroid of the composite beam; d is the distance from the interface between the concrete bridge deck and the steel beam to the centroid of the composite beam; T(y) is the arbitrary vertical nonlinear temperature gradient acting on the composite beam; b(y) is the distribution of the cross-sectional width of the steel-concrete composite beam along the beam height; l l0 is the total length of the continuous steel-concrete composite beam on the arch; the local coordinate system (x, y) of the beam on the arch and the local coordinate system (x F ,y F ) is as follows: y=y F ;

[0108] Step S107: Establishing a hingeless arch analysis model under the action of overall temperature rise and fall (see Figure 7 ), the arch ring is classically analyzed using elastic theory to determine the bending moment M on any section of the arch ring. g and axial force N g :

[0109]

[0110]

[0111] Where H x is the axial force acting on the elastic center; Y g is the vertical coordinate of the arch axis corresponding to any section of the arch ring in the overall rectangular coordinate system of the arch ring; gs is the vertical coordinate of the elastic center of the hingeless arch in the overall rectangular coordinate system of the arch ring; is the horizontal inclination angle of the arch axis; α c is the linear expansion coefficient of concrete; E c is the elastic modulus of concrete; I g is the moment of inertia of the arch ring section; t0 is the amplitude of the arch ring temperature change; l g is the calculated span of the arch ring; μ is the compression coefficient, A gis the area of ​​the arch cross section;

[0112] Step S108: The bending moment M on any section of the arch ring is g and axial force N g Substitute the differential equation of the arch ring's deflection curve under the action of overall temperature increase and decrease and integrate it to obtain the vertical deformation equation of the arch ring under the action of overall temperature increase and decrease w g :

[0113]

[0114] Where, In the formula, the numerator d represents the differential, l g Calculate the span for the arch ring, is the horizontal inclination angle of the arch axis, E c is the elastic modulus of concrete, I g is the moment of inertia of the arch ring section, M g is the bending moment on any section of the arch ring, N g is the axial force on any section of the arch ring, A g is the area of ​​the arch cross section, ε t is the strain generated by the micro-segment of the arch ring under the action of temperature;

[0115] Step S109: Based on the bridge structure deformation coordination analysis method, the temperature-induced deformation of the arch ring is transferred to the arch beam in the form of forced displacement at the continuous beam support position (see Figure 8 ), solve the redundant support reaction force, and determine the vertical deformation equation w of the continuous steel-concrete composite beam on the arch when forced displacement occurs v :

[0116]

[0117] Where, Δ A and Δ B are the forced displacements of the supports on both sides, is the vertical deformation equation of the continuous steel-concrete composite beam under the reaction of the jth redundant support, and where,

[0118]

[0119] F vj is the support reaction force of the jth intermediate support when forced displacement occurs, l l0 is the full length of the continuous steel-concrete composite beam on the arch, E s is the elastic modulus of steel, I l is the converted section moment of inertia of the steel-concrete composite beam, a j is the jth unknown redundant support reaction F j The horizontal axis, b j is the horizontal coordinate a jThe relevant value, b j =l l0 -a j ;

[0120] Step S110: The vertical deformation equation w of the continuous steel-concrete composite beam on the arch under the action of nonlinear temperature gradient in step S106 is converted to T The vertical deformation equation w when the continuous steel-concrete composite beam on the arch is forced to displace in step S109 is v The total temperature-induced deformation of the arch bridge under the action of temperature is determined by superposition, and the temperature-induced deformation function of the arch bridge w is used y Describe the bridge deck line shape of a long-span arch bridge when temperature-induced deformation occurs, and get w y =w T +w v ;

[0121] Step S111: Based on the fastener force and the deformation between the rail and the roadbed plate, the rail deformation matrix can be expressed as: W r =k f C r (E+C s -C r ) -1 B s , where E is the unit matrix; C s is the influence matrix of fastener force on trackbed plate deformation; C r is the influence matrix of fastener force on rail deformation; B s k is the influence matrix of the deformation of the arch beam on the roadbed plate deformation; f is the vertical stiffness of the fastener; the expression for each element in the above matrix can be determined according to the relevant formulas in step S103 and step S104;

[0122] Step S112: By observing the expression of each element of the matrix in step S110, it can be found that only the deformation of the arch beam affects the deformation of the trackbed plate. s Contains the vertical deformation of the arch beam y b (x b ), the values ​​of the other matrices are the same during the analysis process, so the arch bridge temperature-induced deformation function w y Input the influence matrix B of the deformation of the arch beam on the roadbed deformation s , and the vertical deformation of the arch beam y b (x b ) is replaced to realize the mapping of temperature-induced deformation of arch bridge to rail surface.

[0123] The model construction method of the present invention can rapidly determine the track surface geometry and fastener force changes of ballastless track on long-span arch bridges undergoing temperature-induced deformation under temperature loads. Specifically, by programming the construction method through a computer program, relatively accurate track surface geometry data for CRTS I type twin-block ballastless track on long-span arch bridges undergoing temperature-induced deformation can be obtained simply by inputting known parameters. This significantly reduces the labor and material resources consumed by on-site measurements, saving both cost and time.

[0124] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for constructing a mapping model between temperature-induced deformation and rail surface geometry of an arch bridge, wherein the arch bridge comprises an arch ring, an arch upper beam, a trackbed plate, and rails arranged in order from bottom to top, wherein the rails are connected to the trackbed plate via fasteners, characterized in that: The construction method specifically includes the following steps: Step S101, obtaining the overall temperature rise and fall amplitude of the arch ring and the vertical temperature gradient pattern of the continuous steel-concrete composite beam on the arch according to on-site measurement or through formulation; Step S102, establishing the overall rectangular coordinate system of the arch ring, the arch beam, the track bed plate and the rail respectively; Step S103: Establish the rail deformation matrix W r , the rail deformation at all fastener positions within the calculation range is organized into a matrix form: W r =C r P f , where W r is the rail deformation matrix at all fastener positions within the calculation range, C r is the influence matrix of fastener force on rail deformation, P f is the fastener force matrix; Step S104: Establish the local roadbed plate deformation matrix W sm , the deformation of the roadbed slab at all fastener positions of the mth roadbed slab is organized into a matrix form: W sm =B sm +C sm P fm , Where W sm is the deformation matrix of the roadbed slab at all fastener positions on the mth roadbed slab; B sm is the influence matrix of the deformation of the arch beam on the deformation of the mth trackbed plate; C sm is the influence matrix of the fastener force on the deformation of the mth trackbed plate; P fm is the fastener force matrix on the mth track slab; Step S105: transform the local roadbed plate deformation matrix W sm Expand to all fasteners on the MN block roadbed within the calculation range to obtain the expanded roadbed deformation matrix W s =B s +C s P f , Among them, W s is the deformation matrix of the MN slab at all fastener positions; B s is the influence matrix of the deformation of the arch beam on the deformation of the MN block roadbed plate; C s is the influence matrix of the fastener force on the deformation of the MN block roadbed plate; P f is the fastener force matrix; Step S106: Analyze the temperature-induced deformation of the continuous steel-concrete composite beam on the arch under any vertical temperature gradient load to obtain the vertical deformation equation w of the continuous steel-concrete composite beam on the arch under nonlinear temperature gradient. T ; Step S107: Establish an analytical model of the hingeless arch under the action of overall temperature rise and fall, and use elasticity theory to perform classical analysis on the arch ring to determine the bending moment M on any section of the arch ring. g and axial force N g ; Step S108: The bending moment M on any section of the arch ring is g and axial force N g Substitute the differential equation of the arch ring's deflection curve under the action of overall temperature increase and decrease and integrate it to obtain the vertical deformation equation of the arch ring under the action of overall temperature increase and decrease w g ; Step S109: Based on the bridge structure deformation coordination analysis method, the temperature-induced deformation of the arch ring is transferred to the beam body on the arch in the form of forced displacement at the continuous beam support position, and the redundant support reaction force is solved to determine the vertical deformation equation w of the continuous steel-concrete composite beam on the arch when forced displacement occurs. v ; Step S110: The vertical deformation equation w of the continuous steel-concrete composite beam on the arch under the action of nonlinear temperature gradient in step S106 is converted to T The vertical deformation equation w when the continuous steel-concrete composite beam on the arch is forced to displace in step S109 is v The total temperature-induced deformation of the arch bridge under the action of temperature is determined by superposition, and the temperature-induced deformation function of the arch bridge w is used y Describe the bridge deck line shape of a long-span arch bridge when temperature-induced deformation occurs, and get w y =w T +w v ; Step S111: Based on the fastener force and the deformation between the rail and the trackbed plate, the rail deformation matrix is ​​expressed as: W r =k f C r (E+C s -C r ) -1 B s , Where E is the unit matrix; C s k is the influence matrix of fastener force on trackbed plate deformation; f is the vertical stiffness of the fastener; Step S112: The temperature-induced deformation function w of the arch bridge is y Input the influence matrix B of the deformation of the arch beam on the roadbed deformation s , and replace the vertical deformation of the arch beam to realize the mapping of the temperature-induced deformation of the arch bridge to the track surface.

2. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 1, characterized in that: The track surface is a CRTSⅠ type double-block ballastless track track surface.

3. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 2, characterized in that: In step S102, when establishing the overall rectangular coordinate system, the origin of the horizontal coordinate axis of the overall rectangular coordinate system of the arch beam, the roadbed plate and the rail is set at the neutral axis position of the calculation starting point of the interlayer structure, the origin of the horizontal coordinate axis of the overall rectangular coordinate system of the arch ring is set at the arch top, and the origin of the vertical coordinate axis of the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail is set at the gravity balance position before the deformation of the corresponding structure. The coordinate axes are all rightward and downward as positive, satisfying the right-hand spiral rule, and the coordinate axes of the overall rectangular coordinate system of the arch ring, the arch beam, the roadbed plate and the rail are obtained as (X g ,Y g ,Z g )、(X b ,Y b ,Z b )、(X s ,Y s ,Z s )、(X r ,Y r ,Z r ).

4. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 3, characterized in that: In step S103, the rail deformation matrix W is established r Specifically, the rail deformation at the t-th fastener position is expressed as rt Expressed, we get: Where, L ri is the horizontal coordinate of the rail at the position of the ith fastener in the overall rectangular coordinate system; L r(total+1) is the total length of the rail; P f_i is the fastener force of the i-th fastener on the rail; E r I r is the vertical bending stiffness of the rail; total is the total number of fasteners on the MN block roadbed within the calculation range; and the following solution expression is obtained: W r (t,1)=Y rt P f (i,1)=P f_i 。 5. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 4, characterized in that: In step S104, the local roadbed plate deformation matrix W is established sm Specifically, establish the local coordinate system x of the trackbed plate b Oy s , the deformation of the roadbed plate at the position of the ith fastener on the mth roadbed plate is expressed as y si Expressed, we get: Where x b is the local coordinate system x of the trackbed plate b Oy s The horizontal axis below; l si is the local coordinate system x of the trackbed plate b Oy s The horizontal coordinate of the ith fastener on the mth track slab; l s(n+1) is the length of the mth track slab; l su is the local coordinate system x of the trackbed plate b Oy s The horizontal coordinate of the u-th fastener on the m-th trackbed slab; k c is the support line stiffness of the isolation layer composed of geotextile and elastic cushion; E s I s is the vertical bending stiffness of the roadbed slab; ξ is a parameter related to the stiffness of the isolation layer and the roadbed slab, P fm_i is the i-th fastener force on the m-th trackbed slab; y b (x b ) is the vertical deformation of the arch beam; α, β, γ, δ are the horizontal coordinates of the roadbed plate x s The relevant function is specifically expressed by the following formula: α(x s )=ch(ξx s )cos(ξx s ), The overall rectangular coordinate system of the track bed plate (X s ,Y s ) and the local coordinate system of the trackbed plate (x b ,y s ) is: b =X s -L sm , y=Y s ;L sm is the horizontal coordinate of the roadbed slab at the neutral axis position of the starting point of the local coordinate system of the mth roadbed slab in the overall rectangular coordinate system of the roadbed slab; And get the following solution expression: W sm (i,1)=y si P fm (i,1)=P fm_i 。 6. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 5, characterized in that: Step S106 specifically includes the following steps: (1) Remove the middle support of the continuous steel-concrete composite beam, take the simply supported beam as the basic statically determinate system, establish the local coordinate system xOy of the basic statically determinate system of the simply supported beam, and determine the vertical deformation expression w of the basic statically determinate system of the composite beam under the action of nonlinear temperature gradient. T0 : Where, α c is the linear expansion coefficient of concrete, α s is the linear expansion coefficient of steel, E c is the elastic modulus of concrete, E s is the elastic modulus of steel, I c is the moment of inertia of the concrete bridge deck, I s is the moment of inertia of the steel beam, y a is the distance from the upper edge of the steel-concrete composite beam section to the centroid of the composite beam, b is the distance from the lower edge of the steel-concrete composite beam section to the centroid of the composite beam, d is the distance from the interface between the concrete bridge deck and the steel beam to the centroid of the composite beam, T(y) is the arbitrary vertical nonlinear temperature gradient acting on the composite beam, b(y) is the distribution of the cross-sectional width of the steel-concrete composite beam along the beam height, l l0 is the full length of the continuous steel-concrete composite beam on the arch; the overall rectangular coordinate system of the beam on the arch (X b ,Y b ) and the local coordinate system (x, y) of the arch beam are as follows: y=-Y b , where L M is the horizontal coordinate of the arch beam at the neutral axis position of the basic statically determinate system in the overall rectangular coordinate system of the arch beam; (2) According to the selected basic static system, add the corresponding redundant support reaction at the original intermediate support position to establish the local coordinate system x of the continuous beam on the arch F Oy F , the coordinate origin is set at the basic statically determinate system edge support. Then, according to the geometric compatibility equation of continuous beam deformation and the physical relationship between force and displacement, the supplementary equation is determined, the redundant support reaction is solved, and the vertical deformation expression w of the basic statically determinate system of the composite beam under the action of the redundant support reaction is determined. Fj : Where, F j is the jth redundant support reaction force; E s is the elastic modulus of steel; I l Convert the section moment of inertia for the steel-concrete composite beam; l l0 is the full length of the continuous steel-concrete composite beam on the arch; a j is the jth unknown redundant support reaction F j The horizontal axis of b j is the horizontal coordinate a j The relevant value, b j =l l0 -a j ; (3) Superimpose the vertical deformation of the basic statically determinate system of the composite beam under the action of the redundant support reaction force obtained in step (2) and the vertical deformation of the basic statically determinate system of the composite beam under the action of the nonlinear temperature gradient obtained in step (1) to determine the vertical deformation equation w of the continuous steel-concrete composite beam on the arch under the action of the nonlinear temperature gradient T : Where w T0 is the vertical deformation equation of the basic statically determinate system of the composite beam under the action of nonlinear temperature gradient; w Fj is the vertical deformation equation of the basic statically determinate system of the composite beam under the reaction of the jth unknown redundant support; c is the linear expansion coefficient of concrete; α s is the linear expansion coefficient of steel; E c is the elastic modulus of concrete; I c is the moment of inertia of the concrete bridge deck; I s is the moment of inertia of the steel beam; y a y is the distance from the upper edge of the steel-concrete composite beam section to the centroid of the composite beam; b y is the distance from the lower edge of the steel-concrete composite beam section to the centroid of the composite beam; d is the distance from the interface between the concrete bridge deck and the steel beam to the centroid of the composite beam; T(y) is the arbitrary vertical nonlinear temperature gradient acting on the composite beam; b(y) is the distribution of the cross-sectional width of the steel-concrete composite beam along the beam height; the local coordinate system (x, y) of the arch beam and the local coordinate system (x F ,y F ) is as follows: y=y F .

7. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 6, characterized in that: The bending moment M on any section of the arch ring in step S107 g and axial force N g The specific calculation is as follows: Where H x is the axial force acting on the elastic center; Y g is the vertical coordinate of the arch axis corresponding to any section of the arch ring in the overall rectangular coordinate system of the arch ring; gs is the vertical coordinate of the elastic center of the hingeless arch in the overall rectangular coordinate system of the arch ring; is the horizontal inclination angle of the arch axis; α c is the linear expansion coefficient of concrete; E c is the elastic modulus of concrete; I g is the moment of inertia of the arch ring section; t0 is the amplitude of the arch ring temperature change; l g is the calculated span of the arch ring; μ is the compression coefficient, A g is the area of ​​the arch cross section.

8. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 7, characterized in that: In step S108, the vertical deformation equation of the arch ring under the action of overall temperature increase and decrease is w g The calculation process is as follows: Where, l g Calculate the span for the arch ring, is the horizontal inclination angle of the arch axis, E c is the elastic modulus of concrete, I g is the moment of inertia of the arch ring section, M g is the bending moment on any section of the arch ring, N g is the axial force on any section of the arch ring, A g is the area of ​​the arch cross section, ε t is the strain generated in the micro segment of the arch ring under the action of temperature.

9. The method for constructing a mapping model between temperature-induced deformation of an arch bridge and rail surface geometry according to claim 8, characterized in that: In step S109, the vertical deformation equation of the continuous steel-concrete composite beam on the arch when forced displacement occurs is w v The calculation process is as follows: Where, Δ A and Δ B are the forced displacements of the supports on both sides, is the vertical deformation equation of the continuous steel-concrete composite beam under the reaction of the jth redundant support, and where, F vj is the support reaction force of the jth intermediate support when forced displacement occurs, I l is the converted section moment of inertia of the steel-concrete composite beam, a j is the jth unknown redundant support reaction F j The horizontal axis, b j is the horizontal coordinate a j The relevant value, b j =l l0 -a j .

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