Non-linear analysis method in multi-arch structural plane considering translation and rotation of arch feet

Abstract

The invention discloses a non-linear analysis method in a multi-arch structural plane considering translation and rotation of arch feet. The non-linear analysis method comprises the following steps: 1) deducing a geometric non-linear equilibrium equation in the multi-arch structural plane under the action of a vertical uniformly distributed load; 2) introducing a multi-arch boundary condition, and deducing to obtain a vertical displacement approximate analysis in a balance path considering translation and rotation nonlinearity of an adjacent arch structure; 3) further obtaining a relational expression between the dimensionless load of the multiple arches and the dimensionless load parameters; 4) deducing a multi-arch nonlinear branch point buckling equilibrium differential equation, and obtaining a critical load approximate analysis according to an internal force condition of multi-arch branch point buckling; 5) deducing to obtain approximate analysis of the multi-arch nonlinear jump buckling critical load.By means of the method, complex solving through finite element software can be avoided, and the calculation cost and workload of multi-span arch structure analysis are reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of in-plane geometric stability analysis of arch structures, and specifically to a nonlinear analytical method for in-plane of multi-arch structures considering the translation and rotation of arch feet. Background Art

[0002] In the prior art, the research on the nonlinear buckling critical load of multi-arch structures mainly focuses on single-arch models and simple multi-arch models, which can be specifically divided into the following two types of methods: 1. Analytical methods for single-arch models. Based on classical elastic stability theory, the boundary conditions of a single arch are simplified to two-hinged or fixed supports. By establishing the equilibrium differential equations in polar coordinates or rectangular coordinates, an approximate analytical solution of the in-plane nonlinear buckling critical load is derived. Such methods are computationally simple in single-arch structures, but ignore the cooperative effect of adjacent arches in multi-span multi-arch structures, resulting in significant deviations in the prediction of the buckling behavior of continuous multi-arch structures.

[0003] 2. Numerical methods for simple multi-arch structures. For multi-arch structures, the finite element method or analytical method is used. The boundary conditions of adjacent arch seats are assumed to be fully hinged or fully fixed, and the nonlinear buckling equation is solved through discretized modeling. Although such methods can partially reflect the mechanical characteristics of multi-arch structures, they have high computational complexity and are difficult to generalize to other layout scenarios of multi-span continuous multi-arch structures. In addition, their boundary conditions are relatively simple, without considering the translation and rotation of adjacent arch structures, which does not conform to the boundary conditions in actual engineering, further limiting the accuracy.

[0004] The above methods all have certain difficulties in solving the nonlinear stability problems of multi-span continuous multi-arch structures, which are mainly manifested in the following points: 1. The single-arch model cannot characterize the inter-span constraint effect of multi-span multi-arch structures and is insufficiently adaptable to complex multi-arch systems.

[0005] 2. Uniformly simplifying adjacent arch seats to fully hinged or fully fixed ignores the local constraint characteristics between arch structures in the actual structure, resulting in a significant deviation between the analytical solution of the critical load and the measured value.

[0006] 3. Existing methods mostly rely on numerical iteration or empirical formulas and lack explicit analytical expressions considering the nonlinear translation and rotation of adjacent arch structures, making it difficult to support actual engineering design. Summary of the Invention

[0007] The object of the present invention is to provide a method for in-plane non-linear analysis of a multi-arch structure considering the translation and rotation of the arch feet, taking the multi-arch under vertical uniform load as the research object, based on the displacement expression of the multi-arch in the Cartesian rectangular coordinate system, describing the deformation coordination of the fixed nodes, and deducing an approximate analytical solution for the in-plane non-linear buckling critical load of the multi-span continuous arch structure, so that engineers can avoid using finite element software for complex solutions, reducing the calculation cost and workload of the analysis of multi-span arch structures.

[0008] To achieve the above object, the present invention adopts the following technical solutions.

[0009] A method for in-plane non-linear analysis of a multi-arch structure considering the translation and rotation of the arch feet, comprising the following steps: Step S1: Taking the multi-arch under vertical uniform load as the research object, based on the in-plane non-linear strain-displacement expression of the multi-arch structure in the Cartesian rectangular coordinate system, deduce the in-plane geometric non-linear equilibrium equation of the multi-arch structure under uniform load; Step S2: According to the obtained in-plane geometric non-linear equilibrium equation of the multi-arch structure, considering the cooperative action of adjacent arches in the multi-span multi-arch structure, introduce the boundary conditions of the multi-arch, and deduce an approximate analytical solution for the vertical displacement in the equilibrium path considering the non-linearity of the translation and rotation of adjacent arch structures; Step S3: According to the approximate analytical solution for the vertical displacement in the non-linear equilibrium path of the multi-arch obtained, further obtain the relationship between the non-dimensional load of the multi-arch and the non-dimensional load parameter; Step S4: Based on the principle of the invariance of the buckling axis force at the non-linear bifurcation point, deduce the non-linear bifurcation buckling equilibrium differential equation of the multi-arch, and further obtain an approximate analytical solution for the critical load according to the internal force condition of the multi-arch at the bifurcation buckling; Step S5: Based on the characteristic that the non-linear snap-through buckling critical load corresponds to the extreme point in the buckling behavior curve at the time of its instability, deduce an approximate analytical solution for the non-linear snap-through buckling critical load of the multi-arch.

[0010] Specifically, the boundary conditions of the multi-arch in Step S2 are expressed as: ; In the above formula, is the vertical displacement of the curve element of the th arch structure in the Cartesian rectangular coordinate system, i = 1, 2, 3; is the vertical displacement of the curve element of the r th arch structure in the Cartesian rectangular coordinate system, r= 4, 5, 6,..., n, where n is a positive integer; is the horizontal coordinate corresponding to the th arch in the Cartesian rectangular coordinate system, with the right direction being positive; is the span of the cross-arch structure; ; ; ; The approximate analytical expression of the vertical displacement in the nonlinear equilibrium path considering the translation and rotation of adjacent arch structures is: ; In the above formula, is the vertical displacement of the curve element of the th cross-arch structure in the Cartesian rectangular coordinate system; is the horizontal coordinate corresponding to the th cross-arch Cartesian rectangular coordinate system, with the right direction being positive; is the dimensionless axial force stability parameter of the th cross-arch, ; is the axial force parameter of the th cross-arch, , is the horizontal thrust at the arch foot of the th cross-arch, is the elastic modulus of the th cross-arch structure, is the flexural moment of inertia of the cross-section of the main arch ring of the th cross-arch; is the focus parameter of the th parabolic arch, , is the rise of the th cross-arch; is the dimensionless load, , is the vertical uniform load acting on the th cross-arch; is the vertical displacement coefficient, , , , , , , , , , is the ratio of the spans of adjacent arches, , is the ratio of the parabolic focus parameters of adjacent arches, , .

[0011] Specifically, the relationship between the dimensionless load and the dimensionless load coefficient of the continuous arch described in step S3 is: ; In the above formula, , , is the dimensionless load coefficient, expressed as: ; ; ; In the above formula, is the dimensionless axial force stability parameter of the th arch; is the modified slenderness ratio of the th arch, and the expression is: ; In the above formula, is the radius of gyration of the main arch ring section, , is the flexural moment of inertia of the main arch ring section, A is the area of the main arch ring section; is the catenary arch coefficient.

[0012] Specifically, in step S4, based on the principle of constant buckling axial force at the nonlinear bifurcation point, the nonlinear bifurcation buckling equilibrium differential equation of the continuous arch is deduced as follows: The expressions of the in-plane horizontal displacement and vertical displacement after buckling of the nonlinear continuous arch structure considering the translation and rotation of adjacent arch structures in the Cartesian rectangular coordinate system are as follows: ; In the above formula, is the horizontal displacement after buckling of the th arch structure; is the horizontal displacement after deformation of the th arch structure; is the change in horizontal displacement during buckling of the th arch structure; is the vertical displacement after buckling of the th arch structure; is the vertical displacement after deformation of the th arch structure; is the change in vertical displacement during buckling of the th arch structure; Substituting into the equilibrium differential equation in the vertical direction, we get: ; ; In the above formula, is the compressive strain after buckling of the second arch structure; is the vertical coordinate of the corresponding coordinate system of the main arch ring of the th arch; Based on the principle of constant buckling axial force at the nonlinear bifurcation point: ; In the above formula, is the horizontal thrust at the arch springing when the cross-arch structure buckles; The equilibrium differential equation of the nonlinear bifurcation buckling of the continuous arch is obtained as: ; In the above formula, .

[0013] Furthermore, in step S4, the approximate analytical solution of the critical load is obtained according to the internal force condition of the continuous arch undergoing bifurcation buckling, and the process is as follows: Substitute and into the obtained equilibrium differential equation of the nonlinear bifurcation buckling of the continuous arch, and the approximate analytical solution of the critical load of the nonlinear bifurcation buckling of the continuous arch is obtained as: ; In the above formula, , , are dimensionless load coefficients, respectively expressed as: ; ; ; In the above formula, is the ratio of the stability parameters between the side arch and the loaded arch in the continuous arch structure, and the expression is: ; In the above formula, is the elastic modulus of the cross-arch structure; is the flexural moment of inertia of the cross-main arch ring section.

[0014] Specifically, in step S5, the approximate analytical solution of the critical load of the nonlinear snap-through buckling of the continuous arch is deduced as follows: The critical load at which the continuous arch undergoes nonlinear snap-through buckling corresponds to the extreme points on the buckling behavior curve, including the maximum and minimum points of the nonlinear equilibrium path of the continuous arch. The extreme points are obtained by solving the stationary points of the buckling behavior curve, and are expressed as: ; In the above formula, is the in-plane nonlinear snap-through buckling load of the continuous arch structure; is the second-order instability buckling load of the hinged continuous arch; is the focus parameter of the second-span parabolic arch; when the horizontal thrust at the arch springing is greater than zero, the dimensionless load​ It is expressed by the definition of as: ; In the above formula, is the dimensionless load of the second-span arch; Substitute the above formula into the stationary point of the buckling behavior curve and multiply by to obtain: ; Substitute into the above formula, then the horizontal thrust at the springing of the second-span main arch is expressed as: ; According to the critical load formula, rewrite the relationship between the dimensionless load and the dimensionless load coefficient of the continuous arch as an implicit function about : ; In the above formula, is an implicit function about ; The equilibrium equation of the in-plane nonlinear jump buckling of the continuous arch structure is obtained as: ; In the above formula, , , are dimensionless load coefficients, expressed as: ; ; ; In the above formula, is the modified slenderness ratio of the second-span arch; Regarding the load corresponding to the maximum point as the critical load of the in-plane nonlinear jump buckling of the continuous arch, according to the quadratic formula for finding the roots of a quadratic equation: ; Then the approximate analytical expression of the in-plane nonlinear jump buckling load of the three-span continuous arch structure is: .

[0015] Compared with the prior art, the present invention has the following beneficial effects: The method of the present invention applies the in-plane non-linear compression strain-displacement expression and non-linear bending strain-displacement expression of the arch structure, and combines the boundary conditions of the multi-arch to deduce the analysis of the in-plane non-linear mechanical problems of the multi-arch structure; the mechanical concepts in the deduction process are clear and the method is simple, which can quickly determine the analysis of the non-linear mechanical problems of the arch structure in the Cartesian rectangular coordinate system, greatly reducing the workload of bridge designers; and based on the in-plane non-linear equilibrium differential equation and approximate analysis of the multi-arch structure in the Cartesian rectangular coordinate system of the present invention, bridge researchers can further explore the law of in-plane non-linear deformation of the multi-arch structure. Brief Description of the Drawings

[0016] Figure 1 is the flow chart of the in-plane non-linear analysis method of the multi-arch structure considering the translation and rotation of the arch foot of the present invention; Figure 2 is the calculation sketch of the multi-arch structure in the Cartesian rectangular coordinate system of the present invention; Figure 3 is the schematic diagram of the non-linear deformation of the multi-arch structure in the Cartesian rectangular coordinate system of the present invention; Figure 4 is the schematic diagram of the numerical analysis verification result of the in-plane non-linear equilibrium equation of the multi-arch of the present invention; In the figure: 1. Arch axis; 2. Multi-arch structure; 3. Arch foot; 4. Vertical uniform load. Detailed Embodiment

[0017] To facilitate the understanding and implementation of the present invention by those of ordinary skill in the art, the following will detail each step of the method proposed by the present invention. It should be understood that these embodiments are only for illustrating the present invention and not for limiting the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms also fall within the scope defined by the appended claims of this application.

[0018] Embodiment As Figure 1 shown, the present invention discloses an in-plane non-linear analysis method of a multi-arch structure considering the translation and rotation of the arch foot, including the following steps: Step S1: Taking the multi-arch under the action of the vertical uniform load 4 as the research object, based on the in-plane non-linear strain-displacement expression of the multi-arch structure in the Cartesian rectangular coordinate system, deduce the in-plane geometric non-linear equilibrium equation of the multi-arch structure under the action of the uniform load; As Figure 2 and Figure 3 shown, for the multi-arch structure 2 in the Cartesian rectangular coordinate system, the equation of the arch axis 1 is , , being the vertical coordinate of the corresponding coordinate system of the main arch ring of the th span arch, For the corresponding The horizontal coordinate of the Cartesian coordinate system across the arch, For the The focal parameters of the spanning parabolic arch, For the The span of the arch structure, For the The rise of the arch, i =1,2,3; Assume that no load is applied on both sides of the arch, the arch feet 3 at both ends are hinged, the middle arch feet 3 are mutually consolidated, only the deformation in the structural plane is considered, and all out-of-plane displacements and torsion are not considered. The cross section of the arch structure before and after deformation is always perpendicular to the central axis, and the deformation of the two side arches of the arch structure is symmetrical before buckling. Figure 2 Where O1 and O2 are the Cartesian coordinate origins of the corresponding span arches, and the in-plane nonlinear strain-displacement expression of the continuous arch structure based on the Cartesian coordinate system is expressed as: ; In the above formula, For the The compressive strain at any point on the span; For the Bending strain at any point on the span; For the The vertical coordinates of the coordinate system corresponding to the main arch ring of the span arch; For the Horizontal displacement of the span after deformation; is the Cartesian coordinate system Vertical displacement of the infinitesimal element of the arch structure curve.

[0019] The process of deducing the geometric nonlinear equilibrium equation of the in-plane multi-arch structure under the action of uniformly distributed load is as follows: Using the principle of virtual work for the arch structure, the nonlinear strain-displacement expression of the arch structure in the plane is substituted into the calculation to obtain the vertical equilibrium differential equation: ; In the above formula, For the Horizontal thrust across the arch foot; ; Introducing axial force parameters μ and dimensionless loads : ; In the above formula, H is the horizontal thrust of the arch foot; E is the elastic modulus of the arch structure; is the bending moment of inertia of the main arch section; pis the focus parameter of the parabolic arch; q is the vertical uniformly distributed load on the multi - arch; Convert the equilibrium differential equation in the vertical direction to: ; In the above formula, is the axial force parameter of the th - span arch; is the focus parameter of the th - span parabolic arch; is the dimensionless load of the second - span arch; ; ; Step S2: According to the obtained in - plane geometrically nonlinear equilibrium equation of the multi - arch structure, considering the cooperative action of adjacent arches in the multi - span multi - arch structure, introduce the multi - arch boundary conditions, and deduce the approximate analytical solution of the vertical displacement in the equilibrium path considering the nonlinearity of the translational and rotational motions of adjacent arch structures; Step S3: According to the approximate analytical solution of the vertical displacement in the obtained multi - arch nonlinear equilibrium path, further obtain the relationship between the dimensionless load of the multi - arch and the dimensionless load parameter; Step S4: Based on the principle of constant flexural axis force at the nonlinear bifurcation point, deduce the equilibrium differential equation of the multi - arch nonlinear bifurcation buckling, and further obtain the approximate analytical solution of the critical load according to the internal force condition of the multi - arch occurring bifurcation buckling; Step S5: Based on the characteristic that the critical load of nonlinear snap - through buckling corresponds to the extreme point in the buckling behavior curve at the time of its instability, deduce the approximate analytical solution of the critical load of the multi - arch nonlinear snap - through buckling.

[0020] Specifically, the multi - arch boundary conditions in Step S2 are expressed as: ; In the above formula, is the vertical displacement of the curve element of the th - span arch structure in the Cartesian rectangular coordinate system, i = 1, 2, 3; is the vertical displacement of the curve element of the r th - span arch structure in the Cartesian rectangular coordinate system, r= 4, 5, 6,..., n, where n is a positive integer; is the horizontal coordinate of the Cartesian rectangular coordinate system corresponding to the th - span arch, with the right direction being positive; is the span of the th - span arch structure; ; ; The expression of the approximate analytical solution of the vertical displacement in the equilibrium path considering the nonlinearity of the translational and rotational motions of adjacent arch structures is: ; In the above formula, is the vertical displacement of the infinitesimal element of the curve of the cross-arch structure; corresponds to the horizontal coordinate of the Cartesian rectangular coordinate system of the th cross-arch, with the right direction being positive; is the dimensionless axial force stability parameter of the th cross-arch, ; is the axial force parameter of the th cross-arch, is the horizontal thrust at the arch springing of the th cross-arch, is the elastic modulus of the th cross-arch structure, is the flexural moment of inertia of the th cross-main arch ring section; is the focus parameter of the th parabolic arch, is the rise of the th cross-arch; is the dimensionless load, is the uniform vertical load acting on the th cross-arch, is the vertical displacement coefficient, ; ; ; ; ; ; ; ; is the ratio of adjacent arch spans, ; is the ratio of the parabolic focus parameters of adjacent arches, ; .

[0021] Based on the principle that the curve integral of the compressive strain along the arch axis is equal to the shortening amount of the arch axis, the vertical displacements of each arch of the multi-arch are substituted into the following deformation compatibility condition: ; In the above formula, is the cross-sectional area of the main arch ring of the th cross-arch; is the compressive strain at any point on the th cross-arch structure, Integrating both sides of the deformation compatibility condition, the in-plane nonlinear equilibrium control equation of the nonlinear multi-arch structure considering the translation and rotation of adjacent arch structures in the Cartesian rectangular coordinate system is obtained. Furthermore, the relationship between the dimensionless load and the dimensionless load coefficient of the multi-arch is obtained: ; In the above formula, 、 、 are dimensionless load coefficients, expressed as: ; ; ; In the above formula, is the dimensionless axial force stability parameter of the th arch; is the modified slenderness ratio of the th arch, and the expression is: ; In the above formula, is the radius of gyration of the main arch ring section, , is the flexural moment of inertia of the main arch ring section, A is the cross-sectional area of the main arch ring; is the catenary arch coefficient.[[ID=4,6]]

[0022] The expressions of the in-plane horizontal displacement and vertical displacement after buckling of the nonlinear multi-arch structure considering the translation and rotation of adjacent arch structures in the Cartesian rectangular coordinate system are known as follows: ; In the above formula, is the horizontal displacement after buckling of the th arch structure; is the horizontal displacement after deformation of the th arch structure; is the change in horizontal displacement during buckling of the th arch structure; is the vertical displacement after buckling of the th arch structure; is the vertical displacement after deformation of the th arch structure; is the change in vertical displacement during buckling of the th arch structure; Substituting into the equilibrium differential equation in the vertical direction, we get: ; ; In the above formula, is the post - buckling compressive strain of the second - span arch structure; is the vertical coordinate of the corresponding coordinate system of the main arch ring of the Since the horizontal thrust of the multi - arch structure with branch - point buckling hardly changes, and for each arch, it satisfies: ; In the above formula, is the horizontal thrust at the arch foot when the Based on the principle of constant axial force at the non - linear branch - point buckling, the equilibrium differential equation of the non - linear branch - point buckling of the multi - arch structure is expressed as: ; In the above formula, ; Substituting the boundary conditions of the branch - point buckling of the multi - arch structure, the approximate analytical solution of the branch - point buckling is obtained: ; The above approximate analytical solution of the branch - point buckling is the critical value when the loaded arch in the structure loses stability; therefore, for the case where the parameters are different when the remaining arches of the multi - arch structure have branch - point buckling, the approximate analytical solution of the branch - point buckling is expressed as: ; In the above formula, is the ratio of the stability parameters between the side arch and the loaded arch in the multi - arch structure, and the expression is: ; Substitute and into the in - plane non - linear equilibrium differential equation of the multi - arch, and the approximate analytical solution of the critical load of the non - linear branch - point buckling of the multi - arch is obtained as: ; In the above formula, , , are dimensionless load coefficients respectively, and the expressions are: ; ; .

[0023] Specifically, the process of deriving the approximate analytical solution of the critical load of the non - linear jump buckling of the multi - arch in step S5 is as follows: The critical load of the non - linear jump buckling of the multi - arch corresponds to the extreme points on the buckling behavior curve, including the maximum and minimum points of the non - linear equilibrium path of the multi - arch. The extreme points are obtained by solving the stationary points of the buckling behavior curve, and are expressed as: ; In the above formula, is the non - linear jump buckling load in the multi - arch structure plane; is the second - order instability buckling load of the hinged multi - arch; is the focus parameter of the parabola arch of the second span; when the horizontal thrust at the arch foot is greater than zero, the dimensionless load is expressed through the definition of as: ; Substitute the above formula into the stationary point of the buckling behavior curve and multiply by to get: ; Substitute into the above formula, then the horizontal thrust at the arch foot of the main arch of the second span is expressed as: ; According to the critical load formula, rewrite the relationship between the dimensionless load and the dimensionless load coefficient of the multi - arch as an implicit function of about : ; In the above formula, is the implicit function of about ; The equilibrium equation of the non - linear jump buckling in the multi - arch structure plane is obtained as: ; In the above formula, , , are dimensionless load coefficients, expressed as: ; ; ; In the above formula, is the modified slenderness ratio of the second - span arch; Regarding the load corresponding to the maximum point as the critical load of the non - linear jump buckling in the multi - arch plane, according to the quadratic formula for finding the roots of a quadratic equation, we have: ; Then the approximately analytical expression of the non - linear jump buckling load in the three - span multi - arch structure plane is: .

[0024] In summary, the approximate analytical solution of the in-plane nonlinear equilibrium equation of the multi-arch structure and the approximate analytical solution of the in-plane nonlinear jump buckling critical load of the multi-arch structure are obtained, thus solving the essence of the in-plane nonlinear mechanical problem of the multi-arch structure in the Cartesian rectangular coordinate system.

[0025] Next, through a practical example, the technical effects of the method of the present invention are further described.

[0026] To verify the accuracy of the approximate analytical results obtained by the method of the present invention, in this example, the mathematical software MATLAB is used to program and calculate the above-obtained approximate analytical expressions under different rise-span ratios, and the corresponding load-displacement curves and buckling behavior curves are plotted according to the calculation results. At the same time, the ANSYS finite element software is used to establish a multi-arch structure model with a solid rectangular cross-section using the Beam4 beam element. The Beam4 beam element is a uniaxial stress element used to withstand tension, compression, bending, and torsion. This element has six degrees of freedom at each node: , , translation in the [[ID=, , axis and angular displacements rotating about the

[0027] This multi-arch structure finite element model simulates the vertical uniformly distributed load by adding nodal forces to the nodes of the loaded arch model, and defines the coupling of the rotation angle and displacement at the nodes of adjacent arch feet. At the same time, all out-of-plane degrees of freedom of the nodes of the multi-arch model are constrained to avoid out-of-plane deformation, and the arc-length method is used to track the geometric nonlinear displacement at the crown of the arch.

[0028] The calculation results of the finite element software are compared with the approximate analytical results calculated by the method of the present invention. For the finite element model, the height D of the multi-arch cross-section is selected as 0.045 m, the width B is 0.4 m, the elastic modulus E of the material is 210 GPa, and the rise-span ratios are selected as and , and the modified slenderness ratio takes values from 10 to 50 to analyze the accuracy of the approximate analytical solution of the in-plane nonlinear equilibrium equation of the multi-arch under different rise-span ratios.

[0029] As Figure 4 shows, it is a schematic diagram of the numerical analysis verification results of the in-plane nonlinear equilibrium equation of the multi-arch. As can be seen from Figure 4 , the calculation results of the approximate analytical solution of the method of the present invention are in good agreement with the finite element results. The overall changing trends of the two fitted curves are the same and the data are close, indicating that the method of the present invention can obtain an approximate analytical solution of high accuracy for the in-plane nonlinear buckling critical load of the multi-arch structure.

[0030] The above are only the preferred embodiments of the present invention, and are not intended to limit the present invention in other forms. Any person skilled in the relevant art may use the technical content disclosed above to make changes or modifications into equivalent embodiments with equivalent changes. However, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the technical solution content of the present invention still fall within the protection scope of the technical solution of the present invention.

Claims

1. An in-plane nonlinear analytical method for a multi-arch structure considering the translation and rotation of the arch springing, characterized in that It includes the following steps: Step S1: Taking the continuous arch under uniformly distributed vertical load as the research object, based on the in-plane nonlinear strain-displacement expression of the continuous arch structure in the Cartesian rectangular coordinate system, deduce the in-plane geometric nonlinear equilibrium equation of the continuous arch structure under uniformly distributed load; Step S2: According to the obtained in-plane geometric nonlinear equilibrium equation of the continuous arch structure, considering the cooperative action of adjacent arches in the multi-span continuous arch structure, introduce the boundary conditions of the continuous arch, and deduce the approximate analytical solution of the vertical displacement in the nonlinear equilibrium path considering the translation and rotation of adjacent arch structures; Step S3: According to the approximate analytical solution of the vertical displacement in the obtained nonlinear equilibrium path of the continuous arch, further obtain the relationship between the dimensionless load and the dimensionless load parameter of the continuous arch; Step S4: Based on the principle of constant flexural axial force at the nonlinear bifurcation point, deduce the nonlinear bifurcation buckling equilibrium differential equation of the continuous arch, and further obtain the approximate analytical solution of the critical load according to the internal force condition of the continuous arch at the bifurcation buckling; Step S5: Based on the characteristic that the critical load of nonlinear snap-through buckling corresponds to the extreme point in the buckling behavior curve at its instability, deduce the approximate analytical solution of the critical load of nonlinear snap-through buckling of the continuous arch.

2. The in-plane non-linear analytical method for the multi-arch structure considering the translation and rotation of the arch springing according to claim 1, characterized in that, The boundary conditions of the continuous arch described in Step S2 are expressed as: ； In the above formula, is the vertical displacement of the infinitesimal curve element of the cross-arch structure in the Cartesian rectangular coordinate system, i = 1, 2, 3; is the vertical displacement of the infinitesimal curve element of the r cross-arch structure in the Cartesian rectangular coordinate system, r= 4, 5, 6,..., n, where n is a positive integer; is the horizontal coordinate corresponding to the cross-arch Cartesian rectangular coordinate system, with the right direction being positive; is the span of the cross-arch structure; ; ; The expression of the approximate analytical solution of the vertical displacement in the nonlinear equilibrium path considering the translation and rotation of adjacent arch structures is: ； In the above formula, is the vertical displacement of the infinitesimal element of the curved line of the cross-arch structure; corresponds to the horizontal coordinate of the Cartesian rectangular coordinate system of the th cross-arch, with the right direction being positive; ; is the dimensionless axial force stability parameter of the th cross-arch, is the horizontal thrust at the arch foot of the th cross-arch, is the elastic modulus of the cross-main arch ring structure; is the flexural moment of inertia of the cross-main arch ring section; is the focus parameter of the th cross-parabolic arch, ; is the rise of the th cross-arch; is the dimensionless load, is the vertical displacement coefficient, ; ; ; ; ; ; ; ; ; is the ratio of adjacent arch spans, ; is the ratio of adjacent arch parabolic focus parameters, ; .

3. A nonlinear in-plane analytical method for the multi-arch structure considering the translation and rotation of the arch springing according to claim 2, characterized in that, The relationship between the dimensionless load and the dimensionless load coefficient of the continuous arch described in Step S3 is: ； In the above formula, , , are dimensionless load coefficients, expressed as: ； ； ； In the above formula, is the dimensionless axial force stability parameter of the th cross arch; is the modified slenderness ratio of the th cross arch, and the expression is: ； In the above formula, is the radius of gyration of the main arch ring section, , is the flexural moment of inertia of the main arch ring section, A is the area of the main arch ring section; is the catenary arch coefficient.

4. A nonlinear in-plane analytical method for a multi-arch structure considering the translation and rotation of the arch springing, as claimed in claim 3, wherein In Step S4, based on the principle of constant flexural axial force at the nonlinear bifurcation point, deduce the nonlinear bifurcation buckling equilibrium differential equation of the continuous arch. The deduction process is as follows: The expressions of the in-plane horizontal displacement and vertical displacement after buckling of the nonlinear continuous arch structure considering the translation and rotation of adjacent arch structures in the Cartesian rectangular coordinate system are known as follows: ； In the above formula, is the horizontal displacement after buckling of the th cross-arch structure; is the horizontal displacement after deformation of the th cross-arch structure; is the change in horizontal displacement at the time of buckling of the th cross-arch structure; is the vertical displacement after buckling of the th cross-arch structure; is the vertical displacement after deformation of the th cross-arch structure; is the change in vertical displacement at the time of buckling of the th cross-arch structure; Substituting into the equilibrium differential equation in the vertical direction, we get: ； ； In the above formula, is the post - buckling compressive strain of the second - span arch structure; is the vertical coordinate of the main arch ring of the - th span arch in the corresponding coordinate system; Based on the principle of constant flexural axial force at the nonlinear bifurcation point: ； In the above formula, is the horizontal thrust at the arch springing when the cross-arch structure buckles; The nonlinear bifurcation buckling equilibrium differential equation of the continuous arch is obtained and expressed as: ； In the above formula, .

5. A nonlinear in-plane analytical method for a multi-arch structure considering the translation and rotation of the arch springing, as claimed in claim 4, characterized in that In Step S4, the process of obtaining the approximate analytical solution of the critical load according to the internal force condition of the continuous arch at the bifurcation buckling is as follows: Substitute and into the obtained nonlinear bifurcation buckling equilibrium differential equation of the multi-arch, and the approximate analytical expression of the critical load for the nonlinear bifurcation buckling of the multi-arch is obtained as follows: ； In the above formula, , , are dimensionless load coefficients, respectively expressed as: ； ； ； In the above formula, is the ratio of the stability parameter between the side arch and the loaded arch in the multi-arch structure, and the expression is: ； In the above formula, is the elastic modulus of the cross-arch structure; is the flexural moment of inertia of the cross-section of the main arch ring.

6. The in-plane nonlinear analytical method for a multi-arch structure considering the translation and rotation of the arch springing according to claim 5, characterized in that The process of deducing the approximate analytical solution of the critical load of nonlinear snap-through buckling of the continuous arch described in Step S5 is as follows: The critical load of the continuous arch at nonlinear snap-through buckling corresponds to the extreme point in the buckling behavior curve, including the maximum point and the minimum point of the nonlinear equilibrium path of the continuous arch. The extreme point is obtained by solving the stationary point of the buckling behavior curve and is expressed as: ； In the above formula, is the in-plane nonlinear jump buckling load of the multi-arch structure; is the second-order instability buckling load of the hinged multi-arch; is the focus parameter of the parabola arch of the second span; when the horizontal thrust at the arch foot is greater than zero, the dimensionless load is expressed through as follows: ； In the above formula, is the dimensionless load of the second-span arch; Substitute the above equation into the stationary point of the buckling behavior curve and multiply by to obtain: ； Substitute into the above formula, then the horizontal thrust at the arch springing of the second main arch is expressed as: ； According to the critical load formula, rewrite the relationship between the dimensionless load and the dimensionless load coefficient of the multi-arch as Regarding as an implicit function of: ； In the above formula, is an implicit function with respect to The equilibrium equation of the in-plane nonlinear snap-through buckling of the continuous arch structure is obtained and expressed as: ； In the above formula, , , are dimensionless load coefficients, expressed as: ； ； ； In the above formula, is the modified slenderness ratio of the second-span arch; Regarding the load corresponding to the maximum point as the critical load of the in-plane nonlinear snap-through buckling of the continuous arch, according to the quadratic formula for finding the roots of a quadratic equation, we have: ； Then the approximate analytical solution of the in-plane nonlinear snap-through buckling load of the three-span continuous arch structure is expressed as: 。

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