Approximate analysis method for in-plane nonlinear stable load of non-full-span load arch bridge

Abstract

The invention discloses a non-full-span load arch bridge in-plane nonlinear stable load approximate analysis method, which comprises the following steps of: introducing a section integration function to an arch bridge to describe a non-full-span load, and deducing a geometric nonlinear equilibrium differential equation in a structural plane; a vertical displacement approximate solution is obtained by solving the equation, and then a nonlinear balance equation is established; deducing a branch point buckling equilibrium equation by combining strain after buckling, determining critical internal force by utilizing boundary conditions and a constant variation method, and solving a branch point buckling critical load; meanwhile, the critical load of jump buckling is obtained by analyzing the extreme value of the buckling behavior curve. The method is clear in mechanics concept, load distribution is described in a unified mode by introducing a section integration function, critical internal force is determined through boundary conditions and a constant variation method, and high-precision approximate analysis of branch point buckling and jump buckling critical loads is achieved; and a reliable and convenient analytical calculation tool is provided for stability design and evaluation of the non-full-span load arch bridge.

Description

Technical Field

[0001] This invention belongs to the field of in-plane stability analysis technology for arch bridge structures, specifically an approximate analytical method for in-plane nonlinear stability loads on arch bridges under non-full-span loads. Background Technology

[0002] Arch structures, with their excellent load-bearing characteristics, are widely used in bridges, building construction, and other engineering projects. The stability of an arch bridge under external loads depends not only on the magnitude of the load but also on the type of load it bears. Different load types, such as full-span versus partial-span arrangement, uniform versus non-uniform distribution, and concentrated forces, all directly affect the stability of an arch bridge. Under partial-span loads, the stress distribution of an arch bridge is complex, and its nonlinear stability issues are more prominent. Due to the non-uniformity of the load, differences in bending moment distribution occur within the arch bridge, leading to uneven stress distribution in different parts of the arch. This uneven stress inevitably causes nonlinear responses in the arch bridge, thereby increasing the risk of structural instability.

[0003] Currently, for the in-plane nonlinear stability problem of non-circular arch structures, non-full-span loads are basically not considered, and the main methods are as follows: The nonlinear stability theory within the arch surface in polar coordinates is proposed. This theory derives the nonlinear equilibrium equations of a circular arch under non-full-span loading using polar coordinates, analyzes the buckling mode of the circular arch, and obtains the analytical solution for the buckling load under non-full-span loading. However, this method is not applicable to non-circular arch structures. The theory of in-plane nonlinear stability of arches in rectangular coordinates focuses on structures with non-circular arches under full-span loads. The calculation methods and steps used are similar to those in polar coordinates, yielding analytical solutions for buckling loads. However, it lacks a method for solving the in-plane nonlinear stability problem of arch structures under non-full-span loads in rectangular coordinates.

[0004] In summary, the above methods are not applicable to solving the approximate analytical problem of nonlinear stability loads in the deck of a non-circular arch bridge under non-full-span loads, mainly because: (1) The polar coordinate system can only solve problems related to circular arches, not problems related to non-circular arches. Therefore, this method cannot be used to analyze the in-deck stability of non-circular arch bridges under non-full span loads.

[0005] (2) Most studies in the rectangular coordinate system focus on non-circular arch structures under full-span loads, and lack methods for in-plane nonlinear stability problems of arch bridges under non-full-span loads. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of existing technologies and provide an approximate analytical method for nonlinear stability loads in the in-plane of a non-full-span load arch bridge. Based on the principle of virtual work, considering the region on the main arch ring without external loads, a segmental integration function is introduced to derive the nonlinear equilibrium equations in the in-plane of the non-full-span load arch structure. Combined with the post-buckling strain expression, the buckling equilibrium equations at the branch points of the non-full-span load arch structure are derived. Based on boundary conditions and the constant variation method, the critical internal force conditions before and after the buckling at the branch points of the arch bridge are determined, thereby solving for the nonlinear critical load of buckling at the branch points of the non-full-span load arch bridge. The partial derivative of the extreme values ​​of the loads for buckling behavior when the non-full-span load arch bridge becomes unstable is obtained to yield the leap buckling equilibrium equations in the in-plane of the arch bridge. Based on this, an approximate analytical method for solving the leap buckling critical load of the non-full-span load arch bridge is obtained.

[0007] To achieve the above objectives, the present invention provides the following technical solutions.

[0008] An approximate analytical method for nonlinear stability loads in-plane arch bridges under non-full-span loads includes the following steps: Step S1: Taking a parabolic, hingeless arch bridge with a non-full span load as the research object, obtain the expressions for the compressive strain and bending strain of the arch bridge with a non-full span load in the Cartesian coordinate system. Step S2: Based on the expressions for compressive strain and bending strain of the non-full-span load arch bridge in the Cartesian coordinate system obtained in Step S1, the geometric nonlinear equilibrium differential equation in the plane of the non-full-span load arch bridge is derived based on the principle of virtual work and by introducing the segment integration function. Step S3: By solving the in-plane geometric nonlinear equilibrium differential equation of the non-full span load arch bridge obtained in step S2, the approximate analytical vertical displacement of the non-full span load arch bridge is obtained. Step S4: Based on the obtained approximate analysis of the vertical displacement of the non-full span load arch bridge, deduce the nonlinear equilibrium equation of the non-full span load arch bridge to obtain the relationship between the dimensionless internal force coefficients and the dimensionless load parameters. Step S5: Based on the buckling equilibrium equation of the branch point of the arch bridge under non-full span load, determine the critical internal force conditions before and after buckling of the branch point according to the boundary conditions and the constant variation method, and solve for the approximate analytical solution of the critical buckling load of the branch point of the arch bridge under non-full span load. Step S6: Take the partial derivative of the load extremum of the buckling behavior of the arch bridge under instability when it is not fully loaded, and obtain the in-plane buckling equilibrium equation of the arch bridge under instability. Then solve for the approximate analytical solution of the critical load for buckling of the arch bridge under instability.

[0009] Specifically, the expressions for the compressive strain and bending strain of the non-full-span load arch bridge in the Cartesian coordinate system in step S1 are as follows: ; In the above formula, and These represent the compressive strain and bending strain at any point on an arch bridge under non-full span load, respectively. The x-coordinate of the coordinate axis; These are the focal parameters of the arch bridge; and These are the horizontal and vertical displacements of the arch bridge, respectively. The vertical coordinates of the arch axis of the arch bridge after deformation; and They represent respectively to Find the first and second derivatives.

[0010] Specifically, the process of deriving the geometric nonlinear equilibrium differential equations in the deck of the non-full-span load arch bridge based on the principle of virtual work and by introducing the segment integration function in step S2 is as follows: According to the principle of virtual work, for an arch bridge in equilibrium, the virtual work done by external forces is equal to the virtual strain energy generated by the arch bridge. Let's assume the vertical virtual displacement is... Vertical virtual strain becomes Then we have: ; In the above formula, Indicates to Variations; This represents the total potential energy of the arch bridge. The total strain of an arch bridge under non-full-span load; This represents the actual stress. The total volume of the arch bridge , The cross-sectional area of ​​the main arch ring. The vertical coordinates of the arch axis of the arch bridge before deformation. The x-coordinate of the coordinate system; For arch bridges with non-full-span uniformly distributed loads; The span of the arch bridge; Introducing a segment integration function to handle non-full-span uniformly distributed loads on arch bridges The piecewise function expression can be represented by a single global equation as follows: ; In the above formula, For arch bridges with non-full-span uniformly distributed loads; The intensity of a uniformly distributed load; This is a segment integration function; The parameter representing the size of the area on the main arch ring where no external loads are applied; The x-coordinate of the coordinate system; The arch bridge is not under a uniformly distributed load at full span. The piecewise function expression is: ; The integrated non-full-span uniformly distributed load Substituting into the virtual work principle equation and simplifying, the in-plane geometric nonlinear equilibrium differential equation for a non-full-span loaded arch bridge is expressed as follows: ; In the above formula, This represents the vertical displacement of the arch bridge. Indicates to Find the second derivative. Indicates to Find the fourth derivative; It is a dimensionless stability coefficient. , This refers to the horizontal thrust at the arch foot of the arch bridge. The elastic modulus of the arch bridge, The moment of inertia is the bending moment of the main arch section of the arch bridge. The intensity of a uniformly distributed load; These are the focal parameters of the arch bridge; For the integration term of piecewise functions, .

[0011] Specifically, the approximate analytical solution for the vertical displacement of the non-full-span load arch bridge obtained in step S3 is... The formula is expressed as: ; In the above formula, This is the integration term for piecewise functions; The load factor is a dimensionless load factor. , The intensity of a uniformly distributed load. For the focal parameters of the arch bridge, The horizontal thrust at the arch foot of the arch bridge; It is a dimensionless stability coefficient; The x-coordinate of the coordinate system; These are dimensionless internal force coefficients. , The span of the arch bridge; For arch bridges not subjected to full-span loads, the internal force coefficients are... , For non-full span load factors, and 2c is the length of the unloaded area in the middle of the construction load; For dimensionless loads, ; For integration functions, When the non-full span load factor At that time, the length of the unloaded area in the middle of the construction load is 2c=0, and the load on the arch bridge is a uniformly distributed load across the full span.

[0012] Specifically, in step S4, the nonlinear equilibrium equations of the arch bridge under non-full span load are derived to obtain the relationship between dimensionless internal force coefficients and dimensionless load parameters. The process is as follows: Since the deformation of an arch bridge over its entire span is equal to the integral of the compressive strain over the entire span, we have the integral equation: ; In the above formula, The horizontal thrust at the arch foot of the arch bridge; The elastic modulus of the arch bridge; The cross-sectional area of ​​the main arch ring; The vertical coordinates of the arch axis of the arch bridge before deformation. The x-coordinate of the coordinate system; This represents the horizontal displacement of the arch bridge. This represents the vertical displacement of the arch bridge. These are the focal parameters of the arch bridge; Will Substitution and will Substituting into the integral equation, the nonlinear equilibrium equation of the non-full-span loaded arch bridge is derived, yielding the relationship between the dimensionless internal force coefficients and the dimensionless load parameters: ; In the above formula, , , These are the coefficients of the nonlinear equilibrium equation for an arch bridge under non-full span load. This is a dimensionless load factor; ; In the above formula, For non-full span load factors; These are dimensionless internal force coefficients; The corrected slenderness ratio for arch bridges not subjected to full-span loads. , For the arch bridge's rise, The radius of gyration of the main arch section. , The moment of inertia of the main arch section. The cross-sectional area of ​​the main arch ring; It is only compared with the span of the arch bridge. The relevant parameters, The catenary arch coefficient is... The span of the arch bridge; It is a hyperbolic cosine function; It is a hyperbolic sine function.

[0013] Specifically, the buckling equilibrium equation at the branch point of the non-full-span load arch bridge mentioned in step S5 is as follows: ; In the above formula, is the dimensionless stability coefficient of the arch bridge after buckling; This represents the change in vertical displacement that occurs during the buckling process of an arch bridge. Indicates to Find the fourth derivative; This represents the change in compressive strain during the buckling process of the arch bridge. The general solution of the buckling equilibrium equation at the bifurcation point of an arch bridge under non-full-span load is expressed as: ; In the above formula, This refers to the change in vertical displacement that occurs during the buckling process of an arch bridge. is the dimensionless internal force coefficient after buckling of the arch bridge; It is a constant; is the dimensionless stability coefficient of the arch bridge after buckling; The x-coordinate of the coordinate system; Based on the boundary conditions: ; In the above formula, Indicates to Find the first derivative; It is a dimensionless stability coefficient; The critical internal force condition after buckling must have Then, according to the above formula, we get: ; Simplifying the above equation yields Finding the minimum positive solution yields the critical internal force condition after the bifurcation buckling instability of an arch bridge under non-full span load. ; Pre-buckling critical internal force condition Represented as: ; In the above formula, This represents the correlation coefficient of the critical internal force conditions before and after buckling. ; In the above formula, The dimensionless load factor after buckling; This is a dimensionless load factor; The critical internal force condition after buckling instability at the branch point of a non-full-span arch bridge; For non-full span load factors; For dimensionless loads, , The intensity of a uniformly distributed load. For the focal parameters of the arch bridge, The horizontal thrust at the arch foot of the arch bridge; with dimensionless load factor before buckling Relatedly, since the dimensionless load factor cannot be directly obtained without determining the critical internal force condition before buckling at the bifurcation point,... ; Therefore, in order to determine the dimensionless load factor The magnitude of the stability coefficient related to the horizontal thrust at the arch foot in the derived in-plane buckling equilibrium equation for non-full-span load arch structures is determined by the constant variation method. Variation is the variation stability coefficient related to the span-to-vector ratio. , Defined as: ; In the above formula, The span of the arch bridge; The buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load is then transformed into: ; In the above formula, Indicates to Find the second derivative; The x-coordinate of the coordinate system; For the catenary arch parameters; It is a hyperbolic cosine function; This represents the change in compressive strain during the buckling process of the arch bridge. The radius of gyration of the main arch section; This represents the vertical displacement of the arch bridge. This indicates finding the second derivative; Substituting the general solution of the buckling equilibrium equation at the bifurcation point of the arch bridge with constant coefficients before the variation into the homogeneous form corresponding to the buckling equilibrium equation at the bifurcation point of the arch bridge after the variation, we obtain the unbalanced difference function. Represented as: ; In the above formula, It is a constant; is the dimensionless stability coefficient of the arch bridge after buckling; The x-coordinate of the coordinate system; The catenary arch coefficient; is the dimensionless internal force coefficient after buckling of the arch bridge; Represents the hyperbolic secant function; The variable stability coefficient; Because there are unknown terms in the unbalanced difference function. The condition of being identically zero cannot be satisfied. Based on the least squares approach, the unbalanced difference function is integrated along the entire arch. When the integral of the unbalanced difference function along the entire arch is minimized, the impact of the unbalanced difference is minimized. Therefore, we set the integral of this unknown term to zero, and obtain: ; In the above formula, It is a hyperbolic sine function; It is a hyperbolic cosine function; is the base of the natural logarithm; Critical internal force condition before buckling The expression is: ; In the above formula, This indicates the critical internal force condition after a non-full-span arch bridge experiences buckling instability at a branch point. The correlation coefficient of the critical internal force conditions before and after buckling; Thus, the critical internal force condition after the bifurcation buckling instability of a non-full-span, hingeless arch structure is determined. Substituting into the above equation, we obtain the critical internal force condition before buckling. This serves as the basis for determining whether buckling at the branch point has occurred; Critical internal force condition Substituting into the buckling equilibrium equation at the bifurcation point of an arch bridge under non-full span load, the coefficients in the equation simplify to: : ; In the above formula, For non-full span load factors; The corrected slenderness ratio for arch bridges not under full-span load; When satisfied When, the quadratic equation in one variable is solved using the quadratic formula. The dimensionless load factor can then be obtained by solving the buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load. : ; Based on the dimensionless load factor Definition The approximate analytical solution of the critical buckling load at the bifurcation point of an arch bridge under non-full span load is obtained as follows: ; In the above formula, This is an approximate analytical solution for the critical buckling load at the branch point of an arch bridge under non-full-span load. The elastic modulus of the arch bridge; The moment of inertia of the main arch section.

[0014] Specifically, in step S6, the partial derivative of the load extremum of the buckling behavior of the arch bridge under instability with a non-full span load is obtained to obtain the in-plane buckling equilibrium equation for the arch bridge under a non-full span load. The process is as follows: The critical load for nonlinear leap buckling of a non-full-span loaded arch bridge is the load corresponding to the extreme points in the buckling behavior curve of the non-full-span loaded arch bridge. These extreme points are the maximum and minimum points of the nonlinear equilibrium path, obtained by solving for the stationary points of the buckling behavior curve. The nonlinear leap buckling equilibrium equation of the non-full-span loaded arch bridge is expressed as follows: ; In the above formula, , , These are the coefficients of the nonlinear buckling equilibrium equation for an arch bridge under non-full-span load. ; In the above formula, For non-full span load factors; These are dimensionless internal force coefficients; The corrected slenderness ratio for arch bridges not under full-span load; By solving the quadratic formula, the dimensionless internal force coefficients at the maximum and minimum points can be obtained. The corresponding dimensionless load parameters Through dimensionless load parameters The approximate analytical critical load for buckling of an arch bridge under non-full span load is obtained. : ; In the above formula, These are dimensionless internal force coefficients; The elastic modulus of the arch bridge; The moment of inertia of the main arch section; The span of the arch bridge; Based on the approximate analytical solution of the critical load for buckling of arch bridges under non-full span load, the critical value of the buckling load is obtained.

[0015] Compared with the prior art, the present invention has the following beneficial effects: 1. This invention integrates the piecewise expression of non-full-span load into a single continuous function by introducing a segment integration function, avoiding the complexity of piecewise processing. This makes it possible to derive the in-plane nonlinear equilibrium equation of the non-full-span load arch structure in a rectangular coordinate system, laying the foundation for subsequent theoretical analysis.

[0016] 2. This invention clarifies the critical internal force conditions of non-full-span load arch structures before and after buckling at the branch point, thereby enabling accurate calculation of the critical buckling load at the branch point, improving the accuracy and applicability of theoretical analysis, and is especially applicable to non-circular arch types such as parabolic arches.

[0017] 3. This invention has clear mechanical concepts, well-defined objectives, simple methods, and clear expressions. It can not only solve for the critical buckling load at the bifurcation point, but also derive the leap buckling equilibrium equation through extreme value analysis of the buckling behavior curve, and then obtain the approximate analytical solution of the leap buckling critical load. It forms a complete nonlinear stability load analysis method, covering the two main modes of in-plane instability of arch structures, and provides a high-precision theoretical tool for the stability design of non-full span load arch structures. Attached Figure Description

[0018] To provide a more intuitive understanding of the technical implementation of this invention, the accompanying drawings involved in the embodiments of this invention are briefly described below. These drawings are used to assist in illustrating the implementation methods and are not intended to limit the invention. Those skilled in the art can make derivative designs based on the drawings without creative effort.

[0019] Figure 1 This is a flowchart of an approximate analytical method for nonlinear stable loads in the deck of an arch bridge under non-full-span load, according to the present invention. Figure 2 This is a schematic diagram of the theoretical model of the non-full-span parabolic hingeless arch bridge of the present invention. Figure 3 This is a schematic diagram of buckling deformation at the branch point within the deck of the non-full-span load arch bridge according to the present invention. Figure 4 This presents a comparison of the approximate solutions for nonlinear leap buckling loads in the in-plane of arch bridges under different loading coefficients according to the present invention. Figure 5 This presents a comparison of the approximate solutions for nonlinear leap buckling loads of arch bridges under different span parameters according to the present invention.

[0020] In the figure: 1. Span of the arch bridge; 2. Elevation of the arch bridge; 3. Magnitude of the non-full span load distributed on the main arch ring near the two ends of the arch foot; 4. Length of the area on the main arch ring without external load. Detailed Implementation

[0021] To facilitate understanding and implementation of the present invention by those skilled in the art, the various steps of the method proposed in this invention are described in detail below. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various modifications or alterations to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0022] Example 1 like Figure 1 As shown in the figure, this embodiment discloses an approximate analytical method for nonlinear stable loads in the deck of a non-full-span arch bridge, including the following steps: Step S1: Taking a parabolic, hingeless arch bridge with a non-full span load as the research object, obtain the expressions for the compressive strain and bending strain of the arch bridge with a non-full span load in the Cartesian coordinate system. Step S2: Based on the expressions for compressive strain and bending strain of the non-full-span load arch bridge in the Cartesian coordinate system obtained in Step S1, the geometric nonlinear equilibrium differential equation in the plane of the non-full-span load arch bridge is derived based on the principle of virtual work and by introducing the segment integration function. Step S3: By solving the in-plane geometric nonlinear equilibrium differential equation of the non-full span load arch bridge obtained in step S2, the approximate analytical vertical displacement of the non-full span load arch bridge is obtained. Step S4: Based on the obtained approximate analysis of the vertical displacement of the non-full span load arch bridge, deduce the nonlinear equilibrium equation of the non-full span load arch bridge to obtain the relationship between the dimensionless internal force coefficients and the dimensionless load parameters. Step S5: Based on the buckling equilibrium equation of the branch point of the arch bridge under non-full span load, determine the critical internal force conditions before and after buckling of the branch point according to the boundary conditions and the constant variation method, and solve for the approximate analytical solution of the critical buckling load of the branch point of the arch bridge under non-full span load. Step S6: Take the partial derivative of the load extremum of the buckling behavior of the arch bridge under instability when it is not fully loaded, and obtain the in-plane buckling equilibrium equation of the arch bridge under instability. Then solve for the approximate analytical solution of the critical load for buckling of the arch bridge under instability.

[0023] Specifically, such as Figure 2 As shown, taking a parabolic, hingeless arch bridge under partial full-span load as the research object, the expressions for the compressive strain and bending strain of the arch bridge under partial full-span load in the Cartesian coordinate system in step S1 are as follows: ; In the above formula, and These represent the compressive strain and bending strain at any point on an arch bridge under non-full span load, respectively. The x-coordinate of the coordinate axis; These are the focal parameters of the arch bridge; and These are the horizontal and vertical displacements of the arch bridge, respectively. The vertical coordinates of the arch axis of the arch bridge after deformation; and They represent respectively to Find the first and second derivatives; Furthermore, the process of deriving the geometric nonlinear equilibrium differential equations in the in-span load arch bridge based on the principle of virtual work and by introducing the segment integration function in step S2 is as follows: According to the principle of virtual work, for an arch bridge in equilibrium, the virtual work done by external forces is equal to the virtual strain energy generated by the arch bridge. Let's assume the vertical virtual displacement is... Vertical virtual strain becomes Then we have: ; In the above formula, Indicates to Variations; This represents the total potential energy of the arch bridge. The total strain of an arch bridge under non-full-span load; This represents the actual stress. The total volume of the arch bridge , The cross-sectional area of ​​the main arch ring. The vertical coordinates of the arch axis of the arch bridge before deformation. The x-coordinate of the coordinate system; For arch bridges with non-full-span uniformly distributed loads; The span of the arch bridge; Introducing a segment integration function to handle non-full-span uniformly distributed loads on arch bridges The piecewise function expression can be represented by a single global equation as follows: ; In the above formula, For arch bridges with non-full-span uniformly distributed loads; The intensity of a uniformly distributed load; This is a segment integration function; The parameter representing the size of the area on the main arch ring where no external loads are applied; The x-coordinate of the coordinate system; The arch bridge is not under a uniformly distributed load at full span. The piecewise function expression is: ; The integrated non-full-span uniformly distributed load Substituting into the virtual work principle equation and simplifying, the in-plane geometric nonlinear equilibrium differential equation for a non-full-span loaded arch bridge is expressed as follows: ; In the above formula, This represents the vertical displacement of the arch bridge. Indicates to Find the second derivative. Indicates to Find the fourth derivative; It is a dimensionless stability coefficient. , This refers to the horizontal thrust at the arch foot of the arch bridge. The elastic modulus of the arch bridge, The moment of inertia is the bending moment of the main arch section of the arch bridge. The intensity of a uniformly distributed load; These are the focal parameters of the arch bridge; For the integration term of piecewise functions, .

[0024] Specifically, the approximate analytical solution for the vertical displacement of the non-full-span load arch bridge obtained in step S3 is... The formula is expressed as: ; In the above formula, This is the integration term for piecewise functions; The load factor is a dimensionless load factor. , The intensity of a uniformly distributed load. For the focal parameters of the arch bridge, The horizontal thrust at the arch foot of the arch bridge; It is a dimensionless stability coefficient; The x-coordinate of the coordinate system; These are dimensionless internal force coefficients. , The span of the arch bridge; For arch bridges not subjected to full-span loads, the internal force coefficients are... , For non-full span load factors, and 2c is the length of the unloaded area in the middle of the construction load; For dimensionless loads, ; For integration functions, When the non-full span load factor At that time, the length of the unloaded area in the middle of the construction load is 2c=0, and the load on the arch bridge is a uniformly distributed load across the full span.

[0025] Specifically, in step S4, the nonlinear equilibrium equations of the arch bridge under non-full span load are derived to obtain the relationship between dimensionless internal force coefficients and dimensionless load parameters. The process is as follows: Since the deformation of an arch bridge over its entire span is equal to the integral of the compressive strain over the entire span, we have the integral equation: ; In the above formula, The horizontal thrust at the arch foot of the arch bridge; The elastic modulus of the arch bridge; The cross-sectional area of ​​the main arch ring; The vertical coordinates of the arch axis of the arch bridge before deformation. The x-coordinate of the coordinate system; This represents the horizontal displacement of the arch bridge. This represents the vertical displacement of the arch bridge. These are the focal parameters of the arch bridge; Will Substitution and will Substituting into the integral equation, the nonlinear equilibrium equation of the non-full-span loaded arch bridge is derived, yielding the relationship between the dimensionless internal force coefficients and the dimensionless load parameters: ; In the above formula, , , These are the coefficients of the nonlinear equilibrium equation for an arch bridge under non-full span load. This is a dimensionless load factor; ; In the above formula, For non-full span load factors; These are dimensionless internal force coefficients; The corrected slenderness ratio for arch bridges not subjected to full-span loads. , For the arch bridge's rise, The radius of gyration of the main arch section. , The moment of inertia of the main arch section. The cross-sectional area of ​​the main arch ring; It is only compared with the span of the arch bridge. The relevant parameters, The catenary arch coefficient is... The span of the arch bridge; It is a hyperbolic cosine function; It is a hyperbolic sine function.

[0026] Specifically, such as Figure 3 As shown, by Figure 3 (a) to Figure 3 The process shown in (d) clearly shows the in-plane deformation pattern of the arch bridge at the bifurcation point buckling. In order to obtain the approximate analytical solution of the critical load for bifurcation point buckling of the arch bridge under non-full span load, the critical internal force conditions before and after bifurcation point buckling are determined in this embodiment based on the boundary conditions and the constant variation method. The process is as follows: The buckling equilibrium equation at the branch point of the non-full-span load arch bridge mentioned in step S5 is as follows: ; In the above formula, is the dimensionless stability coefficient of the arch bridge after buckling; This represents the change in vertical displacement that occurs during the buckling process of an arch bridge. Indicates to Find the fourth derivative; This represents the change in compressive strain during the buckling process of the arch bridge. The general solution of the buckling equilibrium equation at the bifurcation point of an arch bridge under non-full-span load is expressed as: ; In the above formula, This refers to the change in vertical displacement that occurs during the buckling process of an arch bridge. is the dimensionless internal force coefficient after buckling of the arch bridge; It is a constant; is the dimensionless stability coefficient of the arch bridge after buckling; The x-coordinate of the coordinate system; Based on the boundary conditions: ; In the above formula, Indicates to Find the first derivative; It is a dimensionless stability coefficient; The critical internal force condition after buckling must have Then, according to the above formula, we get: ; Simplifying the above equation yields Finding the minimum positive solution yields the critical internal force condition after the bifurcation buckling instability of an arch bridge under non-full span load. ; Pre-buckling critical internal force condition Represented as: ; In the above formula, This represents the correlation coefficient of the critical internal force conditions before and after buckling. ; In the above formula, The dimensionless load factor after buckling; This is a dimensionless load factor; The critical internal force condition after buckling instability at the branch point of a non-full-span arch bridge; For non-full span load factors; For dimensionless loads, , The intensity of a uniformly distributed load. For the focal parameters of the arch bridge, The horizontal thrust at the arch foot of the arch bridge; with dimensionless load factor before buckling Relatedly, the dimensionless load factor cannot be directly obtained without determining the critical internal force condition before buckling at the bifurcation point. ; Therefore, in order to determine the dimensionless load factor The magnitude of the stability coefficient related to the horizontal thrust at the arch foot in the derived in-plane buckling equilibrium equation for non-full-span load arch structures is determined by the constant variation method. Variation is the variation stability coefficient related to the span-to-vector ratio. , Defined as: ; In the above formula, The span of the arch bridge; The buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load is then transformed into: ; In the above formula, Indicates to Find the second derivative; The x-coordinate of the coordinate system; For the catenary arch parameters; It is a hyperbolic cosine function; This represents the change in compressive strain during the buckling process of the arch bridge. The radius of gyration of the main arch section; This represents the vertical displacement of the arch bridge. This indicates finding the second derivative; Substituting the general solution of the buckling equilibrium equation at the bifurcation point of the arch bridge with constant coefficients before the variation into the homogeneous form corresponding to the buckling equilibrium equation at the bifurcation point of the arch bridge after the variation, we obtain the unbalanced difference function. Represented as: ; In the above formula, It is a constant; is the dimensionless stability coefficient of the arch bridge after buckling; The x-coordinate of the coordinate system; The catenary arch coefficient; is the dimensionless internal force coefficient after buckling of the arch bridge; Represents the hyperbolic secant function; The variable stability coefficient; Because there are unknown terms in the unbalanced difference function. The condition of being identically zero cannot be satisfied. Based on the least squares approach, the unbalanced difference function is integrated along the entire arch. When the integral of the unbalanced difference function along the entire arch is minimized, the impact of the unbalanced difference is minimized. Therefore, we set the integral of this unknown term to zero, and obtain: ; In the above formula, It is a hyperbolic sine function; It is a hyperbolic cosine function; is the base of the natural logarithm; Critical internal force condition before buckling The expression is: ; In the above formula, This indicates the critical internal force condition after a non-full-span arch bridge experiences buckling instability at a branch point. The correlation coefficient of the critical internal force conditions before and after buckling; Thus, the critical internal force condition after the bifurcation buckling instability of a non-full-span, hingeless arch structure is determined. Substituting into the above equation, we obtain the critical internal force condition before buckling. This serves as the basis for determining whether buckling at the branch point has occurred; Critical internal force condition Substituting into the buckling equilibrium equation at the bifurcation point of an arch bridge under non-full span load, the coefficients in the equation simplify to: : ; In the above formula, For non-full span load factors; The corrected slenderness ratio for arch bridges not under full-span load; When satisfied When, the quadratic equation in one variable is solved using the quadratic formula. The dimensionless load factor can then be obtained by solving the buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load. : ; Based on the dimensionless load factor Definition The approximate analytical solution of the critical buckling load at the bifurcation point of an arch bridge under non-full span load is obtained as follows: ; In the above formula, This is an approximate analytical solution for the critical buckling load at the branch point of an arch bridge under non-full-span load. The elastic modulus of the arch bridge; The moment of inertia of the main arch section.

[0027] Specifically, in step S6, the partial derivative of the load extremum of the buckling behavior of the arch bridge under instability with a non-full span load is obtained to obtain the in-plane buckling equilibrium equation for the arch bridge under a non-full span load. The process is as follows: The critical load for nonlinear leap buckling of a non-full-span loaded arch bridge is the load corresponding to the extreme points in the buckling behavior curve of the non-full-span loaded arch bridge. These extreme points are the maximum and minimum points of the nonlinear equilibrium path, obtained by solving for the stationary points of the buckling behavior curve. The nonlinear leap buckling equilibrium equation of the non-full-span loaded arch bridge is expressed as follows: ; In the above formula, , , These are the coefficients of the nonlinear buckling equilibrium equation for an arch bridge under non-full-span load. ; In the above formula, For non-full span load factors; These are dimensionless internal force coefficients; The corrected slenderness ratio for arch bridges not under full-span load; By solving the quadratic formula, the dimensionless internal force coefficients at the maximum and minimum points can be obtained. The corresponding dimensionless load parameters Through dimensionless load parameters The approximate analytical critical load for buckling of an arch bridge under non-full span load is obtained. : ; In the above formula, These are dimensionless internal force coefficients; The elastic modulus of the arch bridge; The moment of inertia of the main arch section; The span of the arch bridge; Based on the approximate analytical solution of the critical load for buckling of arch bridges under non-full span load, the critical value of the buckling load is obtained.

[0028] The following example compares the results of the nonlinear leap buckling critical load of an arch bridge under non-full span load obtained by the method of the present invention with those obtained by the traditional finite element analysis method, in order to verify the accuracy of the method of the present invention.

[0029] In this example, the arch structure is modeled using ANSYS software. A solid rectangular cross-section structural model is established using 200 Beam4 beam elements evenly divided in the horizontal direction. The out-of-plane degrees of freedom of all nodes are constrained. The influence of the loading coefficient on the jump buckling load is considered. The non-full span load with symmetrical distribution at both ends is simulated by adding nodal forces. All geometric nonlinearity options are enabled. The model is analyzed, and the nonlinear load-displacement curve is traced using the arc length method.

[0030] In this example, the specific structural parameters of the arch bridge model under non-full-span load are: arch bridge rise-to-span ratio f / L = 1 / 9, and corrected slenderness ratio. =13, height of rectangular section ,Width Material elastic modulus Poisson's ratio .

[0031] Maintaining the rise-to-span ratio f / L = 1 / 9, and taking the non-full span load factors k = 0, 0.3, and 0.5 respectively (corresponding to the proportion of the unloaded area in the middle of the non-full span load), nonlinear leap buckling analysis is performed: For example... Figure 4 (a) Figure 4 (c) and Figure 4As shown in (e), when the non-full-span load factor k takes values ​​of 0, 0.3, and 0.5 respectively, the error between the critical buckling load obtained by the method of this invention and the finite element analysis results is within 3%. Figure 4 (b) Figure 4 (d) and Figure 4 As shown in (f), when the non-full span load factor k takes 0, 0.3 and 0.5 respectively, the load-displacement variation trends in the pre-buckling and post-buckling stages calculated by the method of the present invention are in high agreement with the finite element results, verifying the applicability of the method of the present invention to different load distribution forms.

[0032] Next, we consider the effect of the slenderness ratio on the jump buckling load and control the modified slenderness ratio. =12, the non-full span load factor k is fixed at 0.2, and other arch bridge structural parameters remain unchanged. The arch bridge rise-to-span ratio f / L is set to 1 / 12.5 and 1 / 8 respectively. All geometric nonlinearity options are enabled to perform finite element analysis on the model, and the arc length method is used to trace the nonlinear load-displacement curve. The calculation results of the method of this invention are compared with the finite element analysis results, such as... Figure 5 As shown, the analysis yields a comparison of the approximate solutions for nonlinear leap buckling of arch bridge structures under different span parameters.

[0033] Depend on Figure 5 (a) and Figure 5 As shown in (c), when the span-to-span ratio f / L is 1 / 12.5 and 1 / 8 respectively, the relative errors of the critical buckling load obtained by the method of the present invention are 2.33% and 1.20% respectively; Figure 5 (b) and Figure 5 As can be seen from (d) in the figure, when the span-to-span ratio f / L is 1 / 12.5 and 1 / 8 respectively, the overall trend of the pre-buckling and post-buckling stages obtained by the method of the present invention has a high degree of agreement with the finite element method.

[0034] In summary, the method of this invention has a clear mechanical concept and well-defined steps. By introducing a segment integration function to uniformly describe the load distribution, it establishes nonlinear equilibrium and buckling equations applicable to arch bridges with non-full span loads. Furthermore, it uses boundary conditions and the constant variation method to determine the critical internal forces, achieving a high-precision approximate analytical approach to the critical loads of bifurcation buckling and leap buckling. Under different loading coefficients and span-to-span ratio parameters, the calculation results of this invention agree well with the finite element analysis results, with small errors and consistent trends in the critical loads. This indicates that the method of this invention has high theoretical accuracy and engineering applicability, providing a reliable and convenient analytical calculation tool for the stability design and evaluation of arch bridges with non-full span loads.

[0035] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. 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 scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. An approximate analytical method for in-deck nonlinear stability loads on a non-full-span arch bridge, characterized in that, Includes the following steps: Step S1: Taking a parabolic, hingeless arch bridge with a non-full span load as the research object, obtain the expressions for the compressive strain and bending strain of the arch bridge with a non-full span load in the Cartesian coordinate system. Step S2: Based on the expressions for compressive strain and bending strain of the non-full-span load arch bridge in the Cartesian coordinate system obtained in Step S1, the geometric nonlinear equilibrium differential equation in the plane of the non-full-span load arch bridge is derived based on the principle of virtual work and by introducing the segment integration function. Step S3: By solving the in-plane geometric nonlinear equilibrium differential equation of the non-full span load arch bridge obtained in step S2, the approximate analytical vertical displacement of the non-full span load arch bridge is obtained. Step S4: Based on the obtained approximate analysis of the vertical displacement of the non-full span load arch bridge, deduce the nonlinear equilibrium equation of the non-full span load arch bridge to obtain the relationship between the dimensionless internal force coefficients and the dimensionless load parameters. Step S5: Based on the buckling equilibrium equation of the branch point of the arch bridge under non-full span load, determine the critical internal force conditions before and after buckling of the branch point according to the boundary conditions and the constant variation method, and solve for the approximate analytical solution of the critical buckling load of the branch point of the arch bridge under non-full span load. Step S6: Take the partial derivative of the load extremum of the buckling behavior of the arch bridge under instability when it is not fully loaded, and obtain the in-plane buckling equilibrium equation of the arch bridge under instability. Then solve for the approximate analytical solution of the critical load for buckling of the arch bridge under instability.

2. The approximate analytical method for in-deck nonlinear stability loads of a non-full-span arch bridge according to claim 1, characterized in that, The process of deriving the geometric nonlinear equilibrium differential equations in the deck of the non-full-span load arch bridge based on the principle of virtual work and by introducing the segment integration function in step S2 is as follows: Introducing a segment integration function to handle non-full-span uniformly distributed loads on arch bridges The piecewise function expression can be represented by a single global equation as follows: ； In the above formula, For arch bridges with non-full-span uniformly distributed loads; The intensity of a uniformly distributed load; This is a segment integration function; The parameter representing the size of the area on the main arch ring where no external loads are applied; The x-coordinate of the coordinate system; The integrated arch bridge non-full span uniformly distributed load Substituting into the virtual work principle equation and simplifying, the in-plane geometric nonlinear equilibrium differential equation for a non-full-span loaded arch bridge is expressed as follows: ； In the above formula, This represents the vertical displacement of the arch bridge. Indicates to Find the second derivative. Indicates to Find the fourth derivative; It is a dimensionless stability coefficient. , This refers to the horizontal thrust at the arch foot of the arch bridge. The elastic modulus of the arch bridge, The moment of inertia is the bending moment of the main arch section of the arch bridge. The intensity of a uniformly distributed load; These are the focal parameters of the arch bridge; For the integration term of piecewise functions, .

3. The approximate analytical method for in-deck nonlinear stability loads of a non-full-span arch bridge according to claim 2, characterized in that, The approximate analytical solution for the vertical displacement of the non-full-span load arch bridge obtained in step S3. The formula is expressed as: ； In the above formula, This is the integration term for piecewise functions; The load factor is a dimensionless load factor. , The intensity of a uniformly distributed load. For the focal parameters of the arch bridge, The horizontal thrust at the arch foot of the arch bridge; It is a dimensionless stability coefficient; The x-coordinate of the coordinate system; These are dimensionless internal force coefficients. , The span of the arch bridge; For arch bridges not subjected to full-span loads, the internal force coefficients are... , For non-full span load factors, and 2c is the length of the unloaded area in the middle of the construction load; For dimensionless loads, ; For integration functions, ; When the non-full span load factor At that time, the length of the unloaded area in the middle of the construction load is 2c=0, and the load on the arch bridge is a uniformly distributed load across the full span.

4. The approximate analytical method for in-deck nonlinear stability load of a non-full-span arch bridge according to claim 3, characterized in that, In step S4, the nonlinear equilibrium equations for the non-full-span load arch bridge are derived, yielding the relationship between dimensionless internal force coefficients and dimensionless load parameters: ； In the above formula, , , These are the coefficients of the nonlinear equilibrium equation for an arch bridge under non-full span load. This is a dimensionless load factor; ； In the above formula, For non-full span load factors; These are dimensionless internal force coefficients; The corrected slenderness ratio for arch bridges not subjected to full-span loads. , For the arch bridge's rise, The radius of gyration of the main arch section. , The moment of inertia of the main arch section. The cross-sectional area of ​​the main arch ring; It is only compared with the span of the arch bridge. The relevant parameters, The catenary arch coefficient is... The span of the arch bridge; It is a hyperbolic cosine function; It is a hyperbolic sine function.

5. The approximate analytical method for in-plane nonlinear stability load of a non-full-span arch bridge according to claim 4, characterized in that, The general solution of the buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load in step S5 is expressed as follows: ； In the above formula, This refers to the change in vertical displacement that occurs during the buckling process of an arch bridge. is the dimensionless internal force coefficient after buckling of the arch bridge; It is a constant; is the dimensionless stability coefficient of the arch bridge after buckling; The x-coordinate of the coordinate system; Based on the boundary conditions: ； In the above formula, Indicates to Find the first derivative; It is a dimensionless stability coefficient; The critical internal force condition after buckling must have Then, according to the above formula, we get: ； Simplifying the above equation yields Finding the minimum positive solution yields the critical internal force condition after the bifurcation buckling instability of an arch bridge under non-full span load. ; Pre-buckling critical internal force condition Represented as: ； In the above formula, This represents the correlation coefficient of the critical internal force conditions before and after buckling. ； In the above formula, The dimensionless load factor after buckling; This is a dimensionless load factor; The critical internal force condition after buckling instability at the branch point of a non-full-span arch bridge; For non-full span load factors; For dimensionless loads, , The intensity of a uniformly distributed load. For the focal parameters of the arch bridge, The horizontal thrust at the arch foot of the arch bridge; with dimensionless load factor before buckling Relatedly, since the dimensionless load factor cannot be directly obtained without determining the critical internal force condition before buckling at the bifurcation point,... ; Therefore, in order to determine the dimensionless load factor The magnitude of the stability coefficient related to the horizontal thrust at the arch foot in the derived in-plane buckling equilibrium equation for non-full-span load arch structures is determined by the constant variation method. Variation is the variation stability coefficient related to the span-to-vector ratio. , Defined as: ； In the above formula, The span of the arch bridge; The buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load is then transformed into: ； In the above formula, Indicates to Find the second derivative; The x-coordinate of the coordinate system; For the catenary arch parameters; It is a hyperbolic cosine function; This represents the change in compressive strain during the buckling process of the arch bridge. The radius of gyration of the main arch section; This represents the vertical displacement of the arch bridge. This indicates finding the second derivative; Substituting the general solution of the buckling equilibrium equation at the bifurcation point of the arch bridge with constant coefficients before the variation into the homogeneous form corresponding to the buckling equilibrium equation at the bifurcation point of the arch bridge after the variation, we obtain the unbalanced difference function. Represented as: ； In the above formula, It is a constant; is the dimensionless stability coefficient of the arch bridge after buckling; The x-coordinate of the coordinate system; The catenary arch coefficient; is the dimensionless internal force coefficient after buckling of the arch bridge; Represents the hyperbolic secant function; The variable stability coefficient; Because there are unknown terms in the unbalanced difference function. The condition of being identically zero cannot be satisfied. Based on the least squares approach, the unbalanced difference function is integrated along the entire arch. When the integral of the unbalanced difference function along the entire arch is minimized, the impact of the unbalanced difference is minimized. Therefore, we set the integral of this unknown term to zero, and obtain: ； In the above formula, It is a hyperbolic sine function; It is a hyperbolic cosine function; is the base of the natural logarithm; Critical internal force condition before buckling The expression is: ； In the above formula, This indicates the critical internal force condition after a non-full-span arch bridge experiences buckling instability at a branch point. The correlation coefficient of the critical internal force conditions before and after buckling; Thus, the critical internal force condition after the bifurcation buckling instability of a non-full-span, hingeless arch structure is determined. Substituting into the above equation, we obtain the critical internal force condition before buckling. This serves as the basis for determining whether buckling at the branch point has occurred; Critical internal force condition Substituting into the buckling equilibrium equation at the bifurcation point of an arch bridge under non-full span load, the coefficients in the equation simplify to: : ； In the above formula, For non-full span load factors; The corrected slenderness ratio for arch bridges not under full-span load; When satisfied When, the quadratic equation in one variable is solved using the quadratic formula. The dimensionless load factor can then be obtained by solving the buckling equilibrium equation at the bifurcation point of the arch bridge under non-full span load. : ； Based on the dimensionless load factor Definition The approximate analytical solution of the critical buckling load at the bifurcation point of an arch bridge under non-full span load is obtained as follows: ； In the above formula, This is an approximate analytical solution for the critical buckling load at the branch point of an arch bridge under non-full-span load. The elastic modulus of the arch bridge; The moment of inertia of the main arch section.

6. The approximate analytical method for in-deck nonlinear stability loads of a non-full-span arch bridge according to claim 5, characterized in that, In step S6, the partial derivative of the load extremum of the buckling behavior of the arch bridge under instability under non-full span load is obtained to obtain the in-plane buckling equilibrium equation for the arch bridge under non-full span load. The process is as follows: The critical load for nonlinear leap buckling of a non-full-span loaded arch bridge is the load corresponding to the extreme points in the buckling behavior curve of the non-full-span loaded arch bridge. These extreme points are the maximum and minimum points of the nonlinear equilibrium path, obtained by solving for the stationary points of the buckling behavior curve. The nonlinear leap buckling equilibrium equation of the non-full-span loaded arch bridge is expressed as follows: ； In the above formula, , , These are the coefficients of the nonlinear buckling equilibrium equation for an arch bridge under non-full-span load. ； In the above formula, For non-full span load factors; These are dimensionless internal force coefficients; The corrected slenderness ratio for arch bridges not under full-span load; By solving the quadratic formula, the dimensionless internal force coefficients at the maximum and minimum points can be obtained. The corresponding dimensionless load parameters Through dimensionless load parameters Find the approximate analytical solution for the critical load of an arch bridge under non-full span load to overcome buckling. : ； In the above formula, These are dimensionless internal force coefficients; The elastic modulus of the arch bridge; The moment of inertia of the main arch section; The span of the arch bridge; Based on the approximate analytical solution of the critical load for buckling of arch bridges under non-full span load, the critical value of the buckling load is obtained.

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