Approximate analytical method for in-plane buckling load of arch structure in rectangular coordinate system

Abstract

The application provides an approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system. The method is based on the in-plane bending strain expression of the arch structure in the rectangular coordinate system and the internal force balance condition after buckling of the arch structure to obtain the in-plane buckling deformation coefficient balance differential equation of the arch structure in the rectangular coordinate system. Firstly, the buckling deformation curve expression of the corresponding constant coefficient differential equation satisfying the boundary condition is solved, and the buckling deformation curve coefficient is obtained through the unbalance difference analysis. Finally, the approximate analytical solution of the in-plane buckling load of the arch structure is obtained. The method can quickly obtain the approximate result of the critical horizontal thrust of the arch foot of the arch structure, has clear mechanical concept, clear target, simple method, and enables engineering and technical personnel to avoid using finite element software for solving, reduces the workload, is favorable for accurately formulating the measures for controlling the arch deformation, and has obvious technical and economic rationality.

Description

Technical Field

[0001] This invention relates to the field of civil engineering technology, and in particular to an approximate analytical method for calculating the in-plane buckling load of an arch structure in a rectangular coordinate system, used for calculating the critical buckling load in the in-plane structure of an arch bridge. Background Technology

[0002] There are generally two methods for calculating the in-plane buckling load of a parabolic arch structure: the analytical method for buckling load of a circular arch structure in polar coordinates and the numerical method for buckling load of an arch structure in rectangular coordinates.

[0003] The first method is the analytical method for in-plane buckling of a circular arch structure in polar coordinates. This method is based on static equilibrium conditions. It establishes equilibrium differential equations in polar coordinates on the deformed circular arch structure, solves the equilibrium differential equations using boundary conditions, and then obtains the analytical buckling load in-plane of the circular arch in polar coordinates. This method is simple and feasible, but because the strain expressions for arch structures differ between rectangular and polar coordinates, it cannot be used for the analytical buckling load analysis of parabolic arches in rectangular coordinates.

[0004] The second method is a numerical calculation method for the in-plane buckling of a parabolic arch in a Cartesian coordinate system. This method is based on the fundamental principles of the finite element method, establishing nodes, elements, load vectors, and boundary conditions. It establishes matrix equations using the element stiffness matrix and load vectors that include axial force contributions, and then solves for the eigenvalues ​​and eigenvectors of these equations to obtain a numerical solution for the stability of the arch structure in a Cartesian coordinate system. While this numerical method has good problem adaptability, it can only provide numerical solutions to practical engineering problems and cannot yield practical formulas suitable for engineers. Summary of the Invention

[0005] Based on this, in order to solve the problem of approximate analytical buckling load in the parabolic arch surface under rectangular coordinate system, an approximate analytical method for buckling load in the arch structure surface under rectangular coordinate system is proposed.

[0006] This invention provides an approximate analytical method for buckling load in the in-plane of an arch structure in a rectangular coordinate system. The method is characterized by: obtaining the post-buckling moment-displacement expression for a parabolic arch based on the bending strain-displacement relationship in the in-plane of the arch structure in a rectangular coordinate system; obtaining the variable-coefficient equilibrium differential equation for buckling in the in-plane of the parabolic arch in a rectangular coordinate system based on the internal force equilibrium conditions after buckling of the arch structure; since the variable-coefficient differential equation has no analytical solution, firstly solving the buckling deformation curve expression of its corresponding constant-coefficient differential equation under the boundary conditions; substituting this buckling deformation curve expression into the variable-coefficient equilibrium differential equation for buckling in the in-plane of the arch structure to obtain the unbalance difference of the differential equation under this buckling deformation expression; integrating the square of the unbalance difference term over the entire arch, and taking the partial derivative of the integral buckling deformation curve coefficient; when the first-order partial derivative is equal to zero and the second-order partial derivative is less than zero, the unbalance difference term is at its minimum value, and the buckling deformation curve coefficient is obtained, finally yielding an approximate analytical method for buckling load in the in-plane of the parabolic arch structure.

[0007] Furthermore, the equilibrium differential equation for the buckling coefficient of the parabolic arch in the rectangular coordinate system is as follows:

[0008]

[0009] Where v” is the second derivative of v, v is the vertical displacement of the infinitesimal element deformation of the arch structure, and k is a parameter (defined as...). H cr , where EI is the horizontal thrust at the arch foot, a is the arch shape coefficient, and ch() is the hyperbolic cosine function.

[0010] Furthermore, the imbalance difference in the differential equation under the buckling deformation expression is:

[0011]

[0012] Where m is defined as

[0013] Furthermore, the integral of the square of the imbalance term over the entire arch is as follows:

[0014]

[0015] Where Π is the sum of unbalanced differences, L is the horizontal distance between the two arch feet of the arch structure, and sh() is a hyperbolic sine function.

[0016] Furthermore, the partial derivative of the integral of the square of the error is:

[0017]

[0018] in:

[0019]

[0020] Furthermore, the buckling deformation curve coefficient is:

[0021]

[0022] Furthermore, the in-plane buckling load of the arch structure is approximately analytically defined as follows:

[0023]

[0024] Compared with the prior art, the advantages of the present invention are as follows:

[0025] The advantage of the approximate analytical method for in-plane buckling load of arch structure in the rectangular coordinate system of the present invention compared with the prior art is that it can quickly obtain the approximate result of the critical horizontal thrust of the arch foot of the arch structure. The mechanical concepts are clear, the objectives are well-defined, and the method is simple, which allows engineers to avoid using finite element software to solve the problem, reducing the workload and facilitating the accurate formulation of measures to control arch deformation. It has obvious technical and economic rationality. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the load-bearing arch bridge of the present invention;

[0027] 1 represents the arch axis; 2 represents the uniformly distributed load q acting on the arch crown; 3 represents the arch foot boundary constraints; 4 represents the horizontal thrust H at the arch foot. cr ; 5 represents the span L between the arch feet; 6 represents the shape of the arch axis after buckling. Detailed Implementation

[0028] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. The described examples are merely one aspect of the present invention, and not all of the embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0029] Please see Figure 1This invention provides an approximate analytical method for buckling load in the plane of an arch structure in a rectangular coordinate system. The method is based on the bending strain-displacement relationship in the plane of an arch structure in a rectangular coordinate system, yielding the post-buckling moment-displacement expression for a parabolic arch. Based on the post-buckling internal force equilibrium conditions of the arch structure, a variable-coefficient buckling equilibrium differential equation in the plane of a parabolic arch in a rectangular coordinate system is obtained. Since the variable-coefficient differential equation has no analytical solution, the corresponding constant-coefficient differential equation is first solved to obtain the buckling deformation curve expression under the boundary conditions. This buckling deformation curve expression is substituted into the variable-coefficient buckling equilibrium differential equation in the plane of the arch structure to obtain the unbalance difference of the differential equation under this buckling deformation expression. The square of the unbalance difference term is integrated over the entire arch, and the partial derivative of the integral buckling deformation curve coefficient is calculated. When the first-order partial derivative is equal to zero and the second-order partial derivative is less than zero, the unbalance difference will be at its minimum value, and the buckling deformation curve coefficient is obtained. Finally, an approximate analytical method for buckling load in the plane of a parabolic arch structure is obtained.

[0030] Please see Figure 1 Above the arch axis 1, there is a uniformly distributed load q2 acting on the arch crown. Below the arch axis 1, there is an arch foot boundary constraint 3, and one end of the arch foot boundary constraint 3 has a horizontal thrust H at the arch foot. cr 4. Wherein, 5 is the span L between the arch feet; 6 is the shape of the arch axis after buckling.

[0031] (1) The expression for the bending strain of the arch structure in the rectangular coordinate system is:

[0032]

[0033] (2) According to Euler-Bernoulli beam theory, the bending strain of an arch in Cartesian coordinates is expressed as:

[0034]

[0035] (3) The expression for the bending moment after buckling of the arch structure is:

[0036]

[0037] (4) The expression for the buckling deformation curve of the arch structure in the rectangular coordinate system is:

[0038]

[0039] (5) The corresponding constant coefficient differential equation satisfies the buckling deformation curve expression under the boundary conditions as follows:

[0040]

[0041] Where n is the buckling mode order of the arch structure.

[0042] (6) Substituting step (5) into step (4), the imbalance difference of the differential equation under the buckling deformation expression is obtained as follows:

[0043]

[0044] in,

[0045] (7) By integrating the square of the imbalance term over the entire arch, the sum of the imbalances is obtained as follows:

[0046]

[0047] (8) The partial derivative of the integral of the square of the error is:

[0048]

[0049] in:

[0050]

[0051] (9) To minimize the imbalance difference, take the minimum value of the sum of the imbalance differences. At this time:

[0052]

[0053] (10) The buckling deformation curve coefficient is:

[0054]

[0055] (11) will Substituting into step (10), we get:

[0056]

[0057] in:

[0058] (12) The approximate analytical solution of the buckling load in the in-plane of the arch structure in the rectangular coordinate system is thus obtained as follows:

[0059]

[0060] The approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system of the present invention can quickly obtain the approximate result of the critical horizontal thrust at the arch foot of the arch structure. It has clear mechanical concepts, well-defined objectives, and simple methods, which allows engineers to avoid using finite element software to solve the problem, reducing workload and facilitating the accurate formulation of measures to control arch deformation. It has obvious technical and economic rationality.

[0061] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not limited to the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system, characterized in that, The method is based on the in-plane bending strain-displacement relationship of the arch structure in the rectangular coordinate system to obtain the bending moment-displacement expression after buckling of the parabolic arch. Based on the equilibrium conditions of internal forces after buckling of the arch structure, the equilibrium differential equation of buckling variable coefficient in the parabolic arch surface in the rectangular coordinate system is obtained. Since the variable coefficient differential equation is not analytical, we first solve the buckling deformation curve expression of its corresponding constant coefficient differential equation under the boundary conditions. We then substitute this buckling deformation curve expression into the buckling variable coefficient equilibrium differential equation in the arch structure to obtain the unbalanced difference of the differential equation under this buckling deformation expression. We then integrate the square of the unbalanced difference term over the entire arch and take the partial derivative of the buckling deformation curve coefficient of this integral. When the first partial derivative is equal to zero and the second partial derivative is less than zero, the unbalanced difference term is at its minimum value. We then obtain the buckling deformation curve coefficient and finally obtain the approximate analytical buckling load in the parabolic arch structure.

2. The approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system according to claim 1, characterized in that, The equilibrium differential equation for the buckling coefficient of the parabolic arch in the rectangular coordinate system is: Where v” is the second derivative of v, v is the vertical displacement of the infinitesimal element deformation of the arch structure, and k is a parameter (defined as...). H cr , where EI is the horizontal thrust at the arch foot, a is the arch shape coefficient, and ch() is the hyperbolic cosine function.

3. The approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system according to claim 2, characterized in that, The imbalance error of the differential equation under the buckling deformation expression is: Where m is defined as 4. The approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system according to claim 3, characterized in that, The integral of the square of the imbalance term over the entire arch is as follows: Where Π is the sum of unbalanced differences, L is the horizontal distance between the two arch feet of the arch structure, and sh() is the hyperbolic sine function.

5. The approximate analytical method for in-plane buckling load of an arch structure in a rectangular coordinate system according to claim 4, characterized in that, The partial derivative of the integral of the square of the error is: in:

6. The method for the evolution of fractures in the surrounding rock of deep tunnels according to claim 5, characterized in that, The buckling deformation curve coefficient is:

7. The method for the evolution of fractures in the surrounding rock of deep tunnels according to claim 6, characterized in that, The in-plane buckling load of the arch structure is approximately analytically defined as follows:

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