Method and system for calculating out-of-plane instability critical load of variable cross-section arc arch rib
By establishing the axial and radial control equations of the variable cross-section circular arch and using the harmonic differential quadrature algorithm to process the control equations, the problem of low calculation accuracy of out-of-plane instability of the rib of the variable cross-section circular arch was solved, achieving high accuracy and fast calculation, which is applicable to bridge design with various boundary conditions and cross-sectional changes.
Patent Information
- Application Number
- CN202511640870.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-11
- Publication Date
- 2026-01-13
AI Technical Summary
When calculating out-of-plane instability of existing variable cross-section circular arch ribs under external loads, the calculation accuracy is not high and it is not easy to adapt to changes in boundary or conditions. Traditional methods require complex polynomial fitting and repeated construction of displacement functions, which are computationally intensive and inconvenient.
An out-of-plane instability analysis model under uniformly distributed radial load is adopted. Combined with the harmonic differential quadrature algorithm, the axial and radial control equations of the variable cross-section circular arch are established. The in-plane dimensionless displacement matrix is constructed by discretizing the matrix using Lagrange interpolation basis functions and weighting coefficients. The out-of-plane control equations are constructed based on the principle of minimum potential energy. The control equation matrix is processed by the harmonic differential quadrature algorithm to adapt to different boundary conditions and variable cross-section working conditions.
It improves the accuracy and speed of calculating the out-of-plane instability critical load of variable cross-section circular arch ribs, is applicable to various boundary conditions and cross-sectional changes, can quickly integrate matrices, reflects the relationship between influencing parameters, and is suitable for bridge design and construction.
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Figure CN121328231A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of bridge design technology, and in particular to a method and system for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib. Background Technology
[0002] Arches are a common structural form in engineering due to their high load-bearing capacity and good spanning performance. To optimize material distribution, improve structural efficiency, and reduce structural weight, long-span arch bridges or arched roof trusses often employ variable cross-section designs. Variable cross-section arches are primarily subjected to compression, and may suddenly collapse under external loads, exhibiting elastic instability.
[0003] Existing methods for out-of-plane instability critical loads of circular arch ribs with variable cross-sections under radially uniformly distributed radial loads, based on harmonic differential quadrature algorithms, provide theoretical support for engineering structural design and construction. Solving the out-of-plane instability problem of circular arches using the Rayleigh-Litz method typically requires constructing a displacement trial function satisfying specific boundary conditions, substituting it into the out-of-plane governing equations, and taking stationary values for the functional to determine an approximate solution. The form of the displacement function is determined by the boundary conditions and displacement shape, directly affecting the calculation speed and accuracy. When the boundary conditions or arch axis stiffness change, the structure may exhibit non-uniform bending and torsional deformation. Under such conditions, the traditional Rayleigh-Litz method requires complex polynomial fitting of out-of-plane deformation and construction of displacement functions to ensure calculation accuracy, resulting in a large computational load. Furthermore, once the cross-sectional shape or in-plane (out-of-plane) boundary changes, the displacement function must be reconstructed, and the solution process must be repeated, making the method inconvenient to implement. Summary of the Invention
[0004] To address the aforementioned technical problems, the present invention aims to provide a method and system for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib. This method and system are applicable to various in-plane and out-of-plane boundary conditions and variable cross-section working conditions, and improve the accuracy of calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib.
[0005] The first technical solution adopted in this invention is: a method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib, comprising the following steps: Based on the out-of-plane instability analysis model of a variable cross-section arch under uniformly distributed radial load, the axial and radial control equations of the variable cross-section circular arc arch are established. The axial and radial control equations of the variable cross-section circular arch are solved by the harmonic differential quadrature algorithm, and the in-plane dimensionless displacement matrix of the arch under unit load intensity is obtained. Based on the out-of-plane instability state of a variable cross-section circular arc arch, the out-of-plane governing equations are constructed and solved to obtain the out-of-plane instability critical load matrix of the variable cross-section arch. By combining the in-plane dimensionless displacement matrix of the arch under unit load intensity and the out-of-plane instability critical load matrix of the variable cross-section arch, the calculation results of the out-of-plane instability critical load of the variable cross-section circular arc arch rib are obtained.
[0006] Furthermore, the step of establishing the axial and radial control equations of the variable cross-section circular arc arch based on the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load specifically includes: Based on the out-of-plane instability analysis model of a variable cross-section arch under uniformly distributed radial load, the equation of the cross-section is assumed to be linearly changing. Based on the equation of linear change of cross section, the energy equation of the pre-buckling system is constructed based on the principle of minimum potential energy. By performing integral-part calculations on the energy equations of the pre-buckling system, preliminary axial and radial control equations for the variable cross-section circular arch are obtained. Define the axial force and bending moment at any point on the axis of the variable cross-section arch, and substitute them into the preliminary axial and radial control equations of the variable cross-section circular arc arch for expansion, to obtain the axial and radial control equations of the variable cross-section circular arc arch.
[0007] Furthermore, the expressions for the axial and radial control equations of the variable cross-section circular arc arch are as follows: ; ; In the above formula, This represents the variational operator. This represents dimensionless axial displacement. This represents dimensionless radial displacement. Represents angular coordinates. This represents taking the first derivative with respect to the axial force on the arch. This represents the first derivative of the bending moment of the arch. Express the second derivative of the bending moment of the arch. Indicates the radius of the circular arch. This indicates the intensity of a radially uniformly distributed load.
[0008] Furthermore, the step of solving the axial and radial control equations of the variable cross-section circular arch using the harmonic differential quadrature algorithm to obtain the in-plane dimensionless displacement matrix of the arch under unit load intensity specifically includes: The arc length of the variable cross-section circular arch is divided into n discrete elements and n+1 discrete nodes. The coordinates of the discrete nodes are represented by Chebyshev-Gauss-Lobatto polynomials with neighborhoods to construct the coordinates of the discrete nodes. Based on discrete node coordinates, the dimensionless axial and radial displacement equations are described using Lagrange interpolation basis functions. Based on the dimensionless axial and radial displacement equations, the Lagrange polynomial and weighting coefficients are defined. Substituting the Lagrange polynomials and weighting coefficients into the axial and radial control equations of the variable cross-section circular arch, we construct the axial and radial control equations in the form of harmonic differential quadrature. Convert the axial and radial control equations in harmonic differential quadrature form into matrix form; Define the boundary conditions for a circular arch with two fixed ends in a plane, a circular arch with two hinged ends in a plane, and a circular arch with one fixed end and one hinged end in a plane. Substitute these conditions into the matrix form of the axial and radial control equations to obtain the dimensionless displacement matrix of the arch under a unit load intensity.
[0009] Furthermore, the expression for the boundary conditions of the circular arches fixed at both ends in the plane is as follows: ; In the above formula, This represents the dimensionless axial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless axial displacement, This represents the dimensionless radial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement, Indicates the included angle of a circular arch. The total number of discrete units. To accumulate the cyclic counter symbol, This represents the first-order weight coefficient of discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement; The expression for the boundary conditions of the in-plane hinged circular arch at both ends is as follows: ; In the above formula, Represents discrete nodes The corresponding dimensionless axial displacement, Represents discrete nodes The second-order weighting coefficients, Represents discrete nodes The first-order weighting coefficients; The expression for the boundary conditions of the circular arch with one end fixed and the other end hinged in the plane is as follows: ; ; In the above formula, This represents the second-order weight coefficient of discrete node 1.
[0010] Furthermore, the step of constructing and solving the out-of-plane governing equations based on the out-of-plane instability state of the variable cross-section circular arc arch to obtain the out-of-plane instability critical load matrix of the variable cross-section arch specifically includes: Based on the out-of-plane instability of the variable cross-section circular arch, an expression for the total out-of-plane potential energy of the system is constructed. The first variation of the out-of-plane total potential energy expression of the system is obtained, and after integration by parts and rearrangement, the out-of-plane control equations for lateral displacement and out-of-plane torsional angle are obtained. The out-of-plane control equations concerning lateral displacement and out-of-plane torsional angle are transformed into harmonic differential quadrature form, resulting in the out-of-plane control equations in harmonic differential quadrature form. Convert the out-of-plane control equations in harmonic differential quadrature form into matrix form; Define the out-of-plane hinged boundary conditions, the out-of-plane fixed boundary conditions, and the out-of-plane fixed-one-hinged boundary conditions, and substitute them into the matrix form of the out-of-plane governing equations to obtain the out-of-plane instability critical load matrix of the variable cross-section arch.
[0011] Furthermore, the specific expressions for the out-of-plane control equations regarding lateral displacement and out-of-plane torsional angle are as follows: ; ; In the above formula, , and This indicates the lateral bending, lateral torsion, and warping stiffness of the cross-section. Indicates the elastic modulus. Represents Poisson's ratio. The out-of-plane governing equations representing lateral displacements. Out-of-plane governing equations representing out-of-plane torsion angles. Represents angular coordinates. Indicates the radius of the circular arch. This represents the fourth derivative with respect to dimensionless radial displacement. This represents finding the second derivative of the outward twist angle. This represents the second derivative with respect to dimensionless radial displacement. This represents the fourth derivative of the outward twist angle. Indicates the bending moment of the arch. Indicates the radius of gyration of the cross section. This represents the first derivative of the outward twist angle. This represents taking the first derivative with respect to the dimensionless radial displacement. Indicates the out-of-plane torsion angle.
[0012] Furthermore, the expression for the out-of-plane hinged boundary condition is as follows: ; ; In the above formula, This represents the dimensionless radial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement, Indicates the cumulative loop counter symbol. Represents the total number of discrete units. Represents discrete nodes The corresponding dimensionless axial displacement, This represents the out-of-plane torsion angle corresponding to discrete node 1. Represents discrete nodes The corresponding out-of-plane torsion angle, Represents discrete nodes The corresponding out-of-plane torsion angle; The expression for the out-of-plane, two-end fixed boundary condition is as follows: ; ; In the above formula, This represents the first-order weight coefficient of discrete node 1. Represents discrete nodes The first-order weighting coefficients; The expression for the out-of-plane boundary condition with one end fixed and the other hinged is as follows: ; ; In the above formula, Represents discrete nodes The second-order weighting coefficients.
[0013] The second technical solution adopted in this invention is: a calculation system for out-of-plane instability critical load of a variable cross-section circular arc arch rib, comprising: The first module is used to establish the axial and radial control equations of the variable cross-section arch based on the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load. The second module is used to solve the axial and radial control equations of the variable cross-section circular arch using the harmonic differential quadrature algorithm, and obtain the in-plane dimensionless displacement matrix of the arch under unit load intensity. The third module is used to construct and solve the out-of-plane control equations based on the out-of-plane instability state of the variable cross-section circular arc arch, and obtain the out-of-plane instability critical load matrix of the variable cross-section arch. The fourth module is used to combine the in-plane dimensionless displacement matrix of the arch under unit load intensity and the out-of-plane instability critical load matrix of the arch with variable cross-section to obtain the calculation results of the out-of-plane instability critical load of the arch rib with variable cross-section circular arc.
[0014] The beneficial effects of the method and system of this invention are as follows: This invention establishes the axial and radial control equations of a variable cross-section arch based on an out-of-plane instability analysis model of a variable cross-section arch under uniformly distributed radial load. Furthermore, it solves the axial and radial control equations of the variable cross-section arch using a harmonic differential quadrature algorithm to obtain the in-plane dimensionless displacement matrix of the arch under a unit load intensity. By assuming a variable cross-section variation function, it establishes the in-plane control equations of the variable cross-section arch before buckling based on the principle of minimum potential energy. The harmonic differential quadrature algorithm is used to discretize these equations, and arbitrary boundary conditions are substituted to solve for the internal forces of the arch axis under a unit load intensity. Then, based on the out-of-plane instability state of the variable cross-section arch, the out-of-plane control equations are constructed and solved to obtain the out-of-plane instability critical load matrix of the variable cross-section arch. Assuming that the internal forces of the arch axis remain constant at the moment of out-of-plane instability and that the deformation is in-plane before deformation, the out-of-plane total potential energy function of the system is established. After variationalizing the total potential energy, the integrated control equations are processed using the harmonic differential quadrature algorithm. Substituting the out-of-plane boundary conditions, the instability critical force of the variable cross-section arch is obtained. No displacement function needs to be assumed. It is applicable to various in-plane (out-of-plane) boundary conditions and variable cross-section working conditions. It can intuitively reflect the interrelationship between various influencing parameters and improve the calculation accuracy of the out-of-plane instability critical load of the variable cross-section circular arc arch rib. Attached Figure Description
[0015] Figure 1 This is a flowchart illustrating the steps of the out-of-plane instability critical load calculation method for a variable cross-section circular arc arch rib according to the present invention. Figure 2 This is a structural block diagram of an out-of-plane instability critical load calculation system for a variable cross-section circular arc arch rib according to the present invention; Figure 3 This is a schematic diagram of the calculation process for the instability critical force of a variable cross-section arch provided in a specific embodiment of the present invention; Figure 4 This is a schematic diagram of the geometric model provided in a specific embodiment of the present invention; Figure 5 This is a schematic diagram of the convergence of the out-of-plane instability critical load provided in a specific embodiment of the present invention; Figure 6 This is a schematic diagram illustrating the relative error variation of the out-of-plane instability critical load provided in a specific embodiment of the present invention; Figure 7 This is a schematic diagram illustrating the variation of the out-of-plane instability critical load with the opening angle, provided in a specific embodiment of the present invention. Figure 8 This is a schematic diagram illustrating the change of out-of-plane instability critical load with opening angle provided in a specific embodiment of the present invention. Detailed Implementation
[0016] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. The step numbers in the following embodiments are only for ease of explanation and do not limit the order of the steps. The execution order of each step in the embodiments can be adapted according to the understanding of those skilled in the art.
[0017] First, it should be noted that, based on the differences in spatial deformation, the instability behavior of an arch can be divided into two types: in-plane and out-of-plane. When the in-plane stiffness is greater than the out-of-plane stiffness, the arch experiences out-of-plane flexural-torsional instability; conversely, the structure experiences in-plane instability. This invention focuses on the first type of working condition, aiming to propose a practical calculation method for the out-of-plane instability critical load of a circular arch rib that is highly accurate, converges quickly, and is more universal, facilitating use and reference by designers and construction personnel.
[0018] To address the out-of-plane elastic instability of variable cross-section arches, this invention proposes a cross-section variation function to accurately describe in-plane axial compression, in-plane (out-of-plane) bending, torsional, and warping stiffness. It reconstructs the in-plane and out-of-plane instability control equations for the variable cross-section circular arc arch before buckling. A harmonic differential quadrature algorithm is introduced to discretize the control equations. Leveraging the algorithm's high accuracy and good convergence, the calculation matrix is quickly integrated to obtain the critical out-of-plane instability load of the arch. By flexibly adjusting the in-plane and out-of-plane boundary interpolation functions, it can satisfy different analytical conditions, thus possessing a certain degree of universality.
[0019] Reference Figure 1 This invention provides a method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib, the method comprising the following steps: S100. Based on the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load, the axial and radial control equations of the variable cross-section circular arc arch are established. S110. An out-of-plane instability analysis model for a variable cross-section arch under uniformly distributed radial load, assuming a linear cross-section variation equation. In this embodiment, the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load is as follows: Figure 4 As shown, the equation for the linear change of the cross section is assumed to be: ; In the above formula, and These represent the height and width of the cross-section, respectively. and These represent the larger and smaller sides of the cross-section, respectively. Indicates the included angle of a circular arch. Represents angular coordinates.
[0020] S120. Based on the equation of linear change of cross section, construct the energy equation of the pre-buckling system based on the principle of minimum potential energy. In this embodiment, the energy equation of the pre-buckling system is established based on the principle of minimum potential energy, and its expression is: ; In the above formula, This represents the total potential energy of the system. For variational operators, Let be the radius of the circular arch. and These are dimensionless axial and radial displacements, respectively. The intensity of the radially uniformly distributed load. , .
[0021] S130. Perform partial integration on the energy equation of the pre-buckling system to obtain the preliminary axial and radial control equations for the variable cross-section circular arch. In this embodiment, the energy equations of the pre-buckling system are integrated by parts, and the axial and radial control equations are obtained: ; ; In the above formula, the axial force at any point on the axis of the variable cross-section arch is... and bending moment Defined as: ; ; In the above formula, The normal stress of the cross section, For the area of the variable cross section, and These are the compressive and bending stiffnesses of the variable cross-section, respectively.
[0022] S140. Define the axial force and bending moment at any point on the axis of the variable cross-section arch, and substitute them into the preliminary axial and radial control equations of the variable cross-section circular arc arch for expansion, to obtain the axial and radial control equations of the variable cross-section circular arc arch.
[0023] In this embodiment, substituting the expressions for axial force and bending moment into the axial and radial control equations, and then rearranging and expanding, we get: ; ; In the above formula, This represents dimensionless axial displacement. This represents dimensionless radial displacement.
[0024] S200. The axial and radial control equations of the variable cross-section circular arch are solved by the harmonic differential quadrature algorithm to obtain the in-plane dimensionless displacement matrix of the arch under unit load intensity. S210. Divide the arc length of the variable cross-section circular arc arch into n discrete elements and n+1 discrete nodes. The coordinates of the discrete nodes are represented by Chebyshev-Gauss-Lobatto polynomials with neighborhoods to construct the coordinates of the discrete nodes. In this embodiment, as can be seen from the expanded axial and radial control equations, this system of equations consists of high-order differential equations with variable coefficients, which cannot be solved directly. This embodiment of the invention proposes to use a harmonic differential quadrature algorithm to process the above equations. Considering the function... In the arch axis region Differentiable, the arc length is divided into n discrete units and n+1 discrete nodes, and the node coordinates use... Chebyshev-Gauss-Lobatto polynomial representation of the neighborhood: ; In the above formula, These are the interpolation points.
[0025] S220. Based on discrete node coordinates, the dimensionless axial and radial displacement equations are described using Lagrange interpolation basis functions. In this embodiment, the dimensionless axial displacement is... and radial displacement Described using Lagrange interpolation basis functions, its expression is: In the above formula, This represents dimensionless axial displacement. This represents dimensionless radial displacement.
[0026] S230. Based on the dimensionless axial and radial displacement equations, define the Lagrange polynomial and weighting coefficients. In this embodiment, the Lagrange polynomial The definition is as follows: ; when Weighting coefficient and Represented as: ; ; In the above formula: ; when Weighting coefficient Represented as: ; Also: ; In the above formula, , , This represents the weighting coefficient.
[0027] S240. Substitute the Lagrange polynomial and weighting coefficients into the axial and radial control equations of the variable cross-section circular arch to construct the axial and radial control equations in the form of harmonic differential quadrature. In this embodiment, the weighting coefficients are... Substituting the expanded axial and radial control equations and rewriting them in harmonic differential quadrature form, we get: ; ; In the above formula, It represents the elastic modulus.
[0028] S250. Convert the axial and radial control equations in harmonic differential quadrature form into matrix form; In this embodiment, the axial and radial control equations in the harmonic differential quadrature form are further rearranged into matrix form as follows: ; In the formula: ; ; matrix The elements are as follows: ; ; ; ; In the above formula, Represents the in-plane elastic stiffness matrix. Represents the in-plane nodal displacement matrix. This represents the in-plane load coefficient matrix.
[0029] S260. Define the boundary conditions for a circular arch with two fixed ends in the plane, a circular arch with two hinged ends in the plane, and a circular arch with one fixed end and one hinged end in the plane. Substitute these conditions into the matrix form of the axial and radial control equations to obtain the dimensionless displacement matrix of the arch under unit load intensity.
[0030] In this embodiment, the boundary conditions for the circular arches fixed at both ends in the plane can be expressed as: ; The boundary conditions for a circular arch with hinged ends in a plane can be expressed as: ; The boundary conditions for a circular arch with one end fixed and the other hinged in a plane can be expressed as: ; ; Substituting any boundary condition—fixed at both ends, hinged at both ends, or fixed at one end and hinged at the other—into the matrix form of the axial and radial control equations yields the dimensionless in-plane displacement matrix of the arch under unit load intensity, expressed as: ; In the above formula, This represents the dimensionless axial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless axial displacement, This represents the dimensionless radial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement, Indicates the included angle of a circular arch. The total number of discrete units. To accumulate the cyclic counter symbol, This represents the first-order weight coefficient of discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement.
[0031] S300. Based on the out-of-plane instability state of the variable cross-section circular arc arch, the out-of-plane control equations are constructed and solved to obtain the out-of-plane instability critical load matrix of the variable cross-section arch. S310. Based on the out-of-plane instability of the variable cross-section circular arc arch, construct the expression for the total out-of-plane potential energy of the system; In this embodiment, as the uniformly distributed radial load continues to increase, at a certain moment the circular arch suddenly transitions from compression-bending deformation to bending-torsional deformation, resulting in out-of-plane instability. At this time, the total out-of-plane potential energy of the system can be expressed as: ; In the above formula, , , The lateral bending, lateral torsion, and warping stiffness of the cross section are defined as follows: ; ; ; In the above formula, For elastic modulus, Poisson's ratio, This is the warping function.
[0032] S320. Find the first variation of the total out-of-plane potential energy expression of the system, integrate by parts and rearrange to obtain the out-of-plane control equations for lateral displacement and out-of-plane torsional angle. In this embodiment, the first variation of the total out-of-plane potential energy is calculated, and after integration by parts and rearrangement, the result is obtained regarding the lateral displacement. Outward twist angle The out-of-plane governing equations are expressed as follows: ; ; In the above formula, , and This indicates the lateral bending, lateral torsion, and warping stiffness of the cross-section. Indicates the elastic modulus. Represents Poisson's ratio. The out-of-plane governing equations representing lateral displacements. Out-of-plane governing equations representing out-of-plane torsion angles. Represents angular coordinates. Indicates the radius of the circular arch. This represents the fourth derivative with respect to dimensionless radial displacement. This represents finding the second derivative of the outward twist angle. This represents the second derivative with respect to dimensionless radial displacement. This represents the fourth derivative of the outward twist angle. Indicates the bending moment of the arch. Indicates the radius of gyration of the cross section. This represents the first derivative of the outward twist angle. This represents taking the first derivative with respect to the dimensionless radial displacement. Indicates the out-of-plane torsion angle.
[0033] S330. The out-of-plane control equations concerning lateral displacement and out-of-plane torsional angle are converted into harmonic differential quadrature form to obtain the out-of-plane control equations in harmonic differential quadrature form. In this embodiment, the lateral displacement will be... Outward twist angle The out-of-plane governing equations can be written in harmonic differential quadrature form, and their expression is: ; ; , and This indicates the lateral bending, lateral torsion, and warping stiffness of the cross-section. It represents the elastic modulus.
[0034] S340. Convert the out-of-plane control equations in harmonic differential quadrature form into matrix form; In this embodiment, the out-of-plane governing equations in the harmonic differential quadrature form are further written in matrix form, and their expression is: ; In the above formula: ; matrix and The elements are as follows: ; ; In the above formula, and They represent the first Axial force and bending moment of the arch under a uniform radial load at each interpolation point.
[0035] S350. Define the out-of-plane hinged boundary conditions at both ends, the out-of-plane fixed boundary conditions at both ends, and the out-of-plane fixed and hinged boundary conditions at one end. Substitute these conditions into the out-of-plane control equations in matrix form to obtain the out-of-plane instability critical load matrix of the variable cross-section arch.
[0036] In this embodiment, the out-of-plane hinged boundaries are represented as follows: ; ; The out-of-plane, two-end fixed boundary condition is expressed as: ; ; The out-of-plane boundary condition with one end fixed and the other hinged is expressed as follows: ; ; Substituting any of the out-of-plane boundary conditions—hinged at both ends, fixed at both ends, or fixed at one end and hinged at the other—into the matrix form of the out-of-plane governing equations and solving the matrix equations yields the out-of-plane instability critical load matrix for the variable cross-section arch. .
[0037] S400, combining the in-plane dimensionless displacement matrix of the arch under unit load intensity and the out-of-plane instability critical load matrix of the variable cross-section arch, the calculation results of the out-of-plane instability critical load of the variable cross-section circular arc arch rib are obtained.
[0038] In summary, as Figure 3 As shown, this embodiment of the invention first assumes a variable cross-section variation function and establishes the in-plane control equations of the variable cross-section arch before buckling based on the principle of minimum potential energy. The equations are then discretized using a harmonic differential quadrature algorithm. Arbitrary boundary conditions are substituted to solve for the internal forces of the arch axis under a unit load intensity. Furthermore, assuming that the internal forces of the arch axis remain constant at the moment of out-of-plane instability and that the system is undergoing in-plane deformation before deformation, the out-of-plane total potential energy function of the system is established. After variational analysis of the total potential energy, the control equations are integrated. The control equation matrix is then processed using the harmonic differential quadrature algorithm, and the out-of-plane boundary conditions are substituted to obtain the critical buckling force of the variable cross-section arch. A cross-section variation function is proposed to accurately describe the in-plane axial compression, in-plane (out-of-plane) bending, torsion, and warping stiffness. The in-plane and out-of-plane buckling control equations of the variable cross-section circular arc arch before buckling are reconstructed. The harmonic differential quadrature algorithm is introduced to discretize the control equations. Leveraging the algorithm's high accuracy and good convergence, the calculation matrix is quickly integrated to obtain the out-of-plane critical load for the arch's buckling. By flexibly adjusting the interpolation functions of the inner and outer boundaries, it can meet different analytical conditions, thus possessing a certain degree of universality.
[0039] Finally, the embodiments of the present invention will be described in conjunction with the accompanying drawings: like Figure 5 and Figure 6 The curves showing the relationship between the first-order dimensionless critical load for instability and its relative error of an out-of-plane hinged variable cross-section circular arch and the number of grid points are presented. The arch has an I-shaped cross-section with a total height of 400 mm, a flange width of 180 mm, a flange thickness of 13.5 mm, and a web thickness of 8.6 mm. The elastic modulus is 2e11, Poisson's ratio is 0.3, and the in-plane boundary conditions are: fixed at both ends (black line), hinged at both ends (red line), and fixed at one end and hinged at the other (blue line). From the figure, it can be seen that as the number of grid points increases, the instability critical load converges rapidly, and the relative error drops sharply to 1%. The harmonic differential quadrature algorithm with a small grid number (n = 10) still shows good adaptability when dealing with variable coefficient differential equations. Furthermore, such as... Figure 7 and Figure 8 The relationship between the out-of-plane first-order dimensionless critical load for instability of a circular arch under different in-plane boundary conditions and variable cross-section forms and the opening angle is given. It can be seen that the graph is smooth and can intuitively describe the influence of the in-plane boundary on the critical load for instability.
[0040] To verify the accuracy of the analysis results, numerical simulation analysis was conducted using the commercial finite element analysis software ANSYS, and the extracted results were compared with the analysis results. The simulation used BEAM188 three-dimensional beam elements to create a circular arch axis, activated the section warping option using the KEYOPT keyword, and implemented the section profile change using the Taper command. The entire arch axis was divided into 200 segments. By constraining the degrees of freedom of the arch foot nodes, the in-plane and out-of-plane boundary conditions were accurately simulated. SFBEAM was used to apply a uniformly distributed radial unit load, and eigenvalue buckling analysis was conducted. Modal extension was performed to determine the out-of-plane instability critical force of the variable cross-section arch.
[0041] Tables 1 to 3 present the first-order out-of-plane instability critical loads of variable cross-section out-of-plane hinged circular arches obtained using the harmonic differential quadrature method and the finite element method under different in-plane boundary and variable cross-section forms. The data in the tables show good agreement between the two methods. The harmonic differential quadrature method has sufficient accuracy in handling the out-of-plane instability problem of variable cross-section arches and can intuitively reflect the influence of various parameters on the instability critical load.
[0042] Table 1. Comparison of out-of-plane instability critical loads for variable cross-section circular arches under different in-plane boundary conditions (×10) 2 N / m) ; Where: Error = |Invention Embodiment - Finite Element| / Finite Element × 100%, .
[0043] Table 2 Comparison of critical instability loads for circular arches with internally fixed joints and externally hinged joints under different variable cross-section forms (×10) 2 N / m) ; Where: Error = |Invention Embodiment - Finite Element| / Finite Element × 100%.
[0044] Table 3. Comparison of critical instability loads for circular arches with internally fixed joints and externally hinged joints under different variable cross-section forms (×10) 2 N / m) ; Where: Error = |Invention Embodiment - Finite Element| / Finite Element × 100%.
[0045] Therefore, the embodiments of the present invention have the following advantages compared with the prior art: 1) No displacement function needs to be assumed, applicable to various in-plane (outside) boundary conditions and variable cross-section conditions, and has a certain degree of universality.
[0046] 2) The algorithm has high accuracy and fast convergence speed, and can quickly integrate matrices, making it especially suitable for computer computing.
[0047] 3) Similar to analytical methods, it can intuitively reflect the interrelationships between various influencing parameters, making it convenient for design and construction personnel to use and refer to.
[0048] Reference Figure 2 A calculation system for out-of-plane instability critical load of a variable cross-section circular arc arch rib, comprising: The first module 201 is used to establish the axial and radial control equations of the variable cross-section arch based on the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load. The second module 202 is used to solve the axial and radial control equations of the variable cross-section circular arch using the harmonic differential quadrature algorithm, and obtain the in-plane dimensionless displacement matrix of the arch under unit load intensity. The third module 203 is used to construct and solve the out-of-plane control equations based on the out-of-plane instability state of the variable cross-section circular arc arch, and obtain the out-of-plane instability critical load matrix of the variable cross-section arch. Module 4, 204, is used to combine the in-plane dimensionless displacement matrix of the arch under unit load intensity and the out-of-plane instability critical load matrix of the variable cross-section arch to obtain the calculation results of the out-of-plane instability critical load of the variable cross-section circular arc arch rib.
[0049] The content of the above method embodiments is applicable to this system embodiment. The specific functions implemented in this system embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those achieved in the above method embodiments.
[0050] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.
Claims
1. A method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib, characterized in that, Includes the following steps: Based on the out-of-plane instability analysis model of a variable cross-section arch under uniformly distributed radial load, the axial and radial control equations of the variable cross-section circular arc arch are established. The axial and radial control equations of the variable cross-section circular arch are solved by the harmonic differential quadrature algorithm, and the in-plane dimensionless displacement matrix of the arch under unit load intensity is obtained. Based on the out-of-plane instability state of a variable cross-section circular arc arch, the out-of-plane governing equations are constructed and solved to obtain the out-of-plane instability critical load matrix of the variable cross-section arch. By combining the in-plane dimensionless displacement matrix of the arch under unit load intensity and the out-of-plane instability critical load matrix of the variable cross-section arch, the calculation results of the out-of-plane instability critical load of the variable cross-section circular arc arch rib are obtained.
2. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 1, characterized in that, The step of establishing the axial and radial control equations for the variable cross-section circular arc arch based on the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load specifically includes: Based on the out-of-plane instability analysis model of a variable cross-section arch under uniformly distributed radial load, the equation of the cross-section is assumed to be linearly changing. Based on the equation of linear change of cross section, the energy equation of the pre-buckling system is constructed based on the principle of minimum potential energy. By performing integral-part calculations on the energy equations of the pre-buckling system, preliminary axial and radial control equations for the variable cross-section circular arch are obtained. Define the axial force and bending moment at any point on the axis of the variable cross-section arch, and substitute them into the preliminary axial and radial control equations of the variable cross-section circular arc arch for expansion, to obtain the axial and radial control equations of the variable cross-section circular arc arch.
3. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 2, characterized in that, The specific expressions for the axial and radial control equations of the variable cross-section circular arch are as follows: In the above formula, This represents the variational operator. This represents dimensionless axial displacement. This represents dimensionless radial displacement. Represents angular coordinates. This represents taking the first derivative with respect to the axial force on the arch. This represents the first derivative of the bending moment of the arch. Express the second derivative of the bending moment of the arch. Indicates the radius of the circular arch. This indicates the intensity of a radially uniformly distributed load.
4. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 3, characterized in that, The step of solving the axial and radial control equations of the variable cross-section circular arch using the harmonic differential quadrature algorithm to obtain the in-plane dimensionless displacement matrix of the arch under unit load intensity specifically includes: The arc length of the variable cross-section circular arch is divided into n discrete elements and n+1 discrete nodes. The coordinates of the discrete nodes are represented by Chebyshev-Gauss-Lobatto polynomials with neighborhoods to construct the coordinates of the discrete nodes. Based on discrete node coordinates, the dimensionless axial and radial displacement equations are described using Lagrange interpolation basis functions. Based on the dimensionless axial and radial displacement equations, the Lagrange polynomial and weighting coefficients are defined. Substituting the Lagrange polynomials and weighting coefficients into the axial and radial control equations of the variable cross-section circular arch, we construct the axial and radial control equations in the form of harmonic differential quadrature. Convert the axial and radial control equations in harmonic differential quadrature form into matrix form; Define the boundary conditions for a circular arch with two fixed ends in a plane, a circular arch with two hinged ends in a plane, and a circular arch with one fixed end and one hinged end in a plane. Substitute these conditions into the matrix form of the axial and radial control equations to obtain the dimensionless displacement matrix of the arch under a unit load intensity.
5. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 4, characterized in that, The expression for the boundary conditions of the circular arch fixed at both ends in the plane is as follows: In the above formula, This represents the dimensionless axial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless axial displacement, This represents the dimensionless radial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement, Indicates the included angle of a circular arch. The total number of discrete units. To accumulate the cyclic counter symbol, This represents the first-order weight coefficient of discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement; The expression for the boundary conditions of the in-plane hinged circular arch at both ends is as follows: In the above formula, Represents discrete nodes The corresponding dimensionless axial displacement, Represents discrete nodes The second-order weighting coefficients, Represents discrete nodes The first-order weighting coefficients; The expression for the boundary conditions of the circular arch with one end fixed and the other end hinged in the plane is as follows: In the above formula, This represents the second-order weight coefficient of discrete node 1.
6. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 5, characterized in that, The step of constructing and solving the out-of-plane governing equations for the out-of-plane instability state of the variable cross-section circular arc arch to obtain the out-of-plane instability critical load matrix of the variable cross-section arch specifically includes: Based on the out-of-plane instability of the variable cross-section circular arch, an expression for the total out-of-plane potential energy of the system is constructed. The first variation of the out-of-plane total potential energy expression of the system is obtained, and after integration by parts and rearrangement, the out-of-plane control equations for lateral displacement and out-of-plane torsional angle are obtained. The out-of-plane control equations concerning lateral displacement and out-of-plane torsional angle are transformed into harmonic differential quadrature form, resulting in the out-of-plane control equations in harmonic differential quadrature form. Convert the out-of-plane control equations in harmonic differential quadrature form into matrix form; Define the out-of-plane hinged boundary conditions, the out-of-plane fixed boundary conditions, and the out-of-plane fixed-one-hinged boundary conditions, and substitute them into the matrix form of the out-of-plane governing equations to obtain the out-of-plane instability critical load matrix of the variable cross-section arch.
7. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 6, characterized in that, The specific expressions for the out-of-plane governing equations concerning lateral displacement and out-of-plane torsional angle are as follows: In the above formula, , and This indicates the lateral bending, lateral torsion, and warping stiffness of the cross section. Indicates the elastic modulus. Represents Poisson's ratio. The out-of-plane governing equations representing lateral displacements. Out-of-plane governing equations representing out-of-plane torsion angles. Represents angular coordinates. Indicates the radius of the circular arch. This represents the fourth derivative with respect to dimensionless radial displacement. This represents finding the second derivative of the outward twist angle. This represents the second derivative with respect to dimensionless radial displacement. This represents the fourth derivative of the outward twist angle. Indicates the bending moment of the arch. Indicates the radius of gyration of the cross section. This represents the first derivative of the outward twist angle. This represents taking the first derivative with respect to the dimensionless radial displacement. Indicates the out-of-plane torsion angle.
8. The method for calculating the out-of-plane instability critical load of a variable cross-section circular arc arch rib according to claim 7, characterized in that, The expression for the out-of-plane hinged boundary condition is as follows: In the above formula, This represents the dimensionless radial displacement corresponding to discrete node 1. Represents discrete nodes The corresponding dimensionless radial displacement, Indicates the cumulative loop counter symbol. Represents the total number of discrete units. Represents discrete nodes The corresponding dimensionless axial displacement, This represents the out-of-plane torsion angle corresponding to discrete node 1. Represents discrete nodes The corresponding out-of-plane torsion angle, Represents discrete nodes The corresponding out-of-plane torsion angle; The expression for the out-of-plane, two-end fixed boundary condition is as follows: In the above formula, This represents the first-order weight coefficient of discrete node 1. Represents discrete nodes The first-order weighting coefficients; The expression for the out-of-plane boundary condition with one end fixed and the other hinged is as follows: In the above formula, Represents discrete nodes The second-order weighting coefficients.
9. A calculation system for out-of-plane instability critical load of a variable cross-section circular arc arch rib, characterized in that, Includes the following modules: The first module is used to establish the axial and radial control equations of the variable cross-section arch based on the out-of-plane instability analysis model of the variable cross-section arch under uniformly distributed radial load. The second module is used to solve the axial and radial control equations of the variable cross-section circular arch using the harmonic differential quadrature algorithm, and obtain the in-plane dimensionless displacement matrix of the arch under unit load intensity. The third module is used to construct and solve the out-of-plane control equations based on the out-of-plane instability state of the variable cross-section circular arc arch, and obtain the out-of-plane instability critical load matrix of the variable cross-section arch. The fourth module is used to combine the in-plane dimensionless displacement matrix of the arch under unit load intensity and the out-of-plane instability critical load matrix of the arch with variable cross-section to obtain the calculation results of the out-of-plane instability critical load of the arch rib with variable cross-section circular arc.