Elastic stability calculation method

Abstract

The invention relates to an elastic stability calculation method. The method comprises the following steps: acquiring component stress under a variable load and component stress under an invariable load; calculating a reference matrix based on the component stress under the invariable load; calculating a tension member geometric stiffness matrix and a compression member geometric stiffness matrix generated by the variable load based on the member stress under the variable load; solving the first elastic stability equation to obtain a characteristic value and a characteristic vector; solving the second elastic stability equation to obtain a characteristic value and a characteristic vector; judging whether the convergence precision is smaller than the convergence precision ROL or not; if yes, taking the characteristic value as a stability coefficient of the ith order, and taking the characteristic vector as a yield mode of the ith order; and if not, solving the second elastic stability equation again according to the characteristic value. According to the elastic stability calculation method provided by the invention, the load is divided into the variable load and the invariable load, and the stress component of the variable load is also divided into the tension component and the compression component, so that the occurrence of a non-positive definite matrix in the elastic stability equation is avoided.

Description

Technical Field

[0001] This invention relates to the field of linear stability analysis technology, and more specifically to an elastic stability calculation method. Background Technology

[0002] Elastic stability, also known as linear stability analysis, is used to predict the buckling behavior of structures under load. Related techniques typically involve formulating an elastic stability equation based on linear stiffness and geometric stiffness, followed by eigenvalue analysis to determine the structure's stability coefficient (critical load factor) and buckling modes. However, when a structure is subjected to both tensile and compressive loads (i.e., both compression and tension), the geometric stiffness matrix may become non-positive definite, making the elastic stability equation unsolvable. Summary of the Invention

[0003] This application provides an elastic stability calculation method to solve the technical problem in related technologies where the elastic stability equation cannot be solved when a structure is simultaneously subjected to tensile and compressive loads.

[0004] This application provides a method for calculating elastic stability, which includes the following steps: Obtaining the stress of components under variable loads Component stress under constant load ; Component stress under constant load Calculate the benchmark matrix ; Component stress under variable load Calculate the geometric stiffness matrix of the tension member caused by the variable load. and the geometric stiffness matrix of the compression member generated by the variable load. ; Solve the first elastic stability equation to obtain the eigenvalues. and eigenvectors The first elastic stability equation is expressed as: ; Solve the second elastic stability equation to obtain the eigenvalues. and eigenvectors The second elastic stability equation is expressed as: ; judge Is it less than the convergence precision ROL? If so, then by eigenvalue For the first i The stability coefficient of order, in terms of eigenvectors For the first i The yield mode of the order; If not, then use the eigenvalues Then, the second elastic stability equation is solved again.

[0005] In one embodiment, the component stress based on constant load Calculate the benchmark matrix include: Component stress under constant load Calculate the geometric stiffness matrix under constant load ; Calculate the linear stiffness matrix of the component ; Determine the reference matrix ; The calculation formula is: .

[0006] In one embodiment, the geometric stiffness matrix under constant load The calculation formula is: ; in, The displacement gradient matrix is ​​composed of shape functions; V The unit volume is denoted as 'unit volume'.

[0007] In one embodiment, the linear stiffness matrix of the component The calculation formula is: ; in, D Let be the material constitutive matrix of the component.

[0008] In one embodiment, the component stress based on variable load... Calculate the geometric stiffness matrix of the tension member caused by the variable load. and the geometric stiffness matrix of the compression member generated by the variable load. include: The stress of the component under the variable load Decompose into stress of tension member and stress in compression members ; Calculate the geometric stiffness matrix of the tension member generated by the variable load. and the geometric stiffness matrix of the compression member generated by the variable load. .

[0009] In one embodiment, the geometric stiffness matrix of the tension member The calculation formula is: ; in, is the displacement gradient matrix, composed of shape functions; V is the element volume.

[0010] In one embodiment, the geometric stiffness matrix of the compression member The calculation formula is: .

[0011] In one embodiment, the method for solving the first elastic stability equation and the second elastic stability equation is the subspace iteration method.

[0012] In one embodiment, the method for solving the first elastic stability equation and the second elastic stability equation is the Lanczos method.

[0013] In one implementation, the convergence accuracy ROL It is 0.001-0.01.

[0014] The beneficial effects of the technical solutions provided in this application include: This application provides an elastic stability calculation method that decomposes the load into variable loads and constant loads. A reference matrix is ​​calculated using the constant load (such as a dead load) to provide the initial stress state. The reference matrix is ​​always a positive definite matrix, and the instability effect is only manifested by the variable load. The load-bearing members under the variable load are further decomposed into tension members and compression members, and the geometric stiffness matrices of the tension / compression members are calculated independently, avoiding numerical distortion caused by mixed effects and clearly distinguishing the positive / negative contributions of the tension and compression regions to the stiffness. Furthermore, iterative solving of the dual elastic stability equations enhances convergence, thereby solving the problem in related technologies where non-positive definite matrices appear when structures are simultaneously subjected to tension and compression loads, making it impossible to solve the elastic stability equations. This method better reflects the dynamic combination characteristics of loads in actual engineering and effectively calculates the structural stability-related eigenvalues ​​and eigenvectors. Attached Figure Description

[0015] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0016] Figure 1 This is a flowchart of an elastic stability calculation method in one embodiment of the present invention. Detailed Implementation

[0017] To enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present application, and not all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present application.

[0018] This application provides an elastic stability calculation method that can solve the technical problem in related technologies where equations cannot be solved when a structure is simultaneously subjected to tensile and compressive loads.

[0019] Linear stiffness matrix: describes the elastic stiffness (material stiffness) of a structure.

[0020] Geometric stiffness matrix: describes the weakening effect of load-induced stress on structural stiffness (stress stiffening or softening), reflects the influence of initial stress (caused by external load) on structural stiffness, and is used to analyze structural stability (such as buckling).

[0021] like Figure 1 As shown, Figure 1 This is a flowchart of an elastic stability calculation method in one embodiment of the present invention.

[0022] This embodiment provides a method for calculating elastic stability, which includes the following steps: Step S1: Obtain the stress of the component under variable load. Component stress under constant load ; Step S2: Component stress under constant load Calculate the benchmark matrix ; Step S3: Component stress under variable load Calculate the geometric stiffness matrix of a tension member under variable load. Geometric stiffness matrix of compression members generated by variable loads ; Step S4: Solve the first elastic stability equation to obtain the eigenvalues. and eigenvectors The first elastic stability equation is expressed as: ; Step S5: Solve the second elastic stability equation to obtain the eigenvalues. and eigenvectors The second elastic stability equation is expressed as: ; Step S6, Judgment Is it less than the convergence precision ROL? If so, then by eigenvalue For the first i The stability coefficient of order, in terms of eigenvectors For the first i The yield mode of the order; If not, then use the eigenvalues Then, the second elastic stability equation is solved again.

[0023] This embodiment provides an elastic stability calculation method that decomposes the load into variable loads and constant loads. A reference matrix is ​​calculated using the constant load (such as a dead load) to provide the initial stress state. The reference matrix is ​​always a positive definite matrix, and the instability effect is only manifested by the variable load. The load-bearing members under the variable load are further decomposed into tension members and compression members, and the geometric stiffness matrices of the tension / compression members are calculated independently, avoiding numerical distortion caused by mixed effects and clearly distinguishing the positive / negative contributions of the tension and compression regions to the stiffness. Furthermore, iterative solving of the dual elastic stability equations enhances convergence, thereby solving the problem in related technologies where non-positive definite matrices appear when structures are simultaneously subjected to tension and compression loads, making it impossible to solve the elastic stability equations. This method better reflects the dynamic combination characteristics of loads in actual engineering, effectively calculates structural stability-related eigenvalues ​​and eigenvectors, and improves solution efficiency.

[0024] The following provides a detailed explanation of each step.

[0025] In one embodiment, step S1 involves obtaining the stress of the component under variable load. Component stress under constant load At that time, the solutions were obtained separately using finite element software.

[0026] In one embodiment, step S2, based on the component stress under constant load Calculate the benchmark matrix include: Step S21: Component stress under constant load Calculate the geometric stiffness matrix under constant load The calculation formula is: ; in, The displacement gradient matrix is ​​composed of shape functions; V The unit volume is denoted as 'unit volume'.

[0027] Step S22: Calculate the linear stiffness matrix of the component. ; The calculation formula is: ; in, D The material constitutive matrix of the component; Step S23: Determine the reference matrix ; The calculation formula is: .

[0028] The above scheme quantifies the weakening or strengthening effect of stress caused by constant load (such as self-weight) on structural stiffness (compression is negative, tension is positive); the linear stiffness matrix describes the elastic deformation stiffness of the component (dependent only on material properties and geometry, independent of load). The reference matrix, composed of the geometric stiffness matrix and the linear stiffness matrix under constant load, is a constant matrix that provides initial reference stiffness and is numerically stable, laying the foundation for subsequent stability analysis under variable loads.

[0029] In one embodiment, step S3, based on the component stress under variable load Calculate the geometric stiffness matrix of a tension member under variable load. Geometric stiffness matrix of compression members generated by variable loads include: Step S31: Calculate the stress of the component under variable load. Decompose into stress of tension member and stress in compression members ; Step S32: Calculate the geometric stiffness matrix of the tension member generated by the variable load. Geometric stiffness matrix of compression members generated by variable loads .

[0030] In one embodiment, the geometric stiffness matrix of the tension member The calculation formula is: ; in, is the displacement gradient matrix, composed of shape functions; V is the element volume.

[0031] In one embodiment, the geometric stiffness matrix of the compression member The calculation formula is: .

[0032] The above scheme considers the different mechanisms by which tension and compression affect stiffness. Specifically, in tension members, axial tension enhances the member's lateral stiffness (e.g., a taut rope is more difficult to bend), and the corresponding term in the geometric stiffness matrix is ​​positive, reflecting the "stress stiffening" effect. In compression members, axial pressure may weaken the member's stiffness or even cause buckling (e.g., slender columns are prone to instability under compression), and the corresponding term in the geometric stiffness matrix is ​​negative, reflecting the "stiffness degradation" effect. Calculating the geometric stiffness matrices of tension members and compression members separately for variable loads avoids conflating the stiffness effects of tension and compression members. Separate calculations accurately reflect stiffness changes in different regions and avoid errors caused by linear assumptions.

[0033] In one embodiment, the method for solving the first elastic stability equation in step S4 and the second elastic stability equation in step S5 is the subspace iteration method.

[0034] The subspace iteration method is an efficient numerical algorithm for solving eigenvalue problems of large sparse matrices, particularly suitable for calculating the first few eigen pairs (eigenvalues ​​and eigenvectors) in structural dynamics (e.g., modal analysis) and stability analysis (e.g., buckling analysis). The basic idea is to construct an m-dimensional subspace (m slightly larger than the number of required eigenvalues ​​r), and iteratively approximate the target eigenvalues ​​and eigenvectors within the subspace, avoiding the direct solution of the full matrix's eigenvalues ​​and reducing computational complexity. Through an iterative logic of "construction-projection-update," the subspace iteration method achieves a balance between computational efficiency and accuracy in large eigenvalue problems.

[0035] In one embodiment, the method for solving the first elastic stability equation in step S4 and the second elastic stability equation in step S5 is the Lanczos method.

[0036] The Lanczos method is an efficient iterative algorithm for solving the eigenvalue problem of large symmetric matrices, proposed by mathematician Cornelius Lanczos in 1950. This method approximates the eigenvalues ​​of the original matrix by progressively transforming the symmetric matrix into a tridiagonal matrix and utilizing efficient methods for solving the eigenvalue problem of tridiagonal matrices. The basic idea is to iteratively construct a set of orthogonal vectors (Lanczos vectors) to project the original symmetric matrix into a tridiagonal matrix. Taking advantage of the ease of finding the eigenvalues ​​of tridiagonal matrices (such as the QR algorithm), the method progressively approximates the target eigenvalues. Its core relies on Krylov subspace theory.

[0037] In one embodiment, convergence accuracy ROL It is 0.001-0.01.

[0038] It should be noted that the sequence numbers of the embodiments in this application are merely for descriptive purposes and do not represent the superiority or inferiority of the embodiments. The terms "comprising" and "having," and any variations thereof, in the specification, claims, and accompanying drawings of this application are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or device that includes a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units not listed, or may optionally include other steps or units inherent to these processes, methods, products, or devices. The terms "first," "second," and "third," etc., are used to distinguish different objects, etc., and do not represent a sequential order, nor do they limit "first," "second," and "third" to different types.

[0039] In the description of the embodiments of this application, terms such as "exemplary," "for example," or "for instance" are used to indicate examples, illustrations, or explanations. Any embodiment or design described as "exemplary," "for example," or "for instance" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of terms such as "exemplary," "for example," or "for instance" is intended to present the relevant concepts in a concrete manner.

[0040] In the description of the embodiments of this application, unless otherwise stated, " / " means "or". For example, A / B can mean A or B. The "and / or" in the text is merely a description of the relationship between related objects, indicating that there can be three relationships. For example, A and / or B can mean: A exists alone, A and B exist simultaneously, and B exists alone. In addition, in the description of the embodiments of this application, "multiple" means two or more.

[0041] In some processes described in the embodiments of this application, multiple operations or steps are included in a specific order. However, it should be understood that these operations or steps may not be executed in the order they appear in the embodiments of this application, or they may be executed in parallel. The sequence number of the operation is only used to distinguish the different operations, and the sequence number itself does not represent any execution order. In addition, these processes may include more or fewer operations, and these operations or steps may be executed sequentially or in parallel, and these operations or steps may be combined.

[0042] The above are merely preferred embodiments of this application and do not limit the patent scope of this application. Any equivalent structural or procedural transformations made using the content of this application's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of this application.

Claims

1. A method for calculating elastic stability, characterized in that, It includes the following steps: Obtaining the stress of components under variable loads Component stress under constant load ; Component stress under constant load Calculate the benchmark matrix ; Component stress under variable load Calculate the geometric stiffness matrix of the tension member caused by the variable load. and the geometric stiffness matrix of the compression member generated by the variable load. ; Solve the first elastic stability equation to obtain the eigenvalues. and eigenvectors The first elastic stability equation is expressed as: ; Solve the second elastic stability equation to obtain the eigenvalues. and eigenvectors The second elastic stability equation is expressed as: ; judge Is it less than the convergence precision? ROL ; If so, then by eigenvalue For the first i The stability coefficient of order, in terms of eigenvectors For the first i The yield mode of the order; If not, then use the eigenvalues Then, the second elastic stability equation is solved again.

2. The elastic stability calculation method as described in claim 1, characterized in that, The stress of the component under constant load Calculate the benchmark matrix include: Component stress under constant load Calculate the geometric stiffness matrix under constant load ; Calculate the linear stiffness matrix of the component ; Determine the reference matrix ; The calculation formula is: .

3. The elastic stability calculation method as described in claim 2, characterized in that, The geometric stiffness matrix under constant load The calculation formula is: ; in, The displacement gradient matrix is ​​composed of shape functions; V The unit volume is denoted as 'unit volume'.

4. The elastic stability calculation method as described in claim 3, characterized in that, The linear stiffness matrix of the component The calculation formula is: ; in, D Let be the material constitutive matrix of the component.

5. The elastic stability calculation method as described in claim 1, characterized in that, The stress of the component under variable load Calculate the geometric stiffness matrix of the tension member caused by the variable load. and the geometric stiffness matrix of the compression member generated by the variable load. include: The stress of the component under the variable load Decompose into stress of tension member and stress in compression members ; Calculate the geometric stiffness matrix of the tension member generated by the variable load. and the geometric stiffness matrix of the compression member generated by the variable load. .

6. The elastic stability calculation method as described in claim 5, characterized in that, The geometric stiffness matrix of the tension member The calculation formula is: ; in, is the displacement gradient matrix, composed of shape functions; V is the element volume.

7. The elastic stability calculation method as described in claim 6, characterized in that, The geometric stiffness matrix of the compression member The calculation formula is: .

8. The elastic stability calculation method as described in claim 1, characterized in that, The method for solving the first and second elastic stability equations is the subspace iteration method.

9. The elastic stability calculation method as described in claim 1, characterized in that, The method for solving the first and second elastic stability equations is the Lanczos method.

10. The elastic stability calculation method as described in claim 1, characterized in that, The convergence accuracy ROL It is 0.001-0.01.

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