Stress finite element method for solving structural stress and deformation

Abstract

The invention discloses a stress finite element method for solving structural stress and deformation, which comprises the following steps of: dividing a research object into quadrilateral units according to regions of different physical and mechanical parameters of the research object; selecting node stress components and node displacement components of all units as independent variables, and constructing a unit balance equation expressed by the node stress components; a node strain component expressed by the node stress component; a unit deformation equation expressed by the node strain component and the node displacement component; a unit deformation equation expressed by the node stress component and the node displacement component; node stress components have known boundary conditions; the node displacement component has known boundary conditions; obtaining an overall control equation; solving the overall control equation to obtain a node stress component and a node displacement component; and carrying out stress and deformation analysis on the structure. The method is firm in theoretical foundation and high in universality and practicability, has important value for finite element theoretical research, and promotes application of a finite element method with stress as a basic variable in engineering.

Description

Technical Field

[0001] The present invention relates to the technical field of finite element analysis, and is particularly suitable for a stress finite element method for solving structural stress and deformation. Background Art

[0002] The finite element method (FEM) originated from the matrix displacement method of bar structures developed in the 1940s and 1950s. In 1956, Turner and others extended the idea of ​​the matrix displacement method of bar structures to solve plane problems in elastic mechanics. In 1960, Clough named this method for solving plane problems in elastic mechanics "finite element method". Since then, the finite element method has developed rapidly and gradually matured. The finite element method is considered to be the most effective numerical method at present, with its outstanding advantages such as solid theoretical foundation, versatility and strong practicality. The research paper "Difference Format Based on Variational Principle" by Chinese mathematician Academician Feng Kang laid a rigorous mathematical foundation for the finite element method and provided a reliable theoretical guarantee for the practical application of the finite element method. The paper is also considered to be a symbol of the independent creation of the finite element method by Chinese scientists from Western countries.

[0003] In elasticity problems, the governing equations include equilibrium equations, constitutive equations, geometric equations, stress boundary conditions, and displacement boundary conditions. When using the finite element method to solve elasticity problems, the steps generally include structural discretization, unit analysis, overall analysis, and numerical solution.

[0004] In specific applications, displacement is currently used as the basic variable, and it is assumed that displacement has satisfied the displacement boundary conditions, strain is determined by displacement according to the geometric equation, and stress is determined by strain according to the constitutive equation. Displacement as the basic variable still needs to satisfy the equilibrium equation and stress boundary conditions, and the establishment of the equilibrium equation is based on the variational principle in elastic mechanics. This finite element method that takes displacement as the basic variable and satisfies additional conditions is usually called the displacement finite element method.

[0005] Theoretically, stress can also be used as the basic variable, and it is assumed that stress has satisfied the equilibrium equation and stress boundary conditions. Strain is determined by stress according to the constitutive equation. Stress as the basic variable still needs to satisfy the geometric equation and displacement boundary conditions. This finite element method with stress as the basic variable and satisfying additional conditions is usually called the stress finite element method. However, in practical applications, it is very difficult to express the internal stress of the unit with undetermined parameters, so the finite element method with stress as the basic variable is rarely used.

[0006] The present invention solves this problem and proposes a finite element method with stress as the basic variable, which has a solid theoretical basis, strong versatility and practicability, is of great value to finite element theory research, and has positive significance to engineering applications. Summary of the invention

[0007] The present invention aims to provide a stress finite element method for solving structural stress and deformation, so as to solve the problem that the finite element method with stress as the basic variable is difficult to promote and apply due to the difficulty in expressing the internal stress of the current unit.

[0008] To achieve the above object, the present invention adopts the following technical solutions: The stress finite element method for solving structural stress and deformation of the present invention comprises the following steps: S1, determine the geometric range of the research object and the areas with different physical and mechanical parameters within the geometric range; S2, for plane strain problems, combined with different physical and mechanical parameter regions, the research object is divided into m quadrilateral elements, totaling n nodes; S3, select all unit node stress components and node displacement components as independent variables, and construct the unit equilibrium equation expressed by the node stress component; the node strain component expressed by the node stress component; the unit deformation equation expressed by the node strain component and the node displacement component; the unit deformation equation expressed by the node stress component and the node displacement component; the node stress component has known boundary conditions; the node displacement component has known boundary conditions; S4, constructing the overall control equation; solving the overall control equation to obtain the node stress component and the node displacement component; S5, based on the solved nodal stress components and nodal displacement components, perform stress and deformation analysis on the structure.

[0009] Furthermore, in step S1, the physical and mechanical parameters include bulk density , elastic modulus E Poisson's ratio .

[0010] Furthermore, in step S3, the unit equilibrium equation expressed by the node stress component is: , is the coefficient matrix, 2 m Row, 3 n List; is a constant matrix, 2 m row, 1 column; is the node stress component, 3 n row, 1 column; and The calculation process is: S3.1.1, the element equilibrium equation for each element expressed in terms of nodal stress components is: ;

[0011]

[0012]

[0013]

[0014]

[0015]

[0016]

[0017] and They are the nodes 1, 2, 3 and 4 of the quadrilateral element respectively. x Coordinates and y coordinate; and The units are x Direction and y physical strength of direction; S3.1.2, integrate all the unit equilibrium equations expressed by the nodal stress components to obtain and .

[0018] Furthermore, in step S3, the node strain component expressed by the node stress component is: , is the node strain component, 3 n row, 1 column; is the coefficient matrix, 3 n Row, 3 n List; The calculation process is: S3.2.1, the nodal strain component expressed by the nodal stress component at any node is:

[0019] Where: E is the elastic modulus; is Poisson’s ratio; S3.2.2, integrate all the nodal strain components expressed by the nodal stress components and obtain .

[0020] Furthermore, in step S3, the element deformation equation expressed by the node strain component and the node displacement component is: ,in: is a coefficient matrix with 3m rows and 3n columns; is a coefficient matrix with 3m rows and 2n columns; and The calculation process is: S3.3.1, the element deformation equation expressed in terms of nodal strains and nodal displacements for each element is: ;

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] in, S is the unit area; and , respectively, of the quadrilateral unit nodes 1, 2, 3, and 4 x To positive strain, y Normal strain and shear strain; and They are unit nodes 1, 2, 3 and 4 respectively. x Coordinates and y coordinate; and They are unit nodes 1, 2, 3 and 4 respectively. x Displacement and y Toward displacement; S3.3.2, integrate all the unit deformation equations expressed by node strains and node displacements to obtain and .

[0027] Furthermore, in step S3, the unit deformation equation expressed by the node stress component and the unit displacement component is: ,in .

[0028] Furthermore, in step S3, the known boundary conditions of the node stress components are: ,in: j represents the number of known boundary conditions for stress components; is the coefficient matrix, j rows, 3n columns; The column elements in the rows corresponding to the known stress components are 1, and the other elements are 0; is the stress boundary condition matrix, j row, 1 column; The elements in the corresponding rows of are the known stress components.

[0029] Furthermore, in step S3, the known boundary conditions of the node displacement components are:

[0030] in: k represents the number of known boundary conditions for the displacement components; is the coefficient matrix, k Row, 2 n List; The column elements in the rows corresponding to the known displacement components are 1, and the other elements are 0; is the displacement boundary condition matrix, k row, 1 column; The elements in the corresponding rows of are the known stress components.

[0031] Furthermore, in step S4, the overall control equation is: , in:

[0032] By solving the minimum norm least squares solution of the overall control equation, the independent variable The value of ; O represents the zero matrix.

[0033] Furthermore, in step S2, for the three-dimensional problem, the research object is divided into hexahedral units, the overall control equation of the three-dimensional problem is established, the solution of the overall control equation is solved, the node stress components and node displacement components are obtained, and the stress and deformation of the research object are analyzed.

[0034] The advantage of the present invention is that it provides a stress finite element method for solving structural stress and deformation, which has a solid theoretical basis, strong versatility and practicality, is of great value to finite element theory research, and promotes the application of finite element methods with stress as the basic variable in engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is a schematic diagram of the cantilever beam and unit division in the embodiment of the present invention.

[0036] Figure 2 is a flow chart of the method of the present invention.

[0037] Figure 3 It is a schematic diagram of the horizontal normal stress distribution of the middle section of the cantilever beam in the embodiment of the present invention.

[0038] Figure 4 It is a schematic diagram of the vertical displacement distribution of the bottom edge line of the cantilever beam in the embodiment of the present invention. DETAILED DESCRIPTION

[0039] The technical solutions in the embodiments of the present invention are described clearly and completely below. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0040] Example 1 The following uses a cantilever beam as an example to illustrate the steps of the stress finite element method for solving structural stress and deformation described in the present invention in a specific engineering application.

[0041] like Figure 1 As shown in the figure, the cantilever beam is 1.0m high vertically and 2.0m long horizontally. The left end face of the cantilever beam is fixed to the rigid boundary, and the vertical and horizontal displacements of the left end face are 0. The upper and lower surfaces of the cantilever beam are free, and the vertical normal stress and shear stress of the upper and lower surfaces are both zero. The right end face of the cantilever beam is subject to vertical shear force, and the vertical downward shear stress of the right end face is 10MPa, and the horizontal normal stress of the right end face is 0. The bulk density of the cantilever beam is , elastic modulus Poisson's ratio The stress finite element method for solving structural stress and deformation described in the present invention is used to calculate the stress and displacement of the cantilever beam, and its flow chart is as follows: Figure 2 shown.

[0042] S1, the geometric range of the cantilever beam under study is: the vertical height of the cantilever beam is 1.0m and the horizontal length is 2.0m. The physical and mechanical parameters of the cantilever beam include bulk density , elastic modulus Poisson's ratio According to the physical and mechanical parameters, the entire cantilever beam of the research object is a region.

[0043] S2, the cantilever beam is divided into 10 units vertically and 20 units horizontally, with a total of 200 units and 231 nodes, i.e. m=200, n=231.

[0044] S3, select all unit node stress components and node displacement components to construct independent variables, the independent variables are:

[0045] Then, the unit equilibrium equation expressed by the node stress component is constructed in sequence; the node strain component expressed by the node stress component; the unit deformation equation expressed by the node strain component and the node displacement component; the unit deformation equation expressed by the node stress component and the node displacement component; the node stress component has known boundary conditions; the node displacement component has known boundary conditions.

[0046] (1) The unit equilibrium equation expressed by the node stress components: , is the coefficient matrix, 2 m Row, 3 n List; is a constant matrix, 2 m row, 1 column; is the node stress component, there are 3 n rows, 1 column. and The calculation process is: S3.1.1, the element equilibrium equation for each element expressed in terms of nodal stress components is:

[0047] in:

[0048]

[0049]

[0050]

[0051]

[0052] and They are the nodes 1, 2, 3 and 4 of the quadrilateral element respectively. x Coordinates and y coordinate.

[0053]

[0054]

[0055] and The units are x Direction and y Direction of physical strength. are the nodal stress components of the quadrilateral element nodes 1, 2, 3, and 4.

[0056] In this embodiment

[0057]

[0058]

[0059]

[0060] S3.1.2, integrate all the unit equilibrium equations expressed by the nodal stress components to obtain and , the unit equilibrium equation expressed by the node stress components is obtained .

[0061] (2) The nodal strain component expressed by the nodal stress component is: , is the node strain component, 3 n row, 1 column; is the coefficient matrix, 3 n Row, 3 n List; The calculation process is: S3.2.1, the nodal strain component expressed by the nodal stress component at any node is:

[0062] Where: E is the elastic modulus; is Poisson’s ratio; In this embodiment

[0063] S3.2.2, integrate all the nodal strain components expressed by the nodal stress components and obtain The nodal strain components expressed by the nodal stress components .

[0064] (3) The unit deformation equation expressed by the node strain component and the node displacement component is: ,in: is a coefficient matrix with 3m rows and 3n columns; is a coefficient matrix with 3m rows and 2n columns; and The calculation process is: S3.3.1, the element deformation equation expressed in terms of nodal strains and nodal displacements for each element is: ;

[0065]

[0066]

[0067]

[0068]

[0069]

[0070] in, S is the unit area; and , respectively, of the quadrilateral unit nodes 1, 2, 3, and 4 x To positive strain, y Normal strain and shear strain; and They are unit nodes 1, 2, 3 and 4 respectively. x Coordinates and y coordinate; and They are unit nodes 1, 2, 3 and 4 respectively. x Displacement and y Toward displacement.

[0071] In this embodiment

[0072]

[0073]

[0074]

[0075]

[0076]

[0077] S3.3.2, integrate all the unit deformation equations expressed by node strains and node displacements to obtain and Then the unit deformation equation expressed by the node strain component and the node displacement component in this embodiment is

[0078] (4) The unit deformation equation expressed by the node stress component and the node displacement component is: ,in .

[0079] In this embodiment:

[0080] in:

[0081] (5) The known boundary conditions of the point stress component are: ,in: j represents the number of known boundary conditions for stress components; is the coefficient matrix, j rows, 3n columns; is the stress boundary condition matrix, j rows, 1 column.

[0082] In this embodiment, the vertical normal stress and shear stress on the upper and lower surfaces of the cantilever beam are both zero; the right end face of the cantilever beam has a vertical shear force, the vertical downward shear stress on the right end face is 10MPa, and the horizontal normal stress on the right end face is 0; there are a total of 100 node stress components with known boundary conditions. So let The column elements in the corresponding rows corresponding to the known stress components of the nodes on the upper surface, lower surface and right end face of the cantilever beam are 1. The elements in other rows are all 0. The row elements corresponding to the Y-direction normal stress and shear stress at the nodes on the upper and lower surfaces of the cantilever beam are set to 0. The row element corresponding to the X-direction normal stress at the right end node of the cantilever beam is 0; The row element corresponding to the tangent stress at the node on the right end of the cantilever beam is -10.

[0083] (6) The known boundary conditions of the node displacement components are: ; in: k represents the number of known boundary conditions for the displacement components; is the coefficient matrix, k Row, 2 n List; is the displacement boundary condition matrix, k row, 1 column; In this embodiment, the vertical and horizontal displacements of the left end face of the cantilever beam are 0; there are a total of 22 node displacement components with known boundary conditions. The column elements in the row corresponding to the known displacement component of the left end node of the cantilever beam are 1. All other elements are 0.

[0084] make The row element corresponding to the known displacement component of the node on the left end of the cantilever beam is 0. The known boundary conditions of the node displacement component are obtained:

[0085] S4, construct the overall control equation; solve the overall control equation to obtain the node stress components and node displacement components.

[0086] The overall governing equation is: , in:

[0087] By solving the minimum norm least squares solution of the overall control equation, the independent variable The value of . All are 0 matrices.

[0088] In this embodiment, ,in:

[0089]

[0090] By solving the minimum norm least squares solution of the overall control equation, the independent variable The value of .

[0091] S5, based on the solved nodal stress components and nodal displacement components, perform stress and deformation analysis on the structure.

[0092] Example 2 For three-dimensional problems, the research object is divided into hexahedral units, and based on the above ideas, the overall control equation of the three-dimensional problem is established. The solution of the overall control equation is solved to obtain the node stress components and node displacement components, and the stress and deformation of the research object are analyzed.

[0093] In summary, the stress finite element method provided in this application provides a stress finite element method with a solid theoretical basis, strong applicability, and stress as the basic variable. It is of great value to finite element theory research and has positive significance for engineering applications.

Claims

1. A stress finite element method for solving structural stress and deformation, characterized in that: The following steps are involved: S1, determine the geometric range of the research object and the areas with different physical and mechanical parameters within the geometric range; S2, for plane strain problems, combined with different physical and mechanical parameter regions, the research object is divided into m quadrilateral elements, totaling n nodes; S3, select all element node stress components and node displacement components as independent variables, and construct the element equilibrium equation expressed by the node stress components; Nodal strain components expressed by nodal stress components; The element deformation equation expressed by nodal strain components and nodal displacement components; The element deformation equation expressed by nodal stress components and nodal displacement components; The boundary conditions of the nodal stress components are known; The boundary conditions of the node displacement components are known; S4, construct the overall control equation; Solve the overall control equation to obtain the nodal stress components and nodal displacement components; S5, based on the solved nodal stress components and nodal displacement components, perform stress and deformation analysis on the structure.

2. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S1, the physical and mechanical parameters include bulk density , elastic modulus E Poisson's ratio .

3. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S3, the unit equilibrium equation expressed by the node stress components is: , is the coefficient matrix, 2 m Row, 3 n List; is a constant matrix, 2 m row, 1 column; is the node stress component, 3 n row, 1 column; and The calculation process is: S3.1.1, the element equilibrium equation for each element expressed in terms of nodal stress components is: ; , , and They are the nodes 1, 2, 3 and 4 of the quadrilateral element respectively. x Coordinates and y coordinate; and The units are x Direction and y physical strength of direction; S3.1.2, integrate all the unit equilibrium equations expressed by the nodal stress components to obtain and .

4. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S3, the node strain components expressed by the node stress components are: , is the node strain component, 3 n row, 1 column; is the coefficient matrix, 3 n Row, 3 n List; The calculation process is: S3.2.1, the nodal strain component expressed by the nodal stress component at any node is Where: E is the elastic modulus; is Poisson’s ratio; S3.2.2, integrate all the nodal strain components expressed by the nodal stress components and obtain .

5. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S3, the element deformation equation expressed by the node strain component and the node displacement component is: ,in: is a coefficient matrix with 3m rows and 3n columns; is a coefficient matrix with 3m rows and 2n columns; and The calculation process is: S3.3.1, the element deformation equation expressed in terms of nodal strains and nodal displacements for each element is: ; in, S is the unit area; and , respectively, of the quadrilateral unit nodes 1, 2, 3, and 4 x Positive strain, y Normal strain and shear strain; and They are unit nodes 1, 2, 3 and 4 respectively. x Coordinates and y coordinate; and They are unit nodes 1, 2, 3 and 4 respectively. x Displacement and y Toward displacement; S3.3.2, integrate all the unit deformation equations expressed by node strains and node displacements to obtain and .

6. The stress finite element method for solving structural stress and deformation according to claims 4 and 5, characterized in that: In step S3, the element deformation equation expressed by the node stress component and the element displacement component is: ,in: .

7. The stress finite element method for solving structural stress and deformation according to claim 1 is characterized in that: In step S3, the known boundary conditions of the node stress components are: ,in: j represents the number of known boundary conditions for stress components; is the coefficient matrix, j rows, 3n columns; The column elements in the rows corresponding to the known stress components are 1, and the other elements are 0; is the stress boundary condition matrix, j row, 1 column; The elements in the corresponding rows of are the known stress components.

8. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S3, the known boundary conditions of the node displacement components are: in: k represents the number of known boundary conditions for the displacement components; is the coefficient matrix, k Row, 2 n List; The column elements in the rows corresponding to the known displacement components are 1, and the other elements are 0; Displacement boundary condition matrix, k row, 1 column; The elements in the corresponding rows of are the known stress components.

9. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S4, the overall control equation is: ,in: ， By solving the minimum norm least squares solution of the overall control equation, the independent variable The value of ; O represents the zero matrix.

10. The stress finite element method for solving structural stress and deformation according to claim 1, characterized in that: In step S2, for three-dimensional problems, the research object is divided into hexahedral units, the overall control equation of the three-dimensional problem is established, the solution of the overall control equation is solved, the node stress components and node displacement components are obtained, and the stress and deformation of the research object are analyzed.

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