A beam element applicable to finite element analysis and a design method thereof

Abstract

The present application relates to a beam element applicable to finite element analysis and a design method. The beam element calculates a shape function matrix through a rigid body displacement matrix and a flexibility displacement matrix, so that the cross-sectional coordinate axes of the beam element can be set in any direction and the two end nodes of the beam element can be set at any position in the cross-sectional coordinate system. The beam element designed by the present invention can be applicable to finite element modeling of various complex beam structures, improving the accuracy of finite element simulation calculation of complex beam structures.

Description

Technical Field

[0001] The present invention relates to the technical field of finite element, and particularly relates to a beam element applicable to finite element analysis and a design method thereof. Background Art

[0002] Currently, structural calculations are mainly implemented through finite element analysis software. As one of the basic finite element units, beam elements are widely used in finite element analysis calculations.

[0003] When using finite element calculation and analysis software for modeling in related technologies, simplification processing is often carried out, including: using a user coordinate system when establishing a cross-section, resulting in difficulty in accurately finding the principal inertia axis of the cross-section for complex cross-sections, thus causing the problem that the custom cross-section user coordinate system is not parallel to the principal inertia axis of the cross-section; for variable cross-section beam elements, an approximate cross-section coordinate system is usually used to uniquely determine the element coordinate system, resulting in the problem that the y and z coordinate axes of the beam element are not parallel to the true principal inertia axis of the beam element cross-section.

[0004] In addition, in related technologies, an eccentricity is set for the cross-section, and the eccentric point is the element node. When calculating, a rigid arm is set between the node and the eccentric point to ensure that the element node is located at the centroid of the cross-section. The problem with this setting is that if an axial force or lateral force in a direction different from the offset direction is applied to the element node, additional bending moments or torques will occur when the load is transferred to the centroid position of the cross-section.

[0005] Therefore, for complex structures (the principal inertia axes of the cross-section centroids at both ends of the component are not parallel, the cross-section characteristics can vary arbitrarily along the length of the component, or the centroids of the components are not connected, etc.), if the related technology is used for simplification processing and then calculation, the calculation results will have errors. Summary of the Invention

[0006] The embodiments of the present invention provide a beam element applicable to finite element analysis and a design method thereof to solve the problem of the accuracy of finite element simulation calculations for complex beam structures in related technologies.

[0007] In a first aspect, a beam element design method is provided, characterized in that the beam element calculates a shape function matrix through a rigid body displacement matrix and a flexibility displacement matrix, so that the cross-section coordinate axes of the beam element satisfy the setting in any direction and the two end nodes of the beam element satisfy the setting at any position in the cross-section coordinate system.

[0008] In some embodiments, the calculating the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix includes the steps of:

[0009] Calculating the shape function matrix according to a first formula, and the first formula includes:

[0010]

[0011]

[0012] {δ(x)} = {δ d (x)} + {δ s (x)} = [H x {B},

[0013]

[0014]

[0015]

[0016]

[0017] {δ(x)} = [H x [C] -1 {δ} e ,

[0018] [N] = [H x [C] -1 , where,

[0019] {δ d (x)} is the displacement of any point in the element caused by the rigid body displacement at the fixed end, is the rigid body displacement matrix, is composed of . is the displacement of the i-th degree of freedom at the coordinate x position when the j-th degree of freedom at the fixed end undergoes a unit rigid body displacement, {B2} is the rigid body displacement of the fixed end in 6 degrees of freedom, {δ s (x)} is the displacement of any point in the element caused by the load acting at the free end, is the flexibility displacement matrix, is composed of . is the displacement of the i-th degree of freedom at the coordinate x position when a unit load acts on the j-th degree of freedom at the free end, {B1} is the unit load acting on the free end in 6 degrees of freedom, [H x , {B}, [C] are intermediate variables, [N] is the said shape function matrix, [H1] is [H x when x = 0, [H2] is [H x when x = L, L is the length of the beam element, {δ(0)} is the fixed end displacement, {δ(L)} is the free end displacement, {δ} e is the element node displacement, {δ(x)} is the displacement at any position of the element.

[0020] In some embodiments, the beam element also calculates the linear stiffness matrix through the rigid body displacement strain matrix and the flexibility strain matrix, including the steps,

[0021] Obtain the rigid body displacement strain matrix of the beam element according to the relationship between the rigid body displacements occurring at the fixed end of the beam element at each degree of freedom when one end of the beam element is fixed and the linear strain at any point within the element;

[0022] Obtain the flexibility strain matrix of the beam element according to the relationship between the loads acting on each degree of freedom at the free end of the beam element when one end of the beam element is fixed and the linear strain at any point within the element;

[0023] Calculate the linear stiffness matrix based on the rigid body displacement strain matrix and the flexibility strain matrix.

[0024] In some embodiments, the rigid body displacement strain matrix is obtained according to a second formula, and

[0025] The second formula is:

[0026] {ε d (x)} = [N 02 {B2},

[0027]

[0028] where, {ε d (x)} is the strain caused by the rigid body displacement occurring at the fixed end at any point within the element, {B2} is the rigid body displacements occurring at the fixed end at 6 degrees of freedom, and [N 02 is the rigid body displacement strain matrix.

[0029] The flexibility strain matrix is obtained according to a third formula, and

[0030] The third formula is: {ε s (x)} = [N 01 {B1},

[0031] where, {ε s (x)} is the strain caused by the load acting at the free end at any point within the element, {B1} is the unit loads acting at 6 degrees of freedom at the free end, and [N 01 is composed of [N 01 (i, j), where [N 01 (i, j) is the strain at the i-th degree of freedom at the coordinate x position when a unit load acts at the j-th degree of freedom at the free end, and [N 01 is the flexibility strain matrix.

[0032] In some embodiments, the calculating the linear stiffness matrix based on the rigid body displacement strain matrix and the flexibility strain matrix includes the steps of:

[0033] Calculate the linear stiffness matrix according to a fourth formula, and the fourth formula includes:

[0034]

[0035] [B0] = [N0][C] -1 , [N0] = [N 01 N 02

[0036]

[0037] where {δ} e is the nodal displacement formulation of the beam element, [K0] is the linear stiffness matrix, [D] is the constitutive matrix, [N0] is an intermediate variable, and [B0] is the total strain matrix.

[0038] In some embodiments, the beam element also calculates the geometric stiffness matrix through a rigid body displacement nonlinear strain matrix and a flexibility nonlinear strain matrix, including the steps of:

[0039] Obtaining the rigid body displacement nonlinear strain matrix of the beam element according to the relationship between the rigid body displacements occurring at the fixed end in each degree of freedom when one end of the beam element is fixed and the derivative of the displacement of any point in the element with respect to the axial coordinate x;

[0040] Obtaining the flexibility nonlinear strain matrix of the beam element according to the relationship between the loads acting on each degree of freedom at the free end when one end of the beam element is fixed and the derivative of the displacement of any point in the element with respect to the axial coordinate x;

[0041] Calculating the geometric stiffness matrix based on the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix.

[0042] In some embodiments, the rigid body displacement nonlinear strain matrix is obtained according to the fifth formula, and

[0043] the fifth formula is:

[0044]

[0045] where {B2} is the rigid body displacement occurring at the fixed end in 6 degrees of freedom,

[0046] is the rigid body displacement nonlinear strain matrix,

[0047] represents the derivative of the displacement of any point in the element with respect to the axial coordinate x;

[0048] The flexibility nonlinear strain matrix is obtained according to the sixth formula, and

[0049] the sixth formula is:

[0050]

[0051] ​Among them, {B1} is the load acting on the free end in 6 degrees of freedom.

[0052] It represents the derivative of the displacement at any point in the element with respect to the axial coordinate x.

[0053] is the flexibility nonlinear strain matrix.

[0054] In some embodiments, calculating the geometric stiffness matrix based on the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix includes the steps of:

[0055] Calculating the geometric stiffness matrix according to the ninth formula, and the ninth formula includes:

[0056]

[0057]

[0058]

[0059]

[0060] [B n1 = [H′1(x)][C] -1 , [B n2 = [H′2(x)][C] -1 , [B n3 = [H′3(x)][C] -1 ,

[0061] dε n = [B n1 {δ} e [B n1 d{δ} e + [B n2 {δ} e [B n2 d{δ} e + [B n3 {δ} e [B n3 d{δ} e ,

[0062] [k σ = N∫([B n1 T [B n1 + [B n2 T [B n2 + [B n3 T [B n3 )dx = N([C]​​​-1 ) T [Q][C] -1

[0063] [Q]=∫([H′1(x)] T [H′1(x)]+[H′2(x)] T [H′2(x)]+[H′3(x)] T [H′3(x)])dx, where,

[0064] [k σ is the geometric stiffness matrix, N is the axial force of the beam element, and H′1(x), H′2(x), H′3(x), [C], and [Q] are all defined intermediate variables.

[0065] In a second aspect, a beam element is provided, characterized in that the beam element calculates a shape function matrix through a rigid body displacement matrix and a flexibility displacement matrix, so that the cross-sectional coordinate axes of the beam element satisfy the setting in any direction and the two end nodes satisfy the setting at any position in the cross-sectional coordinate system.

[0066] In some embodiments, the beam element is defined according to the design method described in any one of the above.

[0067] The beneficial effects brought by the technical solutions provided by the present invention include: the shape function obtained in the beam element design method does not impose any restrictions on conditions such as the cross-sectional form, node position, and element coordinate system of the beam element, so that the beam element defined by the shape function can be applied to finite element modeling of various complex beam structures, avoiding the errors caused by simplified processing and then calculation in the related art, and improving the accuracy of finite element simulation calculation of complex beam structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for description in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0069] Figure 1 It is a schematic flowchart of a beam element design method provided by an embodiment of the present invention;

[0070] Figure 2 It is a schematic flowchart of a beam element design method provided by an embodiment of the present invention;

[0071] Figure 3 It is a schematic flowchart of a beam element design method provided by an embodiment of the present invention;

[0072] Figure 4 Schematic diagram of the beam element coordinate system provided by the embodiment of the present invention;

[0073] Figure 5 Schematic diagram of the cross-section at the I end of the beam element provided by the embodiment of the present invention;

[0074] Figure 6 Schematic diagram of the interface at the J end of the beam element provided by the embodiment of the present invention;

[0075] Figure 7 Schematic diagram of the appearance of the beam element provided by the embodiment of the present invention. Detailed implementation manners

[0076] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0077] As Figure 1 shown, the embodiment of the present invention provides a beam element design method, including the steps:

[0078] S100: The beam element calculates the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix, so that the cross-section coordinate axes of the beam element satisfy the setting in any direction and the two end nodes of the beam element satisfy the setting at any position in the cross-section coordinate system.

[0079] It should be noted that the rigid body displacement matrix is a matrix composed of the displacements of any point in the beam element caused by unit rigid body displacements of each degree of freedom at the fixed end when one end of the element is fixed and the other end is free, and the flexibility displacement matrix is a matrix composed of the displacements of any point in the beam element caused by unit loads acting on each degree of freedom at the free end when one end of the element is fixed and the other end is free.

[0080] In the embodiment of the present invention, the shape function obtained in the beam element design method only calculates the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix, without any restrictions on the cross-section form, node position, and element coordinate system of the beam element, so that the beam element defined by the shape function can be applied to finite element modeling of various complex beam structures, avoiding the errors caused by simplified processing and then calculation in the related art, and improving the accuracy of finite element simulation calculation of complex beam structures.

[0081] Specifically, in some embodiments, any point within the beam element has six degrees of freedom, and there are a total of twelve degrees of freedom at both ends. Fix one end of the element and leave the other end free. In S100: Calculate the shape function matrix according to the first formula, and the first formula includes:

[0082]

[0083]

[0084] {δ(x)} = {δ d (x)} + {δ s (x)} = [H x {B},

[0085]

[0086]

[0087]

[0088] {δ(x)} = [H x [C] -1 {δ} e ,

[0089] [N] = [H x [C] -1 , where,

[0090] {δ d (x)} is the displacement of any point within the element caused by the rigid body displacement at the fixed end, is the rigid body displacement matrix, is composed of , is the displacement of the i-th degree of freedom at the coordinate position x when the j-th degree of freedom at the fixed end undergoes a unit rigid body displacement, {B2} is the rigid body displacement of the fixed end in six degrees of freedom, {δ s (x)} is the displacement of any point within the element caused by the load acting on the free end, is the flexibility displacement matrix, is composed of , is the displacement of the i-th degree of freedom at the coordinate position x when a unit load acts on the j-th degree of freedom at the free end, {B1} is the unit load acting on the free end in six degrees of freedom, [H x , {B}, [C] are intermediate variables, [N] is the shape function matrix, [H1] is [H x when x = 0, [H2] is [H x when x = L, L is the length of the beam element, {δ(0)} is the displacement at the fixed end, {δ(L)} is the displacement at the free end, {δ}e {δ} is the displacement of the unit node, and {δ(x)} is the displacement at any position of the unit.

[0091] As Figure 2 shown, in some embodiments, it further includes the steps of:

[0092] S200: The beam element also calculates the linear stiffness matrix through the rigid body displacement strain matrix and the flexibility strain matrix. S200 includes the steps of:

[0093] S201: Obtain the rigid body displacement strain matrix of the beam element according to the relationship between the rigid body displacements occurring at the fixed end in each degree of freedom when one end of the beam element is fixed and the linear strain at any point within the element;

[0094] S202: Obtain the flexibility strain matrix of the beam element according to the relationship between the loads acting on each degree of freedom at the free end when one end of the beam element is fixed and the linear strain at any point within the element;

[0095] S203: Calculate the linear stiffness matrix based on the rigid body displacement strain matrix and the flexibility strain matrix.

[0096] This embodiment can be applied to various complex beam structures, including cases where the centroidal inertia principal axes of the cross-sections at both ends of the component are not parallel, the cross-section properties can vary arbitrarily along the length of the component, and the centroids of the components are not connected, etc. Since the derivation of the shape function matrix and the linear stiffness matrix does not impose any restrictions on the form of the beam element (including the cross-section form of the element, the node positions of the element, and the element coordinate system, etc.), the beam element defined by this shape function can be applied to finite element modeling of various complex beam structures, further improving the accuracy of finite element simulation calculations for complex beam structures.

[0097] In some embodiments, the rigid body displacement strain matrix in S201 is obtained according to the second formula, and

[0098] the second formula is:

[0099] {ε d (x)} = [N 02 {B2},

[0100]

[0101] where {B2} is the rigid body displacements occurring at the fixed end in 6 degrees of freedom, denoted as {B2} = {A7 A8 A9 A 10 A 11 A 12}, T [N 02 is the rigid body displacement strain matrix;

[0102] The flexibility strain matrix in S202 is obtained according to the third formula, and the third formula is: {εs (x)} = [N 01 {B1}, where {B1} is the unit load acting on the 6 degrees of freedom at the free end, and [N 01 is composed of [N 01 (i, j). [N 01 (i, j) is the strain of the i-th degree of freedom at the coordinate x position when a unit load acts on the j-th degree of freedom at the free end. [N 01 is the flexibility strain matrix.

[0103] In some embodiments, in S203, the linear stiffness matrix is calculated according to the fourth formula, and the fourth formula includes:

[0104]

[0105] [B0] = [N0][C] -1 , [N0] = [N 01 N 02

[0106]

[0107] where {δ} e is the nodal displacement formulation of the beam element, [K0] is the linear stiffness matrix, [D] is the constitutive matrix, and [B0] is the total strain matrix.

[0108] As Figure 3 shown, in some embodiments, it further includes the steps:

[0109] S300: The beam element also calculates the geometric stiffness matrix through the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix. S300 includes the steps:

[0110] S301: Obtain the rigid body displacement nonlinear strain matrix of the beam element according to the relationship between the rigid body displacements occurring at the fixed end in each degree of freedom when one end of the beam element is fixed and the derivative of the displacement of any point in the element with respect to the axial coordinate x;

[0111] S302: Obtain the flexibility nonlinear strain matrix of the beam element according to the relationship between the loads acting on each degree of freedom at the free end when one end of the beam element is fixed and the derivative of the displacement of any point in the element with respect to the axial coordinate x;

[0112] S303: Calculate the geometric stiffness matrix based on the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix.

[0113] ​In this embodiment, in order to make the beam element applicable to various complex beam structures, no restrictions are imposed on the form of the beam element (including the form of the unit cross-section, the position of the unit nodes, and the unit coordinate system, etc.) when deriving the geometric stiffness matrix, which can further improve the accuracy of the finite element simulation calculation of the complex beam structure.

[0114] In some embodiments, in step S301, the rigid body displacement nonlinear strain matrix is obtained according to the fifth formula, and the fifth formula is:

[0115]

[0116] where {B2} is the rigid body displacement occurring at the fixed end in six degrees of freedom,

[0117] is the rigid body displacement nonlinear strain matrix,

[0118] represents the derivative of the displacement of any point in the element with respect to the axial coordinate x;

[0119] In S302, the flexibility nonlinear strain matrix is obtained according to the sixth formula, and the sixth formula is:

[0120]

[0121] where {B1} is the load acting on the free end in six degrees of freedom,

[0122] represents the derivative of the displacement of any point in the element with respect to the axial coordinate x,

[0123] is the flexibility nonlinear strain matrix.

[0124] In some embodiments, in S303, the geometric stiffness matrix is calculated according to the ninth formula, and the ninth formula includes:

[0125]

[0126]

[0127]

[0128]

[0129] [B n1 = [H′1(x)][C] -1 [B n2 = [H′2(x)][C] -1 [B n3 = [H′3(x)][C] -1 ,

[0130] dε n =[B n1 {δ} e [B n1 d{δ} e +[B n2 {δ} e [B n2 d{δ} e +[B n3 {δ} e [B n3 d{δ} e ,

[0131] [k σ =N∫([B n1 T [B n1 +[B n2 T [B n2 +[B n3 T [B n3 )dx=N([C] -1 ) T [Q][C] -1

[0132] [Q]=∫([H′1(x)] T [H′1(x)]+[H′2(x)] T [H′2(x)]+[H′3(x)] T [H′3(x)])dx, where

[0133] [k σ is the geometric stiffness matrix, N is the axial force of the beam element, and H′1(x), H′2(x), H′3(x), [C], and [Q] are all defined intermediate variables.

[0134] In a specific embodiment, the beam element coordinate system is as Figure 4 shown. The length of the beam element is L = 10 m. The I-end cross-section is as Figure 5 shown, and the J-end cross-section is as Figure 6 shown. (y d , z d ) are the coordinates of the node position in the cross-section coordinate system, (y0, z0) are the coordinates of the shear center in the cross-section coordinate system, (y c , z c ) are the cross-section centroid positions, y s , z s are the cross-section coordinate system axes, y e , z e ​​​is the axis of the element coordinate system. The axes of the section coordinate system are parallel to those of the element coordinate system, but the origins can be different. The origin of the section coordinate system is at the centroid of the section, and the origin of the element coordinate system is at the I-end node. The axes of the section coordinate system may not be the principal axes of inertia of the section. The shape of the beam element is as Figure 7 shown.

[0135] The section characteristic parameters include: A x 、A y 、A z 、A yz 、I x 、I y 、I z 、I yz 、y d 、z d 、y0, z0 represent the axial area, the shear area in the y direction, the shear area in the z direction, the shear influence coefficient, the torsional constant, the moment of inertia about the y axis, the moment of inertia about the z axis, the product of inertia, the coordinates of the node position on the y and z axes in the section coordinate system, and the coordinates of the shear center position on the y and z axes in the section coordinate system. The section characteristic values at the I-end and J-end are shown in Table 1:

[0136] Table 1: (Unit: m system)

[0137] Characteristic I-end cross-section J-end cross-section <![CDATA[A x > 1.560 0.558 <![CDATA[A y > 1.246 0.423 <![CDATA[A z > 1.277 0.446 <![CDATA[A yz > 0.026 0.046 <![CDATA[I x > 0.317 0.037 <![CDATA[I y > 0.278 0.039 <![CDATA[I z > 0.160 0.021 <![CDATA[I y z]]> 0.044 0.010 <![CDATA[y d > -0.285 0.792 <![CDATA[z d > -0.31 0.41 <![CDATA[y0]]> 0.009 -0.009 <![CDATA[z0]]> -0.014 0.030

[0138] Among them, the section characteristics at the internal coordinate x of the element are: n is the number of variation. Preferably, the number of variation of all characteristics in Table 1 is taken as 1, T I is the section characteristic at the I-end, and T J is the section characteristic at the J-end. The elastic modulus E = 34500 MPa, and the shear modulus G = 13800 MPa.

[0139] When deriving the shape function matrix, it can be divided into four steps: S1, obtaining the rigid body displacement matrix; S2, obtaining the flexibility displacement matrix; S3, combining the rigid body displacement matrix and the flexibility displacement matrix to form the displacement mode matrix; S4, calculating the shape function matrix. The specific steps are as follows:

[0140] In S1, any point in the element has 6 degrees of freedom, and there are 12 degrees of freedom at the I and J ends. Fix the I-end of the element. The 6 degrees of freedom of the fixed end undergo rigid body displacements of {B2} = {A7 A8 A9 A 10 A 11 A 12}, T then the displacement of any point in the element is expressed by the rigid body displacement matrix as Among them, is the rigid body displacement matrix, The displacement of the $i$-th degree of freedom at the coordinate $x$ when a unit rigid body displacement occurs in the $j$-th degree of freedom at the I end. It can be obtained that:

[0141]

[0142] In S2, fix the I end of the element, and apply a unit load $\{B1\}=\{A1\ A2\ A3\ A4\ A5\ A6\}$ to the 6 degrees of freedom at the J end T , then the displacement of any point in the element is expressed by the flexibility displacement matrix as where is the flexibility displacement matrix, is the displacement of the $i$-th degree of freedom at the coordinate $x$ when a unit load acts on the $j$-th degree of freedom at the J end. The flexibility displacement matrix is a function matrix of $x$. In this example, there is no analytical solution, and the numerical solution can be obtained by Gaussian integration. The flexibility displacement matrix when $x = L$ is shown in Table 2:

[0143] Table 2 (unit: mm / KN, 0.001rad / KN)

[0144] 1.8E-03 6.1E-03 -6.2E-03 0.0E+00 1.6E-03 1.8E-03 6.1E-03 8.8E-02 -1.4E-02 -2.9E-03 2.6E-03 1.5E-02 -6.2E-03 -1.4E-02 5.0E-02 -1.7E-03 -8.7E-03 -2.6E-03 0.0E+00 -2.9E-03 -1.7E-03 5.5E-03 0.0E+00 0.0E+00 1.6E-03 2.6E-03 -8.7E-03 0.0E+00 2.5E-03 8.6E-04 1.8E-03 1.5E-02 -2.6E-03 0.0E+00 8.6E-04 4.5E-03

[0145] In S3, superimpose and combine the rigid body displacement matrix and the flexibility displacement matrix:

[0146]

[0147] In S4, when $x = 0$, $[H x = [H1]

[0148]

[0149] When $x = L$, $[H x = [H2]$, and $[H2]$ is shown in Table 3:

[0150] Table 3:

[0151]

[0152] Let the nodal displacement at the I end $\{\delta(0)\} = [H1]\{B\}$, the nodal displacement at the J end $\{\delta(L)\} = [H2]\{B\}$, the nodal displacement and The displacement inside the element $\{\delta x \} = [H x [C] -1 \{\delta\} e , then the shape function matrix is $[N] = [H x [C] -1 .

[0153] When deriving the linear stiffness matrix, it can be divided into four steps: S5, obtaining the rigid body displacement strain matrix; S6, obtaining the flexibility strain matrix; S7, combining the rigid body displacement strain matrix and the flexibility strain matrix to form the total strain matrix; S8, calculating the linear stiffness matrix.

[0154] In S5, fix the I end of the beam element. The six degrees of freedom at the fixed end have a rigid body displacement of {B2} = {A7 A8 A9 A 10 A 11 A 12} T , then the strain at any point within the element is: According to {ε d (x)} = [N 02 {B2}, and since rigid body displacement does not produce strain, the rigid body displacement strain matrix is:

[0155]

[0156] In S6, fix the I end of the element and apply a load {B1} = {A1 A2 A3 A4 A5 A6} at the six degrees of freedom at the J end T , then the strain at any point within the element The six strains are: axial strain, y shear strain, z shear strain, torsional curvature, bending moment curvature about the y-axis, and bending moment curvature about the z-axis. Expressed in the flexibility strain matrix as: {ε s (x)} = [N 01 {B1}, where [N 01 is the flexibility strain matrix, and [N 01 (i, j) is the i-th strain at the coordinate x position when a unit load is applied to the j-th degree of freedom at the J end. The specific flexibility strain matrix is:

[0157]

[0158] Among them, A x , A y , A z , A yz , I x , I y , I z , I yz , y d , z d , y0, z0 are the cross-sectional properties at the coordinate x.

[0159] In S7, form the total strain matrix by combining the rigid body displacement strain matrix and the flexibility strain matrix. Let:

[0160]

[0161] Among them, [B0] = [N0][C] -1 is the total strain matrix, {δ} e is the displacement formulation of the beam element nodes.

[0162] In S8, let:

[0163]

[0164] Among them, [D] is the constitutive matrix and can be expressed as:

[0165]

[0166] K 1324 = K1K3 - K2K4, A x , A y , A z , A yz , I x , I y , I z , I yz , y d , z d , y0, z0 are the cross-sectional properties at the coordinate x.

[0167] It should be noted that in this example, [K0] of the beam element has no analytical solution and the numerical solution can be obtained through Gaussian integration. Specifically:

[0168] [K 12 = [K 21 T , where

[0169] [K 11 is shown in Table 4:

[0170] Table 4:

[0171] 3.4E+06 9.1E+03 1.3E+05 4.3E+04 -2.7E+06 -9.5E+05 9.1E+03 3.0E+04 1.0E+04 1.9E+04 -6.9E+04 2.0E+05 1.3E+05 1.0E+04 5.9E+04 2.3E+04 -4.5E+05 2.5E+04 4.3E+04 1.9E+04 2.3E+04 2.0E+05 -1.7E+05 1.1E+05 -2.7E+06 -6.9E+04 -4.5E+05 -1.7E+05 4.8E+06 2.9E+05 -9.5E+05 2.0E+05 2.5E+04 1.1E+05 2.9E+05 1.8E+06

[0172] [K 12 is shown in Table 5:

[0173] Table 5:

[0174] -3.4E+06 -9.1E+03 -1.3E+05 -4.3E+04 1.4E+06 1.0E+06 -9.1E+03 -3.0E+04 -1.0E+04 -1.9E+04 -3.4E+04 1.1E+05 -1.3E+05 -1.0E+04 -5.9E+04 -2.3E+04 -1.3E+05 7.7E+04 -4.3E+04 -1.9E+04 -2.3E+04 -2.0E+05 -5.8E+04 7.9E+04 2.7E+06 6.9E+04 4.5E+05 1.7E+05 -3.0E+05 -9.8E+05 9.5E+05 -2.0E+05 -2.5E+04 -1.1E+05 -5.5E+05 1.5E+05

[0175] [K 22 is shown in Table 6:

[0176] Table 6:

[0177] 3.4E+06 9.1E+03 1.3E+05 4.3E+04 -1.4E+06 -1.0E+06 9.1E+03 3.0E+04 1.0E+04 1.9E+04 3.4E+04 -1.1E+05 1.3E+05 1.0E+04 5.9E+04 2.3E+04 1.3E+05 -7.7E+04 4.3E+04 1.9E+04 2.3E+04 2.0E+05 5.8E+04 -7.9E+04 -1.4E+06 3.4E+04 1.3E+05 5.8E+04 1.6E+06 2.1E+05 -1.0E+06 -1.1E+05 -7.7E+04 -7.9E+04 2.1E+05 9.1E+05 ​

[0178] When deriving the geometric stiffness matrix, it can be divided into four steps: S9, obtaining the rigid body displacement nonlinear strain matrix; S10, obtaining the flexibility nonlinear strain matrix; S11, combining the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix to form the total nonlinear strain matrix; S12, calculating the geometric stiffness matrix.

[0179] In S9, fix the I end of the element. The six degrees of freedom at the fixed end have a rigid body displacement of {B2} = {A7 A8 A9 A 10 A 11 A 12}, then the derivative of the displacement at any point in the element with respect to the axial coordinate x is: T

[0180]

[0181]

[0182] In S10, fix the I end of the element, and apply the load {B1} = {A1 A2 A3 A4 A5 A6}T to the six degrees of freedom at the J end. Then the derivative of the displacement at any point in the element with respect to the axial coordinate x is:

[0183]

[0184]

[0185] Where: ε x1 、ε x2 、ε x3 、ε x4 、ε x5 、ε x6 are respectively the axial strains in the beam element caused by unit loads of F x 、F y 、F z 、M x 、M y 、M z acting at the J end, ε x4 = 0; θ z1 、θ z2 、θ z3 、θ z4 、θ z5 、θ z6 are respectively the rotations about the z-axis at the coordinate x caused by unit loads of F x 、F y 、F z 、M x 、M y 、M z acting at the J end, θ z4 = 0; θ y1 、θy2 、 θ y3 、 θ y4 、 θ y5 、 θ y6 are respectively F x 、 F y 、 F z 、 M x 、 M y 、 M z The rotation angle about the y-axis at the coordinate x caused by a unit load acting at the J end, θ y4 = 0; γ y1 、 γ y2 、 γ y3 、 γ y4 、 γ y5 、 γ y6 are respectively F x 、 F y 、 F z 、 M x 、 M y 、 M z The shear deformation in the y-axis direction at the coordinate x caused by a unit load acting at the J end, where γ z1 、 γ z2 、 γ z3 、 γ z4 、 γ z5 、 γ z6 are respectively F x 、 F y 、 F z 、 M x 、 M y 、 M z The shear deformation in the z-axis direction at the coordinate x caused by a unit load acting at the J end, where γ z1 γ z4 γ z5 γ z6 are all 0.

[0186] In S11, the combined rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix form the total nonlinear strain matrix. Let:

[0187]

[0188]

[0189] In S12, the nonlinear term of the axial strain is:

[0190]

[0191]

[0192] Let [B n1= [H′1(x)][C] -1 , [B n2 = [H′2(x)][C] -1 , [B n3 = [H′3(x)][C] -1 ,

[0193] dε n = [B n1 {δ} e [B n1 d{δ} e + [B n2 {δ} e [B n2 d{δ} e + [B n3 {δ} e [B n3 d{δ} e ,

[0194]

[0195] where [Q]= ∫([H′1(x)] T [H′1(x)]+ [H′2(x)] T [H′2(x)]+ [H′2(x)] T [H′3(x)])dx, and N is the axial force of the element. In this example, there is no analytical solution for the geometric stiffness matrix [k σ , and the numerical solution can be obtained by Gaussian integration. Specifically:

[0196]

[0197] [K σ12 = [K σ21 T ,

[0198] When N = 10000 KN, [K σ11 is shown in Table 7:

[0199] Table 7:

[0200] 2570.0 -55.2 175.0 23.8 -1760.0 -733.0 -55.2 1210.0 -22.3 627.0 190.0 1720.0 175.0 -22.3 1250.0 366.0 -2010.0 -207.0 23.8 627.0 366.0 438.0 -507.0 836.0 -1760.0 190.0 -2010.0 -507.0 22600.0 1670.0 -733.0 1720.0 -207.0 836.0 1670.0 20400.0

[0201] [K σ12 is shown in Table 8:

[0202] Table 8:

[0203] -2570.0 55.2 -175.0 -23.8 -76.0 40.2 55.2 -1210.0 22.3 -627.0 31.6 417.0 -175.0 22.3 -1250.0 -366.0 -519.0 -12.5 -23.8 -627.0 -366.0 -438.0 -140.0 214.0 1760.0 -190.0 2010.0 507.0 -2460.0 294.0 733.0 -1720.0 207.0 -836.0 424.0 -3030.0

[0204] [K σ22 is shown in Table 9: ​

[0205] Table 9:

[0206] 2570.0 -55.2 175.0 23.8 76.0 -40.2 -55.2 1210.0 -22.3 627.0 -31.6 -417.0 175.0 -22.3 1250.0 366.0 519.0 12.5 23.8 627.0 366.0 438.0 140.0 -214.0 76.0 -31.6 519.0 140.0 7580.0 -206.0 -40.2 -417.0 12.5 -214.0 -206.0 7190.0

[0207] The present invention also provides a beam element, which calculates a shape function matrix through a rigid body displacement matrix and a flexibility displacement matrix, so that the cross-section coordinate axes of the beam element satisfy the setting in any direction and the two end nodes satisfy the setting at any position in the cross-section coordinate system.

[0208] It can be understood that the beam element is defined by the shape function obtained in the beam element design method involved in the foregoing embodiments. Since the shape function only calculates the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix, and does not impose any restrictions on conditions such as the cross-section form, node position, and element coordinate system of the beam element, the beam element defined by the shape function can be applied to finite element modeling of various complex beam structures, avoiding the errors caused by simplified processing and then calculation in the related art, and improving the accuracy of finite element simulation calculation of complex beam structures.

[0209] It can be understood that for the beam element mentioned in the present invention, the shape function matrix, linear stiffness matrix, and / or geometric stiffness matrix are derived according to the methods involved in the foregoing method embodiments, and the beam element can satisfy the technical effects achievable in the foregoing embodiments.

[0210] Those of ordinary skill in the art can understand that all or some of the steps in the methods disclosed above, and the functional modules / units in the systems and devices, can be implemented as software, firmware, hardware, and their appropriate combinations. In the hardware implementation, the division of the functional modules / units mentioned above does not necessarily correspond to the division of physical components; for example, one physical component can have multiple functions, or one function or step can be executed by several physical components in cooperation. Some physical components or all physical components can be implemented as software executed by a processor, such as a central processing unit, a digital signal processor, or a microprocessor, or be implemented as hardware, or be implemented as an integrated circuit, such as an application-specific integrated circuit. Such software can be distributed on a computer-readable storage medium, which can include a computer-readable storage medium (or non-transitory medium) and a communication medium (or transitory medium).

[0211] It should be noted that in the present invention, relational terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the term "comprising", "including" or any other variant thereof is intended to cover non-exclusive inclusion, such that a process, method, article or device comprising a series of elements not only includes those elements but also includes other elements not expressly listed, or further includes elements inherent to such process, method, article or device. Without further limitation, an element defined by the statement "comprising an..." does not exclude the presence of additional identical elements in the process, method, article or device comprising the said element.

[0212] The above are only specific embodiments of the present invention, enabling those skilled in the art to understand or implement the present invention. Various modifications to these embodiments will be obvious to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to these embodiments shown herein, but rather to the broadest scope consistent with the principles and novel features claimed herein.

Claims

1. A beam element design method, characterized in that The beam element calculates the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix, so that the cross-sectional coordinate axes of the beam element satisfy the setting in any direction and the two end nodes of the beam element satisfy the setting at any position in the cross-sectional coordinate system; The calculation of the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix includes the steps of: Calculating the shape function matrix according to the first formula, and the first formula includes: ， ， ， ， ， ， ， , wherein, is the displacement caused by the rigid body displacement of any point within the element from the fixed end, is the rigid body displacement matrix, which is composed of and is the displacement of the ith degree of freedom at the coordinate x position when the jth degree of freedom of the fixed end undergoes a unit rigid body displacement, is the rigid body displacement occurring at the 6 degrees of freedom of the fixed end, is the displacement caused by the load acting on the free end at any point within the element, is the flexibility displacement matrix, which is composed of and is the displacement of the ith degree of freedom at the coordinate x position when a unit load acts on the jth degree of freedom of the free end, , are intermediate variables, is the said shape function matrix, is the when x = 0, is the when x = L, where L is the length of the beam element, is the fixed end displacement, is the free end displacement, is the element nodal displacement, is the displacement at any position within the element.

2. The design method of a beam element according to claim 1, characterized in that The beam element also calculates the linear stiffness matrix through the rigid body displacement strain matrix and the flexibility strain matrix, including the steps of Obtaining the rigid body displacement strain matrix of the beam element according to the relationship between the rigid body displacements occurring at the fixed end in each degree of freedom when one end of the beam element is fixed and the linear strain of any point in the element; Obtaining the flexibility strain matrix of the beam element according to the relationship between the loads acting on each degree of freedom at the free end when one end of the beam element is fixed and the linear strain of any point in the element; Calculating the linear stiffness matrix based on the rigid body displacement strain matrix and the flexibility strain matrix.

3. A beam element design method according to claim 2, characterized in that The rigid body displacement strain matrix is obtained according to the second formula, and The second formula is: ， ， Among them, is the strain caused by the rigid body displacement of any point in the unit from the fixed end, is the rigid body displacement of the fixed end in six degrees of freedom, is the rigid body displacement strain matrix; The flexibility strain matrix is obtained according to the third formula, and The third formula is as follows: , Among them, is the strain caused by the load acting on the free end at any point within the unit, is the unit load acting on the 6 degrees of freedom at the free end, consists of , is the strain of the ith degree of freedom at the coordinate x position when a unit load acts on the jth degree of freedom at the free end, is the flexibility strain matrix.

4. The design method of a beam element according to claim 3, characterized in that The calculation of the linear stiffness matrix based on the rigid body displacement strain matrix and the flexibility strain matrix includes the steps of: Calculating the linear stiffness matrix according to the fourth formula, and the fourth formula includes: ， ， , where is the displacement formulation of the beam element node, is the said linear stiffness matrix, [D] is the constitutive matrix, is the intermediate variable, is the total strain matrix.

5. A beam element design method according to claim 1, characterized in that The beam element also calculates the geometric stiffness matrix through the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix, including the steps of: Obtaining the rigid body displacement nonlinear strain matrix of the beam element according to the relationship between the rigid body displacements occurring at the fixed end in each degree of freedom when one end of the beam element is fixed and the derivative of the displacement of any point in the element with respect to the axial coordinate x; Obtaining the flexibility nonlinear strain matrix of the beam element according to the relationship between the loads acting on each degree of freedom at the free end when one end of the beam element is fixed and the derivative of the displacement of any point in the element with respect to the axial coordinate x; Calculating the geometric stiffness matrix based on the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix.

6. A beam element design method according to claim 5, characterized in that The rigid body displacement nonlinear strain matrix is obtained according to the fifth formula, and The fifth formula is: ， Among them, is the rigid body displacement that occurs at the fixed end in six degrees of freedom. is the nonlinear strain matrix for rigid body displacement, Denote the derivative of the displacement of any point within the element with respect to the axial coordinate x; The flexibility nonlinear strain matrix is obtained according to the sixth formula, and The sixth formula is: ， Among them, is a load acting on the free end with six degrees of freedom, Denote the derivative of the displacement at any point within the element with respect to the axial coordinate x, is the flexibility nonlinear strain matrix.

7. A beam element design method according to claim 6, characterized in that, The calculation of the geometric stiffness matrix based on the rigid body displacement nonlinear strain matrix and the flexibility nonlinear strain matrix includes the steps of: Calculating the geometric stiffness matrix according to the ninth formula, and the ninth formula includes: ， ， ， ， ， ， ， ， , wherein, is the geometric stiffness matrix, is the axial force of the beam element, are all defined intermediate variables, is the non - linear term of the axial strain.

8. A beam element, characterized in that, The beam element calculates the shape function matrix through the rigid body displacement matrix and the flexibility displacement matrix, so that the cross-sectional coordinate axes of the beam element satisfy the setting in any direction and the two end nodes satisfy the setting at any position in the cross-sectional coordinate system, which is defined according to the design method described in any one of claims 1 to 7.

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