Quantitative analysis method for plane structure performance based on deformation energy decomposition of orthotropic rectangular elements

Abstract

The present invention belongs to the field of mechanical analysis and material technology, and discloses a method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element, comprising the following steps: constructing the planar deformation of an orthotropic rectangular element in a plane rectangular coordinate system, and obtaining a deformation energy decomposition basis matrix based on mathematical orthogonality and mechanical equilibrium and considering the material properties of the element; establishing a planar structure model, dividing the planar structure by using orthotropic rectangular elements, and obtaining the nodal displacement vector, the basic deformation energy and the basic displacement projection coefficient vector after arbitrary displacement and deformation are generated by the orthotropic rectangular elements under any load condition; obtaining the basic deformation information of the orthotropic rectangular element under any load condition, identifying the main deformation and the secondary deformation of the orthotropic rectangular element under any load condition, and further realizing the decomposition of the deformation energy and the quantitative analysis of the deformation performance of the orthotropic planar structure.

Description

Technical Field

[0001] The present invention belongs to the fields of mechanical analysis and materials, and relates to a method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element. Background Art

[0002] A planar orthotropic material means that there are two mutually perpendicular symmetry planes passing through any point of this material. The directions perpendicular to the symmetry planes are called the elastic principal directions, and different elastic moduli exist in the elastic principal directions in the transverse and vertical directions. With the development of the economy and the progress of materials science, more and more orthotropic materials, such as wood, bamboo, orthotropic alloys, etc., are applied to engineering structures. Deformation information is an important basis for structural analysis and design. Currently, the design methods of structures are aimed at their macroscopic deformations, such as separately designing for bending resistance, shear resistance, torsion resistance, etc., while common structural analysis methods are mostly oriented towards comprehensive responses, such as the maximum lateral displacement, inter-story drift angle, etc. The single basic deformation information is included in the comprehensive deformation, and it is difficult to separate the basic deformation from the comprehensive deformation. Therefore, decomposing the total deformation of an orthotropic rectangular element and quantitatively identifying the basic deformation is of great significance for the performance analysis and optimal design of structures.

[0003] The existing deformation decomposition methods are mainly for isotropic elements and cannot consider material properties such as the elastic modulus and Poisson's ratio of the elements. Therefore, it is necessary to propose a new deformation decomposition method for orthotropic rectangular elements, and by introducing the physical parameters of the material, further conduct in-depth mechanical analysis of orthotropic structures from the perspective of basic deformation energy, and identify the quantitative information of the macroscopic deformation of orthotropic planar structures. Currently, there is no report on the method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element, which can identify the main basic deformations and secondary basic deformations of the orthotropic rectangular element, so as to realize the quantitative analysis of the deformation performance of the orthotropic planar structure.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions:

[0006] A method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element, comprising the following steps:

[0007] Step 1: Construct the plane deformation of an orthotropic rectangular element in the plane rectangular coordinate system. Based on mathematical orthogonality and mechanical equilibrium and considering the material properties of the element, obtain the deformation energy decomposition basis matrix of the orthotropic rectangular element.

[0008] Step 2: Establish a plane structure model. Use orthotropic rectangular elements to divide the plane structure and perform finite element solution to obtain the node coordinate values of the orthotropic rectangular element in the plane rectangular coordinate system and the node coordinate values after arbitrary displacement and deformation of the orthotropic rectangular element under any load condition, and then obtain the node displacement vector of the orthotropic rectangular element after arbitrary displacement and deformation under any load condition.

[0009] Step 3: Project the node displacement vector of the orthotropic rectangular element after arbitrary displacement and deformation under any load condition onto the deformation energy decomposition basis matrix to obtain the basic deformation energy and the basic displacement projection coefficient vector of the orthotropic rectangular element after arbitrary displacement and deformation under any load condition.

[0010] Step 4: Based on the basic deformation energy projection coefficient of the orthotropic rectangular element after arbitrary displacement or deformation under any load condition, obtain the basic deformation information of the orthotropic rectangular element under any load condition, identify the main deformation and secondary deformation generated by the orthotropic rectangular element under any load condition, and then realize the deformation energy decomposition and performance quantification analysis of the orthotropic plane structure.

[0011] Furthermore, the orthotropic rectangular element has four nodes, and the coordinates of the four nodes in the X and Y directions are x1, y1, x2, y2, x3, y3, x4, y4. The length of the orthotropic rectangular element is 2l, the height is 2h, and the thickness is b. The plane deformation of the orthotropic rectangular element is composed of 8 basic deformations and basic displacements, namely axial tension and compression deformation in the X direction, axial tension and compression deformation in the Y direction, bending deformation in the X direction, bending deformation in the Y direction, shear deformation in the XOY plane, rigid body translation in the X direction, rigid body translation in the Y direction, and rigid body rotation in the XOY plane.

[0012] For the tensile and compressive deformation of the orthotropic rectangular element, let its nodal loads be F X1 , F X2 , F X3 , F X4 . According to the force equilibrium condition, we have:

[0013]

[0014] The solution is:

[0015]

[0016] Let F X4 be the unit 1, and the nodal displacement base vector d1 for the axial tension and compression deformation of the orthotropic rectangular element in the X-axis direction can be obtained as follows:

[0017] d1 = (1 0 -1 0 -1 0 1 0) T ;

[0018] Then, the unit energy q1 corresponding to the axial tension and compression deformation of the orthotropic rectangular element in the X-axis direction can be calculated by the following method:

[0019]

[0020] Where: B is the strain matrix of the element; D is the elastic matrix of the element:

[0021]

[0022]

[0023] Where E X and E Y are the elastic moduli of the orthotropic rectangular element in the X-axis and Y-axis directions respectively, and μ XY is the main Poisson's ratio of the orthotropic rectangular element;

[0024] Thus, the unit energy q1 corresponding to the axial tension and compression deformation of the orthotropic rectangular element in the X-axis direction is:

[0025]

[0026] Furthermore, the deformation energy base vector u1 for the axial tension and compression deformation of the orthotropic rectangular element in the X-axis direction is obtained as:

[0027]

[0028] According to the force balance, the nodal displacement base vector d2 for the axial tension and compression deformation of the orthotropic rectangular element in the Y-axis direction can be obtained:

[0029] d2 = (0 1 0 1 0 -1 0 -1) T ;

[0030] Then, the unit energy q2 corresponding to the axial tension and compression deformation of the orthotropic rectangular element in the Y-axis direction is:

[0031]

[0032] Furthermore, the deformation energy base vector u2 for the axial tension and compression deformation of the orthotropic rectangular element in the Y-axis direction is obtained as:

[0033]

[0034] Similarly, the remaining deformation energy basis vectors and displacement basis vectors u3 - u8 of the orthotropic rectangular element can be obtained:

[0035] u3 is the bending deformation energy basis vector of the orthotropic rectangular element in the X - axis direction:

[0036]

[0037] u4 is the bending deformation energy basis vector of the orthotropic rectangular element in the Y - axis direction:

[0038]

[0039] u5 is the shear deformation energy basis vector of the orthotropic rectangular element in the XOY plane:

[0040]

[0041] u6 is the rigid - body translational basis vector of the orthotropic rectangular element in the X - axis direction:

[0042] u6=(1 0 1 0 1 0 1 0) T ;

[0043] u7 is the rigid - body translational basis vector of the orthotropic rectangular element in the Y - axis direction:

[0044] u7=(0 1 0 1 0 1 0 1) T ;

[0045] u8 is the rigid - body rotational basis vector of the orthotropic rectangular element in the XOY plane:

[0046]

[0047] The above basic deformation energy and basic displacement basis vectors satisfy mathematical orthogonality, that is:

[0048]

[0049] From the above basic deformation energy of the orthotropic rectangular element and the basis vectors of the basic displacements u1 - u8, the deformation energy decomposition basis matrix U of the orthotropic rectangular element is constructed as

[0050] U=(u1 u2 u3 u4 u5 u6 u7 u8) T .

[0051] Furthermore, the node coordinate vector of the orthotropic rectangular element in the plane rectangular coordinate system is u p ,

[0052] u p=(x1 y1 x2 y2 x3 y3 x4 y4),

[0053] The nodal coordinate vector of the orthotropic rectangular element after arbitrary displacements and deformations under any load condition is u q ,

[0054] u q =(x’1 y’1 x’2 y’2 x’3 y’3 x’4 y’4),

[0055] From u q -u p The nodal displacement vector u of the orthotropic rectangular element after arbitrary displacements and deformations under any load condition can be obtained r ,

[0056] u r =(x’1 - x1 y’1 - y1 x’2 - x2 y’2 - y2 x’3 - x3 y’3 - y3 x’4 - x4 y’4 - y4).

[0057] Furthermore, step 3 is specifically as follows:

[0058] Project the nodal displacement vector u of the orthotropic rectangular element after arbitrary displacements and deformations under any load condition onto the deformation energy decomposition basis matrix U, that is r w = u

[0059] ·U r ·U -1 ,

[0060] w is the basic deformation energy and the basic displacement projection coefficient vector of the orthotropic rectangular element:

[0061] w = (w1 w2 w3 w4 w5 w6 w7 w8),

[0062] where: w1 is the projection coefficient of the axial tension and compression deformation energy in the X direction of the element, w2 is the projection coefficient of the axial tension and compression deformation energy in the Y direction, w3 is the projection coefficient of the bending deformation energy in the X direction, w4 is the projection coefficient of the bending deformation energy in the Y direction, w5 is the projection coefficient of the shear deformation energy in the XOY plane, w6 is the projection coefficient of the rigid body translation in the X direction, w7 is the projection coefficient of the rigid body translation in the Y direction, and w8 is the projection coefficient of the rigid body rotation in the XOY plane.

[0063] Furthermore, step 4 specifically includes:

[0064] Perform rigid-flexible separation on the decomposition results, that is, ignore the influence of the basic displacement projection coefficients w6 - w8, and only compare the absolute values of the basic deformation energy projection coefficients w1 - w5 in the above component information. The deformation corresponding to the maximum absolute value of the deformation energy is determined as the main deformation of the orthotropic rectangular element, and so on. The second largest is determined as the secondary deformation of the orthotropic rectangular element, and then the proportion p of any deformation under any load condition of the orthotropic plane structure can be calculated. k , realizing the quantitative analysis of the structural deformation performance:

[0065]

[0066] Where: n k is the number of orthotropic rectangular elements that undergo main or secondary deformation, and n t is the total number of orthotropic rectangular elements.

[0067] Furthermore, use the positivity and negativity of the tension-compression deformation energy projection coefficients of the orthotropic rectangular element to further judge the tension-compression state of the element. If w1 is positive, it means that the orthotropic rectangular element is in the X-axis tensile deformation state; if w1 is negative, it means that the orthotropic rectangular element is in the X-axis compressive deformation state.

[0068] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0069] Based on mathematical complete orthogonality, mechanical force balance conditions, and considering the material properties of the element, the present invention proposes a method for quantitative analysis of structural performance based on energy decomposition. The present invention decomposes the comprehensive deformation energy of the orthotropic rectangular element into basic deformation energies to obtain the quantitative deformation information of the orthotropic rectangular element. The present invention realizes the refined analysis of the structural deformation performance by identifying the main and secondary deformations of the orthotropic rectangular element, and then guides the targeted reinforcement and optimization design of the structure; more importantly, the present invention considers the material properties of the element, and can not only perform quantitative analysis of the performance of isotropic structures, but also be applicable to orthotropic structures with more complex deformation performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 is a schematic flow chart of the method for quantitative analysis of the performance of a plane structure based on the decomposition of the deformation energy of an orthotropic rectangular element according to the present invention.

[0071] Figure 2 is a schematic diagram of a four-node orthotropic rectangular element in a plane rectangular coordinate system according to the present invention.

[0072] Figure 3Schematic diagram of tensile and compressive deformation and force conditions in the X-axis direction of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0073] Figure 4 Schematic diagram of tensile and compressive deformation and force conditions in the Y-axis direction of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0074] Figure 5 Schematic diagram of bending deformation and force conditions in the X-axis direction of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0075] Figure 6 Schematic diagram of bending deformation and force conditions in the Y-axis direction of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0076] Figure 7 Schematic diagram of shear deformation and force conditions in the XOY plane of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0077] Figure 8 Schematic diagram of rigid body translation in the X-axis direction and force conditions of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0078] Figure 9 Schematic diagram of rigid body translation in the Y-axis direction and force conditions of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0079] Figure 10 Schematic diagram of rigid body rotation in the XOY plane and force conditions of an orthotropic rectangular element under orthogonal in the plane rectangular coordinate system in the present invention.

[0080] Figure 11 Schematic diagram of counterclockwise rotation of an orthotropic rectangular element in the plane rectangular coordinate system in the present invention.

[0081] Figure 12 Schematic diagram of an orthotropic column in the plane rectangular coordinate system in the present invention. Detailed implementation manners

[0082] The following embodiments are used to illustrate the present invention, but are not used to limit the protection scope of the present invention. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art.

[0083] Figure 1 The flow schematic diagram of the method for quantitatively analyzing the performance of a plane structure based on the decomposition of the strain energy of an orthotropic rectangular element in the present invention is shown. Let any four-node orthotropic rectangular element, and its schematic diagram in the plane rectangular coordinate system is as Figure 2 shown, and its eight basic deformations and basic displacements in the plane rectangular coordinate system are asFigures 3 to 10 As shown in the figure. The coordinates of the four nodes of the orthotropic rectangular element in the X and Y directions are x1, y1, x2, y2, x3, y3, x4, y4. The length of the orthotropic rectangular element is 2l, the height is 2h, and the thickness is b. The plane deformation of the orthotropic rectangular element is composed of the linear superposition of 8 basic deformations and basic displacements, namely tensile and compressive deformation in the X-axis direction, tensile and compressive deformation in the Y-axis direction, bending deformation in the X-axis direction, bending deformation in the Y-axis direction, shear deformation in the XOY plane, rigid body translation in the X-axis direction, rigid body translation in the Y-axis direction, and rigid body rotation in the XOY plane.

[0084] For the tensile and compressive deformation of the orthotropic rectangular element, let the nodal loads be F X1 , F X2 , F X3 , F X4 . According to the force balance condition, we have:

[0085]

[0086] The solution is:

[0087]

[0088] Let F X4 be 1, and the nodal displacement base vector d1 of the orthotropic rectangular element in the X-axis tensile and compressive deformation can be obtained as:

[0089] d1 = (1 0 -1 0 -1 0 1 0) T ;

[0090] Then the unit energy q1 corresponding to the X-axis tensile and compressive deformation of the orthotropic rectangular element can be calculated by the following method:

[0091]

[0092] Where: B is the strain matrix of the element; D is the elastic matrix of the element:

[0093]

[0094]

[0095] Where E X and E Y are the elastic moduli of the orthotropic rectangular element in the X-axis and Y-axis directions respectively, and μ XY is the main Poisson's ratio of the orthotropic rectangular element;

[0096] Thus, the unit energy q1 corresponding to the X-axis tensile and compressive deformation of the orthotropic rectangular element is:

[0097]

[0098] Furthermore, the base vector u1 of the tensile and compressive deformation energy in the X-axis direction of the orthotropic rectangular element is obtained as follows:

[0099]

[0100] According to the force balance, the nodal displacement base vector d2 of the tensile and compressive deformation in the Y-axis direction of the orthotropic rectangular element can be obtained:

[0101] d2 = (0 1 0 1 0 -1 0 -1) T ;

[0102] Then, the unit energy q2 corresponding to the tensile and compressive deformation in the Y-axis direction of the orthotropic rectangular element is:

[0103]

[0104] Furthermore, the base vector u2 of the tensile and compressive deformation energy in the Y-axis direction of the orthotropic rectangular element is obtained as follows:

[0105]

[0106] Similarly, the remaining deformation energy base vectors and displacement base vectors u3 - u8 of the orthotropic rectangular element can be obtained:

[0107] u3 is the base vector of the bending deformation energy in the X-axis direction of the orthotropic rectangular element:

[0108]

[0109] u4 is the base vector of the bending deformation energy in the Y-axis direction of the orthotropic rectangular element:

[0110]

[0111] u5 is the base vector of the shear deformation energy in the XOY plane of the orthotropic rectangular element:

[0112]

[0113] u6 is the base vector of the rigid body translation in the X-axis direction of the orthotropic rectangular element:

[0114] u6 = (1 0 1 0 1 0 1 0) T ;

[0115] u7 is the base vector of the rigid body translation in the Y-axis direction of the orthotropic rectangular element:

[0116] u7 = (0 1 0 1 0 1 0 1) T ;

[0117] u8 is the rigid body rotation base vector in the XOY plane of the orthotropic rectangular element:

[0118]

[0119] where E X and E Y are the elastic moduli of the orthotropic rectangular element in the X-axis and Y-axis directions respectively, and μ XY is the major Poisson's ratio of the orthotropic rectangular element.

[0120] The above basic strain energy and basic displacement base vectors satisfy mathematical orthogonality, that is:

[0121]

[0122] Based on the above basic strain energy of the orthotropic rectangular element and the base vectors of the basic displacements u1 to u8, the strain energy decomposition base matrix U of the orthotropic rectangular element is constructed,

[0123] U = (u1 u2 u3 u4 u5 u6 u7 u8) T .

[0124] The node coordinate vector of the orthotropic rectangular element in the plane rectangular coordinate system is u p ,

[0125] u p = (x1 y1 x2 y2 x3 y3 x4 y4),

[0126] Then, after the orthotropic rectangular element generates arbitrary displacements and deformations under any load condition, the node coordinate vector is u q ,

[0127] u q = (x’1 y’1 x’2 y’2 x’3 y’3 x’4 y’4),

[0128] From u q - u p the node displacement vector u r after the orthotropic rectangular element generates arbitrary displacements and deformations under any load condition can be obtained,

[0129] u r = (x’1 - x1 y’1 - y1 x’2 - x2 y’2 - y2 x’3 - x3 y’3 - y3 x’4 - x4 y’4 - y4).

[0130] The node displacement vector u rProjected onto the decomposed basis matrix U of the strain energy, i.e.,

[0131] w = u r ·U -1 ,

[0132] where w is the vector of the basic strain energy and the basic displacement projection coefficients of the orthotropic rectangular element:

[0133] w = (w1 w2 w3 w4 w5 w6 w7 w8),

[0134] where: w1 is the projection coefficient of the axial tension and compression strain energy in the X direction of the element, w2 is the projection coefficient of the axial tension and compression strain energy in the Y direction, w3 is the projection coefficient of the bending strain energy in the X direction, w4 is the projection coefficient of the bending strain energy in the Y direction, w5 is the projection coefficient of the shear strain energy in the XOY plane, w6 is the projection coefficient of the rigid body translation in the X direction, w7 is the projection coefficient of the rigid body translation in the Y direction, and w8 is the projection coefficient of the rigid body rotation in the XOY plane.

[0135] Perform rigid-flexible separation on the decomposition result, that is, ignore the influence of the basic displacement projection coefficients w6 - w8, and only compare the absolute values of the basic strain energy projection coefficients w1 - w5 in the above component information. The deformation corresponding to the maximum absolute value of the strain energy is determined as the main deformation of the orthotropic rectangular element, and so on. The second largest is determined as the secondary deformation of the orthotropic rectangular element. Furthermore, the proportion p of any deformation under any load condition of the orthotropic plane structure can be calculated k , realizing the quantitative analysis of the structural deformation performance:

[0136]

[0137] where n k is the number of orthotropic rectangular elements with main or secondary deformation, and n t is the total number of orthotropic rectangular elements.

[0138] Furthermore, use the positive and negative nature of the projection coefficients of the axial tension and compression strain energy of the orthotropic rectangular element to further judge the tension and compression state of the element. If w1 is positive, it means that the orthotropic rectangular element is in the state of axial tension deformation in the X direction; if w1 is negative, it means that the orthotropic rectangular element is in the state of axial compression deformation in the X direction.

[0139] Rigid body rotation displacement error analysis

[0140] Since the rotational displacement is a non - linear displacement, errors will occur during linear decomposition. That is, the element rotational displacement vector not only has projection coefficients in the rigid - body rotation basis vectors, but may also have projection coefficients in other basic deformation energies and rigid - body displacement basis vectors. Therefore, it is necessary to analyze and calculate the errors caused by the rigid - body rotational displacement to determine whether they affect the calculation accuracy.

[0141] As Figure 11 shown, let the length of the four - node orthotropic rectangular element m be 2l and the height be 2h. After rotating the element j counter - clockwise by an angle θ around the centroid, the coordinate displacement vectors of the four nodes of the element are:

[0142] u rm =(φlsin(σ - θ)-lφlcos(σ - θ)-l l - φlsin(σ + θ)

[0143] φlcos(σ + θ)-h l - φlsin(σ - θ)h - φlcos(σ - θ)

[0144] φlsin(σ + θ)-l h - φlcos(σ + θ)),

[0145] where:

[0146] When the nodal displacement vector of the element m during rigid - body rotational displacement is linearly represented by the basic deformation energy and the basic displacement basis vectors, the rotational displacement vector of the element only has projections on the rigid - body rotation basis vectors, the X - axis axial tension - compression deformation energy basis vectors, and the Y - axis axial tension - compression deformation energy basis vectors, and the projection coefficients on other basic deformation energies and basic displacement basis vectors are 0. Therefore, the solution results of the 8 constraint equations obtained by projecting the coordinate displacement vector of the element rigid - body rotation are as follows:

[0147]

[0148] Perform Taylor expansion on w1, w2, w8 at θ = 0:

[0149]

[0150] It can be seen that when the element undergoes rigid - body rotation, there are still projection coefficients of the rigid - body rotational displacement on the X - axis and Y - axis axial tension - compression deformation energy basis vectors, resulting in certain errors. However, when θ approaches 0, w1 and w2 are higher - order infinitesimals of w8. Therefore, when small deformations occur, the errors caused by the rigid - body rotational displacement can be ignored, that is, the constructed deformation - energy decomposition basis matrix has sufficient accuracy.

[0151] Implementation case

[0152] As Figure 12As shown in the figure, taking an orthotropic cantilever column as an example, the column height is 2m, the column cross-section size is 0.4×0.4m, and the elastic modulus E X = 30GPa, E Y = 6GPa; the main Poisson's ratio μ XY = 0.2; the density is 2500kg / m 3 .

[0153] Apply an inverted triangular load to the cantilever column, Figure 12 The nodal displacement vectors of the 1-3 orthotropic rectangular elements in it are:

[0154] u r1 = (5.412E+02, 3.455E+01, 5.429E+02, 6.905E+01, 5.146E+02, 6.879E+01, 5.139E+02, 3.440E+01)×mm;

[0155] u r2 = (4.962E+01, -3.027E+01, 4.493E+01, -1.403E+01, 3.112E+01, -1.141E+01, 3.618E+01, -2.507E+01)×mm;

[0156] u r3 = (4.577E+02, -3.285E+01, 4.579E+02, 5.876E-01, 4.301E+02, 5.432E-01, 4.300E+02, -3.271E+01)×mm.

[0157] The decomposition results of the strain energy of the 1-3 orthotropic rectangular elements are shown in Table 1.

[0158] Table 1 Decomposition results of the strain energy of the 1-3 orthotropic rectangular elements

[0159]

[0160]

[0161] Furthermore, the proportions p k of the basic deformations of the cantilever column under the action of the inverted triangular load are obtained as shown in Table 2.

[0162] Table 2 Proportions of the basic deformations of the cantilever column

[0163]

[0164] As can be seen from Table 1, the projection coefficient of the tensile and compressive deformation energy of Unit 1 in the X-axis direction is -4.731E+00 J. The coefficient is negative and has the largest absolute value, indicating that the main deformation in the area where Unit 1 is located is compression in the X-axis direction; the value of the shear deformation energy is the second largest, indicating that the secondary deformation in the area where Unit 1 is located is shear deformation.

[0165] Similarly, it can be known that the main deformation in the area where Unit 2 is located is compression in the Y-axis direction, and the secondary deformation is shear deformation; the main deformation in the area where Unit 3 is located is shear deformation, and the secondary deformation is compression in the Y-axis direction.

[0166] As can be seen from Table 2, under the action of the inverted triangular load, the main deformation of the cantilever column as a whole is shear deformation, with a proportion of 65%; while the tensile and compressive deformation in the Y-axis direction is the secondary deformation, with a proportion of 31%.

[0167] The above-described embodiments are only the preferred embodiments of the present invention, and are only used to explain the present invention, not to limit the scope of implementation of the present invention. For those skilled in the art of this technology, of course, according to the technical content disclosed in this specification, other implementation manners can be easily made by means of substitution or change. Therefore, all changes and improvements made on the principle of the present invention should be included within the scope of the patent application of the present invention.

Claims

1. A method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element, characterized in that Including the following steps: Step 1: Construct the plane deformation of an orthotropic rectangular element in a plane rectangular coordinate system. Based on mathematical orthogonality and mechanical equilibrium and considering the material properties of the element, obtain the deformation energy decomposition basis matrix of the orthotropic rectangular element; the orthotropic rectangular element has four nodes, and the coordinates of the four nodes in the X and Y directions are x1, y1, x2, y2, x3, y3, x4, y4. The length of the orthotropic rectangular element is 2l, the height is 2h, and the thickness is b; the plane deformation of the orthotropic rectangular element is composed of 8 basic deformations and basic displacements, namely tensile and compressive deformation in the X-axis direction, tensile and compressive deformation in the Y-axis direction, bending deformation in the X-axis direction, bending deformation in the Y-axis direction, shear deformation in the XOY plane, rigid body translation in the X-axis direction, rigid body translation in the Y-axis direction, and rigid body rotation in the XOY plane; The basis vectors of the basic deformation energy and basic displacement of the orthotropic rectangular element are u1 to u8, specifically as follows: u1 is the basis vector of the tensile and compressive deformation energy of the orthotropic rectangular element in the X-axis direction: u2 is the basis vector of the tensile and compressive deformation energy of the orthotropic rectangular element in the Y-axis direction: u3 is the basis vector of the bending deformation energy of the orthotropic rectangular element in the X-axis direction: u4 is the basis vector of the bending deformation energy of the orthotropic rectangular element in the Y-axis direction: u5 is the basis vector of the shear deformation energy of the orthotropic rectangular element in the XOY plane: u6 is the basis vector of the rigid body translation of the orthotropic rectangular element in the X-axis direction: u6＝(1 0 1 0 1 0 1 0) T ； u7 is the basis vector of the rigid body translation of the orthotropic rectangular element in the Y-axis direction: u7＝(0 1 0 1 0 1 0 1) T ； u8 is the basis vector of the rigid body rotation of the orthotropic rectangular element in the XOY plane: Among them, E X and E Y are the elastic moduli of the orthotropic rectangular element in the X-axis and Y-axis directions respectively, and μ XY is the major Poisson's ratio of the orthotropic rectangular element; The above basic deformation energy and basic displacement basis vectors satisfy mathematical orthogonality, that is: Construct the deformation energy decomposition basis matrix U of the orthotropic rectangular element from the basis vectors u1 to u8 of the basic deformation energy and basic displacement of the orthotropic rectangular element, U = (u1 u2 u3 u4 u5 u6 u7 u8) T ; Step 2: Establish a plane structure model, divide the plane structure using orthotropic rectangular elements and perform finite element solution to obtain the node coordinate values of the orthotropic rectangular elements in the plane rectangular coordinate system and the node coordinate values after the orthotropic rectangular elements generate arbitrary displacements and deformations under any load condition, and then obtain the node displacement vector of the orthotropic rectangular elements after generating arbitrary displacements and deformations under any load condition; Step 3: Project the node displacement vector of the orthotropic rectangular element after generating arbitrary displacements and deformations under any load condition onto the deformation energy decomposition basis matrix to obtain the basic deformation energy and basic displacement projection coefficient vector of the orthotropic rectangular element after generating arbitrary displacements and deformations under any load condition; Step 4: Based on the basic deformation energy projection coefficient of the orthotropic rectangular element after generating arbitrary displacements or deformations under any load condition, obtain the basic deformation information of the orthotropic rectangular element under any load condition, distinguish the main deformation and secondary deformation generated by the orthotropic rectangular element under any load condition, and then realize the deformation energy decomposition and performance quantitative analysis of the orthotropic plane structure.

2. The method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element according to claim 1, characterized in that, The nodal coordinate vector of the orthotropic rectangular element in the plane rectangular coordinate system is u p , u p = (x1 y1 x2 y2 x3 y3 x4 y4), The nodal coordinate vector of the orthotropic rectangular element after generating arbitrary displacements and deformations under any load condition is u q , u q = (x1'y1'x'2y'2x'3y'3x'4y'4), From u q -u p The nodal displacement vector u after any displacement and deformation of the orthotropic rectangular element under any load condition can be obtained r , u r =(x1'-x1 y1'-y1 x'2-x2 y'2-y2 x'3-x3 y'3-y3 x'4-x4 y'4-y4).

3. The method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element according to claim 2, wherein Specifically, Step 3 is: The nodal displacement vector \(u\) of an orthotropic rectangular element after generating arbitrary displacements and deformations under any load condition r is projected onto the deformation energy decomposition basis matrix \(U\), that is w = u r ·U -1 , $w$ is the basic deformation energy and the vector of basic displacement projection coefficients of an orthotropic rectangular element: $w=(w_1\ w_2\ w_3\ w_4\ w_5\ w_6\ w_7\ w_8)$ where: $w_1$ is the projection coefficient of the tensile and compressive deformation energy in the X-axis direction of the element, $w_2$ is the projection coefficient of the tensile and compressive deformation energy in the Y-axis direction, $w_3$ is the projection coefficient of the bending deformation energy in the X-axis direction, $w_4$ is the projection coefficient of the bending deformation energy in the Y-axis direction, $w_5$ is the projection coefficient of the shear deformation energy in the XOY plane, $w_6$ is the projection coefficient of the rigid body translation in the X-axis direction, $w_7$ is the projection coefficient of the rigid body translation in the Y-axis direction, and $w_8$ is the projection coefficient of the rigid body rotation in the XOY plane.

4. The method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element according to claim 1 or 3, characterized in that, The specific steps of step 4 include: Perform rigid-flexible separation on the decomposition results, that is, ignore the influence of the basic displacement projection coefficients w6 - w8, and only compare the absolute values of the basic deformation energy projection coefficients w1 - w5 in the component information. The deformation corresponding to the maximum absolute value of the deformation energy is determined as the main deformation of the orthotropic rectangular element, and so on. The second largest is determined as the secondary deformation of the orthotropic rectangular element. Furthermore, the proportion p of any deformation under any load condition of the orthotropic plane structure can be calculated. k , realizing the quantitative analysis of the structural deformation performance: where: n k is the number of orthotropic rectangular elements that undergo major or minor deformations, and n t is the total number of orthotropic rectangular elements.

5. The method for quantitatively analyzing the performance of a planar structure based on the decomposition of the deformation energy of an orthotropic rectangular element according to claim 3, characterized in that Further judge the tensile and compressive states of the element by using the positive and negative properties of the projection coefficients of the tensile and compressive deformation energies of the orthotropic rectangular element. If $w_1$ is positive, it means that the orthotropic rectangular element is in the state of tensile deformation in the X-axis direction; if $w_1$ is negative, it means that the orthotropic rectangular element is in the state of compressive deformation in the X-axis direction.

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