Structural deformation decomposition method based on three-dimensional 6-node rectangular element

Abstract

The application belongs to the field of mechanical analysis, and discloses a structure deformation decomposition method based on a three-dimensional 6-node rectangular element, which comprises the following steps: constructing basic displacement and basic deformation base vectors of the 6-node rectangular element to obtain an orthogonal mechanical base matrix of the three-dimensional 6-node rectangular element; establishing a space structure model to obtain displacement vectors of each node of the 6-node rectangular element; projecting the displacement vectors of the 6-node rectangular element on the orthogonal mechanical base matrix of the three-dimensional 6-node rectangular element to obtain a projection coefficient vector of the basic deformation and the basic displacement of each rectangular element; and obtaining the main basic deformation and the secondary basic deformation of the rectangular element according to the size of the projection coefficients in the projection coefficient vector, so that the deformation decomposition and deformation identification of the structure model can be realized. The application can accurately identify the in-plane deformation of a single-phase stressed thin plate component with a small thickness compared with the width and length, and can also accurately identify the out-of-plane bending deformation and other out-of-plane deformations.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of mechanical analysis, and particularly relates to a structural deformation decomposition method based on a three-dimensional 6-node rectangular element. BACKGROUND

[0002] The deformation bases decomposed by the three-dimensional 4-node rectangular element are mainly the in-plane basic deformations, and the out-of-plane deformation is only the warping deformation, which is not accurate for the decomposition of the out-of-plane basic deformation. Considering the shortcomings of the 4-node rectangular element and the single-phase stress characteristics of the special-shaped column, the coupling beam and other components with a large span-depth ratio, the three-dimensional 6-node rectangular element is constructed.

[0003] The currently published deformation decomposition methods are all for the decomposition of a component, and the computational efficiency is particularly important when the deformation decomposition is performed on a large structural model. The three-dimensional 6-node rectangular element can accurately identify the in-plane and out-of-plane deformation conditions of the component, and the 6-node rectangular element has fewer degrees of freedom and faster calculation speed, so that the calculation of the complex structure can be quickly completed, and the in-plane and out-of-plane deformation conditions of the components at various positions can be identified. SUMMARY

[0004] The application aims to provide a structural deformation decomposition method based on a three-dimensional 6-node rectangular element, which can not only accurately identify the in-plane deformation condition of the single-phase stress thin plate component with a small thickness and a large width and length, but also accurately identify the out-of-plane bending deformation and other out-of-plane deformations.

[0005] To achieve the above object, the application adopts the following technical scheme:

[0006] The application provides a structural deformation decomposition method based on a three-dimensional 6-node rectangular element, which comprises the following steps:

[0007] Step 1: in a three-dimensional rectangular coordinate system, the basic displacement and basic deformation base vectors of the 6-node rectangular element are constructed according to the force balance condition and the orthogonal decomposition theory, so as to obtain the orthogonal mechanical base matrix P of the three-dimensional 6-node rectangular element;

[0008] Step 2: a spatial structure model is established in a spatial rectangular coordinate system, the spatial structure model is divided by using a quadrilateral shell element, the three-dimensional 6-node rectangular element is formed by selecting the four corner points and the midpoints of two opposite sides of the quadrilateral shell element, and the displacement vector s of each node of the 6-node rectangular element is obtained;

[0009] Step 3: the displacement vector s of each node of the 6-node rectangular element is projected onto the orthogonal mechanical base matrix P of the three-dimensional 6-node rectangular element, so as to obtain the projection coefficient vector r of the basic deformation and the basic displacement of each rectangular element;

[0010] Step 4: According to the size of the projection coefficient in the projection coefficient vector r, the main and secondary basic deformations of each quadrilateral element are obtained, and the deformation decomposition and deformation identification of the spatial structure model are realized.

[0011] In one technical solution, the basic displacement and basic deformation of the 6-node quadrilateral element include: X-axis rigid translation displacement, Y-axis rigid translation displacement, Z-axis rigid translation displacement, X-axis rigid rotation displacement, Y-axis rigid rotation displacement, Z-axis rigid rotation displacement, X-axis tensile and compressive deformation of XOY plane, Y-axis tensile and compressive deformation of XOY plane, X-axis in-plane bending deformation of XOY plane, Y-axis in-plane bending deformation of XOY plane, shear deformation of XOY plane, warping deformation of XOY plane, out-of-plane bending deformation of XOY plane, expansion and contraction deformation of XOY plane, punching deformation of XOY plane, reverse warping deformation of XOY plane, reverse asymmetric bending deformation of XOY plane, and reverse asymmetric tensile and compressive deformation of XOY plane.

[0012] In one technical solution, the step 1 specifically includes the following steps:

[0013] Step 1.1: In the spatial rectangular coordinate system, 18 kinds of basic displacement and basic deformation basis vectors p1-p18 of the element are constructed according to the force balance condition, moment balance condition and orthogonal theory for the 6-node quadrilateral element. 18 , specifically as follows:

[0014] p1 represents the X-axis rigid translation displacement basis vector:

[0015]

[0016] p2 represents the Y-axis rigid translation displacement basis vector:

[0017]

[0018] p3 represents the Z-axis rigid translation displacement basis vector:

[0019]

[0020] p4 represents the X-axis rigid rotation displacement basis vector:

[0021]

[0022] p5 represents the Y-axis rigid rotation displacement basis vector:

[0023] p5 = (0, 0, -0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0) T ,

[0024] p6 represents a rotation displacement basis vector around the Z axis of the rigid body:

[0025]

[0026] p7 represents a tension-compression deformation basis vector in the XOY plane in the X axis direction:

[0027] p7 = (-0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0, 0, 0) T ,

[0028] p8 represents a tension-compression deformation basis vector in the XOY plane in the Y axis direction:

[0029]

[0030] p9 represents an in-plane bending deformation basis vector in the XOY plane in the X axis direction:

[0031] p9 = (-0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0, 0, 0, 0, 0, 0) T ,

[0032] p 10 represents an in-plane bending deformation basis vector in the XOY plane in the Y axis direction:

[0033] p 10 = (0, -0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0, 0, 0, 0, 0) T ,

[0034] p 11 represents a shear deformation basis vector in the XOY plane:

[0035]

[0036] p 12 represents a warping deformation basis vector in the XOY plane:

[0037] p 12 = (, 0, 0, -0.5 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0, 0,, 00) T ,

[0038] p 13 represents an out-of-plane bending deformation basis vector in the XOY plane:

[0039]

[0040] p 14 represents a dilation deformation basis vector in the XOY plane:

[0041]

[0042] p 15 presents the XOY plane shear deformation base vector:

[0043]

[0044] p 16 presents the XOY plane reverse warping deformation base vector:

[0045]

[0046] p 17 presents the XOY plane anti-symmetric bending deformation base vector:

[0047]

[0048] p 18 presents the XOY plane anti-symmetric stretching-compression deformation base vector:

[0049]

[0050] wherein m is the length of the rectangular element in the Y-axis direction, and n is the length of the rectangular element in the X-axis direction;

[0051]

[0052] Step 1.2: combining the 6 basic displacement base vectors and 12 basic deformation base vectors shown in step 1.1 to construct a complete orthogonal mechanical base matrix P of the 6-node rectangular element:

[0053] P = [p1 p2 p3 p4 p5 p6 p7 p8 p9 p 10 p 11 p 12 p 13 p 14 p 15 p 16 p 17 p 18 ].

[0054] In one technical solution, step 2 specifically comprises the following steps:

[0055] Step 2.1: using a finite element software to establish a space structure model in a three-dimensional rectangular coordinate system, and dividing the space structure model by using a quadrilateral shell element, then selecting 4 corner points and two midpoints in the X-axis direction of any quadrilateral shell element to form a 6-node rectangular element, then the node coordinates d i of the i th 6-node rectangular element are:

[0056] d i= (x i1 ,y i1 ,z i1 ,x i2 ,y i2 ,z i2 ,x i3 ,y i3 ,z i3 ,x i4 ,y i4 ,z i4 ,x i5 ,y i5 ,z i5 ,x i6 ,y i6 ,z i6 ) where:

[0057] d i represents the node coordinates of the i-th rectangular element in the spatial model, i = 1, 2, 3,..., k, k is the number of rectangular elements;

[0058] x ij represents the coordinate value on the X-axis of the j-th node of the i-th rectangular element;

[0059] y ij represents the coordinate value on the Y-axis of the j-th node of the i-th rectangular element;

[0060] z ij represents the coordinate value on the Z-axis of the j-th node of the i-th rectangular element; j = 1, 2, 3, 4, 5, 6;

[0061] Step 2.2: The spatial model is displaced and deformed under the action of any load, generating the deformed node coordinates d i ' of the i-th 6-node rectangular element:

[0062] d' i = (x' i1 ,y' i1 ,z' i1 ,x' i2 ,y' i2 ,z' i2 ,x' i3 ,y' i3 ,z' i3 ,x' i4 ,y' i4 ,z' i4 ,x' i5 ,y' i5 ,z' i5 ,x' i6 ,y' i6 ,z' i6 ),

[0063] wherein:

[0064] d' i represents the node coordinates of the i-th rectangular element after the spatial model is deformed, i = 1, 2, 3, …, k, k is the number of rectangular elements; x' ij represents the coordinate value on the X-axis of the j-th node of the i-th rectangular element after deformation;

[0065] y' ij represents the coordinate value on the Y-axis of the j-th node of the i-th rectangular element after deformation;

[0066] z' ij represents the coordinate value on the Z-axis of the j-th node of the i-th rectangular element after deformation; j = 1, 2, 3, 4, 5, 6;

[0067] Step 2.3: Subtract the node coordinates d i of the rectangular element before deformation from the node coordinates d i of the rectangular element after deformation to obtain the node displacement vector s i of the i-th rectangular element:

[0068] s i = (x' i1 -x i1 , y' i1 -y i1 , z' i1 -z i1 , x' i2 -x i2 , y' i2 -y i2 , z' i2 -z i2 , x' i3 -x i3 , y' i3 -y i3 , z' i3 -z i3 , x' i4 -x i4 , y' i4 -y i4 , z' i4 -z i4 , x' i5 -x i5 , y' i5 -y i5 , z' i5 -z i5 , x' i6 -x i6 , y' i6 -y i6 , z' i6 -zi6 ).

[0069] In one technical solution, step 3 specifically includes:

[0070] Step 3.1: Transfer the nodal displacement vector s of the i-th 6-node rectangular element. i Projected onto the corresponding complete orthogonal mechanical basis matrix P i Above, we get:

[0071] s i =r i ·p i ;

[0072] Step 3.2: Transform the above formula to obtain the projection coefficient vector r of the basic displacement and basic deformation of the i-th 6-node rectangular element in the spatial model. i :

[0073]

[0074] in:

[0075] s i P is the nodal displacement vector of the i-th 6-node rectangular element; i Let be the complete orthogonal mechanical basis matrix of the i-th 6-node rectangular element. For P i The transpose of the matrix, (p i ) -1 For p i The inverse matrix of r; i The projection coefficient vector of the basic deformation and basic displacement of the i-th 6-node rectangular element;

[0076] p i =[p i1 ,p i2 ,p i3 ,p i4 ,p i5 ,p i6 ,p i7 ,p i8 ,p i9 ,p i10 ,p i11 ,p i12 ,p i13 ,p i14 ,p i15 ,p i16 ,p i17 ,p i18 ], r i =(r i1 ,r i2 ,r i3 ,r i4 ,ri5 ,r i6 ,r i7 ,r i8 ,r i9 ,r i10 ,r i11 ,r i12 ,r i13 ,r i14 ,r i15 ,r i16 ,r i17 ,r i18 );

[0077] wherein, p il is the lth basic displacement and deformation base vector of the ith quadrilateral element;

[0078] r ij is the projection coefficient corresponding to the lth basic displacement and deformation base vector of the ith quadrilateral element, l = 1, 2,..., 18.

[0079] In one technical solution, the step 4 specifically comprises:

[0080] removing the projection coefficients r i ~ r i1 corresponding to the 6 basic displacements in each quadrilateral element projection coefficient vector r i6 , and comparing the absolute values of the projection coefficients r i7 ~ r i18 corresponding to the remaining 12 basic displacements of each quadrilateral element in the structure one by one, wherein the largest absolute value is the main basic deformation of the quadrilateral element, and the smaller absolute value is the secondary basic deformation of the quadrilateral element, so as to obtain the main basic deformation and various secondary basic deformations of each quadrilateral element, thereby realizing the deformation decomposition and deformation identification of the structural model.

[0081] Compared with the prior art, the present application has the beneficial effects that:

[0082] The space structure deformation decomposition method based on the three-dimensional 6-node quadrilateral element obtained based on the orthogonal decomposition theory, the force balance condition, the moment balance condition, etc., can not only accurately identify the in-plane deformation of the single-phase stress thin plate component with a small thickness compared with the width and length, but also accurately identify the out-of-plane bending deformation and other out-of-plane deformations; in addition, the 6-node quadrilateral element has fewer degrees of freedom and fast calculation speed, and can quickly and accurately identify the deformation of each part of the complex building structure. BRIEF DESCRIPTION OF DRAWINGS

[0083] Figure 1 is the flowchart of the structural deformation decomposition method based on the three-dimensional 6-node quadrilateral element of the present application.

[0084] Figure 2 A schematic diagram of a horizontal 6-node rectangular element in a spatial rectangular coordinate system.

[0085] Figure 3 A schematic diagram of a vertical 6-node rectangular element in a spatial rectangular coordinate system.

[0086] Figure 4 A schematic diagram of an arbitrary deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0087] Figure 5 A schematic diagram of an X-axis rigid body translation displacement of a 6-node rectangular element in a spatial rectangular coordinate system.

[0088] Figure 6 A schematic diagram of a Y-axis rigid body translation displacement of a 6-node rectangular element in a spatial rectangular coordinate system.

[0089] Figure 7 A schematic diagram of a Z-axis rigid body translation displacement of a 6-node rectangular element in a spatial rectangular coordinate system.

[0090] Figure 8 A schematic diagram of an X-axis rigid body rotation displacement of a 6-node rectangular element in a spatial rectangular coordinate system.

[0091] Figure 9 A schematic diagram of a Y-axis rigid body rotation displacement of a 6-node rectangular element in a spatial rectangular coordinate system.

[0092] Figure 10 A schematic diagram of a Z-axis rigid body rotation displacement of a 6-node rectangular element in a spatial rectangular coordinate system.

[0093] Figure 11 A schematic diagram of an X-axis tension-compression deformation of a 6-node rectangular element in an XOY plane in a spatial rectangular coordinate system.

[0094] Figure 12 A schematic diagram of a Y-axis tension-compression deformation of a 6-node rectangular element in an XOY plane in a spatial rectangular coordinate system.

[0095] Figure 13 A schematic diagram of an X-axis in-plane bending deformation of a 6-node rectangular element in an XOY plane in a spatial rectangular coordinate system.

[0096] Figure 14 A schematic diagram of a Y-axis in-plane bending deformation of a 6-node rectangular element in an XOY plane in a spatial rectangular coordinate system.

[0097] Figure 15 A schematic diagram of a shear deformation of a 6-node rectangular element in an XOY plane in a spatial rectangular coordinate system.

[0098] Figure 16A schematic diagram of the XOY plane warping deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0099] Figure 17 A schematic diagram of the XOY plane out-of-plane bending deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0100] Figure 18 A schematic diagram of the XOY plane expansion and contraction deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0101] Figure 19 A schematic diagram of the XOY plane punching shear deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0102] Figure 20 A schematic diagram of the XOY plane reverse warping deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0103] Figure 21 A schematic diagram of the XOY plane reverse asymmetric bending deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0104] Figure 22 A schematic diagram of the XOY plane reverse asymmetric tensile-compressive deformation of a 6-node rectangular element in a spatial rectangular coordinate system.

[0105] Figure 23 A schematic diagram of a finite element model of a shear wall structure.

[0106] Figure 24 A schematic diagram of the rightmost two-span shear walls of a shear wall structure and the selection of elements. DETAILED DESCRIPTION

[0107] The following examples are intended to illustrate the present application and are not intended to limit the scope of the present application. Unless otherwise indicated, the techniques employed in the examples are standard techniques commonly used in the art. The test methods used in the following examples are standard methods unless otherwise indicated.

[0108] Figure 1 A flow chart of the structural deformation decomposition method based on a three-dimensional 6-node rectangular element.

[0109] Figure 2 A schematic diagram of a three-dimensional 6-node rectangular element, in which the 6 nodes are the 4 corners of a rectangle and the midpoints of two opposite sides, the 4 corners are sequentially numbered as node 1, node 2, node 3 and node 4 from the lower left corner in a counterclockwise direction, and the midpoints of the two opposite sides parallel to the X axis are sequentially numbered as node 5 and node 6. In addition, the 6-node rectangular element has another form: the 5th and 6th nodes are located at the midpoints of the opposite sides parallel to the Y axis (as shown in FIG. 1B). Figure 3

[0110] ​In spatial structures, some components are vertically loaded (such as irregularly shaped columns). These components have a relatively long vertical dimension and a relatively short horizontal dimension, making them more suitable for decomposition using 3D vertical 6-node rectangular elements. Other components are horizontally loaded (such as connecting beams), making them more suitable for decomposition using horizontal 6-node elements. The difference between vertical and horizontal 6-node rectangular elements lies in the positions of nodes 5 and 6. Specifically, in a horizontal 6-node rectangular element, nodes 5 and 6 are located at the midpoints of the top and bottom sides of the rectangle. Figure 2 ); Nodes 5 and 6 of the vertical 6-node rectangular element are located at the midpoints of the left and right sides of the rectangle. Figure 3 Since the analysis principles and calculation processes of vertical 6-node rectangular elements and horizontal 6-node rectangular elements are basically the same, this invention only describes them in detail. Figure 2 The horizontal 6-node unit is shown.

[0111] Figure 4 This represents any deformation that occurs when a three-dimensional 6-node rectangular element is subjected to force. Figures 5-22 The diagram illustrates the six basic displacements and twelve basic deformations of a three-dimensional 6-node rectangular element, namely: rigid body translational displacement along the X-axis, rigid body translational displacement along the Y-axis, rigid body translational displacement along the Z-axis, rigid body rotational displacement about the X-axis, rigid body rotational displacement about the Y-axis, rigid body rotational displacement about the Z-axis, tensile and compressive deformation along the X-axis of the XOY plane, tensile and compressive deformation along the Y-axis of the XOY plane, in-plane bending deformation along the X-axis of the XOY plane, in-plane bending deformation along the Y-axis of the XOY plane, shear deformation, warping deformation, out-of-plane bending deformation, expansion and contraction deformation, punching deformation, reverse warping deformation, anti-symmetric bending deformation, and anti-symmetric tensile-compression deformation of the XOY plane.

[0112] The basis vectors for the basic displacements and basic deformations are p1 to p1, respectively. 18 :

[0113] p1 represents the basis vector of rigid body translational displacement along the X-axis:

[0114]

[0115] p2 represents the basis vector of rigid body translational displacement along the Y-axis:

[0116]

[0117] p3 represents the basis vector of rigid body translational displacement along the Z-axis:

[0118]

[0119] p4 represents the basis vector of rigid body rotational displacement about the X-axis:

[0120]

[0121] p5 represents a displacement basis vector for rigid body rotation about the Y axis:

[0122] p5 = (0, 0, -0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0) T ,

[0123] p6 represents a displacement basis vector for rigid body rotation about the Z axis:

[0124]

[0125] p7 represents a displacement basis vector for XOY plane X axis tensile / compressive deformation:

[0126] p7 = (-0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0, 0, 0) T ,

[0127] p8 represents a displacement basis vector for XOY plane Y axis tensile / compressive deformation:

[0128]

[0129] p9 represents a displacement basis vector for XOY plane X axis in-plane bending deformation:

[0130] p9 = (-0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0, 0, 0, 0, 0, 0) T ,

[0131] p 10 10 represents a displacement basis vector for XOY plane Y axis in-plane bending deformation:

[0132] p 10 11 = (0, -0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0, 0, 0, 0, 0) T ,

[0133] p 11 12 represents a displacement basis vector for XOY plane shear deformation:

[0134]

[0135] p 12 13 represents a displacement basis vector for XOY plane warping deformation:

[0136] p 12 14 = (, 0, 0, -0.5 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0, 0,, 00) T ,

[0137] p 13 The out-of-plane bending deformation base vector of XOY plane is represented as:

[0138]

[0139] p 14 The dilatation deformation base vector of XOY plane is represented as:

[0140]

[0141] p 15 The shear deformation base vector of XOY plane is represented as:

[0142]

[0143] p 16 The reverse warping deformation base vector of XOY plane is represented as:

[0144]

[0145] p 17 The anti-symmetric bending deformation base vector of XOY plane is represented as:

[0146]

[0147] p 18 The anti-symmetric tensile-compression deformation base vector of XOY plane is represented as:

[0148]

[0149] Wherein, m is the length of the rectangular element in the Y-axis direction, and n is the length of the rectangular element in the X-axis direction;

[0150]

[0151] The combined base vectors p1-p 18 The complete orthogonal mechanical base matrix P of the 6-node rectangular element is constructed as:

[0152] P=[p1 p2 p3 p4 p5 p6 p7 p8 p9 p 10 p 11 p 12 p 13 p 14 p 15 p 16 p 17 p 18 ]。

[0153] By using finite element software, a space structure model is established in three-dimensional rectangular coordinate system, and quadrilateral shell element is used to divide the space structure model, then 4 corner points and two middle points in X axis direction of any quadrilateral shell element are selected to form a 6-node rectangular element, then node coordinates d of the i-th 6-node rectangular element are i :

[0154] d i = (x i1 ,y i1 ,z i1 ,x i2 ,y i2 ,z i2 ,x i3 ,y i3 ,z i3 ,x i4 ,y i4 ,z i4 ,x i5 ,y i5 ,z i5 ,x i6 ,y i6 ,z i6 ),

[0155] wherein:

[0156] d i represents node coordinates of the i-th rectangular element in the space model, i = 1, 2, 3,..., k, k is the number of rectangular elements;

[0157] x ij represents coordinate value of the i-th rectangular element in X axis;

[0158] y ij represents coordinate value of the i-th rectangular element in Y axis;

[0159] z ij represents coordinate value of the i-th rectangular element in Z axis; j = 1, 2, 3, 4, 5, 6.

[0160] The space model is displaced and deformed under any load, and the discrete elements in the space model are also displaced and deformed, then node coordinates d of the i-th 6-node rectangular element after deformation are i :

[0161] d' i = (x' i1 ,y' i1 ,z' i1 ,x' i2 ,y' i2 ,z' i2 ,x' i3 ,y'i3 ,z' i3 ,x' i4 ,y' i4 ,z' i4 ,x' i5 ,y' i5 ,z' i5 ,x' i6 ,y' i6 ,z' i6 ),

[0162] in:

[0163] d' i Represents the node coordinates of the i-th rectangular element after the spatial model is deformed by force, i = 1, 2, 3, ..., k, where k is the number of rectangular elements;

[0164] x' ij This represents the X-axis coordinate value of the j-th node after the i-th rectangular element has been deformed;

[0165] y' ij This represents the Y-axis coordinate value of the j-th node after the i-th rectangular element has been deformed;

[0166] z' ij This represents the Z-axis coordinate of the j-th node after the i-th rectangular element has been deformed; j = 1, 2, 3, 4, 5, 6.

[0167] The coordinates d' of the deformed rectangular element nodes i Subtract the node coordinates d of the rectangular element before deformation i Obtain the nodal displacement vector s of the i-th rectangular element. i :

[0168] s i =(x' i1 -x i1 ,y' i1 -y i1 ,z' i1 -z i1 ,x' i2 -x i2 ,y' i2 -y i2 ,z' i2 -z i2 ,x' i3 -x i3 ,y' i3 -y i3 ,z' i3 -z i3 ,x' i4 -x i4 ,

[0169] y' i4 -y i4 ,z' i4 -z i4 ,x' i5 -x i5 ,y' i5 -y i5 ,z' i5 -z i5 ,x' i6 -x i6 ,y' i6 -y i6 ,z' i6 -z i6 )。

[0170] Then the node coordinate displacement vector s i of the i-th 6-node quadrilateral element is projected onto the corresponding complete orthogonal mechanics basis matrix P i , and the following is obtained:

[0171] s i = r i · p i ;

[0172] Transforming the above equation, the projection coefficient vector r i of the basic displacement and basic deformation of the i-th 6-node quadrilateral element in the space model is obtained:

[0173]

[0174] Wherein:

[0175] s i is the node displacement vector of the i-th 6-node quadrilateral element; P i is the complete orthogonal mechanics basis matrix of the i-th 6-node quadrilateral element, is the transpose matrix of P i , (p i ) -1 is the inverse matrix of p i ; r i is the projection coefficient vector of the basic deformation and basic displacement of the i-th 6-node quadrilateral element;

[0176] p i = [p i1 , p i2 , p i3 , p i4 , p i5 , p i6 , p i7 , p i8 , p i9 , p i10 , p i11 , pi12 ,p i13 ,p i14 ,p i15 ,p i16 ,p i17 ,p i18 ],

[0177] r i =(r i1 ,r i2 ,r i3 ,r i4 ,r i5 ,r i6 ,r i7 ,r i8 ,r i9 ,r i10 ,r i11 ,r i12 ,r i13 ,r i14 ,r i15 ,r i16 ,r i17 ,r i18 );

[0178] where p il is the l-th basic displacement and deformation basis vector of the i-th rectangular element;

[0179] r ij is the projection coefficient corresponding to the l-th basic displacement and deformation basis vector of the i-th rectangular element, l = 1, 2,..., 18.

[0180] Since the rigid body displacement does not cause stress and strain of the structure and members, the projection coefficients r i ~ r i1 corresponding to the 6 basic displacements in the projection coefficient vector r i6 of each rectangular element are removed, and the absolute values of the projection coefficients r i7 ~ r i18 corresponding to the remaining 12 basic displacements of each rectangular element are compared one by one, wherein the largest absolute value is the main basic deformation of the rectangular element, and the smaller absolute value is the secondary basic deformation of the rectangular element. By comparing the absolute values of the projection coefficients r i7 ~ r i18 corresponding to each rectangular element in the spatial model in turn, the main basic deformation and the secondary basic deformation of each rectangular element are obtained, thereby realizing the deformation recognition of the structure.

[0181] In addition, it should be noted that the projection coefficient can be negative, positive and negative represent the direction of deformation, such as X direction tensile deformation base p7 corresponding projection coefficient r7 is positive, indicating the tensile deformation, r7 is negative, indicating the compression deformation; such as X direction bending deformation base p9 corresponding projection coefficient r9 is positive, indicating that the bending is the bending of the upper side of the unit under compression and the lower side under tension, and if it is negative, it is just the opposite.

[0182] Rigid body rotation displacement error analysis

[0183] Rigid body rotation displacement is a nonlinear displacement, and linear decomposition will produce errors, that is, the rigid body rotation displacement of the three-dimensional 6-node rectangular element will not only exist in the projection coefficient of the rigid body rotation displacement base vector, but also exist in the projection coefficient of other displacement and deformation base vectors, which will cause errors compared with the theoretical situation, so it is also necessary to analyze the error of the projection coefficient value of the rigid body rotation displacement of the three-dimensional 6-node rectangular element on other deformation and displacement base vectors, and to judge whether the rigid body rotation displacement affects the calculation accuracy.

[0184] Set the long side of the 6-node rectangular element as 2l and the short side as l, when the element rotates counterclockwise around the center point by θ, the displacement vector of the 6 nodes of the element is:

[0185]

[0186] Project the rotation displacement vector of the three-dimensional 6-node rectangular element onto the complete orthogonal mechanical base which has been constructed, the rotation displacement vector of the 6-node rectangular element only has non-zero projection coefficients on the Z-axis rigid body rotation displacement base vector, X-axis tensile deformation base vector, Y-axis tensile deformation base vector and shear deformation base vector, and the projection coefficients on other basic displacement and basic deformation base vectors are all 0; therefore, the 12 constraint equations obtained by projecting the coordinate displacement vector of the rigid body rotation of the 6-node rectangular element are simplified into 4 independent constraint equations, as shown below:

[0187]

[0188] Wherein: r6 is the projection coefficient of the Z-axis direction rigid body rotation displacement on the rigid body rotation displacement base, r7 and r8 are the projection coefficients of the Z-axis direction rigid body rotation displacement on the X tensile deformation base and Y tensile deformation base respectively, and r 11 is the projection coefficient thereof on the shear deformation base.

[0189] Solving the above equations can obtain:

[0190]

[0191] Perform Taylor expansion on sinθ and cosθ in the formula and ignore high-order infinitesimals to obtain:

[0192]

[0193] Then we have: It is known that the projection coefficient of the rotational displacement around the Z-axis direction on p7 and p8 is an infinitesimal quantity of the projection coefficient of the rotational displacement around the Z-axis direction on p6 with respect to θ, so the projection coefficient of the rotational displacement around the Z-axis direction on the X tension-compression deformation base and the Y tension-compression deformation base can be ignored, and the error is within the allowable range of the method.

[0194] Application Example

[0195] As Figure 23 shown in one shear wall structure, the concrete elastic modulus is 3×10 11 pa, and the Poisson's ratio is 0.2, the deformation conditions of the shear wall and the coupling beam in the structure are analyzed.

[0196] With the help of finite element software modeling, the model is divided by quadrilateral shell elements, and the deformation conditions of the structure are obtained by modal analysis. The coordinates and relative displacement information of each node and element in the vibration mode of the structure are extracted; the file is imported into mathematical software for further analysis, and the element information and node information are reassembled to enter the deformation decomposition calculation process. Due to the large size of the structure model, only the leftmost two-span shear wall is selected for analysis Figure 23 . Among them, since the coupling beam is a transverse flexural member and the shear wall is a vertical flexural member, the deformation decomposition of the coupling beam is carried out by using a transverse 6-node element, and the deformation decomposition of the shear wall is carried out by using a vertical 6-node element. No. 1 element is selected at the bottom of the shear wall, and No. 2 element is selected on the coupling beam Figure 24 . The basic stress conditions of No. 1 element and No. 2 element under the action of load are calculated. Since the element calculation process is repetitive, only the calculation process of the basic deformation condition of No. 1 element is listed here, and the results of others are listed.

[0197] After establishing the finite element model in the finite element software, the coordinate vector d1 and displacement vector s1 of each node of No. 1 element under the first-order vibration mode are obtained by modal analysis:

[0198] d1=(14.8,0,0,14.8,1.8,0,14.8,1.8,2.4,14.8,0,2.4,14.8,1.8,1.2,14.8,0,1.2);

[0199] s1=(0,0,0,0,0,0,2.2308×10 -4 ,-5.11×10 -6 ,-4.123×10 -5 ,2.3163×10 -4 ,-8.15×10 -6 ,-4.918×10 -5, 6.656 x 10 -5 , 1.64 x 10 -6 , -1.853 x 10 -5 , 7.624 x 10 -5 , -5.27 x 10 -6 , -3.09 x 10 -5 ).

[0200] No. 1 unit is the wall stem of shear wall, which is suitable for analysis by vertical 6-node rectangular unit, so the deformation decomposition base of vertical 6-node rectangular unit is adopted. Since the plane where the unit is located is YOZ plane, it is necessary to convert the coordinates and convert the deformation decomposition base of XOY plane into that of YOZ plane. From the node coordinate vector d1, it can be known that the length of No. 1 unit in Y axis direction is n = 1.8 m and the length in Z axis direction is m = 2.4 m, so the corresponding complete orthogonal mechanical base matrix P can be calculated:

[0201] P = [p1 p2 p3 p4 p5 p6 p7 p8 p9 p 10 p 11 p 12 p 13 p 14 p 15 p 16 p 17 p 18 ],

[0202] wherein,

[0203] p1 = (0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0) T ,

[0204] p2 = (0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082) T , p3 = (0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0) T ,

[0205] p4 = (-0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0, 0, 0, 0, 0, 0) T ,

[0206] p5 = (-0.4082, 0, 0, 0.4082, 0, 0, 0.4082, 0, 0, -0.4082, 0, 0, 0.4082, 0, 0, -0.4082, 0, 0) T ,

[0207] p6 = (0, -0.3682, 0.2762, 0, -0.3682, -0.2762, 0, 0.3682, -0.2762, 0, 0.3682, 0.2762, 0, 0, -0.2762, 0, 0, 0.2762) T ,

[0208] p7 = (0, 0.4082, 0, 0, -0.4082, 0, 0, -0.4082, 0, 0, 0.4082, 0, 0, -0.4082, 0, 0, 0.4082, 0) T ,

[0209] p8 = (0, 0, -0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, 0.5, 0, 0, 0, 0, 0, 0) T ,

[0210] p9 = (0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0, 0) T ,

[0211] p 10 = (0, 0, 0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0) T ,

[0212] p 11 = (0, -0.3382, -0.3007, 0, -0.3382, 0.3007, 0, 0.3382, 0.3007, 0.3382, -0.3007, 0, 0, 0.3007, 0, -0.3007) T ,

[0213] p 12 = (0.5, 0, 0, -0.5, 0, 0, 0.5, 0, 0, -0.5, 0, 0, 0, 0, 0, 0, 0, 0) T ,

[0214] p 13 = (0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, -0.5774, 0, 0, -0.5774, 0, 0) T ,

[0215] p 14 =(0, -0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, -0.2887, 0, 0, -0.5774, 0, 0, 0.5774, 0) T ,

[0216] p 15 =(0, 0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, -0.5774, 0, 0, -0.5774, 0) T ,

[0217] p 16 =(-0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, -0.2887, 0, 0, -0.5774, 0, 0, 0.5774, 0, 0) T ,

[0218] p 17 =(0, 0, -0.2887, 0, 0, 0.2887, 0, 0, 0.2887, 0, 0, -0.2887, 0, 0, -0.5774, 0, 0, 0.5774) T ,

[0219] p 18 =(0, 0, -0.2887, 0, 0, -0.2887, 0, 0, -0.2887, 0, 0, -0.2887, 0, 0, 0.5774, 0, 0, 0.5774) T ,

[0220] The 1st unit displacement vector s1 is projected onto the orthogonal deformation basis matrix P to obtain the projection coefficient vector r1 of the 1st unit:

[0221] r1=(-6.89×10 -6 , -5.71×10 -5 , 2.439×10 -4 , 2.2736×10 -4 , -7.44×10 -6 , -1.05×10 -5 , -4.06×10 -6 , -4.52×10 -5 , 1.52×10 -6 , 3.98×10 -6 , 1.62×10 -6 , -4.28×10 -6 , 4.88×10 -5 , -3.11×10-6 -1.73 x 10 -6 3.12 x 10 -6 -4.85 x 10 -6 -2.44 x 10 -6 ).

[0222] Similarly, the projection coefficient vectors of other units can be calculated. Since the rigid body displacement does not cause stress or strain of the unit, the six projection coefficients related to the rigid body displacement can be removed from the projection coefficient vector, and the projection coefficients of various basic deformations of the first unit and the second unit and the proportions of the deformations are obtained, as shown in Table 1.

[0223] Table 1 Proportions of projection coefficients of basic deformations of units under the action of loads in total deformation

[0224]

[0225] According to the decomposition results of the deformations, it can be seen that when the structure is subjected to the top load, the main deformation of the first unit is the out-of-plane bending deformation and the tensile and compressive deformation in the Z direction; the main deformation of the second unit is the shear deformation, the anti-symmetric bending deformation, the out-of-plane bending deformation and the expansion and contraction deformation. The main deformations of the units analyzed by the deformation decomposition method can guide the structural design, and the corresponding design scheme is proposed on the corresponding component.

[0226] The above-described embodiments are only preferred embodiments of the present application, merely used to explain the present application, and are not intended to limit the scope of the present application. For those skilled in the art, of course, other embodiments can be easily made by substitution or change according to the technical content disclosed in the present specification, and therefore, any changes and improvements made on the principles of the present application shall be included in the scope of the patent application of the present application.

Claims

1. A method for decomposition of structural deformation based on three-dimensional 6-node rectangular element, characterized by, comprising the steps of: Step 1: in a three-dimensional rectangular coordinate system, basic displacement and basic deformation base vectors of a 6-node rectangular element are constructed according to force balance condition and orthogonal decomposition theory, so as to obtain an orthogonal mechanical base matrix of the three-dimensional 6-node rectangular element P ; the basic displacement and basic deformation of the 6-node rectangular element include X-axis rigid translation displacement, Y-axis rigid translation displacement, Z-axis rigid translation displacement, X-axis rigid rotation displacement, Y-axis rigid rotation displacement, Z-axis rigid rotation displacement, XOY plane X-axis tension-compression deformation, XOY plane Y-axis tension-compression deformation, XOY plane X-axis in-plane bending deformation, XOY plane Y-axis in-plane bending deformation, XOY plane shear deformation, XOY plane warping deformation, XOY plane out-of-plane bending deformation, XOY plane expansion-contraction deformation, XOY plane punching deformation, XOY plane reverse warping deformation, XOY plane reverse asymmetric bending deformation and XOY plane reverse asymmetric tension-compression deformation. said step 1 specifically comprises the steps of: Step 1.1: In the Cartesian coordinate system, according to the force balance condition, the moment balance condition and the orthogonal theory, 18 kinds of basic displacement and basic deformation basis vectors of the 6-node rectangular element are constructed as follows:​ represents the X-axis translational displacement basis vector of the rigid body: ， represents the Y-axis translational displacement basis vector of the rigid body: ， represents the Z-axis rigid body translation basis vector: ， represents the displacement base vector of the rotation of the rigid body about the X axis: ， represents the displacement basis vector around the Y axis to the rigid body: ， denotes the rotation displacement basis vector around the Z axis of the rigid body: ， XOY plane X-axis tensile and compressive deformation base vector: ， represents the Y-axis directional stretch and compression deformation basis vector of the XOY plane: ， represents the XOY plane X-axis in-plane bending deformation basis vector: ， represents the in-plane bending deformation basis vector of the XOY plane in the Y-axis direction ， represents the XOY plane shear deformation base vector: ， represents the XOY plane warping deformation basis vector: ， represents the out-of-plane bending deformation basis vector of the XOY plane: ， denotes the vector of the expansion-contraction deformation in the XOY plane: ， represents the XOY plane shear deformation basis vector: ， represents the inverse warping deformation basis vector of XOY plane: ， denotes the anti-symmetric bending deformation basis vector in XOY plane: ， denotes the anti-symmetric stretch-compression deformation basis vector in XOY plane: ， wherein L is the length of the rectangular element in the Y-axis direction, L is the length of the rectangular element in the X-axis direction; ； ； Step 1.2: The 6 basic displacement base vectors shown in step 1.1 are combined with 12 basic deformation base vectors to construct a complete orthogonal mechanics base matrix for a 6-node rectangular element : ； Step 2: in a space rectangular coordinate system, a space structure model of the stressed thin plate type component is established, and the space structure model is divided by using quadrilateral shell elements, 4 corner points and midpoints of two opposite sides of the quadrilateral shell elements are selected to form a three-dimensional 6-node rectangular element, and a displacement vector of each node of the 6-node rectangular element is obtained ; Step 3: The displacement vector of each node of the 6-node quadrilateral element Projection to the orthogonal mechanical basis matrix of the 3D 6-node quadrilateral element P The projection coefficient vector of the basic deformation and basic displacement of each quadrilateral element is obtained ; Step 4: According to the projection coefficient vector The main and secondary basic deformation of each quadrilateral element can be obtained by the size of the projection coefficient, and the deformation decomposition and deformation identification of the spatial structure model can be realized.

2. The three-dimensional 6-node rectangular element-based structural deformation decomposition method according to claim 1, wherein, step 2 specifically comprises the steps of: Step 2.1: A spatial structure model is established in a three-dimensional rectangular coordinate system by using finite element software, and the spatial structure model is divided by using quadrilateral shell elements, then four corner points and two midpoints in the X-axis direction of any quadrilateral shell element are selected to form a 6-node rectangular element, and the node coordinates of the first 6-node rectangular element are : : ， wherein: node coordinates representing a 2D rectangular cell in the spatial model, , Nc is the number of rectangular cells;​ represents the coordinate value on the X-axis of the first node of the first rectangular unit represents the coordinate value on the X-axis of the first node of the first rectangular unit represents the coordinate value on the X-axis of the first node of the first rectangular unit represents the coordinate value on the Y axis of the first node of the first rectangular unit represents the coordinate value on the Y axis of the first node of the first rectangular unit represents the coordinate value on the Y axis of the first node of the first rectangular unit represents the coordinate value on the Z-axis of the first node of the first rectangular unit; represents the coordinate value on the Z-axis of the first node of the first rectangular unit; represents the coordinate value on the Z-axis of the first node of the first rectangular unit; ; Step 2.2: The space model is displaced and deformed under the action of any load, producing the deformed 6-node rectangular element node coordinates Xdeformed : ， wherein: representing node coordinates of the i-th rectangular element after the spatial model is deformed by the force, , is the number of rectangular elements;​ represents the coordinate value on the X-axis of the first node of the first rectangular unit after deformation represents the coordinate value on the X-axis of the first node of the first rectangular unit after deformation represents the coordinate value on the X-axis of the first node of the first rectangular unit after deformation represents the coordinate value on the Y axis of the first node of the first rectangular unit after deformation represents the coordinate value on the Y axis of the first node of the first rectangular unit after deformation represents the coordinate value on the Y axis of the first node of the first rectangular unit after deformation represents the coordinate value on the Z axis of the first node of the first rectangular unit after deformation; ;​​ Step 2.3: Subtract the pre-deformed rectangular element node coordinates from the deformed rectangular element node coordinates Step 2.4: Obtain the node displacement vector for the first :​​ 。 3. The decomposition method of structural deformation based on three-dimensional 6-noded rectangular element according to claim 2, wherein, said step 3 specifically is: Step 3.1: The nodal displacement vector of the 6-node rectangular element is projected onto the corresponding complete orthogonal mechanical basis matrix which gives​​ ； Step 3.2: Transforming the above equation gives the projection coefficient vector of the basic displacement and basic deformation of the 6-node rectangular element in the spatial model :​ ； wherein: For the first The nodal displacement vectors of a 6-node rectangular element; For the first The complete orthogonal mechanical basis matrix of a 6-node rectangular element. for The transpose of the matrix, for The inverse matrix; For the first The projection coefficient vector of the basic deformation and basic displacement of a 6-node rectangular element; ， ； wherein is the first basic displacement and basic deformation basis vector for the th rectangular element of the th rectangular element of the for the first basic displacement and the deformation basis vector in the first projection coefficient corresponding to the first basic displacement and the deformation basis vector in the first 4. The decomposition method of structural deformation based on three-dimensional 6-node rectangular element according to claim 3, characterized in that, said step 4 specifically comprises: Remove the projection coefficient vector of each rectangular unit The projection coefficients corresponding to the six basic displacements in And compare the projection coefficients corresponding to the remaining 12 basic displacements of each rectangular element in the structure one by one. The absolute values ​​of the values ​​are used to determine the main basic deformations of the rectangular element, with the largest absolute value representing the primary basic deformation and the smaller absolute value representing the secondary basic deformations. This allows for the determination of the primary and secondary basic deformations of each rectangular element, thereby enabling deformation decomposition and identification of the structural model.

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