Plate shell unit construction method for eliminating shearing self-locking effect

Abstract

The invention discloses a plate shell unit construction method for eliminating a shear self-locking effect, and relates to the technical field of ship engineering, and the method comprises the steps: constructing a thick plate bending unit under a local coordinate system, and correcting the stiffness characteristic of the thick plate unit through employing a hypothesis shear strain method, thereby effectively avoiding the numerical ill-conditioned condition when the shear strain tends to zero. A plane stress unit under a local coordinate system is constructed, and a non-coordinated mode is introduced to improve the deformation capability of the plane stress unit. The unit stiffness matrixes of a thick plate bending unit and a non-coordinated plane stress unit are integrated, the synergistic effect of a hypothetical shear strain field and a non-coordinated displacement field is achieved, and a unified plate shell unit capable of effectively eliminating the shear self-locking effect is efficiently constructed. According to the method, the requirement for modeling of the plate shell unit in ship structure simulation can be met, and the progress of ship structure finite element simulation is promoted.

Description

Technical Field

[0001] The present invention relates to the technical field of ship engineering, and particularly to a method for constructing a plate-shell element that eliminates the shear locking effect. Background Art

[0002] The plate-shell element is an important element type in the finite element method, specifically used to simulate the mechanical behavior of thin plates and shell structures. Such elements can simultaneously describe bending deformation and in-plane membrane deformation. Compared with three-dimensional solid elements, when simulating structures where the thickness is much smaller than the dimensions in the other two directions, plate-shell elements can not only ensure high simulation accuracy but also significantly reduce the computational cost. Existing plate-shell elements are usually constructed based on the Kirchhoff thin plate theory and the Mindlin thick plate theory, and can meet the modeling requirements of plate-shell structures with different thicknesses. Plate-shell elements are widely used in ship finite element simulations. The hull structure is usually composed of complex thin plates and shells, including the outer shell plate, deck, bulkhead, and bottom plate, etc. These structures need to withstand wave loads, wind loads, and dynamic loads generated during ship operation. Accurate modeling and analysis of their stress and deformation behaviors are the key to ensuring the safety of ship design and operation.

[0003] Shear locking is a common numerical pathology in the finite element method, usually occurring in plate-shell elements based on the Mindlin thick plate theory. When the structure thickness is relatively thin, the shear strain tends to zero. However, due to the shape functions of conventional plate-shell elements not accurately describing this deformation characteristic, the element generates excessive shear stiffness in the thin plate limit case, thereby significantly restricting the deformation ability of the element. In ship structure simulations, the ship's outer shell plate, deck, and bulkhead are usually thin plate structures. If shear locking occurs when simulating the stress and deformation of these components using conventional plate-shell elements, it will lead to a relatively high overall stiffness. In areas with stress concentration or large local deformation, shear locking may mask the true structural response. Summary of the Invention

[0004] In view of the above problems and technical requirements, the inventor of the present invention proposes a method for constructing a plate-shell element that eliminates the shear locking effect. By integrating the assumed shear strain method and the incompatible mode on the basis of traditional plate-shell elements, a plate-shell element algorithm that can eliminate the shear locking effect is constructed to meet the modeling requirements of plate-shell elements in ship structure simulations and promote the progress of ship structure finite element simulations. The technical solution of the present invention is as follows:

[0005] A method for constructing a plate-shell element that eliminates the shear locking effect, comprising the following steps:

[0006] Construct a thick plate bending element in the local coordinate system, and modify the element stiffness matrix of the thick plate bending element using the assumed shear strain method;

[0007] Construct a plane stress element in the local coordinate system, introduce the incompatible mode and calculate the element stiffness matrix of the incompatible plane stress element;

[0008] Combine the element stiffness matrix of the thick plate bending element obtained by correction and the element stiffness matrix of the incompatible plane stress element, and obtain the element stiffness matrix of the plate-shell element in the global coordinate system after coordinate transformation.

[0009] A further technical solution thereof is to correct the element stiffness matrix of the thick plate bending element by using the assumed shear strain method, including:

[0010] On the edges of the thick plate bending element in the x-direction and y-direction, m sampling points are respectively selected for interpolation to construct a new transverse shear strain;

[0011] According to the relationship equation between the shear strain and displacement of the nodes of the thick plate bending element, obtain the shear strain matrix applying the assumed shear strain method;

[0012] Based on the shear strain matrix applying the assumed shear strain method, construct the element stiffness matrix of the corrected thick plate bending element.

[0013] A further technical solution thereof is to introduce the incompatible mode and calculate the element stiffness matrix of the incompatible plane stress element, including:

[0014] Obtain the displacement function of the plane stress element of the incompatible mode;

[0015] According to the relationship equation between the strain and displacement of the nodes of the plane stress element of the incompatible mode, obtain the standard strain-displacement matrix and the additional term of the strain-displacement matrix in the incompatible mode;

[0016] Based on the standard strain-displacement matrix and the additional term of the strain-displacement matrix, calculate the element stiffness matrix of the incompatible plane stress element.

[0017] The beneficial technical effects of the present invention are:

[0018] This method corrects the stiffness characteristics of the thick plate bending element by using the assumed shear strain method, which can effectively avoid the numerical ill-conditioning when the shear strain approaches zero. The incompatible mode is also introduced to improve the deformation ability of the quadrilateral plane stress element. Finally, by integrating the element stiffness matrices of the thick plate bending element and the incompatible plane stress element, the synergistic effect of the assumed shear strain field and the incompatible displacement field is realized, and a unified plate-shell element that effectively eliminates the shear locking effect is efficiently constructed. It is verified that this method is superior to the traditional plate-shell element in terms of accuracy, and the calculation results are exactly the same as those of the commercial software Nastran, significantly improving the reliability and applicability of the plate-shell element in the finite element simulation of ship structures. Description of the Drawings

[0019] Figure 1 It is a schematic flow chart of the method for constructing a plate-shell element to eliminate the shear self-locking effect provided by this application.

[0020] Figure 2 It is a schematic diagram of the global coordinate system and local coordinate system of the plate-shell element provided by this application.

[0021] Figure 3 It is a schematic diagram of the verification model working condition setting provided by this application.

[0022] Figure 4 It is a schematic diagram of the quadrilateral plate-shell element in the local coordinate system provided by this application.

[0023] Figure 5 It is a schematic diagram of the sampling points of the assumed shear strain method provided by this application.

[0024] Figure 6 It is a schematic diagram of the calculation result of the plate-shell element considering only the assumed shear strain method provided by this application.

[0025] Figure 7 It is a schematic diagram of the calculation result of the plate-shell element considering the assumed shear strain method and the superposition of incompatible modes provided by this application.

[0026] Figure 8 It is a comparison diagram of the calculation results using the commercial software Nastran under the same conditions Specific embodiments

[0027] The following further describes the specific embodiments of the present invention with reference to the accompanying drawings.

[0028] In one embodiment of this application, a method for constructing a plate-shell element to eliminate the shear self-locking effect is provided. The plate-shell element is combined by a plane stress element and a thick plate bending element, and can be applied to the analysis of both thin plates and thick shell structures at the same time. The plate-shell element includes 4 nodes, and each node has 5 degrees of freedom (u x , u y , u z , θ x , θ y ), where u x , u y , u z represent the linear displacements in three directions, which are used to describe the translation of the node in three-dimensional space; θ x , θ y represent the rotational degrees of freedom about the x-axis and y-axis, which are used to describe the rotation angle of the node in the bending deformation. As Figure 1 shown, the method specifically includes the following contents:

[0029] Step 1: Construct a thick plate bending element in the local coordinate system o'x'y'z', and use the assumed shear strain method to correct the element stiffness matrix of the thick plate bending element to optimize the stiffness characteristics.

[0030] Step 2: Construct a quadrilateral plane stress element in the local coordinate system o'x'y'z', introduce a non - conforming mode and calculate the element stiffness matrix of the non - conforming plane stress element. The non - conforming mode can improve the shear locking problem while ensuring the efficient calculation of first - order elements.

[0031] Step 3: Combine the element stiffness matrix of the corrected thick plate bending element and the element stiffness matrix of the non - conforming plane stress element, and obtain the element stiffness matrix of the plate - shell element in the global coordinate system oxyz after coordinate transformation.

[0032] In this embodiment, the assumed shear strain field provides a more reasonable strain distribution, and the non - conforming displacement field further releases the element constraints. The two work together to efficiently and accurately eliminate the shear locking problem. The following will supplement the description of the solution of the embodiment of the present application with specific formulas:

[0033] In Step 1, constructing the thick plate bending element specifically includes:

[0034] The displacement function of the thick plate is

[0035] where: I is a third - order identity matrix, and the element shape function ξ and η represent dimensionless parameters defining the position within the element in the local coordinate system, with a value range of [-1, 1]; the element nodal displacement vector is

[0036]

[0037] The strain equation of the thick plate bending element is expressed as:

[0038]

[0039] where ε x , ε y respectively represent the normal strains in the x and y directions; γ xy represents the in - plane shear strain; γ xz , γ yz represent the transverse shear strains; κ x , κ y respectively represent the curvatures about the x and y axes.

[0040] Combining the thick plate displacement function and the strain equation gives: κ = B b a e , γ = B s a e ,

[0041] Where: Bending strain matrix B b =[B b1 B b2 B b3 B b4 ], shear strain matrix B s =[B s1 B s2 B s3 B s4 ],

[0042]

[0043] Element stiffness matrix of thick plate bending element Expressed as:

[0044]

[0045] in, is the bending strain stiffness matrix, is the shear strain stiffness matrix, A represents the unit area;

[0046]

[0047] Where E is Young's modulus, v is Poisson's ratio, is the shear modulus, t is the plate thickness, and k is the influence coefficient considering the uneven distribution of shear stress, which can be taken as 1.2.

[0048] In step 1, the element stiffness matrix of the thick plate bending element is modified using the assumed shear strain method, which includes:

[0049] In order to avoid the shear self-locking phenomenon of the Mindlin plate when thinning, m sampling points are selected on the x-direction and y-direction edges of the thick plate bending unit to interpolate and construct new transverse shear strains. and Expressed as:

[0050]

[0051] in, and is the interpolation function of the i-th sampling point;

[0052] and are the assumed transverse shear strains at the sampling points in the x and y directions, respectively.

[0053] Then, according to the relationship equation between shear strain and displacement of thick plate bending unit node The shear strain matrix B obtained by applying the assumed shear strain method is s , substitute into formula (1) and recalculate Furthermore, the element stiffness matrix modified by the assumed shear strain method is obtained.

[0054] In step 2, constructing the plane stress element specifically includes:

[0055] The displacement function of the plane stress element is expressed as:

[0056]

[0057] where I is the second-order identity matrix, and the shape function N of this element i is the same as the shape function of the thick plate bending element, and the element nodal displacement vector is:

[0058]

[0059] In step 2, the displacement function of the quadrilateral plane stress element introducing the non-conforming mode is expressed as:

[0060]

[0061] where, α = [α1 α2 α3 α4] T .

[0062] Let N5 = 1 - ξ 2 N6 = 1 - η 2 . Compared with the plane stress element in the standard form, the displacement function of the plane stress element in the non-conforming mode adds the quadratic term N of the shape function corresponding to the pure bending state I and the additional degree of freedom α.

[0063] According to the relationship equation between the strain and displacement of the nodes of the plane stress element in the non-conforming mode, the strain of the plane stress element in the non-conforming mode is obtained as:

[0064]

[0065] where:

[0066] B C is the standard strain-displacement matrix, and B I is the additional term of the strain-displacement matrix in the non-conforming mode.

[0067] Finally, B C and B I are used to calculate the element stiffness matrix of the non-conforming plane stress element which is expressed as:

[0068]

[0069] where:

[0070] In step 3, the element stiffness matrix of the plate and shell element in the local coordinate system can be obtained by combining the above two parts of the element stiffness matrices. So far, the situations related to in-plane deformation involved in the calculation of the plate and shell element are the same as those of the plane stress problem, and the situations related to bending deformation are the same as those of the thick plate bending problem.

[0071] For the quadrilateral plate and shell element in the local coordinate system, each node has 5 degrees of freedom, and its stiffness matrix is a 20-order square matrix. However, in the global coordinate system, each node of the plate and shell element has 6 degrees of freedom. To facilitate the transformation from the local coordinate system to the global coordinate system, the nodal angular displacement θ z , that is, the rotation angle in the plane of the element or the rotation angle of the normal about the z-axis, is added in the local coordinate system of the element. In this way, the element stiffness matrix of the plate and shell element is expanded to a 24-order square matrix, in which the rows and columns corresponding to θ z are all zero elements.

[0072] The global coordinate system and the local coordinate system of the plate and shell element are as Figure 2 shown. The coordinate transformation between the global coordinate system oxyz and the local coordinate system o'x'y'z' is carried out through the matrix T, where φ is the direction cosine matrix between the local coordinate system and the global coordinate system.

[0073] The basic equation of the plate and shell element is expressed as:

[0074]

[0075] where F i is the load applied at the i-th node of the plate and shell element, and a i is the displacement of the i-th node of the plate and shell element;

[0076] After coordinate transformation, the element stiffness matrix of the plate and shell element in the global coordinate system:

[0077]

[0078] where, is the element stiffness matrix of the incompatible plane stress element in the global coordinate system, is the element stiffness matrix of the thick plate bending element obtained by correction in the global coordinate system. In actual solution, to avoid singularity problems, arbitrary virtual stiffness coefficients are usually given at the corresponding positions of the element stiffness matrix In this way, the equilibrium equation in the direction of θ z is to satisfy the condition of having a unique solution.

[0079] The following will conduct experimental verification on a method for constructing a plate-shell element that eliminates the shear self-locking effect provided by the examples of this application. As Figure 3 shown in the cantilever plate, its plate thickness is 0.5 (dimensionless), divided into 16 square plate-shell elements in total, with each element side length of 5, end load F = 100, Young's modulus E = 2700000, and Poisson's ratio ν = 0.3. The displacements of the cantilever plate are solved by using two methods: the plate-shell element that only considers the assumed shear strain correction and the plate-shell element that superimposes the incompatible mode.

[0080] Taking Figure 4 the quadrilateral plate-shell element in the local coordinate system shown as an example, an assumed shear strain field is constructed. Based on the Mindlin thick plate theory, the originally obtained shear strain field according to the strain and displacement relationship is replaced by the assumed shear strain field. As Figure 5 shown, on the x-direction side of the thick plate bending element, 2 sampling points A and B are selected, and on the y-direction side, 2 sampling points C and D are selected. The new transverse shear strains and

[0081]

[0082] are constructed by interpolation.

[0083] Among them, zi a and b respectively represent the length and width of the quadrilateral plate-shell element. In this example, a = b = 5, u zi is the z-direction linear displacement of the i-th node, and θ xi is the rotational degree of freedom of the i-th node around the x-axis.

[0084] The element stiffness matrix corrected by the assumed shear strain method is obtained by using the method introduced in step 1 above. Or, substituting the interpolation function into the formula can obtain the new B si , and the element stiffness matrix corrected by the assumed shear strain method can also be obtained. And the element stiffness matrix is assembled accordingly to solve the basic equation F = Kx, and finally the displacement result of the structure is obtained.

[0085] Using the above plate-shell element algorithm that only considers the assumed shear strain to solve the finite element model shown in Figure 3 , the calculation result is shown in Figure 6 . The maximum displacement of the cantilever plate structure is 1.977e-02. Subsequently, the incompatible mode provided in step 2 is introduced into the assumed shear strain algorithm. After recalculation, the maximum displacement of the cantilever plate structure is shown in Figure 7 , which is 2.000e-02. Comparing the two calculation results with Figure 8Comparing the results with the commercial software Nastran, we found that the results of the plate and shell element calculations using only the assumed shear strain method had a certain error compared to Nastran. However, after introducing the non-compatible mode, the calculation results were completely consistent with Nastran. This shows that the introduction of the non-compatible mode can further improve the calculation accuracy and reliability based on the assumed shear strain method, effectively and accurately eliminate the shear self-locking problem, and achieve the same level of accuracy as the commercial software Nastran.

[0086] The above description is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiment. It is understood that other improvements and variations directly derived or imagined by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included in the scope of protection of the present invention.

Claims

1. A method for constructing a plate and shell element to eliminate the shear self-locking effect, characterized in that The method includes: Construct a thick plate bending element in the local coordinate system, and correct the element stiffness matrix of the thick plate bending element by using the assumed shear strain method; Construct a plane stress element in the local coordinate system, introduce a non - conforming mode and calculate the element stiffness matrix of the non - conforming plane stress element; Combine the corrected element stiffness matrix of the thick plate bending element and the element stiffness matrix of the non - conforming plane stress element, and obtain the element stiffness matrix of the plate - shell element in the global coordinate system after coordinate transformation.

2. The method for constructing a plate and shell element for eliminating the shear self-locking effect according to claim 1, wherein The step of correcting the element stiffness matrix of the thick plate bending element by using the assumed shear strain method includes: On the edges in the x - direction and y - direction of the thick plate bending element, respectively select m sampling points to interpolate and construct a new transverse shear strain; According to the relationship equation between the shear strain and displacement of the nodes of the thick plate bending element, obtain the shear strain matrix for applying the assumed shear strain method; Based on the shear strain matrix for applying the assumed shear strain method, construct the element stiffness matrix of the corrected thick plate bending element.

3. The method for constructing a plate-shell element for eliminating the shear self-locking effect according to claim 2, characterized in that Newly constructed transverse shear strain Expressed as: wherein, and are interpolation functions for the i-th sampling point; and are the assumed transverse shear strains at the sampling points in the x and y directions, respectively.

4. The method for constructing a plate-shell element for eliminating the shear self-locking effect according to claim 2, wherein, The element stiffness matrix of the modified thick plate bending element is expressed as: Among them, is the bending strain stiffness matrix, and B b is the bending strain matrix; is the shear strain stiffness matrix, and B s is the shear strain matrix of the application of the assumed shear strain method, and A represents the element area; Wherein, E is Young's modulus, v is Poisson's ratio, G is shear modulus, t is plate thickness, and k is the influence coefficient considering the non - uniform distribution of shear stress.

5. The method for constructing a plate-shell element for eliminating shear self-locking effect according to claim 1, characterized in that The step of introducing a non - conforming mode and calculating the element stiffness matrix of the non - conforming plane stress element includes: Obtain the displacement function of the plane stress element with non - conforming mode; According to the relationship equation between the strain and displacement of the nodes of the plane stress element with non - conforming mode, obtain the standard strain - displacement matrix and the additional term of the strain - displacement matrix in the non - conforming mode; Based on the standard strain - displacement matrix and the additional term of the strain - displacement matrix, calculate the element stiffness matrix of the non - conforming plane stress element.

6. The method for constructing a plate-shell element for eliminating the shear self-locking effect according to claim 5, characterized in that, The displacement function of the plane stress element with non - conforming mode is expressed as: Among them, N i is the shape function of the i-th node of the plane stress element, i = 1, 2, 3, 4, a e is the nodal displacement vector of the plane stress element; Let N5 = 1 - ξ 2 N6 = 1 - η 2 , the quadratic terms of the shape function corresponding to the pure bending state added by introducing the incompatible mode The additional degrees of freedom α = [α1 α2 α3 α4] T , where ξ and η are dimensionless parameters that define the position within the element in the local coordinate system.

7. The method for constructing a plate-shell element for eliminating the shear self-locking effect according to claim 5, wherein The element stiffness matrix of the incompatible plane stress element is expressed as: Wherein: B C is the standard strain-displacement matrix, B I is the additional term of the strain-displacement matrix in the incompatible mode, and A represents the element area; Where E is Young's modulus, v is Poisson's ratio, and t is the thickness of the plate.

8. The method for constructing a plate and shell element for eliminating the shear self-locking effect according to claim 1, characterized in that, The basic equation of the plate - shell element is expressed as: Among them, F i is the load applied at the i-th node of the plate and shell element, and a i is the displacement of the i-th node of the plate and shell element; The element stiffness matrix of the plate and shell element in the global coordinate system is the element stiffness matrix of the incompatible plane stress element in the global coordinate system, and is the element stiffness matrix of the thick plate bending element obtained by modification in the global coordinate system.

Images