Rapid prediction method for critical bending buckling load of ribbed wallboard

Abstract

The invention discloses a rapid prediction method for critical bending buckling load of a ribbed wallboard, and relates to the field of stiffened plate buckling analysis. The method comprises the following steps: establishing a blade stiffened wallboard simplified model under the action of pure bending moment; determining a stress distribution expression according to the simplified model of the blade stiffened wallboard; finite element simulation is adopted, and an effective width analytic expression of the blade stiffened wall plate is fitted; based on the stress distribution expression, establishing an equilibrium equation and boundary conditions of the simplified model of the blade stiffened wallboard; establishing a generalized characteristic equation by using a differential quadrature method based on the effective width analytic expression according to the balance equation and the boundary condition of the simplified model of the blade stiffened wallboard; and iteratively solving the generalized characteristic equation according to the geometric dimension, the initial value of the buckling stress and the initial value of the half-wave number of the blade stiffened wall plate to obtain a prediction result of the elastic critical bending buckling stress of the blade stiffened wall plate. The method can improve the prediction speed and precision of the buckling load.

Description

Technical Field

[0001] The present application relates to the field of buckling analysis of stiffened panels, and in particular to a method for rapidly predicting critical bending buckling loads of stiffened wall panels. Background Art

[0002] At present, integrally stiffened panels are increasingly used in the aerospace industry. One of the difficulties in forming highly stiffened integrally stiffened panels is that the top of the stiffened panel may buckle during the forming process. Creep aging technology is widely used in processing ribbed panels, which can obtain formed components with smooth curvature and no defects. When the ratio of the rib height to the thickness (height-to-thickness ratio) of the integral panel increases, the top of the rib of the stiffened panel will first buckle and become unstable during the bending forming process because the high pressure stress exceeds the critical buckling stress. Therefore, most of the existing research is on the creep aging forming of shallow rib panels (such as <15mm) or integral panels with the bending moment direction parallel to the rib direction. There is a lack of research on the forming of high-rib integral panels with a rib height-to-thickness ratio greater than 10.

[0003] There are two mainstream manufacturing processes for aluminum alloy ribbed wall panels: 1) First roll the thick plate, then mill and bend the thick plate to obtain the ribbed wall panel component. This process has the disadvantages of low forming accuracy, large residual stress, low processing efficiency, and high manufacturing cost. 2) First cut / extrude the thick plate to obtain a flat plate structure with ribs, and then bend the ribbed flat plate component to form it. For example, using traditional creep aging forming technology, this process has the characteristics of high forming accuracy, small forming stress, and low residual stress. However, during the forming process of the overall ribbed wall panel, the top of the rib is subjected to high pressure stress, which can easily cause the rib to buckle and deform, resulting in failure of the overall wall panel forming.

[0004] At present, the main methods for solving structural buckling problems and predicting buckling stress are the finite element simulation method and the energy method (Ritz method). The finite element simulation method requires multiple simulation calculations, resulting in a slow prediction of the critical stress. The energy method is an approximate analytical solution that has been developed for buckling analysis of stiffened plates. The energy method solves the buckling problem based on the total potential energy, which is the sum of the strain energy stored in the structure (depending on the deflection of the structure) and the work done by the applied load (related to the critical buckling strength). The buckling stress is obtained by minimizing the total potential energy. This method is more easily applied to the buckling calculation of structures with complex geometric conditions (such as stiffened plates), but the deflection of the buckled structure must be assumed before the calculation, which adds some uncertainty to the method and makes it difficult to guarantee the accuracy of the calculation results. Summary of the invention

[0005] The purpose of this application is to provide a rapid prediction method for the critical bending buckling load of a ribbed wall panel, which can improve the prediction speed and accuracy of the buckling load.

[0006] To achieve the above objectives, this application provides the following solutions:

[0007] The present application provides a method for quickly predicting the critical bending buckling load of a stiffened wall panel, comprising: establishing a simplified model of a blade stiffened wall panel under pure bending moment; determining a stress distribution expression based on the simplified model of the blade stiffened wall panel; the stress distribution expression characterizes the relationship between the normal force applied by the blade stiffened wall panel and the buckling stress; fitting an effective width analytical expression of the blade stiffened wall panel using finite element simulation; the effective width analytical expression characterizes the relationship between the effective width and the half-wave number of the blade stiffened wall panel; establishing the equilibrium equation and boundary conditions of the simplified model of the blade stiffened wall panel based on the stress distribution expression; establishing a generalized characteristic equation based on the differential quadrature method based on the equilibrium equation and boundary conditions of the simplified model of the blade stiffened wall panel; iteratively solving the generalized characteristic equation based on the geometric dimensions of the blade stiffened wall panel, the initial value of the buckling stress and the initial value of the half-wave number, and obtaining a prediction result of the elastic critical bending buckling stress of the blade stiffened wall panel.

[0008] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0009] The present application provides a rapid prediction method for the critical bending buckling load of a stiffened wall panel. The effective width analytical formula of the blade stiffened wall panel is fitted by finite element simulation, and the equilibrium equation and boundary conditions of the simplified model of the blade stiffened wall panel are established. Based on the effective width analytical formula, the generalized characteristic equation is established by using the differential quadrature method, and then the generalized characteristic equation is iteratively solved to obtain the prediction result of the elastic critical bending buckling stress of the blade stiffened wall panel. The present application combines the advantages of the equilibrium method and the energy method, applies the differential quadrature method to participate in the iterative calculation, and uses the finite element method to assist the calculation, thereby improving the prediction speed and accuracy of the buckling load. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0011] Figure 1 A schematic diagram of a process flow of a rapid prediction method for critical bending buckling load of a ribbed wall panel in one embodiment of the present application;

[0012] Figure 2 A schematic diagram of a blade stiffening plate provided in another embodiment of the present application;

[0013] Figure 3 A schematic diagram of the buckling of a blade stiffener provided in another embodiment of the present application;

[0014] Figure 4 A schematic diagram of a simplified model of a blade stiffening wall panel provided in another embodiment of the present application;

[0015] Figure 5 A schematic diagram of a rigid skin and a flexible skin provided in another embodiment of the present application;

[0016] Figure 6 A schematic diagram of a finite element model of torsion along the center line during finite element simulation of a flexible skin provided by another embodiment of the present application;

[0017] Figure 7 A schematic diagram of the deflection of the skin on the symmetry line under different skin width conditions according to the finite element simulation results of the blade stiffener provided in another embodiment of the present application;

[0018] Figure 8 A schematic diagram of the rotation angles of a rigid skin and a flexible skin provided in another embodiment of the present application;

[0019] Fig. 9 A schematic diagram of a fitting curve after finite element simulation provided by another embodiment of the present application;

[0020] Fig.10 A schematic diagram of grid point distribution of a simplified model provided in another embodiment of the present application;

[0021] Fig.11 A schematic diagram of a detailed process of iteratively solving the generalized characteristic equation provided in another embodiment of the present application. DETAILED DESCRIPTION

[0022] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0023] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0024] When using the equilibrium method to solve structural buckling problems and predict buckling stresses, the equilibrium differential equations of buckling are directly solved based on the load and deflection functions of the structure and the corresponding boundary constraint equations. The equilibrium differential equations of the plate are established based on the Mindlin-Reissner plate theory (thick plate bending theory), and the first-order shear effect through the plate thickness is considered under a series of assumptions.

[0025] The combination of the equilibrium method and the Levy method introduces the sine half-wave number, which can efficiently and accurately obtain the calculation results of the critical buckling strength. It has been widely used in the calculation of the buckling strength of thin plates under different loads and boundary conditions. Compared with the energy method, the equilibrium method does not require any assumptions about the deflection of the structure and can provide more accurate prediction results.

[0026] In view of this, in an exemplary embodiment, Figure 1 As shown, a method for quickly predicting the critical bending buckling load of a ribbed wall panel is provided, comprising the following steps 101 to 106. Among them:

[0027] Step 101: Establish a simplified model of the blade stiffened wall panel under pure bending moment.

[0028] Step 102: Determine a stress distribution expression based on the simplified model of the blade stiffened wall panel; the stress distribution expression represents the relationship between the normal force applied by the blade stiffened wall panel and the buckling stress.

[0029] Step 103: using finite element simulation to fit an effective width analytical formula of the blade stiffened wall panel; the effective width analytical formula represents the relationship between the effective width of the blade stiffened wall panel and the half-wave number.

[0030] Step 104: Based on the stress distribution expression, establish the equilibrium equation and boundary conditions of the simplified model of the blade stiffened wall panel.

[0031] Step 105: According to the equilibrium equations and boundary conditions of the simplified blade stiffened panel model, based on the effective width analytical expression, the generalized characteristic equation is established using the differential quadrature method.

[0032] Step 106: According to the geometric dimensions, initial value of buckling stress and initial value of half-wave number of the blade stiffened panel, the generalized characteristic equation is solved iteratively to obtain the prediction result of elastic critical bending buckling stress of the blade stiffened panel.

[0033] By implementing the above steps 101 to 106, taking the classic blade stiffened wall panel under pure bending moment as an example, the advantages of the balance method and the energy method are combined, the differential quadrature method (DQ) is used to participate in the iterative calculation, and the finite element method is used to assist the calculation to describe the rapid prediction method of the elastic critical bending buckling load. The problems of slow critical stress prediction speed and complex calculation of elastic buckling of traditional stiffened wall panels are solved.

[0034] In another exemplary embodiment of the present application, the process of establishing the simplified model of the blade stiffened wall panel under the action of pure bending moment in the above step 101 is as follows:

[0035] Figure 2The schematic diagram of the blade stiffener is given, as well as its geometric parameters and boundary conditions under pure bending moment, and the coordinate system is defined in it. Figure 2 As shown in the figure, the geometric parameters that affect the buckling behavior of the stiffened plate include the stiffened plate length a, the stiffened plate width b, and the skin thickness t sk , reinforcement height h and reinforcement plate thickness t st , are given in the figure. Bending moment M Z Applied on the two transverse edges of the stiffened plate, the two transverse edges of the stiffened plate are set as simply supported boundary conditions, and the two longitudinal edges are set as free boundary conditions.

[0036] In order to better analyze the buckling process, a simplified model is established for this stiffened plate.

[0037] The top of the stiffened plate is under compression and the bottom is under tension, so buckling first occurs in the stiffened section, that is, in the xy plane, and the main buckling mode should be stiffened buckling. Figure 3 Part (a) of the paper gives the general buckling mode of the stiffened panel, where w is the deflection of the stiffened panel in the z direction, indicating the occurrence of buckling, u is the deflection of the bottom skin in the y direction caused by the torsional moment applied by the stiffened panel, and φ x and φ y are the rotation angles around the x-axis and y-axis respectively. The stiffened plate and the skin are analyzed separately. For the stiffened plate, the constraint from the skin can be regarded as the bending moment M of the bottom edge. b (x direction), such as Figure 3 Part (b) and Figure 3 The skin section is loaded by tension at the two lateral edges, and due to the buckling of the stiffeners, the moment M along the bottom edge b ' is the torsional moment of the skin. Bending moment M b is the torsional moment M b 'reaction, they are equal in magnitude.

[0038] Since the longitudinal stiffener is the main buckling area, the buckling problem can be simplified to only consider the longitudinal stiffener part. b ') can be represented by an elastic built-in constraint on the bottom edge. Figure 4 The simplified buckling model of a stiffened plate under pure bending moment can be considered as a plate with an elastic built-in bottom edge (y = 0), a free edge (y = h), and two simply supported loading edges (x = 0, x = a) subjected to nonlinear force. X is the normal force applied per unit length of the stiffened panel in the x-direction and η is the loading factor.

[0039] That is, the longitudinal coordinate of the elastic built-in bottom edge in the xy plane is y=0, the longitudinal coordinate of the free edge in the xy plane is y=h, the horizontal coordinate of one simply supported loaded edge loaded by nonlinear force in the xy plane is x=0, and the horizontal coordinate of the other simply supported loaded edge loaded by nonlinear force in the xy plane is x=a; h is the reinforcement height, and a is the length of the reinforcement plate in the blade reinforcement wall panel.

[0040] In another exemplary embodiment of the present application, the stress distribution expression in the above step 102 is:

[0041] F X =σ xx t st .

[0042]

[0043] In the formula, F X is the normal force applied per unit length of the stiffened plate in the x direction, σ xx is the normal stress of the stiffened plate in the x direction, t st is the thickness of the stiffened plate, σ xxy＝h is the buckling stress, η is the loading factor, h is the reinforcement height, y is the ordinate, y0 is the distance between the neutral surface and the bottom edge of the stiffened plate, t sk is the skin thickness in the blade stiffener panel, and b is the stiffener panel width.

[0044] In another exemplary embodiment of the present application, the above step 103 adopts finite element simulation, and the specific process of fitting the effective width analytical formula of the blade stiffened wall panel is as follows:

[0045] According to Saint Venant's torsion theory, M can be calculated analytically under the assumption that the skin is rigid. b ', that is, after deformation, the skin remains straight in the yz plane section. The schematic diagram of the rigid skin is as follows Figure 5 As shown in part (a) of φ y ' is the rotation angle of the rigid skin, φ y is the rotation angle of the stiffened plate after buckling. However, in actual situations, the skin usually rotates flexibly and is accompanied by deformation. The schematic diagram of the flexible skin is shown in Figure 5 As shown in part (b) of .

[0046] (1) For Figure 5 The rigidity condition in part (a) can be solved analytically:

[0047] M b 'The rotation angle φ of the rigid skin y 'Decide, φ y '=φ y .

[0048]

[0049] In the formula, GJ sk is the torsional stiffness of the skin, where G is the shear modulus of the material, J sk is the polar inertia under the rigid skin assumption. J sk It can be expressed as:

[0050]

[0051] In the above formula, b is the real width, which will be replaced by the effective width below.

[0052] M b By φ y and φ x Decide:

[0053]

[0054] In the formula, E is the elastic modulus of the material, β and γ are material constants. According to the plasticity theory, in the elastic case, it can be expressed as:

[0055]

[0056] Where v is the Poisson's ratio of the material.

[0057] (2) For Figure 5 The flexible condition in part (b) is:

[0058] M b It can be modeled as a trigonometric function based on the Levy method:

[0059]

[0060] Where m is the number of sinusoidal half waves in the x direction, It is a function of m and y and is determined by the geometry of the stiffened plate. b ' determines the rotation of the bottom skin, which can also be modeled as a trigonometric function according to the above formula, but it cannot be obtained analytically under flexible conditions. Therefore, this application combines the finite element simulation method to analyze and solve it. A 1 / 4 unit skin model with a half-wave area is selected for finite element simulation, such as Figure 6 As shown, Figure 6 Part (a) shows the moment of the finite element model twisted along the center line. Figure 6 Part (b) shows the boundary conditions of the finite element model with twisting along the center line. The length and width of the model are a / 2m and b / 2 respectively, and the twisting moment is M b ' / 2. x represents the x coordinate in the simplified model of the stiffened plate.

[0061]

[0062] Among them, M b0 ' is the torsional moment at the connection between the bottom edge and the symmetry line of the entire skin. Finite element simulations were performed on the models under different width conditions. The geometric dimensions of the model include a length of 50 mm, a thickness of 1 mm, and a width ranging from 25 mm to 150 mm. Since the deformation of the skin is proportional to the magnitude of the torsional torque in the elastic zone, the magnitude of the torsional torque will not affect the deformation trend of the skin. Therefore, the M in these finite element models is b0 'Set to 20 (N·m)m -1 .

[0063] Figure 7 The finite element simulation results of the blade stiffener show the deflection of the skin on the symmetry line under different skin width conditions, and two semi-analytical results assuming that the skin is rigid are also plotted for comparison. It can be seen that when the width is small, the rotational behavior of the skin is in good agreement with the semi-analytical results that follow the rigid skin assumption, indicating that for skins with smaller widths, the rigid skin assumption is satisfied. However, as the skin width increases, the cross-sectional deformation of the skin is severe, which is significantly different from the semi-analytical results, and the rigid skin assumption is not applicable to this case. In addition, as the skin width increases, the deformed skin tends to converge to the same shape. This phenomenon can be explained by the fact that the torque is only applied to the bottom edge of the skin, and the area of ​​influence on the flexible skin is limited.

[0064] According to the rigidity condition M b ', which is determined only by the rotation angle and torsional stiffness of the skin. From the above, when the skin width is large, φ under flexible conditions y Greater than φ under rigid conditions y This indicates that the actual torsional stiffness is smaller than that under the assumption of rigid conditions.

[0065] Therefore, the effective width b is introduced eff The concept of torsional stiffness is used to modify the torsional stiffness according to the flexibility. eff It is defined as the width of an equivalent rigid skin with the same torsional stiffness as a flexible skin with actual width b. When subjected to the same torsion, the two skins rotate at the same angle, e.g. Figure 8 As shown. eff According to the torsional stiffness GJs of the flexible skin k ' is expressed as:

[0066]

[0067] The torsional stiffness of the flexible skin is calculated by dividing the torque by the angle per unit length according to the analytical formula of the rigidity condition:

[0068]

[0069] In order to obtain the torsional stiffness of the flexible skin from the finite element simulation results, the above formula is rewritten according to the second-order forward difference as follows:

[0070]

[0071] Where Δx is the spatial increment of the adjacent nodes. When it is greater than 0, b eff According to the convergence test of Δx, with an error of 1%, Δx is set to a / 20m. Substitute the above formula into b eff The calculation formula can be used to calculate, where the angle in the denominator is obtained through finite element simulation.

[0072]

[0073] The following simulations are carried out to study the effective width b at different skin aspect ratios mb / a. eff . Effective width to length ratio mb eff The changes of / a and mb / a are as follows Fig. 9 shown.

[0074] Initially, as mb / a increases, mb eff / a shows an approximately proportional increase trend, and then as mb / a increases, the slope gradually decreases until it reaches a saturation level. It is found that this relationship can be well fitted by the hyperbolic tangent equation. For the data shown above, the effective width fitting equation of the blade stiffener is:

[0075]

[0076] The polar moment of inertia J sk b in the sentence eff Alternatively, the non-rigid rotational effect of the flexible skin can be considered to obtain a semi-analytical solution to the buckling analysis.

[0077] In another exemplary embodiment of the present application, the establishment of the equilibrium equations and boundary conditions of the simplified blade stiffened wall panel model in step 104 specifically includes the following steps 201 to 203:

[0078] Step 201: Based on the stress distribution expression, the equilibrium equation of the simplified model of the blade stiffened wall under the action of the linearly varying force in the x direction is established as follows:

[0079]

[0080] In the formula, Q xz , Q yz are the resultant shear forces on the xz plane and the yz plane, M xx 、M xy 、M yyare the moments per unit length caused by the normal stress of the stiffened plate in the x direction, the normal stress of the stiffened plate in the xy direction, and the normal stress of the stiffened plate in the y direction, respectively, and w is the lateral displacement in the z direction.

[0081] Step 202: According to the Mindlin-Reissner theory, the equilibrium equation is converted into the form of the governing differential equation:

[0082]

[0083] In the formula, κ 2 is the shear correction factor in Mindlin theory, G is the intermediate variable, E is the elastic modulus of the material, v is the Poisson's ratio of the material, φ x and φ y are the rotation angles around the x-axis and y-axis respectively.

[0084] Step 203: The boundary conditions for establishing the simplified model of the blade stiffened wall panel are:

[0085]

[0086] In the formula, w| y＝0 is the lateral displacement of the elastic built-in bottom edge in the z direction, w| x＝0 is the lateral displacement in the z direction of a simply supported loading side subjected to nonlinear force, w| x＝a is the lateral displacement of the other simply supported loading side in the z direction under nonlinear force loading, φ x | y＝0 is the rotation angle of the elastic built-in bottom around the x-axis, φ y | x＝0 is the rotation angle of a simply supported side subjected to nonlinear force about the y-axis, φ y | x＝a is the rotation angle of the other simply supported side subjected to nonlinear force around the y-axis, M yy | y＝h is the moment per unit length caused by the normal stress on the free edge of the stiffened plate in the y direction, M xy | y＝h is the moment per unit length caused by the normal stress on the free edge of the stiffened plate in the xy direction, M xx | x＝0 is the moment per unit length caused by the normal stress in the x-direction on a simply supported side of the stiffened plate subjected to nonlinear force, M xx | x＝a Q is the moment per unit length caused by the normal stress in the x-direction on the other simply supported side of the stiffened plate subjected to nonlinear force, yz | y＝h is the resultant shear force on the free edge of the stiffened plate in the yz plane, Mb ′ is the torsional moment of the skin, M b is the bending moment; the ordinate of the elastic built-in bottom edge in the xy plane is y=0, the ordinate of the free edge in the xy plane is y=h, the abscissa of one simply supported loaded edge loaded by a nonlinear force in the xy plane is x=0, and the abscissa of the other simply supported loaded edge loaded by a nonlinear force in the xy plane is x=a.

[0087] In another exemplary embodiment of the present application, the above step 105 using the differential quadrature method to establish the generalized characteristic equation specifically includes the following steps 301 to 303:

[0088] Step 301: Use the differential quadrature method to transform the equilibrium equation in the form of the governing differential equation into:

[0089]

[0090] Where, the buckling stress σ xxy＝h Abbreviated as σ b , are the weighted coefficients of the first-order partial derivative and the second-order partial derivative with respect to x, respectively. are the weighted coefficients of the first-order partial derivative and the second-order partial derivative with respect to y, is the Kronecker inner product operation, I is the identity matrix;

[0091] Step 302: using the differential quadrature method, the boundary conditions of the simplified model of the blade stiffened wall panel are expressed as: boundary conditions of the elastic built-in bottom edge, boundary conditions of the free edge, and boundary conditions of two simply supported loading edges loaded by nonlinear forces;

[0092] The boundary conditions of the elastic built-in bottom edge are:

[0093]

[0094] In the formula, I 1* are all the values ​​in the first row of the identity matrix I, for All values ​​in the first row; b eff is the effective width.

[0095] The boundary conditions of the free edge are:

[0096]

[0097] In the formula, I N* are all the values ​​in the Nth row of the identity matrix I, for All values ​​in row N.

[0098] The boundary conditions of the two simply supported loading sides subjected to nonlinear force are:

[0099]

[0100] Step 303: Integrate the equilibrium equation obtained by differential quadrature method, the boundary conditions of the elastic built-in bottom edge, the boundary conditions of the free edge, and the boundary conditions of the two simply supported loading edges loaded by nonlinear forces to determine the generalized characteristic equation:

[0101]

[0102] In the formula, A 11 , A 12 , A 13 , A 21 , A 22 , A 23 , A 31 , A 32 , A 33 is the coefficient on the left side of the equilibrium equation obtained by differential quadrature method, A 41 , A 42 , A 43 is the boundary condition coefficient obtained by differential quadrature method, B 11 is the buckling stress coefficient of the equilibrium equation obtained by the differential quadrature method.

[0103] This transforms the prediction of buckling stress into a problem of solving generalized eigenvalues ​​and eigenvectors. b and [w, φ x ,φ y ] T are the generalized eigenvectors of the A matrix and the B matrix in the equation respectively.

[0104] The DQ method is described in detail below:

[0105] In the DQ method, the discrete grid points of the model need to be defined in advance. The grid point distribution of the simplified model is as follows: Fig.10 As shown, there are N x points, there are N points in the y direction y points. For this model, numerically, N x =N y , in order to achieve fast convergence.

[0106]

[0107] At each grid point (x i ,y i ) can be approximated as the weighted linear sum of the function values ​​at all grid points in the i row or j column, as follows:

[0108]

[0109] Among them, f(x i ,y j ) is the grid point (x i ,y i ), represents the weighting coefficient of the p-order partial derivative of the function value at the grid point (n, j) with respect to x at the grid point (i, j), Represents the weighting coefficient of the p-order partial derivative of the function value at the grid point (i, n) with respect to y at the grid point (i, j).

[0110] The weighting coefficient of the first-order derivative can be calculated as follows:

[0111]

[0112] In the formula,

[0113]

[0114] The weighting coefficients for the second-order derivatives can be obtained by using the following recursive formula:

[0115]

[0116] Because in this model, N x =N y , so the p-order partial derivative of the function f with respect to x or y at each grid point mentioned above can be converted into a matrix form:

[0117]

[0118] [f]=[f 11 , f 12 ,…,f 1N , f 21 , f 22 ,…,f NN ].

[0119] Where [f] is the value of the function at all grid points, with a dimension of 1×N 2 [I] is the identity matrix, with dimension N×N. are the weighted coefficients of the p-order partial derivatives with respect to x and y, respectively, and their dimensions are N×N. For any two matrices [A] and [B], perform the Kronecker inner product operation Defined as

[0120] In another exemplary embodiment of the present application, Fig.11 As shown, the detailed process of iteratively solving the generalized characteristic equation in step 106 is as follows: Steps 401 to 407:

[0121] Step 401: According to the geometric dimensions of the blade stiffener panel and the initial value of the buckling stress, the normal force applied by the blade stiffener panel is calculated using the stress distribution expression.

[0122] Step 402: Calculate the effective width of the blade stiffener panel according to the geometric dimensions of the blade stiffener panel and the initial value of the half-wave number using the effective width analytical formula.

[0123] Step 403: Solve the generalized characteristic equation according to the calculated normal force and effective width applied by the blade stiffener panel to obtain the buckling stress and half-wave number of this iteration.

[0124] Step 404: If the half-wave number of this iteration is not equal to the initial value of the half-wave number, update the initial value of the half-wave number to the half-wave number of this iteration, and return to the step "calculate the effective width of the blade stiffened wall panel according to the geometric dimensions of the blade stiffened wall panel and the initial value of the half-wave number using the effective width analytical formula".

[0125] Step 405: If the half-wave number of this iteration is equal to the initial value of the half-wave number, then according to the formula Determine whether the buckling stress of this iteration has converged; among them, is the buckling stress of the i+1th iteration, is the buckling stress at the i-th iteration.

[0126] Step 406: When the buckling stress of this iteration does not satisfy the formula , update the initial value of the buckling stress to the buckling stress of this iteration, and return to the step "calculate the normal force applied by the blade stiffened panel using the stress distribution expression according to the geometric dimensions of the blade stiffened panel and the initial value of the buckling stress".

[0127] Step 407: When the buckling stress of this iteration satisfies the formula When , the buckling stress of this iteration is output as the prediction result of the elastic critical bending buckling stress of the blade stiffened panel.

[0128] That is, when the buckling stress and the half-wave number both meet the requirements, the buckling stress is output and the predicted buckling stress result is obtained.

[0129] Fig.11 The plastic constants in refer to material constants such as loading coefficients η, β and γ.

[0130] The key points of this application include:

[0131] 1. Convert the rigid skin condition to a true flexible skin condition, using the effective width b eff Instead of the actual width b, it is used in the calculation.

[0132] 2. Use the DQ method to transform the equilibrium equations and boundary conditions into the form of governing differential equations and integrate them into the same generalized characteristic equation, transforming the predicted buckling stress into the problem of solving generalized eigenvalues ​​and eigenvectors.

[0133] 3. For the first time, the buckling problem of stiffened plates in bending state is solved by theoretical calculation method.

[0134] 4. By combining finite element simulation and DQ method, a faster prediction of the buckling stress of the stiffened plate is achieved.

[0135] 5. Establish a simplified model for the integral stiffened plate to simplify the calculation.

[0136] The advantages of this application are as follows:

[0137] 1. The purpose of this application is to study the buckling behavior of stiffened panels in the elastic and plastic zones during CAF forming, and to establish a method to predict the buckling stress of stiffened panels in a short time by combining it with the DQ method and the finite element method.

[0138] 2. It fills the gap in the prediction of buckling stress of reinforced wall panels under pure bending conditions.

[0139] 3. Convert the rigid skin into a flexible skin to improve the realism of the prediction results.

[0140] The technical features of the above embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0141] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A rapid prediction method for critical bending buckling load of a reinforced wall panel, characterized in that: include: A simplified model of blade stiffened panels under pure bending moment is established; Determine a stress distribution expression according to the simplified blade stiffened panel model; The stress distribution expression characterizes the relationship between the normal force applied by the blade stiffened panel and the buckling stress; Finite element simulation is used to fit the analytical formula of the effective width of the blade stiffened wall panel; The effective width analytical expression represents the relationship between the effective width of the blade stiffened wall plate and the half-wave number; Based on the stress distribution expression, the equilibrium equation and boundary conditions of the simplified model of blade stiffened panels are established; According to the equilibrium equation and boundary conditions of the simplified model of blade stiffened panels, the generalized characteristic equation is established by using the differential quadrature method based on the effective width analytical formula. According to the geometric dimensions, initial values ​​of buckling stress and initial values ​​of half-wave number of the blade stiffened panel, the generalized characteristic equation is solved iteratively to obtain the prediction result of the elastic critical bending buckling stress of the blade stiffened panel.

2. The rapid prediction method for critical bending buckling load of a ribbed wall panel according to claim 1 is characterized in that: The simplified model of the blade stiffened wall panel is a rectangle with an elastic built-in bottom edge, a free edge and two simply supported loaded edges loaded with nonlinear forces; wherein the longitudinal coordinate of the elastic built-in bottom edge in the xy plane is y=0, the longitudinal coordinate of the free edge in the xy plane is y=h, the horizontal coordinate of one simply supported loaded edge loaded with nonlinear forces in the xy plane is x=0, and the horizontal coordinate of the other simply supported loaded edge loaded with nonlinear forces in the xy plane is x=a; h is the stiffening height, and a is the stiffened plate length in the blade stiffened wall panel.

3. The rapid prediction method for critical bending buckling load of a ribbed wall panel according to claim 1 is characterized in that: The stress distribution expression is: F X =s xx t st ; In the formula, F X is the normal force applied per unit length of the stiffened plate in the x direction, σ xx is the normal stress of the stiffened plate in the x direction, t st is the thickness of the stiffened plate, σ xxy＝h is the buckling stress, η is the loading factor, h is the reinforcement height, y is the ordinate, y0 is the distance between the neutral surface and the bottom edge of the stiffened plate, t sk is the skin thickness in the blade stiffener panel, and b is the stiffener panel width.

4. The rapid prediction method for critical bending buckling load of a ribbed wall panel according to claim 1 is characterized in that: The effective width analytical formula of the blade stiffened wall panel is: Where b eff is the effective width, a is the length of the stiffened plate in the blade stiffened wall panel, m is the half-wave number, and b is the stiffened plate width.

5. The rapid prediction method of critical bending buckling load of reinforced wall panels according to claim 3 is characterized in that: Based on the stress distribution expression, the equilibrium equations and boundary conditions of the simplified model of the blade stiffened panel are established, including: Based on the stress distribution expression, the equilibrium equation of the simplified model of the blade stiffened wall under the action of linearly varying force in the x direction is established as follows: In the formula, Q xz , Q yz are the resultant shear forces on the xz plane and the yz plane, M xx 、M xy 、M yy are the moments per unit length caused by the normal stress of the stiffened plate in the x direction, the normal stress of the stiffened plate in the xy direction, and the normal stress of the stiffened plate in the y direction, respectively, and w is the lateral displacement in the z direction; According to the Mindlin-Reissner theory, the equilibrium equation is transformed into the form of the governing differential equation: In the formula, κ 2 is the shear correction factor in Mindlin theory, G is the intermediate variable, E is the elastic modulus of the material, v is the Poisson's ratio of the material, φ x and φ y are the rotation angles around the x-axis and y-axis respectively; The boundary conditions for establishing the simplified model of blade stiffened panels are: In the formula, w| y＝0 is the lateral displacement of the elastic built-in bottom edge in the z direction, w| x＝0 is the lateral displacement in the z direction of a simply supported loading side subjected to nonlinear force, w| x＝a is the lateral displacement of the other simply supported loading side in the z direction under nonlinear force loading, φ x | y＝0 is the rotation angle of the elastic built-in bottom around the x-axis, φ y | x＝0 is the rotation angle of a simply supported side subjected to nonlinear force about the y-axis, φ y | x＝a is the rotation angle of the other simply supported side subjected to nonlinear force around the y-axis, M yy | y＝h is the moment per unit length caused by the normal stress on the free edge of the stiffened plate in the y direction, M xy | y＝h is the moment per unit length caused by the normal stress on the free edge of the stiffened plate in the xy direction, M xx | x＝0 is the moment per unit length caused by the normal stress in the x-direction on a simply supported side of the stiffened plate subjected to nonlinear force, M xx | x＝a Q is the moment per unit length caused by the normal stress in the x-direction on the other simply supported side of the stiffened plate subjected to nonlinear force, yz | y＝h is the resultant shear force on the free edge of the stiffened plate in the yz plane, M b ′ is the torsional moment of the skin, M b is the bending moment; the ordinate of the elastic built-in bottom edge in the xy plane is y=0, the ordinate of the free edge in the xy plane is y=h, the abscissa of one simply supported loaded edge loaded by a nonlinear force in the xy plane is x=0, and the abscissa of the other simply supported loaded edge loaded by a nonlinear force in the xy plane is x=a.

6. The rapid prediction method for critical bending buckling load of a reinforced wall panel according to claim 5 is characterized in that: The generalized characteristic equation is established using the differential quadrature method, including: The differential quadrature method is used to transform the equilibrium equation in the form of the governing differential equation into: Where, the buckling stress σ xxy＝h Abbreviated as σ b , are the weighted coefficients of the first-order partial derivative and the second-order partial derivative with respect to x, respectively. are the weighted coefficients of the first-order partial derivative and the second-order partial derivative with respect to y, is the Kronecker inner product operation, I is the identity matrix; The boundary conditions of the simplified blade stiffened panel model are expressed by differential quadrature method as follows: boundary conditions of the elastic built-in bottom edge, boundary conditions of the free edge and boundary conditions of two simply supported loading edges loaded by nonlinear forces. The boundary conditions of the elastic built-in bottom edge are: In the formula, I 1* are all the values ​​in the first row of the identity matrix I, for All values ​​in the first row; b eff is the effective width; The boundary conditions of the free edge are: In the formula, I N* are all the values ​​in the Nth row of the identity matrix I, for All values ​​in row N; The boundary conditions of the two simply supported loading sides subjected to nonlinear force are: By integrating the equilibrium equation obtained by differential quadrature method, the boundary conditions of the elastic built-in bottom edge, the boundary conditions of the free edge, and the boundary conditions of the two simply supported loading edges loaded by nonlinear forces, the generalized characteristic equation is determined as: In the formula, A 11 , A 12 , A 13 , A 21 , A 22 , A 23 , A 31 , A 32 , A 33 is the coefficient on the left side of the equilibrium equation obtained by differential quadrature method, A 41 , A 42 , A 43 is the boundary condition coefficient obtained by differential quadrature method, B 11 is the buckling stress coefficient of the equilibrium equation obtained by the differential quadrature method.

7. The rapid prediction method for critical bending buckling load of a ribbed wall panel according to claim 1 is characterized in that: According to the geometric dimensions, initial value of buckling stress and initial value of half-wave number of the blade stiffened wall panel, the generalized characteristic equation is iteratively solved to obtain the prediction result of elastic critical bending buckling stress of the blade stiffened wall panel, which specifically includes: According to the geometrical dimensions of the blade stiffened panel and the initial value of the buckling stress, the normal force exerted by the blade stiffened panel is calculated by using the stress distribution expression; According to the geometrical dimensions and initial value of half-wave number of blade stiffening wall plate, the effective width of blade stiffening wall plate is calculated by using the effective width analytical formula; Solving the generalized characteristic equation according to the calculated normal force and effective width applied by the blade stiffener panel to obtain the buckling stress and half-wave number of this iteration; If the half-wave number of this iteration is not equal to the initial value of the half-wave number, the initial value of the half-wave number is updated to the half-wave number of this iteration, and the process returns to step "calculating the effective width of the blade stiffened wall panel using the effective width analytical formula according to the geometric dimensions of the blade stiffened wall panel and the initial value of the half-wave number"; If the half-wave number of this iteration is equal to the initial value of the half-wave number, then according to the formula Determine whether the buckling stress of this iteration has converged; among them, is the buckling stress of the i+1th iteration, is the buckling stress of the i-th iteration; When the buckling stress of this iteration does not satisfy the formula , update the initial value of buckling stress to the buckling stress of this iteration, and return to step "calculate the normal force applied by the blade stiffened wall panel using the stress distribution expression according to the geometric dimensions of the blade stiffened wall panel and the initial value of buckling stress"; When the buckling stress of this iteration satisfies the formula When , the buckling stress of this iteration is output as the prediction result of the elastic critical bending buckling stress of the blade stiffened panel.

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