A method for quickly predicting critical bending buckling load of a wallboard with ribs

Abstract

The application discloses a method for quickly predicting critical bending buckling load of a ribbed wallboard, and relates to the field of buckling analysis of a ribbed plate. The method comprises the following steps: establishing a simplified model of a blade ribbed wallboard under the action of pure bending moment; determining a stress distribution expression according to the simplified model of the blade ribbed wallboard; adopting finite element simulation to fit an effective width analytical expression of the blade ribbed wallboard; establishing a balance equation and boundary conditions of the simplified model of the blade ribbed wallboard based on the stress distribution expression; establishing a generalized characteristic equation by using the differential quadrature method based on the effective width analytical expression according to the balance equation and boundary conditions of the simplified model of the blade ribbed wallboard; and iteratively solving the generalized characteristic equation according to the geometric size, initial buckling stress value and initial half-wave number value of the blade ribbed wallboard to obtain a prediction result of the elastic critical bending buckling stress of the blade ribbed wallboard. The application can improve the prediction speed and accuracy of the buckling load.

Description

Technical Field

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

[0002] At present, integrally reinforced panels are increasingly used in the aerospace industry. One of the difficulties in forming highly reinforced integrally reinforced panels is that the top of the reinforced panel may buckle during the forming process. Creep aging technology is widely used in the processing of 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 reinforced panel will first buckle and fail during the bending forming process because the high pressure stress it is subjected to exceeds the critical buckling stress. Therefore, most of the existing research focuses on the creep aging forming of shallow ribbed panels (such as <15mm) or integral panels with a bending moment direction parallel to the rib direction. There is a lack of research on the forming of high-ribbed integral panels with a rib height-to-thickness ratio greater than 10.

[0003] There are two mainstream manufacturing processes for aluminum alloy ribbed siding: 1) First, roll the thick plate, then mill the bent thick plate to obtain the ribbed siding component. This process has disadvantages such as 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 into shape. For example, using traditional creep aging forming technology, this process has the characteristics of high forming accuracy, low forming stress, and low residual stress. However, during the forming process of the integral ribbed siding, the top of the ribs is subjected to high pressure stress, which can easily cause the ribs to buckle and deform, resulting in failure of the integral siding 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 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 external load (related to the critical buckling strength). The buckling stress is obtained by minimizing the total potential energy. This method is easier to apply to the buckling calculation of structures with complex geometric conditions (such as stiffened plates), but the deflection of the buckling 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 stiffened 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 exerted 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 and the effective width analytical expression; iteratively solving the generalized characteristic equation based on the geometric dimensions, initial buckling stress value and initial half-wave number of the blade stiffened wall panel to obtain 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] This application provides a rapid prediction method for the critical bending buckling load of a stiffened blade panel. Finite element simulation is used to fit the effective width analytical formula for the blade stiffened panel, establishing the equilibrium equations and boundary conditions for a simplified model of the blade stiffened panel. Based on the effective width analytical formula, the differential quadrature method is used to establish a generalized characteristic equation. This generalized characteristic equation is then iteratively solved to obtain the predicted elastic critical bending buckling stress of the blade stiffened panel. This application combines the advantages of the equilibrium method and the energy method, applying the differential quadrature method to iterative calculations and employing the finite element method to assist in the calculations, thereby improving the speed and accuracy of buckling load prediction. 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 following briefly introduces the drawings required for use in the embodiments. 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 creative work.

[0011] Figure 1 This is a flow chart of a method for rapidly predicting the 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 stiffener 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 stiffener 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 centerline 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 a blade stiffener provided in another embodiment of the present application;

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

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

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

[0021] Figure 11 A schematic diagram of a detailed process for 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 this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0023] In order to make the above-mentioned purposes, 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 for buckling are directly solved based on the load and deflection functions of the structure and the corresponding boundary constraint equations. The equilibrium differential equations for the plate are established based on the Mindlin-Reissner plate theory (thick plate bending theory), and a series of assumptions are made to consider first-order shear effects through the plate thickness.

[0025] The combination of the equilibrium method and the Levy method, which incorporates the sine half-wave number, efficiently and accurately calculates the critical buckling strength. This method has been widely used to calculate the buckling strength of thin plates under various loading and boundary conditions. Compared to the energy method, the equilibrium method does not require any assumptions about the deflection of the structure, providing more accurate predictions.

[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 stiffened wall panel is provided, comprising the following steps 101 to 106. In which:

[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 blade stiffened panel model; the stress distribution expression represents the relationship between the normal force exerted by the blade stiffened panel and the buckling stress.

[0029] Step 103: Using finite element simulation, fitting an analytical expression for the effective width of the blade stiffener panel; the analytical expression for the effective width represents the relationship between the effective width of the blade stiffener panel and the half-wave number.

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

[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, a generalized characteristic equation is established using the differential quadrature method.

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

[0033] By implementing steps 101 to 106 above, using the classic case of a blade-stiffened panel subjected to pure bending moment as an example, this method combines the advantages of the equilibrium and energy methods, employs the differential quadrature method (DQ) for iterative calculations, and uses the finite element method to assist in the calculations. This method addresses the slow and complex critical stress prediction for elastic buckling of conventional stiffened panels.

[0034] In another exemplary embodiment of the present application, the process of establishing the simplified model of the blade stiffened wall panel under pure bending moment in 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 occurs first 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 figure 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 at the bottom edge. b (x direction), such as Figure 3 Part (b) and Figure 3 The skin is loaded by tension at the two transverse 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 area where buckling occurs, the buckling problem can be simplified to consider only 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 viewed as a plate with an elastic built-in bottom edge (y = 0), a free edge (y = h), and two simply supported loaded edges (x = 0, x = a) subjected to nonlinear force. X is the normal force applied per unit length of the stiffener in the x-direction and η is the loading factor.

[0039] That is, the vertical coordinate of the elastic built-in bottom edge in the xy plane is y=0, the vertical 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] Where, F X is the normal force applied per unit length of the stiffener 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 coefficient, h is the reinforcement height, y is the vertical coordinate, y0 is the distance between the neutral surface and the bottom edge of the stiffened plate, t sk is the skin thickness of the blade stiffener panel, and b is the stiffener width.

[0044] In another exemplary embodiment of the present application, the above step 103 uses finite element simulation to fit the effective width analytical expression of the blade stiffener panel 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 rigid skin schematic diagram 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 'Decision, 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, and 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 true width, which will be replaced by the effective width below.

[0052] M b By φ y and φ x Decide:

[0053]

[0054] Where 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. According to the above formula, it can also be modeled as a trigonometric function, 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 centerline. 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 models with different widths. The geometric dimensions of the model include a length of 50 mm, a thickness of 1 mm, and widths 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 region, 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 along the symmetry line under different skin width conditions. Two semi-analytical results assuming a rigid skin are also plotted for comparison. It can be seen that when the width is small, the skin's rotational behavior agrees well with the semi-analytical results that follow the rigid skin assumption, indicating that the rigid skin assumption is satisfied for skins with smaller widths. However, as the skin width increases, the skin's cross-section deforms severely, significantly different from the semi-analytical results, indicating that the rigid skin assumption is not applicable in this case. Furthermore, as the skin width increases, the deformed skin tends to converge toward the same shape. This phenomenon can be explained by the fact that the torque is applied only to the bottom edge of the skin, limiting the area of ​​influence on the flexible skin.

[0064] According to the rigidity condition M b ' is calculated by the formula, 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 assumed under 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 the 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 flexible skin torsional stiffness GJs 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 as follows according to the second-order forward difference method:

[0070]

[0071] Where Δx is the spatial increment of adjacent nodes. When it is greater than 0, b eff According to the convergence test of Δx, with an error of 1% as the criterion, set Δx to a / 20m. Substitute the above formula into b eff The calculation formula can be used, 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 under different skin aspect ratios mb / a eff . Effective width to length ratio mb eff The changes between / a and mb / a are as follows: Figure 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 Use b in 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 panel under the action of the linearly varying force in the x-direction is established as follows:

[0079]

[0080] Where Q xz , Q yz are the resultant shear forces on the xz plane and 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 governing differential equation:

[0082]

[0083] Where, 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 panel are:

[0085]

[0086] Where, 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 around 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 in the y direction on the free edge of the stiffened plate, 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 loading, 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 vertical coordinate of the elastic built-in bottom edge in the xy plane is y=0, the vertical coordinate of the free edge in the xy plane is y=h, the horizontal coordinate of one simply supported loaded edge loaded by a nonlinear force in the xy plane is x=0, and the horizontal coordinate 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 of establishing the generalized characteristic equation using the differential quadrature method 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, 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 blade stiffener panel model 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 subjected to nonlinear force loading;

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

[0093]

[0094] Where, 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] Where, 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 forces are:

[0099]

[0100] Step 303: Integrate the equilibrium equation obtained by the 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] Where 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 Figure 10 As shown, there are N in the x direction. 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 row i or column j, 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] Where,

[0113]

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

[0115]

[0116] Because N in this model 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 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, Figure 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: Calculate the normal force exerted by the blade stiffener panel based on the geometric dimensions and initial buckling stress value of the blade stiffener panel using a stress distribution expression.

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

[0123] Step 403: Solve the generalized characteristic equation based on 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, the initial value of the half-wave number is updated to the half-wave number of this iteration, and the process returns to the step "calculating the effective width of the blade stiffened panel based on the geometric dimensions of the blade stiffened panel and the initial value of the half-wave number using the analytical formula for the effective width".

[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 buckling stress value to the buckling stress of this iteration, and return to the step "Calculate the normal force exerted by the blade stiffener panel using the stress distribution expression based on the geometric dimensions of the blade stiffener panel and the initial buckling stress value".

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

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

[0130] 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 It is used in the calculation instead of the actual width b.

[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 prediction of 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 for predicting 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 can be combined arbitrarily. 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 document uses specific examples to illustrate the principles and implementation methods of this application. The description of the above examples is only intended to help understand the method and core concept of this application. At the same time, for those skilled in the art, based on the concept of this application, there may 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 reinforced wall panels, characterized by: include: A simplified model of a blade stiffened wall panel under pure bending moment is established; 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 by nonlinear forces; wherein the vertical coordinate of the elastic built-in bottom edge in the xy plane is y=0, the vertical coordinate of the free edge in the xy plane is y=h, the horizontal coordinate of one simply supported loaded edge loaded by nonlinear forces in the xy plane is x=0, and the horizontal coordinate of the other simply supported loaded edge loaded by nonlinear forces in the xy plane is x=a; h is the stiffening height, and a is the stiffening plate length in the blade stiffened wall panel; Determining a stress distribution expression based on the simplified blade stiffened panel model; the stress distribution expression characterizes the relationship between the normal force exerted by the blade stiffened panel and the buckling stress; Finite element simulation is used to fit an analytical formula for the effective width of the blade stiffener panel; the analytical formula for the effective width represents the relationship between the effective width of the blade stiffener panel and the half-wave number; Based on the stress distribution expression, the equilibrium equations and boundary conditions of the simplified blade stiffened panel model are established; According to the equilibrium equations and boundary conditions of the simplified blade stiffener panel model, the generalized characteristic equation is established using the differential quadrature method based on the effective width analytical expression. According to the geometric dimensions, initial values ​​of the buckling stress and the initial values ​​of the 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 reinforced wall panels according to claim 1 is characterized in that: The stress distribution expression is: F X =s xx t st ; Where, F X is the normal force applied per unit length of the stiffener 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 xx|y＝h is the buckling stress, η is the loading coefficient, h is the reinforcement height, y is the vertical coordinate, y0 is the distance between the neutral surface and the bottom edge of the stiffened plate, t sk is the skin thickness of the blade stiffener panel, and b is the stiffener width.

3. The rapid prediction method for critical bending buckling load of reinforced wall panels according to claim 1 is characterized in that: The analytical formula for the effective width of the blade stiffener 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.

4. The rapid prediction method for critical bending buckling load of reinforced wall panels according to claim 2 is characterized in that: Based on the stress distribution expression, the equilibrium equations and boundary conditions of the simplified blade stiffener panel model are established, including: Based on the stress distribution expression, the equilibrium equation of the simplified model of the blade stiffened panel under the action of linearly varying force in the x-direction is established in the elastic case: Where Q xz , Q yz are the resultant shear forces on the xz plane and 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; w is the lateral displacement in the z direction; According to the Mindlin-Reissner theory, the equilibrium equation is converted into the governing differential equation: Where, 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 stiffener panels are: Where, 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 around 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 in the y direction on the free edge of the stiffened plate, 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 loading, 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 vertical coordinate of the elastic built-in bottom edge in the xy plane is y=0, the vertical coordinate of the free edge in the xy plane is y=h, the horizontal coordinate of one simply supported loaded edge loaded by a nonlinear force in the xy plane is x=0, and the horizontal coordinate of the other simply supported loaded edge loaded by a nonlinear force in the xy plane is x=a.

5. The rapid prediction method for critical bending buckling load of reinforced wall panels according to claim 4 is characterized in that: The generalized characteristic equation is established using the differential quadrature method, including: Using the differential quadrature method, the equilibrium equation in the form of the governing differential equation is transformed into: Where, the buckling stress σ xx|y＝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, 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 stiffener 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 loaded edges subjected to nonlinear force. The boundary conditions of the elastic built-in bottom edge are: Where, 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: Where, 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 forces are: By integrating the equilibrium equation obtained by the 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 loaded edges loaded by nonlinear forces, the generalized characteristic equation is determined as: Where 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.

6. The rapid prediction method for critical bending buckling load of reinforced wall panels according to claim 1 is characterized in that: Based on the geometric dimensions, initial values ​​of the buckling stress, and initial values ​​of the half-wave number of the blade stiffened panel, the generalized characteristic equation is iteratively solved to obtain the prediction results of the elastic critical bending buckling stress of the blade stiffened panel, specifically including: According to the geometric dimensions of the blade stiffener and the initial value of the buckling stress, the normal force exerted by the blade stiffener is calculated using the stress distribution expression. According to the geometric dimensions and initial value of half-wave number of blade stiffener, the effective width of blade stiffener is calculated by using the effective width analytical formula. Solving the generalized characteristic equation based on the calculated normal force and effective width of the blade stiffener 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, update the initial value of the half-wave number to the half-wave number of this iteration, and return to step "Calculate the effective width of the blade stiffener panel using the analytical formula for the effective width based on the geometric dimensions of the blade stiffener 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 buckling stress value to the buckling stress of this iteration, and return to step "Calculate the normal force exerted by the blade stiffener panel using the stress distribution expression based on the geometric dimensions of the blade stiffener panel and the initial buckling stress value"; 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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