A method for explicit fast analysis of buckling and post-buckling under mechanical load

Abstract

The present application relates to the technical field of buckling and post-buckling analysis, and particularly relates to a method for explicit fast analysis of buckling and post-buckling under mechanical load, which simultaneously considers the influences of pre-buckling nonlinear deformation, post-buckling large deflection, initial geometric defects and the influences of the positions of the stiffeners in space. Based on high-order shear deformation theory, the influences of precise curvature relationship are considered, and control equations for buckling problems of a stiffened composite cylindrical shell structure are established, wherein the control equations contain tensile-bending coupling effect, tensile-twist coupling effect, bending-twist coupling effect and thermal effect. The method introduces precise curvature expression, geometric nonlinear relationship and the relationship between the stiffened structure and the coordinate position of the shell into the buckling and post-buckling analysis of the fiber-reinforced anisotropic stiffened laminated cylindrical shell under axial compression, external pressure and torsion working conditions, and can consider different combinations and distribution forms of the stiffeners.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of buckling and post-buckling analysis, and particularly relates to a method for explicit fast analysis of buckling and post-buckling under mechanical load. BACKGROUND

[0002] Composite structures have high specific strength and specific modulus (stiffness) and good fatigue resistance, creep resistance, impact resistance and fracture toughness. With the improvement of manufacturing processes, the performance indicators of composite materials are continuously improved, and more and more main load-bearing components in aerospace and marine engineering structures are made of composite materials, especially composite stiffened cylindrical shell structures are often used as load-bearing components in engineering structures.

[0003] With the increasing application of composite materials in engineering structures, the composite materials may produce buckling under complex internal and external loads, and even failure. Generally, in the process of structure and material design, the stiffness difference caused by different material layups, the effects of tensile-bending, bending-torsion and tensile-torsion coupling deformation, transverse shear effect and thermal effect need to be considered, and the buckling and post-buckling behavior of anisotropic composite stiffened cylindrical shell needs to be discussed in a more general sense. Therefore, the buckling problem of composite structures has attracted widespread attention from engineering designers.

[0004] Composite cylindrical shells are stiffened to improve strength and stiffness, and the overall structure is simple in form and greatly enhanced in load-bearing capacity, and is widely used as a basic structure in engineering. However, it is found in design practice that there is a huge difference between the prediction results of the classical theory and the experimental results under the action of a temperature field; the main reason is that the design personnel have not thoroughly studied the buckling failure mechanism of composite structures and other problems.

[0005] In addition, due to various reasons, some geometric defects will inevitably occur in such structures, and the existence of these defects will seriously affect the load-bearing characteristics of the shell structure. Therefore, the buckling and post-buckling characteristics of cylindrical shells with different boundaries and different initial geometric defects can provide a useful reference for engineering applications.

[0006] In view of the above problems, it is necessary to provide a method for explicit fast analysis of buckling and post-buckling under mechanical load, which is used to solve the buckling problem of composite shell structures, especially stiffened shell structures, to overcome the insufficient qualitative understanding of the buckling behavior near the critical point, and to provide a reliable basis for engineering design. SUMMARY

[0007] The present application aims to provide a method for explicit fast analysis of buckling and post-buckling under mechanical load, to solve the problems raised in the background art.

[0008] To achieve the above object, the present application provides the following technical solutions:

[0009] A buckling and post-buckling explicit fast analysis method under mechanical load, for a composite stiffened shell containing a precise curvature expression, comprising the following steps:

[0010] Step 1: Based on the high-order shear deformation theory, the transverse shear displacement-strain relationship of the cylindrical shell is distributed along the shell thickness direction according to the parabolic law;

[0011] Step 2: The stiffened plate is equivalent to a variable stiffness plate structure, that is, the stiffness of the plate structure corresponding to the local stiffened part is superimposed with the influence of the stiffener stiffness increment;

[0012] Step 3: According to the stress on the bending neutral surface of the plate structure under pure bending condition, the local neutral surface height h0 can be determined;

[0013] Step 4: The stiffness coefficients of the non-stiffener area and the stiffened area are combined into a variable stiffness function;

[0014] At the same time, in order to ensure the derivability of the stiffness coefficient matrix with respect to the position coordinates, the hyperbolic tangent function is introduced to smoothly transition the variable stiffness coefficient matrix;

[0015] Step 5: According to the equivalent constitutive relation of the above composite stiffened plate, the internal force and bending moment expression of the stiffened cylindrical shell structure under general laying conditions is obtained;

[0016] Step 6: According to Hamilton's principle, the Euler-Lagrange equation is used to obtain the equilibrium differential equation of the composite stiffened cylindrical shell;

[0017] Step 7: Introduce the general form of initial defect expression, and perform dimensionless processing on the equilibrium differential equation, and introduce a small parameter ε with obvious physical meaning, that is, it is inversely proportional to the equivalent length geometric parameter of the shell;

[0018] When ε<1, the equilibrium differential equation of the composite stiffened cylindrical shell is the boundary layer equation;

[0019] Step 8: The singular perturbation method is used for solving, and the solution of the equation is divided into regular solution and boundary layer solution;

[0020] Step 9: The initial defect numerical value of the fully anisotropic stiffened cylindrical shell is processed to form the deflection value generally distributed at different positions of the shell;

[0021] At the same time, considering the influence of thermal effect, the thermal bending moment and initial deflection are generated and substituted into the equilibrium differential equation of the composite stiffened cylindrical shell for calculation;

[0022] Step 10: The perturbation equations of each order are obtained by discretizing the equilibrium differential equation of the composite stiffened cylindrical shell with the same order of ε. The regular solution and the boundary layer solution are obtained by solving the perturbation equations of each order and combining them. The large deflection asymptotic solution which strictly satisfies the clamped boundary conditions in the asymptotic sense is obtained.

[0023] On this basis, the quantitative relationship expression of deflection and rotation and the boundary layer width expression of shell buckling are obtained. The equivalent stress σ ij of the shell structure is obtained by using the constitutive relation.

[0024] Step 11: The post-buckling equilibrium path under different load conditions is obtained; wherein:

[0025] The pressure λ p and the shear stress λ s satisfy:

[0026]

[0027] The end shortening δ p :

[0028]

[0029] The pressure λ q and the shear stress λ s satisfy:

[0030]

[0031] The end shortening δ q :

[0032]

[0033] The shear stress λ s and the axial load λ xp satisfy:

[0034]

[0035] The end shortening δ s :

[0036]

[0037] And the relative torsion angle:

[0038]

[0039] In equations (1a)-(3d), the upper right corner bracket parameters correspond to different perturbation orders.

[0040] Step 12: The perturbation parameters in equations (1a)-(3d) are converted by using the second-order perturbation parameters Convert to dimensionless maximum deflection, that is:

[0041]

[0042] Where, W m is the dimensionless maximum deflection, taking (x, y) = (π / 2m, π / 2n) points in the deflection expression, there is:

[0043]

[0044] Step 13: Get the post-buckling equilibrium path of the shear cylindrical shell under the action of torque load with the dimensionless maximum deflection as the perturbation parameter.

[0045] As a further scheme of the application: in step 2, the rectangular cross-section rib is used, and the equivalent stress balance and strain displacement compatibility relationship of the reinforced area is obtained according to the reinforced structure coordinate position and the reinforced structure material or geometric parameters:

[0046]

[0047] In the formula: σ1, σ2, σ6 are in-plane bending related stresses, the right upper corner marks p and s respectively mark the stress variables of the corresponding plate and rib, and the right lower corner represents the in-plane bending deformation;

[0048] Based on the composite material laminated plate theory, the material stiffness coefficients A p , B p , D p …H p of the plate can be obtained.

[0049]

[0050] The rib is simplified as a flexible beam structure, and taking the rectangular cross-section laminated beam as an example, similar derivation can obtain the corresponding stiffness coefficients A s , B s , D s …H s .

[0051] As a further scheme of the application: in step 3, the equivalent stiffness coefficients of the local area are further derived Satisfy the following conditions:

[0052]

[0053] Where, A p ~H p , A s ~H s are the stiffness coefficient matrices of the flat plate and the rib to the neutral plane respectively.

[0054] As a further scheme of the present application: in step 4, considering the oblique stiffener condition, the stiffness coefficient matrix of formula (8) can be rewritten as:

[0055]

[0056] Wherein, a i , b i , c 2i and c 2i-1 are the geometric equations of the i-th rib parallel line; T i is the local-global coordinate transformation matrix of the i-th rib; and λ is the transition area smoothing coefficient.

[0057] Thus, the equivalent variable stiffness coefficient function established by the hyperbolic tangent function is obtained, which takes into account the influence of tensile-torsional, tensile-bending and bending-torsional coupling stiffness.

[0058] As a further scheme of the present application: in step 8, the boundary layer solution is order of magnitude, and the order of magnitude of different buckling boundary layer effect parameters ε (the axial compression boundary layer effect is ε 1 order, the external pressure buckling boundary layer effect is ε 3 / 2 order, and the torsion boundary layer effect is ε 5 / 4 order), under the action of axial compression, external pressure and torque, the small deflection classical solution and initial geometric defect of the completely anisotropic stiffened cylindrical shell can be taken as:

[0059] Axial compression buckling:

[0060]

[0061] External pressure buckling:

[0062]

[0063] Torsional buckling:

[0064]

[0065] Wherein, is the defect parameter.

[0066] As a further scheme of the present application: in step 9, the deflection value is generated by Fourier expansion method to generate the terms corresponding to the coefficients in formula (10), formula (12) and formula (14).

[0067] As a further scheme of the present application: in step 13, the post-buckling equilibrium path is obtained by substituting formula (5) into formula (1a)-(3d).

[0068] As a further scheme of the present application: in step 13, for the perfect shell or μ = 0, let Generally W m ≠ 0;

[0069] For general initial imperfection or μ i ≠ 0, the Fourier expansion method is used to generate the terms corresponding to the coefficients in equations (11), (13) and (15), and by comparison, the minimum buckling load and the corresponding buckling mode (m, n) can be easily obtained.

[0070] Compared with the prior art, the present application has the following advantages:

[0071] (1) The method introduces the precise curvature expression, the geometric nonlinear relationship and the relationship between the stiffened structure and the shell coordinate position into the buckling and post-buckling analysis of the fiber-reinforced anisotropic stiffened laminated cylindrical shell under the axial compression, external pressure and torsion conditions, and can consider different combinations and distribution forms of the stiffeners;

[0072] (2) The method simultaneously considers the influences of the pre-buckling nonlinear deformation, the post-buckling large deflection and the initial geometric imperfection, as well as the influences of the transverse shear deformation and the coupling stiffness, and uses the singular perturbation method to give the post-buckling large deflection asymptotic solution of the completely anisotropic stiffened laminated cylindrical shell under the axial compression, external pressure and torsion loads, which not only satisfies the control equation but also strictly satisfies the boundary conditions in the asymptotic sense, and the singular perturbation method is used to obtain the buckling load and the post-buckling equilibrium path of the shear cylindrical shell under the axial compression;

[0073] (3) The research by the method shows that different laying modes, laying sequences, geometric parameters, stiffener stiffness and distribution forms have significant influences on the axial compression, external pressure and torsion buckling critical load and the post-buckling path of the medium-thickness anisotropic cylindrical shell, and can reflect the local buckling phenomenon of the structure;

[0074] (4) The results obtained by the method prove that: (a) under the action of the axial compression, the post-buckling path of the completely anisotropic stiffened composite laminated cylindrical shell is unstable, the cylindrical shell is sensitive to the initial geometric imperfection, and the completely anisotropic cylindrical shell is accompanied by shear stress and torsion under the action of the axial compression; (b) under the action of the external pressure, the post-buckling path of the medium-length completely anisotropic stiffened composite laminated cylindrical shell is stable, the cylindrical shell is not sensitive to the initial geometric imperfection, and the completely anisotropic cylindrical shell is accompanied by shear stress and torsion under the action of the external pressure; (c) the results obtained by the method prove that under the action of the torque, the post-buckling path of the completely anisotropic stiffened composite laminated cylindrical shell is weakly stable, the cylindrical shell is not sensitive to the initial geometric imperfection, and the completely anisotropic cylindrical shell is accompanied by the pressure stress in addition to the shear stress under the action of the torque;

[0075] (5) The quantitative relationship between the deflection and the rotation angle obtained by the method is of great significance for predicting the energy transfer mechanism of modal stiffening and the buckling propagation. BRIEF DESCRIPTION OF DRAWINGS

[0076] Figure 1 The figure is a schematic diagram of the composite cylindrical shell and its coordinate system under the action of axial pressure in the embodiment of the application.

[0077] Figure 2 The figure is a schematic diagram of the composite cylindrical shell and its coordinate system under the action of external pressure in the embodiment of the application.

[0078] Figure 3 The figure is a schematic diagram of the composite cylindrical shell and its coordinate system under the action of torque in the embodiment of the application.

[0079] Figure 4 The figure is a schematic diagram of the geometric relationship between the shell and the stiffened structure in the embodiment of the application.

[0080] Figure 5 The figure is a schematic diagram of the buckling and post-buckling analysis process of the anisotropic laminated composite cylindrical stiffened shell in the embodiment of the application.

[0081] Figure 6 The figure is a schematic diagram of typical initial geometric deformation defects in the embodiment of the application; wherein (a) is an overall corrugated deformation; (b) is a local convex-concave deformation.

[0082] Figure 7 The figure is a schematic diagram of the post-buckling equilibrium path of the orthotropic laminated composite cylindrical shell under axial pressure in the embodiment of the application.

[0083] Figure 8 The figure is a schematic diagram of the post-buckling equilibrium path of the orthotropic laminated composite cylindrical shell under external pressure in the embodiment of the application.

[0084] Figure 9 The figure is a schematic diagram of the post-buckling equilibrium path of the anisotropic laminated composite cylindrical shell under torque in the embodiment of the application.

[0085] Figure 10 The figure is a schematic diagram of the influence of anisotropic effects on the post-buckling equilibrium path of the shear cylindrical shell in the embodiment of the application; wherein (a) is a schematic diagram of the relationship between torque and end shortening; (b) is a schematic diagram of the relationship between torque and rotation angle. DETAILED DESCRIPTION

[0086] Clearly, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the protection scope of the present application.

[0087] The specific implementation of the present application will be described in detail below in combination with specific embodiments.

[0088] A buckling and post-buckling explicit fast analysis method under mechanical load for a composite stiffened shell including a precise curvature expression, comprising the following steps:

[0089] Step 1: Based on the high-order shear deformation theory, it is assumed that the transverse shear displacement-strain relationship of the cylindrical shell is distributed along the shell thickness direction according to a parabolic law;

[0090] Step 2: The stiffened plate is equivalent to a variable stiffness plate structure, that is, the stiffness of the plate structure corresponding to the local stiffened part is superimposed with the influence of the stiffener stiffness increment;

[0091] Step 3: According to the stress on the bending neutral surface of the plate structure under pure bending condition being zero, the local neutral surface height h0 can be determined;

[0092] Step 4: The stiffness coefficients of the non-stiffener area and the stiffened area are combined into a variable stiffness function;

[0093] At the same time, in order to ensure the derivability of the stiffness coefficient matrix with respect to the position coordinates, a hyperbolic tangent function is introduced to smoothly transition the variable stiffness coefficient matrix;

[0094] Step 5: According to the equivalent constitutive relation of the above composite stiffened plate, the internal force and bending moment expression of the stiffened cylindrical shell structure under general laying conditions is obtained;

[0095] Step 6: According to the Hamilton principle, the Euler-Lagrange equation is used to obtain the equilibrium differential equation of the composite stiffened cylindrical shell;

[0096] Step 7: The general form initial defect expression is introduced, the equilibrium differential equation is dimensionless processed, and a small parameter ε with obvious physical meaning is introduced, that is, it is inversely proportional to the equivalent length geometric parameter of the shell;

[0097] When ε<1, the equilibrium differential equation of the composite stiffened cylindrical shell is the boundary layer type equation;

[0098] Step 8: The singular perturbation method is used for solving, and the solution of the equation is divided into regular solution and boundary layer solution;

[0099] Step 9: The initial defect values of the fully anisotropic stiffened cylindrical shell are processed to form the deflection values generally distributed at different positions of the shell;

[0100] At the same time, the thermal effect is considered to generate the thermal bending moment and initial deflection, which are substituted into the equilibrium differential equation of the composite stiffened cylindrical shell for calculation;

[0101] Step 10: The equilibrium differential equation of the composite stiffened cylindrical shell is discretized according to the same order power of ε to obtain the perturbation equation set of each order, which is solved step by step to synthesize the regular solution and boundary layer solution, and the large deflection asymptotic solution strictly satisfying the clamped boundary condition in the asymptotic sense is obtained;

[0102] On this basis, the quantitative relationship expression of the deflection and the rotation angle and the boundary layer width expression of the shell buckling are obtained, and the equivalent stress σ ij of the shell structure is obtained by using the constitutive relationship.

[0103] Step 11: The post-buckling equilibrium path under different load conditions is obtained; wherein:

[0104] The pressure λ p and the shear stress λ s under the axial compression buckling condition satisfy:

[0105]

[0106]

[0107] And the end shortening δ p :

[0108]

[0109] The pressure λ q and the shear stress λ s under the external pressure buckling condition satisfy:

[0110]

[0111] And the end shortening δ q :

[0112]

[0113] The shear stress λ s and the axial load λ xp under the torsional buckling condition satisfy:

[0114]

[0115] And the end shortening δ s :

[0116]

[0117] and the relative torsion angle:

[0118]

[0119] Step 12: Convert the expressions (1a)-(3d) into dimensionless maximum deflection, i.e.:

[0120]

[0121] where W m is the dimensionless maximum deflection, and at the point (x, y) = (π / 2m, π / 2n) in the deflection expression, we have:

[0122]

[0123] Step 13: Obtain the post-buckling equilibrium path of the shear cylindrical shell under torsion load with the dimensionless maximum deflection as the perturbation parameter.

[0124] In step 2, rectangular cross-section ribs are used, and based on the coordinate position of the stiffened structure and the material or geometric parameters of the stiffened structure, the equivalent stress balance and strain displacement compatibility relationship of the stiffened area is obtained:

[0125]

[0126] In the formula: σ1, σ2, σ6 are in-plane bending related stresses, the right upper corner marks p and s respectively mark the stress variables corresponding to the plate and the rib, and the right lower corner represents the in-plane bending deformation;

[0127] Based on the composite laminated plate theory, the material stiffness coefficients A p , B p , D p …H p of the plate are obtained as

[0128]

[0129] The rib can be simplified as a flexible beam structure. Taking a rectangular cross-section laminated beam as an example, similar derivation can be obtained for the corresponding stiffness coefficients A s , B s , D s …H s .

[0130] In step 3, the equivalent stiffness coefficients of the local area are further derived to satisfy the following conditions:

[0131] ​

[0132] where A p ~H p , A s ~H s are the stiffness coefficient matrices of the plate and the stiffener pair with respect to the neutral surface, respectively.

[0133] In step 4, the stiffness coefficient matrix of equation (8) can be rewritten as:

[0134]

[0135] where a i , b i , c 2i and c 2i-1 are the geometric equations of the parallel edges of the ith stiffener; T i is the local-global coordinate transformation matrix of the ith stiffener; and λ is the transition zone smoothing coefficient.

[0136] Thus, the equivalent variable stiffness coefficient function is established by the hyperbolic tangent function, which takes into account the effects of tensile-torsional, tensile-bending, and bending-torsional coupling stiffness.

[0137] In step 7, for most composite materials, Thus, when we have ε < 1. In particular, for isotropic cylindrical shells, we have where is the Batdorf shell parameter, and for the classical cylindrical shell linear buckling analysis, In practical engineering, usually there is always ε << 1.

[0138] In step 8, the boundary layer solution is of the order of for different buckling boundary layer effect parameters ε, where the axial compression boundary layer effect is of the order of ε 1 , the external pressure buckling boundary layer effect is of the order of ε 3 / 2 , and the torsional boundary layer effect is of the order of ε 5 / 4 For fully anisotropic stiffened cylindrical shells under axial compression, external pressure, and torsion, the small deflection classical solution and initial geometric imperfection can be taken as:

[0139] Axial compression buckling:

[0140]

[0141] External pressure buckling:

[0142]

[0143] Torsional buckling:

[0144]

[0145] wherein, is the imperfection parameter.

[0146] In step 9, the deflection value is used to generate terms corresponding to the coefficients in formula (10), formula (12) and formula (14) by Fourier expansion method.

[0147] In step 13, the post-buckling equilibrium path is obtained by substituting formula (5) into formula (1a)-(3d).

[0148] In step 13, for the perfect shell or μ = 0, let Generally, W m ≠ 0;

[0149] For the general initial imperfection or μ i ≠ 0, terms corresponding to the coefficients in formula (11), formula (13) and formula (15) are generated by Fourier expansion method, and by comparison, the minimum buckling load and the corresponding buckling mode (m, n) are easily obtained.

[0150] Application example:

[0151] The object of the application is a composite stiffened shell containing an accurate curvature expression, which is divided into a skin and a rib two parts, the length of the stiffened shell is L, the radius is R, the skin is composed of N layers of orthogonal single-layer with a thickness of t, and the ribs are uniformly distributed on the skin for local reinforcement;

[0152] Taking the stiffened cylindrical structure as a variable stiffness medium-thick shell, considering the initial geometric imperfection and the effect of non-uniform temperature load ΔT(x, y, z), the displacement components of the shell in the right-hand coordinate system along the X, Y and Z directions are represented by and respectively, and the rotation angles of the normal line of the middle surface relative to the Y axis and the X axis are represented by and respectively, and the displacement field relationship of the anisotropic cylindrical shell is as follows:

[0153]

[0154] The method has similar calculation processes for the buckling and post-buckling analysis of the cylindrical shell under the action of axial compression, external pressure and torsion (as shown in Figure 6 For convenience and comparison with existing test results, three buckling cases are analyzed and introduced respectively. First, the buckling load and the minimum post-buckling load of the cylindrical shell under the action of axial compression, external pressure and torsion are given respectively, and compared with the structure in the literature, and the results are shown in the following tables 1-3.

[0155] Table 1 Buckling critical load of stiffened cylindrical shell under axial compression

[0156]

[0157] 1 Assume simply supported boundary condition; 2 Assume clamped boundary condition; 3 The numbers in the bracket are the axial half wave number and circumferential wave number of the buckling mode of the shell, respectively. 4 The buckling load of the present method with clamped boundary condition; 5 The minimum post-buckling load of the present method; 6 Singer, J. and Abramovich, H. Vibration techniques for definition of practical boundary conditions in stiffened shells. AIAA Journal, 17:762-769, 1979.

[0158] Table 2 Buckling critical load of unstiffened cylindrical shell under hydrostatic pressure (0 / 90) with clamped boundary condition 12T Comparison of buckling critical load (MPa) of cylindrical shell (E 11 = 162.0 GPa, E 22 = 9.6 GPa, G 12 = G 13 = 6.1 GPa, G 23 = 3.5 GPa, v 12 = 0.298)

[0159]

[0160] 1 Hur, S. H., Son, H. J., Kweon, J. H. and Choi, J. H., 2008, "Postbuckling of composite cylinders under external hydrostatic pressure," Composite Structures, 86, pp. 114-124.

[0161] 2 The numbers in the bracket are the axial half wave number and circumferential wave number of the buckling mode of the shell, respectively.

[0162] Table 3 Buckling critical load (N xy ) cr (lbf / in) comparison (R = 7.5 in, L = 15 in)

[0163]

[0164] 1 The numbers in the brackets are the circumferential wave numbers of the shell buckling modes, respectively.

[0165] 2 Wilkins, D. J., Love, T. S. Compression-torsion buckling test of laminated composite cylindrical shells. AIAA paper, pp. 74-379, 1974.

[0166] 3 Simitses G. J., Shaw, D., Sheinman, I. Stability of imperfect laminated cylinders: a comparison between theory and experiment. AIAA Journal, 1985, 23: 1086-1092.

[0167] The detailed procedure is as follows:

[0168] The real boundary condition in the experiment is close to the clamped boundary.

[0169] According to the geometric and material parameters of the cylindrical shell structure, the stiffness coefficient A p ~ H p and the stiffness coefficient A s ~ H s of the stiffened part are calculated. The calculation method of each material stiffness coefficient is similar, and A p is taken as an example here:

[0170] Axial compression case (Table 1-AB6, R = 120.1 mm, n s = 85, Al 7075-76E = 75.0 kg / mm 2 , v = 0.3):

[0171]

[0172] External pressure case (Table 2-CTM3, E 11 = 162.0 GPa, E 22 = 9.6 GPa, G 12 = G 13 = 6.1 GPa, G 23 = 3.5 GPa, v 12 = 0.298):

[0173]

[0174] Twist case (Table 3, Boron / epoxy, E 11 = 30.0 x 10 6 lbf / in 2 , E 22 = 2.7 x 10 6 lbf / in 2 , G 12 = G 13 = G 23 = 0.65 x 10 6 lbf / in 2 , v 12 = 0.21, (45 / -45) S , R = 7.5 in, L = 15 in):

[0175]

[0176] According to the displacement-strain relationship, the strain is calculated, and the expressions and calculation results of the cross-sectional force N and the moment M are given;

[0177] The reduced stiffness coefficient of the cylindrical shell is calculated If there is a longitudinal or hoop stiffening structure, the cross-sectional area A1 of the longitudinal rib, the cross-sectional area A2 of the hoop rib, the moment of inertia I1 of the longitudinal rib, the moment of inertia I2 of the hoop rib, the torsion moment J1 of the longitudinal rib, and the torsion moment J2 of the hoop rib at the stiffening coordinate position need to be calculated according to the geometry (such as the height and thickness of the rib, and the rib-to-rib spacing d1 of the longitudinal rib, the eccentricity e1 of the longitudinal rib, and the rib symbol is negative; or the rib-to-rib spacing d2 of the hoop rib, the eccentricity e2 of the hoop rib, and the rib symbol is negative) and the material parameters (such as the elastic modulus E S1 of the longitudinal rib, the shear modulus G1 of the longitudinal rib, or the elastic modulus E S2 of the hoop rib, and the shear modulus G2 of the hoop rib), and the reduced stiffness coefficient of the stiffener is calculated

[0178] If there is a geodesic line stiffening structure, the cross-sectional area A3 of the geodesic rib, the moment of inertia I3 of the geodesic rib, and the torsion moment J3 of the geodesic rib at the stiffening coordinate position need to be calculated according to the geometry (such as the height and thickness of the rib, and the rib-to-rib spacing d3 of the geodesic rib, the eccentricity e3 of the geodesic rib, and the rib symbol is negative, the number of circumferential geodesic ribs N g of the geodesic rib, and the axial angle T g ) and the material parameters (such as the elastic modulus E S3 of the geodesic rib, and the shear modulus G3 of the geodesic rib), and the reduced stiffness coefficient of the stiffener is calculated

[0179] According to Hamilton principle, the equilibrium differential equations of composite stiffened cylindrical shells are obtained by using Euler-Lagrange equation;

[0180] Introducing the general form of initial geometric imperfection, the equilibrium differential equations are non-dimensionalized, and the parameter ε has obvious physical meaning, which is inversely proportional to the equivalent length of the shell .

[0181] For most composites, Thus, when we have ε < 1.

[0182] In particular, for isotropic cylindrical shells, we have where is the Batdorf shell parameter, for the linear buckling analysis of classical cylindrical shells,

[0183] In practical engineering, usually there is always ε << 1. When ε < 1, the equilibrium differential equations of composite stiffened cylindrical shells are boundary layer type equations.

[0184] The singular perturbation method is used to solve the equations, and the solution of the equations is divided into "external" solution (regular solution) and boundary layer solution, and the boundary layer solution is order of magnitude, the boundary layer effect of axial compression buckling is ε 1 order, the boundary layer effect of external pressure buckling is ε 3 / 2 order, and the boundary layer effect of torsion is ε 5 / 4 order. For fully anisotropic stiffened cylindrical shells, the classical solution of the non-dimensional small deflection can be taken as:

[0185] Axial compression post-buckling:

[0186]

[0187] External pressure post-buckling:

[0188]

[0189] Torsion post-buckling:

[0190]

[0191] Let the initial geometric imperfection of the shell have the following form:

[0192] Axial compression post-buckling:

[0193]

[0194] Post-buckling under external pressure:

[0195]

[0196] Post-buckling under torsion:

[0197]

[0198] where, is the imperfection parameter;

[0199] The initial imperfection of a fully anisotropic stiffened cylindrical shell is processed to form deflection values generally distributed at different locations of the shell (e.g. global wave imperfection, Figure 6 (a) and the Fourier expansion method is used to generate terms corresponding to the coefficients in the expression of the initial geometric imperfection.

[0200] For example, with local dimple-type imperfection (as shown in Figure 6 (b) ), its mathematical model can be characterized by a bidirectional exponential decay function is the imperfection amplitude, and C1 and C2 are half of the axial and circumferential characteristic lengths of the dimple-type imperfection.

[0201] Further, the mathematical expression of the non-dimensional local dimple-type imperfection can be obtained as:

[0202]

[0203] where, μ is the imperfection parameter;

[0204] At the same time, if the influence of thermal effect is considered, the thermal bending moment and initial deflection are generated and substituted into the equilibrium differential equation of the composite stiffened cylindrical shell for calculation.

[0205] The equilibrium differential equation of the composite stiffened cylindrical shell is discretized according to the same order of power of ε to obtain perturbation equation sets of each order, which are solved step by step to synthesize regular solution and boundary layer solution, and we obtain large deflection asymptotic solution that strictly satisfies the clamped boundary condition in the asymptotic sense (analysis process see Figure 5 ).

[0206] The asymptotic solution of post-buckling under axial compression is:

[0207]

[0208]

[0209] The asymptotic solution of post-buckling under external pressure is:

[0210]

[0211]

[0212]

[0213] The asymptotic solution of post-buckling is:

[0214]

[0215]

[0216] All the coefficients in the above solution can be expressed in the form of From the deflection expression, it can be seen that the pre-buckling deformation is nonlinear.

[0217] On this basis, the quantitative relationship expression of deflection and rotation angle under different types of load is obtained, that is:

[0218] The axial compression rotation angle and deflection are respectively:

[0219]

[0220]

[0221] The external pressure rotation angle and deflection are respectively:

[0222] The torsion rotation angle and deflection are respectively:

[0223]

[0224] Under the condition of axial compression: the boundary layer width expression of shell AB6 buckling Where the values of for the shell given by specific geometric and material parameters are all constants. Under the condition of external pressure: the boundary layer width expression of shell CTM3 buckling

[0225] Where the values of for the shell given by specific geometric and material parameters are all constants.

[0226] Under the condition of torsion: boron fiber resin matrix composite (45 / -45) S The boundary layer width expression of shell buckling Where the values of for the shell given by specific geometric and material parameters are all constants.

[0227] And then the post-buckling equilibrium path under different load conditions is obtained.

[0228] The axial compression post-buckling equilibrium path:

[0229]

[0230] and

[0231]

[0232] Post-buckling equilibrium path under external pressure:

[0233]

[0234] and

[0235]

[0236] Post-buckling equilibrium path under torsion:

[0237]

[0238] Correspondingly,

[0239]

[0240] and:

[0241]

[0242] and relative torsion angle:

[0243]

[0244] Thus, the axial compressive stress of the fully anisotropic cylindrical shell under torsion can be obtained as

[0245]

[0246] where the variable k (in this example, the value of k at the maximum deflection after buckling is 1.9062) can be determined by the following formula:

[0247]

[0248] It should be noted here that before torsional buckling occurs, the value of the variable k is zero, and the previous small deflection solution satisfies the boundary conditions.

[0249] Using the quadratic perturbation parameter transformation, the above expression for is converted into the dimensionless maximum deflection, i.e.

[0250]

[0251] where W m is the dimensionless maximum deflection, Θ1=6.6017×10 -2 (axial compression and external pressure), Θ1=1.9138×10 -2(twist).

[0252] Taking the point (x, y) = (π / 2m, π / 2n) in the deflection expression, we have

[0253]

[0254] where C3= 0.8339, Θ2= 0.4157 (axial compression and external pressure); C3= 0.7755, Θ2= 0.2541 (twist).

[0255] Substituting the maximum deflection expression into the post-buckling equilibrium path expression, we obtain the post-buckling equilibrium path of the shear cylindrical shell under axial compression load with the dimensionless maximum deflection as the perturbation parameter.

[0256] For the perfect shell (or μ i = 0, i = 1, 2, 3, 4), let (usually W m ≠ 0); for the general initial imperfection (or μ i ≠ 0), the Fourier expansion method is used to generate the terms corresponding to the coefficients in the initial imperfection expression. After comparison, we can easily obtain the minimum buckling load and the corresponding buckling mode (m, n).

[0257] Axial compression case result analysis:

[0258] The above calculation results are shown in Table 1. It can be seen that the results of the present method (especially the post-buckling minimum load, including the buckling mode) are closer to the experimental results. In order to further verify the correctness of the present method, the buckling loads of the internally stiffened grid shell (0 / 90) S and (-45 / 45 / 90 / 0) S under axial compression are calculated and compared with the analysis results of Gerhard et al. (Gerhard, C. S., Gurdal, Z., Kapania, R. K., 1996. Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory. NASA CR 1471, 1996), as shown in the following Tables 4 and 5:

[0259] Table 4 Buckling load comparison results of geodesically internally stiffened orthotropic symmetrically laid cylindrical shell (0 / 90) S under axial compression (R = 85 inches, L = 100 inches, t = 0.2 inches, 1 x 12 grid shell, ​

[0260]

[0261]

[0262] 6 FEA - Finite Element Analysis, an abbreviation; 7 Shell theory, Table 5 has the same meaning;

[0263] Gerhard, C. S., Gurdal, Z., Kapania, R. K., 1996. Finite element analysis of geodesically stiffened cylindrical composite shells using a layered theory. NASA CR 1471.

[0264] Table 5 Geodesic stiffened orthogonally symmetrically laid cylindrical shells under axial compression (-45 / 45 / 90 / 0) S Buckling load comparison results (R = 85 inches, L = 100 inches, t = 0.2 inches, 1 x 12 grid shell, )

[0265]

[0266] Gerhard, C. S., Gurdal, Z., Kapania, R. K., 1996. Finite element analysis of geodesically stiffened cylindrical composite shells using a layered theory. NASA CR 1471.

[0267] The geometric parameters are taken as R = 85 inches (1 inch = 25.4 mm), L = 100 inches, t = 0.2 inches, 1 x 12 geodesic stiffeners, and the stiffener heights are 0.5, 1.0, 1.5 inches, respectively, with the internal grid stiffeners having a thickness of 0.2 inches. It can be seen that the present results are reasonably consistent with those of Gerhard et al. (1996).

[0268] Figure 7 (0 / 90) SPost-buckling load-shortening curve of a laminated cylindrical shell under axial compression and experimental results of Farhadinia (Fahadinia, Mahmood. Finite element analysis and experimental evaluation of buckling phenomena in laminate composite tubes and plates. Ph.D. Thesis, University of Missouri-Rolla, 1992). The data used in the calculation are: L = 12.0 in (1 in = 254 mm), R / t = 74.5, t = 0.02 in, Young's modulus and shear modulus are: E 11 = 6.56 x 106psi (1 psi = 6.895 kPa), E 22 = 1.167 x 106psi, G 12 = G 13 = 0.886 x 106psi, v 12 = 0.28. The limit point load of the present method is 407.046 lb / in (1 lb = 0.454 kg), which is compared with the experimental result 394.2 lb / in, and the finite element solution is 449.7 lb / in.

[0269] The present method gives a good agreement with the experimental results.

[0270] Results analysis for external pressure cases:

[0271] The above results are shown in Table 2, and it can be seen that the present method (especially the post-buckling minimum load, including the buckling mode) gives a better agreement with the experimental results. To further verify the correctness of the present method, the buckling loads of the internally stiffened grid shells (0 / 90) S and (-45 / 45 / 90 / 0) S under external pressure are calculated and compared with the analytical results of Gerhard et al. (Gerhard, C. S., Gurdal, Z., Kapania, R. K. Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory. NASA CR 1471, 1996), as shown in Tables 6 and 7 below: Table 6 Buckling load comparison results of geodesically stiffened orthotropic symmetrically laminated cylindrical shells (0 / 90) S under external pressure (R = 85 in, L = 100 in, t = 0.2 in, 1 x 12 grid shell,

[0272]

[0273] 3 FEA - Finite Element Analysis, an abbreviation; 4 Shell theory, Table 7 has the same meaning;

[0274] Gerhard, C. S., Gurdal, Z., Kapania, R. K., 1996. Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory. NASA CR 1471.

[0275] Table 7 Geodesic stiffened cylindrical composite shells under external pressure (-45 / 45 / 90 / 0) S Buckling load comparison results (R = 85 inches, L = 100 inches, t = 0.2 inches, 1 x 12 grid shell, )

[0276]

[0277] Gerhard, C. S., Gurdal, Z., Kapania, R. K., 1996. Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory. NASA CR 1471.

[0278] The geometric parameters are taken as R = 85 inches (1 inch = 25.4 mm), L = 100 inches, t = 0.2 inches, 1 x 12 geodesic stiffeners, and the height of the stiffeners is 0.5, 1.0, 1.5, 2.0 inches, respectively, and the thickness of the inner grid stiffeners is 0.2 inches. It can be seen that the current results are reasonably consistent with the results of Gerhard et al. (Gerhard, C. S., Gurdal, Z., Kapania, R. K. Finite element analysis of geodesically stiffened cylindrical composite shells using a layerwise theory. NASA CR 1471, 1996). ​

[0279] Figure 8 The buckling load of the laminated composite cylindrical shell numbered as CTM1 (0 / 90) 12T The post-buckling load-shortening curve of the laminated cylindrical shell under hydrostatic pressure with clamped boundary condition and the experimental results of Hur (2008) are compared. The data used in the calculation are: L = 600.0 mm, R / t = 62.7, t = 2.52 mm, E 11 = 162.0 GPa, E 22 = 9.6 GPa, G 12 = G 13 = 6.1 GPa, G 23 = 3.5 GPa, v 12 = 0.298. The initial imperfection is taken as and compared with the experimental results.

[0280] The results of the present method agree well with the experimental results.

[0281] Analysis of the torsional case results:

[0282] The results are shown in Table 3. It can be seen from the table that the results of the present method are very close to the experimental results. The results of Simitses et al. (Simitses G.J., Shaw, D., Sheinman, I. Stability of imperfect laminated cylinders: a comparison between theory and experiment. AIAA Journal, 23: 1086-1092, 1985) are higher than the results of the present method and the experimental results because the effect of the variable k is not considered in their solution. It can be seen that the results of the present method (especially the post-buckling minimum load, including the buckling mode) are closer to the experimental results.

[0283] To further verify the correctness of the present method, the buckling loads of the multi-layer composite cylindrical shell under torsion obtained by the present method are compared with those obtained by Zhou and Zhou (Zhou C, Zhou J. Buckling of multi-layer composite cylindrical shells under torsion. Thin-Walled Structures, 48: 1-12, 2009) using the energy method and the finite difference method. Table 8 gives the comparison of the present method with the experimental results of Zhou and Zhou, Tennyson (Tennyson, R.C. Buckling of laminated composite cylinders: a review. Composites, 6: 17-24, 1975), and the results of Tennyson based on the Koiter theory. The geometric parameters are given in Table 8. The specimens are glass fiber / epoxy cylindrical shells, and the material constants are: E 11 = 5.5 x 106 psi, E 22= 2.6 × 10⁶ psi, G 12 =G 13 =G 23 =0.7×106psi, ν 12 =0.37. The results show that the results in this paper are generally closer to the experimental results than those analyzed by Zhou Chengti and Zhou Jianping. At the same time, it should be noted that the influence of initial geometric defects should be taken into account when performing torsional buckling analysis.

[0284] Table 8. Buckling load of shear cylinder shell under torque (M) S ) cr (lbf·in) comparison (R=6.26in, t=0.027in, L=12.5in)

[0285]

[0286] Tennyson, RCBuckling oflaminated composite cylinders:areview.Composites,1975,6:17-24;

[0287] Zhou Chengti, Zhou Jianping. Nonlinear instability calculation of multilayer composite cylindrical shell. Applied Mathematics and Mechanics, 1986, 7(1):17-23.

[0288] Figure 4 Comparison of (15 / 0 / 10 / -10 / 0 / -15) S The buckling shear stress-shear strain curves of a cylindrical shell under torque and the experimental results of Derstine (Derstine, MS, Pindera, MJ, Bowles, DE Combined mechanical loading of composite tubes, NASA CR-183012, 1988). The calculated data used are: L = 10.0 inches (1 inch = 254 mm), R / t = 17.17, t = 0.02 inches, and the material used is P75 / 934 resin-based graphite fiber reinforced material E. 11 =35.25×106psi (1psi=6.895kPa), E 22 = 1.04 × 10⁶ psi, G 12 =G 13 = 0.570 × 10⁶ psi, v 12 =0.331, v 23 =0.49. When the effect of initial defects is taken into account. The calculation results obtained by this method agree well with the experimental results.

[0289] Figure 5 Given a shell of the same length under torque. Shearing a cylindrical shell (±45) 4T (±45) 2S and (-452 / -302 / 602 / 152) T Torque-end shortening and torque-turn curves for three layup methods. Assume all layups are of uniform thickness, with a total thickness of t = 4.0 mm, R / t = 40, and material constants for graphite / epoxy composites: E 11 =138.0 GPa, E 22 =8.9 GPa, G 12 =G 13 =5.17 GPa, G 23 =2.89 GPa, ν 12 =0.30, and in all graphs This represents the initial geometric defect value of the shell. From... Figure 5 As can be seen from this, for the shear shells of these three laying methods, (-452 / -302 / 602 / 152) T The shell has the highest torsional stiffness; (±45) 4T The shell has the largest buckling load and post-buckling strength, but its path is relatively gentle.

[0290] To further verify the correctness of this method, the buckling load of the longitudinally stiffened cylindrical shell under torque was calculated. We adopted the geometric material parameters of the SCT1 (unstiffened) and SCT2 (11 longitudinal stiffeners) specimens reported by Kuenzi and Norris (Kuenzi, EW, Norris, CB, 1962. Torsional buckling of longitudinally stiffened, thin-walled, plywood cylinders. No. 1563, United States Department of Agriculture, Forest Service, Forest Products Laboratory, Madison, Wisconsin, In Cooperation with the University of Wisconsin, 1962), as detailed in Table 9 below. The geometric parameters were taken as R = 9.15 inches (1 inch = 25.4 mm), L = 29 inches, t = 0.037 inches, with a stiffener height of 0.029 inches and a stiffener thickness of 0.246 inches for the internally uniformly spaced stiffeners. The material parameters of SCT1 are: E 11 =1.189 × 10⁶ psi 2, E 22 = 1.189 × 10⁶ psi, G12 = G 13 = G 23 = 0.871 x 106psi, v 12 = 0.39; SCT2 material parameters are: E 11 = 1.171 x 106psi, E 22 = 1.171 x 106psi, G 12 = G 13 = G 23 = 0.871 x 106psi, v 12 = 0.37, stringer material parameters are: E 11 = 2.026 x 106psi, G 12 = 0.74 x 106psi.

[0291] Table 9 Buckling loads τ under torque for longitudinally internally stiffened cylindrical shells cr (psi) Comparison results (R = 9.15 inches, L = 29 inches, t = 0.037 inches)

[0292]

[0293] 4 Kuenzi, E. W., Norris, C. B., 1962. Torsional buckling of longitudinally stiffened, thin-walled, plywood cylinders. No. 1563, United States Department of Agriculture Forest Service Forest Products Laboratory, Madison, Wisconsin, In Cooperation with the University of Wisconsin;

[0294] 5 Numbers in parentheses are the circumferential wave numbers of torsional buckling.

[0295] The torsional buckling critical load τ cr (psi) agrees well with the experimental results.

[0296] Further comparison of the calculation time of different calculation methods in the above cases can be known (as shown in Table 10), compared with the commercial finite element method, the calculation time of the present method is greatly reduced, which embodies the great efficiency advantage of semi-analytical explicit solution for this problem.

[0297] Table 10 Average calculation time of buckling of axially compressed, externally pressurized and torsionally stiffened cylindrical shells

[0298]

[0299] Both calculation methods were run on the same workstation (Inter Xeon CPU E5-2697 2.20 GHz processor, 256 GB).

[0300] It should be noted that in the present application, it should be understood that although the present application is described in the form of embodiments, not every embodiment contains only one independent technical solution, and the description manner of the specification is only for the sake of clarity, and the person skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be properly combined to form other embodiments which can be understood by the person skilled in the art.

Claims

1. A method for explicit fast analysis of buckling and post-buckling under mechanical loads for composite stiffened shells containing an exact curvature representation, characterized in that, The method comprises the following steps: Step 1: based on the high-order shear deformation theory, the transverse shear displacement-strain relationship of the cylindrical shell is distributed along the shell thickness direction according to a parabolic law; Step 2: the stiffened plate is equivalent to a variable stiffness plate structure, that is, the plate structure stiffness corresponding to the local stiffened part is superimposed with the influence of the stiffener stiffness increment; Step 3: The local neutral plane height can be determined according to the fact that the stress on the neutral plane of the plate structure under pure bending condition is zero ; Step 4: the stiffness coefficients of the area without the stiffener and the area with the stiffener are combined into a variable stiffness function; Meanwhile, in order to ensure the derivability of the stiffness coefficient matrix with respect to the position coordinates, a hyperbolic tangent function is introduced to smoothly transition the variable stiffness coefficient matrix; Step 5: according to the equivalent constitutive relation of the composite stiffened plate, the internal force and bending moment expressions of the stiffened cylindrical shell structure under general laying conditions are obtained; Step 6: according to the Hamilton principle, the Euler-Lagrange equation is used to obtain the equilibrium differential equation of the composite stiffened cylindrical shell; Step 7: Introducing the general form of initial imperfection expression, the balance differential equation is dimensionless, and a small parameter with obvious physical meaning is introduced , that is, it is inversely proportional to the equivalent length geometric parameter of the shell ; When The equilibrium differential equation of the composite stiffened cylindrical shell is a boundary layer type equation. Step 8: the singular perturbation method is used for solving, and the solution of the equation is divided into a regular solution and a boundary layer solution; Step 9: the numerical value of the initial defect of the fully anisotropic stiffened cylindrical shell is processed to form the deflection value generally distributed at different positions of the shell; At the same time, considering the influence of thermal effect, the thermal bending moment and the initial deflection are generated and substituted into the equilibrium differential equation of the composite stiffened cylindrical shell for calculation; Step 10: Press The equilibrium differential equations of discrete composite stiffened cylindrical shells of the same order power can be used to obtain perturbation equations of various orders. Solve them one by one and synthesize the canonical solution and boundary layer solution to obtain the large deflection asymptotic solution that strictly satisfies the fixed boundary condition in an asymptotic sense. On this basis, the quantitative relationship expression of deflection and rotation angle and the boundary layer width expression of shell buckling are obtained, and the equivalent stress of shell structure is obtained by using the constitutive relationship expression; Step 11: Obtain dimensionless deflection w With pressure And shear stress Expressed post buckling equilibrium path; wherein: Axial compression buckling: ；(1a) ；(1b) And: ；(1c) External pressure buckling: ；(2a) ；(2b) And: ；(2c) Torsional buckling: ；(3a) ；(3b) And: ；(3c) And the relative torsion angle: ；(3d) Step 12: Convert the expressions (1a)-(3d) into dimensionless maximum deflection using the second order perturbation parameters, i.e. Step 12: Convert the expressions (1a)-(3d) into dimensionless maximum deflection using the second order perturbation parameters, i.e. ；(4) wherein is the dimensionless maximum deflection, taken in the deflection expression point, has: ；(5) Step 13: the post-buckling equilibrium path of the shear cylindrical shell under the action of the torsion load is obtained with the dimensionless maximum deflection as the perturbation parameter; The rectangular cross-section stiffener is adopted, and according to the coordinate position of the stiffened structure and the material or geometric parameters of the stiffened structure, the equivalent stress balance and strain displacement compatibility relationship of the stiffened area is obtained: ；(6) wherein: is the in-plane bending related stress, right upper index and denote the stress variables of the corresponding panel and rib, respectively; Based on the composite laminated plate theory, the material stiffness coefficients of the plate can be obtained are: ；(7) The rib is simplified as a flexible beam structure, and the stiffness coefficient is derived by taking a rectangular cross-section laminated beam as an example ; Further derivation gives the equivalent stiffness coefficient of the local region satisfies the following conditions: ； ； ； ； ； ；(8) wherein, Kpand Ksare the stiffness matrices of the plate and the strip, respectively, with respect to the neutral surface.

2. The method of fast explicit analysis of buckling and post-buckling under mechanical loads according to claim 1, characterized in that, Considering the condition of oblique stiffening, the stiffness coefficient matrix of formula (8) can be rewritten as: ； ； ； (9) wherein, and is the geometric equation of the parallel line of the root muscle strip; is the local-global coordinate conversion matrix of the root muscle strip; is the transition area smoothing coefficient; Thus, the equivalent variable stiffness coefficient function established by the hyperbolic tangent function is obtained, which considers the influence of the tensile-torsional, tensile-bending and bending-torsional coupling stiffness.

3. The method of fast explicit analysis of buckling and post-buckling under mechanical loads according to claim 1 or 2, characterized in that, The boundary layer solution is of the order of magnitude, the different buckling boundary layer effect parameters of the order of magnitude, the different buckling boundary layer effect parameters of the order of magnitude, the different buckling boundary layer effect parameters of the order of magnitude, the different buckling boundary layer effect parameters of the order of magnitude, the different buckling boundary layer effect parameters Axial compression buckling: ；(10) ；(11) ；(12) ；(13) ；(14) ；(15) wherein is a defect parameter.

4. The method of fast explicit analysis of buckling and post-buckling under mechanical loads according to claim 3, characterized in that, The deflection value is generated by the Fourier expansion method to generate the term corresponding to the coefficient in formula (10), formula (12) and formula (14).

5. The method of fast explicit analysis of buckling and post-buckling under mechanical loads according to claim 4, characterized in that, The post-buckling equilibrium path is obtained by substituting formula (5) into formula (1a)-(3d).

6. The method of fast explicit analysis of buckling and post-buckling under mechanical loads according to claim 5, characterized in that, For perfecting the shell Or , let , usually ; For the general initial imperfection Or The terms corresponding to the coefficients in equation (11), equation (13) and equation (15) are generated by using the Fourier expansion method, and by comparison, the minimum buckling load and the corresponding buckling mode ( m, n ) are easily obtained.

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