Buckling analysis method for functionally graded cylindrical shell structures with axisymmetric imperfections

Abstract

This invention discloses a buckling analysis method for functionally graded material cylindrical shell structures with axisymmetric defects. The method includes dividing the cylindrical shell structure into multiple regions based on the location and number of axisymmetric defects; proposing continuity assumptions based on these regions to obtain compatibility conditions; these continuity assumptions include displacement continuity, rotational continuity, bending moment continuity, shear force continuity, axial internal force continuity, and equivalent torsional continuity; deriving the buckling control equation for a defect-free cylindrical shell under a Lagrangian system; importing the buckling control equation for the defect-free cylindrical shell into a Hamiltonian system for solution, obtaining expressions for the buckling modes of each region; obtaining the buckling modes of each region based on the proposed buckling mode equivalence assumption, compatibility conditions, fixed boundary conditions, and expressions for the buckling modes of each region; and combining the buckling modes of multiple regions to obtain the buckling modes of the cylindrical shell structure. This invention is simple, rapid, and efficient, and is beneficial for engineering applications.

Description

Technical Field

[0001] This invention belongs to the technical field of buckling strength design and analysis of cylindrical shell structures, specifically relating to a buckling analysis method for cylindrical shell structures made of functionally graded materials with axisymmetric defects. Background Technology

[0002] Functionally graded materials (FGMs) are composites of two or more materials whose composition changes continuously along the thickness direction from one side to the other. This results in a continuous gradient in macroscopic material properties. This design can significantly reduce or even eliminate the problem of property mismatch between material interfaces, largely avoiding stress concentration or discontinuities caused by interfacial discontinuities in traditional laminated composites. This gives FGMs advantages that traditional materials cannot match. FGMs are widely used in aerospace, military, and automotive fields, for example, in aerospace thermal insulation structures, aircraft stealth structures, mortar tube walls, and automotive sound-absorbing and energy-dampening structures. Cylindrical shell structures, due to their good mechanical properties, simple structure, and ease of manufacturing, are widely used in various engineering projects to manufacture primary or secondary load-bearing structures, such as rocket shells, propellant tanks, and support rods and actuators for various components of aerospace vehicles. These structures are characterized by being hollow and having a high aspect ratio; some structures also have a large diameter-to-thickness ratio. Structural stability must be considered when the structure is under load. Various experimental studies have shown that the buckling load of theoretically calculated cylindrical shell structures is significantly lower than the experimental values. The generally accepted reason is that engineering structures inevitably have microscopic defects and dispersion at the material level. Slender or thin-walled structures are highly sensitive to these defects, and this microscopic inhomogeneity causes the actual buckling load to be lower than the theoretical calculation. For such buckling problems, an initial defect is usually introduced to calculate the buckling load under this condition. Currently, three methods are commonly used in engineering for buckling analysis of cylindrical shell structures considering initial defects: 1) using empirically verified formulas and introducing a large safety factor for conservative estimation; 2) using the finite element method to introduce initial modal defects for calculation; 3) using the inverse or semi-inverse method of traditional elasticity, assuming that the structural buckling modes satisfy a certain mathematical form, and solving for the unknown coefficients in the preset modal solution through boundary conditions. While the first method is simple, it is too conservative, and the calculation accuracy varies depending on the type of shell. For example, when used for thick shells, thin shells, or long shells, frequent adjustments of the coefficients are needed to meet engineering accuracy requirements. Furthermore, this method fails to provide crucial features such as structural buckling modes. The second method can calculate structural buckling loads and modes, but the results are affected by the discrete dimensions of the structure, requiring prior verification such as mesh independence to obtain a mesh that meets accuracy requirements before it can be used for structural validation. Additionally, for structures of different sizes, the model needs to be continuously rebuilt. The third method's assumption of the fundamental solution form significantly impacts the calculation results; it is difficult to implement when the form of the solution cannot be accurately known beforehand.In recent years, some universities have proposed structural buckling analysis methods based on Hamiltonian systems and successfully applied them to buckling calculations of cylindrical shell structures. However, the most common applications are for isotropic cylindrical shells, and cases for functionally graded materials mainly include thin shells, Donnell short shells, and Reddy higher-order shear shells. Moreover, there are very few cases that consider the influence of defects. Summary of the Invention

[0003] The purpose of this invention is to overcome the above-mentioned defects and provide a buckling analysis method for functionally graded material cylindrical shell structures with axisymmetric defects. This method solves the technical problems of low accuracy and limited application scenarios of existing analysis methods. This invention is simple, fast and efficient, which is conducive to engineering application.

[0004] To achieve the above-mentioned objectives, the present invention provides the following technical solution:

[0005] This invention, based on Reisner shell theory, proposes an analytical method for the elastic buckling problem of functionally graded material (FGR) cylindrical shell structures considering axisymmetric defects. First, the buckling control equations and solution method for defect-free FGR cylindrical shells are established. Then, based on the modal superposition concept, a solution strategy for the buckling problem of FGR cylindrical structures with axisymmetric defects is proposed. The principle of this method is to decompose the FGR cylindrical shell structure with axisymmetric defects into defect-free regions and defective regions. When the overall structure buckles, it is equivalent to assuming that both defect-free and defective regions buckle simultaneously. Superimposing the buckling modes of the local regions yields the buckling modes of the overall structure. A key consideration is establishing reasonable and accurate region connection conditions to ensure that the results conform to the buckling characteristics of the actual structure. This invention proposes an equivalent linear analysis strategy for analyzing complex buckling problems containing local geometric nonlinearities, thus offering simplicity, speed, and efficiency.

[0006] A buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects, comprising:

[0007] Based on the location and number of axisymmetric defects, the cylindrical shell structure is divided into multiple regions;

[0008] We propose the buckling mode equivalence assumption: the overall buckling mode of a cylindrical shell with axisymmetric defects is equivalent to the superposition state of the buckling modes in each region;

[0009] Based on multiple regions, continuity assumptions are proposed to obtain compatibility conditions; the continuity assumptions include displacement continuity, rotational continuity, bending moment continuity, shear force continuity, axial internal force continuity, and equivalent torsion continuity.

[0010] The buckling control equation for a defect-free cylindrical shell is obtained under the Lagrange system.

[0011] The buckling control equation of a defect-free cylindrical shell is introduced into the Hamiltonian system for solution, and the expressions of the buckling modes in each region are obtained respectively.

[0012] Based on the buckling mode equivalence assumption, compatibility condition, fixed boundary condition, and expressions for the buckling modes in each region, the buckling modes in each region are obtained.

[0013] By superimposing the buckling modes of multiple regions, the overall buckling mode of the cylindrical shell structure is obtained.

[0014] Furthermore, methods for dividing a cylindrical shell structure into multiple regions based on the location and number of axisymmetric defects include:

[0015] When n axisymmetric defects are located at the n ends of a cylindrical shell, the cylindrical shell structure is divided into n+1 regions by n dividing lines, which include n regions containing defects and 1 defect-free region, where n = 1 or 2.

[0016] When m axisymmetric defects are located at the non-ends of a cylindrical shell, the cylindrical shell structure is divided into 2m+1 regions by 2m dividing lines, which include m regions containing defects and m+1 defect-free regions, where m is an integer greater than 1.

[0017] When m axisymmetric defects are located at the non-ends of a cylindrical shell and n axisymmetric defects are located at the n ends of the cylindrical shell, the cylindrical shell structure is divided into 2m+n+1 regions, which include m+n regions containing defects and m+1 regions without defects.

[0018] The dividing line is perpendicular to the axial direction of the cylindrical shell structure.

[0019] Furthermore, when m axisymmetric defects are located at the non-ends of the cylindrical shell, the compatibility condition and fixed boundary condition are as follows:

[0020]

[0021] Let the cylindrical coordinate system be (r, θ, x), where r is the radial direction of the cylindrical shell, θ is the circumferential direction of the cylindrical shell, and x is the axial direction of the cylindrical shell. Y = {M x Q x N x ,T xθ} T The superscripts for X and Y indicate sequential numbering of multiple regions; the positions of the 2m boundary lines are x = L1, L2, ..., L 2m-1 ,L 2m The axial dimension of the cylindrical shell is L; u, v, w represent the displacement of any point on the surface of the shell. M represents a turning angle. x Q represents bending moment. x Represents shear force, Nx T represents the axial internal force. xθ It represents the equivalent torque.

[0022]

[0023] Furthermore, when n axisymmetric defects are located at n ends of a cylindrical shell, the compatibility condition and fixed boundary condition are as follows:

[0024]

[0025] Let the cylindrical coordinate system be (r, θ, x), where r is the radial direction of the cylindrical shell, θ is the circumferential direction of the cylindrical shell, and x is the axial direction of the cylindrical shell. Y = {M x Q x N x ,T xθ} T The superscripts for X and Y indicate sequential numbering of multiple regions; when n=1, the boundary line is located at x=L1, and when n=2, the boundary line is located at x=L1,L2; the axial dimension of the cylindrical shell is L; u, v, w represent displacements. M represents a turning angle. x Q represents bending moment. x Represents shear force, N x T represents the axial internal force. xθ It represents the equivalent torque.

[0026] Furthermore, the buckling governing equation for a defect-free cylindrical shell is:

[0027]

[0028] Let the cylindrical coordinate system be (r, θ, x), where r is the radial direction of the cylindrical shell, θ is the circumferential direction of the cylindrical shell, and x is the axial direction of the cylindrical shell. R is the radius of the cylindrical shell, h is the thickness of the cylindrical shell, and E z For elastic modulus, υ z Let be Poisson's ratio, u, v, w represent the displacement of any point on the mid-surface, and N0 be the uniform compressive load borne by the cross-section of the transverse cylindrical shell.

[0029]

[0030] Where, Φ={q T ,p T} T Let represent the original state variable and its dual variable to be solved in the Hamiltonian system; the original state variable is . The dual variable is p = {p1, p2, p3, p4} T , represents the generalized force corresponding to the original state variable; H is the Hamiltonian operator matrix,

[0031] Furthermore, the expression for the buckling mode is:

[0032]

[0033] Where, ψ n (x) is a symplectic eigenvalue, λ n Let λ be a symplectic eigenvalue. n =in(n=0,±1,±2,...), non-zero λ n The corresponding ψ n (x) represents two categories, namely and

[0034]

[0035] c is the intermediate coefficient. nk There are 8 relatively independent constants, and k takes the value: k = 1, 2... 8.

[0036] Furthermore, the intermediate coefficients are denoted as follows:

[0037]

[0038]

[0039] c nk (k = 1, 2…8) are 8 relatively independent constants;

[0040] ξ nk (k = 1, 2, ..., 8) is the equation a8ξ 8 +a6ξ 6 +a4ξ 4 +a2ξ 2 The root of +a0 = 0:

[0041]

[0042] In the above equation, γ = KR 2 / D=12R 2 / h 2 N cr =N0 / D,N cr This indicates the dimensionless buckling critical load.

[0043] Furthermore, based on the buckling mode equivalence assumption, compatibility condition, fixed boundary condition, and expressions for the buckling modes of each region, the method for obtaining the buckling modes of each region is as follows:

[0044] When an axisymmetric defect is located at a non-end of a cylindrical shell, the cylindrical shell structure is divided into three regions by two dividing lines, including one region containing the defect and two regions without defects.

[0045] The compatibility condition and the fixed boundary condition are as follows:

[0046]

[0047] Let the cylindrical coordinate system be (r, θ, x), where r is the radial direction of the cylindrical shell, θ is the circumferential direction of the cylindrical shell, and x is the axial direction of the cylindrical shell. Y = {M x Q x N x ,T xθ} T The superscripts X and Y indicate sequential numbering of multiple regions; the positions of the two boundary lines are x = L1 and L2, respectively; the axial dimension of the cylindrical shell is L; u, v, and w represent the displacement of any point on the surface of the shell. M represents a turning angle. x Q represents bending moment. x Represents shear force, N x T represents the axial internal force. xθ Indicates equivalent torque;

[0048] The expression Φ for the buckling modes of each region 1 Φ 2 and Φ 3 Substituting into the above formula, after simplification, we obtain a system of linear equations A containing 24 equations. k c = 0, where c is a vector of undetermined constants c = {c n1 ,c n2 ,…,c n24} T A k Here is the corresponding coefficient matrix, k = 1, 2, ..., 24;

[0049] Determine the undetermined constant vector c and the critical buckling load N in the system of equations. cr ;

[0050] Undetermined constant vector c and critical buckling load N cr Once confirmed, Φ 1 Φ 2 and Φ 3 To determine the value.

[0051] Furthermore, the undetermined constant vector c and the critical buckling load N in the equation system are determined.cr The method is as follows:

[0052] For a cylindrical shell structure to buckle, the necessary and sufficient condition for the buckling modes to necessarily not all be zero is |A k | = 0;

[0053] According to equation |A k |=0 determines the dimensionless critical buckling load N containing the defect region. cr ;

[0054] N cr Substitute the value into the system of equations A k When c = 0, we obtain the vector of undetermined constants c.

[0055] Furthermore, the defect includes circumferential stiffening.

[0056] Compared with the prior art, the present invention has at least one of the following advantages:

[0057] (1) This invention creatively proposes a buckling mode analysis method for Reissner-type functionally graded material cylindrical shells that consider axisymmetric defects. This method can realize the rapid solution and parameterized analysis of buckling modes of structurally complex cylindrical shells, which is beneficial to improving the analysis accuracy and efficiency of buckling modes.

[0058] (2) The present invention is simple, fast and efficient, which is conducive to its practical application in engineering.

[0059] (3) This invention is applicable to various situations, including those with multiple axisymmetric defects, those with boundary axisymmetric defects, and those with multiple circumferential stiffeners, and has strong applicability;

[0060] (4) The present invention is based on buckling mode analysis, which is conducive to the stability design of cylindrical shell structure and has broad application prospects in aerospace heat insulation structure, aircraft stealth structure, mortar tube wall, automobile sound absorption and energy absorption structure. Attached Figure Description

[0061] Figure 1 This is a schematic diagram of the cylindrical coordinate system of the present invention;

[0062] Figure 2 This is a schematic diagram of the cylindrical shell containing axisymmetric defects according to the present invention. Detailed Implementation

[0063] The features and advantages of the present invention will become clearer and more apparent from the following detailed description.

[0064] The term “exemplary” as used herein means “serving as an example, embodiment, or illustration.” Any embodiment illustrated herein as “exemplary” is not necessarily to be construed as superior to or better than other embodiments. Although various aspects of embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated otherwise.

[0065] This invention proposes an analytical calculation method for the buckling problem of Reissner-type functionally graded material cylindrical shell structures, considering the influence of axisymmetric defects, based on the Hamiltonian system. The proposed method is applicable to the buckling strength design and analysis of cylindrical shell structures, and is particularly suitable for the analysis and verification of buckling loads and modes of functionally graded material cylindrical shell structures in engineering applications, considering initial defects, uneven material distribution, or local variable cross-sections, providing theoretical support for structural stability design. Specific steps include:

[0066] 1. Solving the buckling governing equations for defect-free functionally graded cylindrical shells

[0067] (1) Under the Hamiltonian system, the buckling control equation of defect-free functional graded material cylindrical shells is derived using Reisner shell theory:

[0068] Consider a radius of R, a length of L, and a thickness of h, with the thickness varying from -h / 2 to h / 2, and an elastic modulus E. z Poisson's ratio υ z The cylindrical shell. Taking the cylindrical coordinate system as (r, θ, x), see... Figure 1 Where r is radial, θ is circumferential, x is axial, and the corresponding displacement is (w, v, u). Assume that when the shell buckles, the cross-section is subjected to a uniform compressive load N0. Taking a two-component functionally graded material as an example, the material's equivalent elastic modulus can be expressed as:

[0069] E z =E1V1+E2(1-V1) (1)

[0070] Where V1 represents the volume content of the first component, (1-V1) represents the volume content of the second component, and E1 and E2 are the elastic moduli of the two components, respectively.

[0071] The equivalent Poisson's ratio of a material can be expressed as:

[0072] υ z =υ1V1+υ2(1-V1) (2)

[0073] Wherein, υ1 and υ2 are the Poisson's ratios of the two components, respectively.

[0074] Based on the Kirchhoff-Love assumption, the strain relationship at any point in the shell is:

[0075]

[0076] Among them, e x e θ e xθ ε represents the strain component at any point on the shell in cylindrical coordinates. x , ε θ , ε xθ κ represents the strain component at the corresponding point on the surface within the shell. x κ θ κ xθ The corresponding curvature component is z, where z is the position of a point on the shell at the r-coordinate (-h / 2≤z≤h / 2).

[0077] The relationship between mid-surface strain, curvature and mid-surface displacement of the shell is as follows:

[0078]

[0079] The stress-strain relationship is

[0080]

[0081] Where, σ x , σ θ , σ xθ These represent the stresses in each coordinate system direction. The subscript xθ indicates that the shear stress lies in the plane with x as the normal and points in the θ direction.

[0082] The resultant force and resultant moment per unit length on the surface of the shell can be obtained by integrating the stress components.

[0083]

[0084] Where, N x N θ N xθ These are the resultant forces in each coordinate system direction.

[0085]

[0086] Among them, M x M θ M xθ These are the resultant moments in each coordinate direction.

[0087] Substituting equation (5) into equations (6) and (7) yields...

[0088]

[0089] Among them, A ij B ij D ij(i,j=1,2,3) are the tensile stiffness, coupling stiffness and bending stiffness matrix components of the functionally graded material cylindrical shell, respectively. remember Then equation (8) can be written as

[0090]

[0091] The strain energy U of a cylindrical shell can be expressed as:

[0092]

[0093] The work done by the external force W can be expressed as

[0094]

[0095] Neglecting kinetic energy, the work-energy equation is:

[0096] L g =UW (12)

[0097] According to the Lagrange variational principle and integration by parts, the Lagrange function can be obtained from (12) as follows:

[0098]

[0099] Substitute equation (4) into equation (13) and define the symbol. Then introduce circumferential rotation angle Then equation (13) can be expressed as

[0100]

[0101] By taking variational results for u, v, and w in equation (14), the buckling control equations for the Lagrange system can be obtained:

[0102]

[0103] (2) Solving the buckling control equation for defect-free functional graded material cylindrical shells:

[0104] When solving the buckling control equation of a cylindrical shell in the Lagrange system, methods such as Gaussian elimination are commonly used. However, this buckling control equation is a fourth-order partial differential equation containing only three variables to be solved. Without introducing some other assumptions, the solution becomes difficult. Therefore, the problem is solved in the Hamiltonian system. The buckling control equation of a defect-free functionally graded material cylindrical shell is imported into the Hamiltonian system for solution. First, the original variable vector is defined. Then, through variational methods, the dual variable vector p = {p1, p2, p3, p4} under the Hamiltonian system is obtained. T They are respectively

[0105]

[0106] The Hamiltonian function can be written as

[0107]

[0108] Let the total state vector Φ = {q} T ,p T} T Using Hamilton's principle, the Hamiltonian canonical equations are obtained by variational analysis of the Hamiltonian function (17). Right now

[0109]

[0110] H is the Hamiltonian operator matrix.

[0111]

[0112] in

[0113]

[0114]

[0115] In the Hamiltonian system, the solution to the equation can be expressed as:

[0116]

[0117] Where λ n For symplectic eigenvalues ​​and ψ n (x) represents the symplectic eigenvalues ​​(vectors), each corresponding to a buckling mode. Based on the continuity of the circumferential end conditions of the cylindrical shell (θ = 0 and θ = 2π), the symplectic eigenvalues ​​can be expressed as λ. n =in(n=0,±1,±2,...), the symplectic eigenvalue can be obtained through the characteristic equation (inI-H)ψ n Given (x) = 0, the symplectic eigenvalues ​​corresponding to non-zero eigenvalues ​​can be divided into two categories: one for cases where n is greater than zero, and the other for cases where n is less than zero. These are denoted as follows: and And there is a relationship: Introducing functionals

[0118]

[0119] Where ψ1 and ψ2 are any two sets of symplectic eigensols, and (q1,p1) and (q2,p2) are the solution vectors of the two sets of solutions, respectively.

[0120] It can be proven that there exists a symplectic conjugate orthogonality relationship between symplectic eigensoles as follows:

[0121]

[0122] Where, δ nm Let the notation be Kronecker, n, m = 1, 2, 3...

[0123] Here, only the following are listed The form of the solution ( (Solution similarly), as shown below.

[0124]

[0125] The intermediate coefficients are denoted as follows:

[0126]

[0127] c nk There are 8 relatively independent constants. The values ​​of k in each intermediate coefficient of the above (23) are: k = 1, 2... 8.

[0128] Intermediate parameter γ = KR 2 / D=12R 2 / h 2 .

[0129] Here ξ nk It is a root of the following equation:

[0130] a8ξ 8 +a6ξ 6 +a4ξ 4 +a2ξ 2 +a0=0 (24)

[0131] Define N cr =N0 / D represents the dimensionless critical buckling load, then the coefficient in equation (24) can be expressed as

[0132]

[0133] The above gives the form of the solution to the buckling problem of a defect-free cylindrical shell, namely equations (20) and (23), which includes the critical buckling load N to be determined. cr .

[0134] 2. Solution of buckling problem of cylindrical shells made of functionally graded materials considering axisymmetric defects.

[0135] When considering structures containing axisymmetric defects, the introduction of local nonlinear geometric factors renders various linear exact solution methods, including those mentioned above, inapplicable. Therefore, some reasonable and necessary assumptions need to be made to simplify the problem. The buckling mode equivalence assumption is proposed: when a cylindrical shell structure containing axisymmetric defects buckles, the axial force N at each location on the shell cross-section... cr All are equal, and buckling occurs separately in the defective and defect-free regions. The mode obtained by combining the buckling modes of each region is the overall buckling mode of the cylindrical shell. Since the defective structure still has axisymmetry, it is foreseeable that the buckling mode of the defective structure is similar to that of the defect-free structure under axial compression, except that the defect affects the magnitude of the buckling deformation and its axial position. Without loss of generality, assuming that the defect is not at the end of the structure, the cylindrical shell can be divided into three regions along the axial direction, as shown in the diagram. Figure 2 As shown. The three regions are denoted as I, II, and III, each with a uniform thickness. Regions I and III are intact, with a thickness of h1. Region II is a missing region with a thickness of h2. In this invention, I, II, and III are equivalent to 1, 2, and 3.

[0136] When the overall structure buckles, assuming that the intact region and the missing region also buckle simultaneously under the same buckling load, the buckling modes of each region can still be solved using the linear method in the previous section. However, the boundary conditions for each region differ. The main difference lies in the fact that the connection between the intact and missing regions is no longer a fixed boundary condition, but rather a deformation and force transmission compatibility condition. Here, the following continuity assumptions are made for the connection region: displacements u, v, w are continuous; rotation angles are continuous. Continuous; bending moment M x Continuous; shear force Q x Continuous; Axial internal force N x and equivalent torque T xθ The specific expression of the buckling modes in each region can be given by substituting equation (23) into equation (20), and denoted as Φ respectively. Ⅰ Φ Ⅱ and Φ Ⅲ By connecting the buckling modes of each region together, the buckling mode Φ of the overall structure is obtained. At this point, Φ... Ⅰ , Φ Ⅱ and Φ Ⅲ The boundary conditions and the previously mentioned compatibility conditions must be satisfied, as listed below:

[0137]

[0138] in, Y = {M x Q x N x ,T xθ}T The superscripts I, II, and III indicate the partition to which each physical quantity belongs.

[0139] The specific expressions for each component of Y are as follows:

[0140]

[0141] Equation (26) can be simplified as follows:

[0142] A k c = 0 (28)

[0143] Where c is a vector of undetermined constants, c = {c n1 ,c n2 ,…,c n24} T A k (k = 1, 2, ..., 24) is the coefficient matrix corresponding to c.

[0144] The system of equations (28) contains 24 equations, including the critical buckling load N. cr And the undetermined constant vector c, a total of 25 parameters to be determined. First, when the structure buckles, the buckling modes must not all be zero, that is, the system of equations (28) has non-zero solutions, the necessary and sufficient condition being that the coefficient matrix A k The determinant of is zero, therefore we get

[0145] |A k |=0 (29)

[0146] From the above, the critical buckling load N of the incomplete structure can be obtained. cr , will N cr Substituting into equation (28) and solving the system of linear equations yields the unknown undetermined constant vector c, which in turn leads to Φ. Ⅰ , Φ Ⅱ and Φ Ⅲ Combining these parameters yields the overall buckling mode Φ of the incomplete structure, thus solving the problem. It should also be noted that the critical loads obtained from (29) are not singular; they can be arranged in ascending order and denoted as... (n = 0, 1, 2...; m = 0, 1, 2...), which means that for different n, the m-th critical buckling load is...

[0147] 3. Other defects

[0148] This invention mainly derives a buckling solution method for functionally graded material cylindrical shells containing axisymmetric defects. The aforementioned solution derivation process only discusses the case where the defect is located at the non-end of the shell and only considers one defect. According to the proposed strategy, it can be fully derived to other special cases.

[0149] First, consider the case where the defect is located at the end of the shell. In this case, the shell can be divided into two regions, I and II, along the axial direction (assuming a boundary at x = L1). The buckling mode expressions for each region are determined by equations (23) and (20). The boundary conditions and compatibility conditions that the buckling modes of each region must satisfy are simplified as follows:

[0150]

[0151] Using the method described in Section 2 of the invention, the modes of each region are obtained from equations (20) and (23), and the linear equation system established by combining equation (30) has a solution condition that the determinant of the matrix of undetermined coefficients of the equation system is zero. Then the critical buckling load is obtained, and the corresponding buckling modes can be obtained by solving the linear equation system.

[0152] Second, consider the case involving multiple axisymmetric defects. Assume there are m (m = 2, 3, 4...) defects, and the defects are not at the ends. Divide the shell into (2m+1) partitions, with the boundary positions of each defect being x = L1, L2, ... L... 2m-1 L, m2 The boundary conditions and compatibility conditions that each buckling mode in each segment must satisfy are as follows:

[0153]

[0154] In equation (31), the superscripts of X and Y represent the m-th (m = 2, 3, 4...) partition. The solution strategy for the system of equations obtained from this equation is the same as that in Section 2, except that the dimension of the system of equations is increased. If the case where the defect is located at the end needs to be considered, the partition can be reduced by one (one end contains the defect) or two (both ends contain the defect), which is the same idea as the evolution of equation (26) to equation (31).

[0155] Third, the circumferentially stiffened cylindrical shell structure. The circumferentially stiffened cylindrical shell structure is still an axisymmetric structure. The only difference from the structure with defects is that the thickness of the structure increases at the stiffening points. Each stiffener divides the cylindrical shell into regions with varying thickness along the axial direction, including unstiffened regions and stiffened regions. The buckling modes of each sub-region still satisfy equation (20). Therefore, the buckling problem of this type of structure can also be solved using the same solution method as in Section 2 of the invention. In the equation, this is reflected in the change of tensile stiffness and bending stiffness in the Hamiltonian operator matrix H in the Hamiltonian canonical equation (18), as detailed in the parameter description in equation (8). Of course, the solution method of this invention is also applicable to the case of multiple axial stiffenings, and the idea is the same as the interpretation of equation (31).

[0156] In summary, this invention derives a method for solving the buckling control equation of a Reissner-type functionally graded material cylindrical shell with axisymmetric defects. This method has the following advantages: (1) It creatively proposes a solution strategy for the buckling problem of a Reissner-type functionally graded material cylindrical shell considering axisymmetric defects. This strategy, through the approximate transformation of the buckling modes of the cylindrical shell structure with defects, can realize the solution of the buckling problem with geometric nonlinear characteristics by solving the linear equation system, thus providing a rational and feasible method for solving such complex buckling problems; (2) The solution process of this invention mainly uses the solution of the linear equation system and the solution of the determinant of its coefficient matrix. It can realize the rapid solution and parameterization analysis of complex buckling problems through a simple program, which has the characteristics of simplicity, speed and efficiency, and is conducive to engineering application; (3) This invention is applicable to various situations such as having multiple axisymmetric defects, having boundary axisymmetric defects and having multiple circumferential stiffeners, and has strong applicability. The buckling analysis method for functionally graded material cylindrical shell structures with axisymmetric defects of this invention can accurately analyze the buckling load and buckling modes of cylindrical shell structures, providing theoretical guidance for the buckling resistance design of similar products and improving the stability and reliability of the products.

[0157] The present invention has been described in detail above with reference to specific embodiments and exemplary examples; however, these descriptions should not be construed as limiting the present invention. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and embodiments of the present invention without departing from the spirit and scope of the invention, and all such modifications and improvements fall within the scope of the present invention. The scope of protection of the present invention is defined by the appended claims.

[0158] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A buckling analysis method for a functionally graded material cylindrical shell structure containing axisymmetric defects, characterized in that, include: Based on the location and number of axisymmetric defects, the cylindrical shell structure is divided into multiple regions; We propose the buckling mode equivalence assumption: the overall buckling mode of a cylindrical shell with axisymmetric defects is equivalent to the superposition state of the buckling modes in each region; Based on multiple regions, continuity assumptions are proposed to obtain compatibility conditions; the continuity assumptions include displacement continuity, rotational continuity, bending moment continuity, shear force continuity, axial internal force continuity, and equivalent torsion continuity. The buckling control equation for a defect-free cylindrical shell is obtained under the Lagrange system. The buckling control equation of a defect-free cylindrical shell is introduced into the Hamiltonian system for solution, and the expressions of the buckling modes in each region are obtained respectively. Based on the buckling mode equivalence assumption, compatibility condition, fixed boundary condition, and expressions for the buckling modes in each region, the buckling modes in each region are obtained. By superimposing the buckling modes of multiple regions, the overall buckling mode of the cylindrical shell structure is obtained; The buckling governing equation for a defect-free cylindrical shell is: Take cylindrical coordinate system as ,in, r For the radial direction of the cylindrical shell, It is a cylindrical shell with a circumferential orientation. x For the axial direction of the cylindrical shell, , , R Where is the radius of the cylindrical shell. h For the thickness of the cylindrical shell, E z For elastic modulus, Poisson's ratio, u , v , w This represents the displacement of any point on the mid-surface. The uniform compressive load borne by the cross section of the transverse cylindrical shell; The expression for the buckling mode is: ； in, For the explanation of Xin Benzheng, Let symmetric eigenvalues ​​be denoted as symmetric eigenvalues. ( ), non-zero corresponding There are two categories, namely and , , , , , , , , , As an intermediate coefficient, They are 8 relatively independent constants. k The possible values ​​are: .

2. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 1, characterized in that, Methods for dividing a cylindrical shell structure into multiple regions based on the location and number of axisymmetric defects include: when n An axisymmetric defect is located in the cylindrical shell. n Each end, with n The dividing line divides the cylindrical shell structure into n +1 area, which includes n One area contains a defective region and one area does not contain a defective region. n =1 or 2; when m An axisymmetric defect is located at the non-end of the cylindrical shell, with 2 m The dividing line divides the cylindrical shell structure into 2 m +1 area, which includes m A region containing defects and m +1 defect-free area m It is an integer greater than 1; when m An axisymmetric defect is located at a non-end of the cylindrical shell. n An axisymmetric defect is located in the cylindrical shell. n The cylindrical shell structure is divided into 2 parts at each end. m+n+ One area, which includes m+n A region containing defects and m+ One defect-free area; The dividing line is perpendicular to the axial direction of the cylindrical shell structure.

3. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 2, characterized in that, when m An axisymmetric defect is located at a non-end of a cylindrical shell. The compatibility conditions and fixed boundary conditions are as follows: Take cylindrical coordinate system as ,in, r For the radial direction of the cylindrical shell, It is a cylindrical shell with a circumferential orientation. x For the axial direction of the cylindrical shell, , The superscripts for X and Y indicate sequential numbering of multiple regions; 2 m The positions of the dividing lines are as follows: The axial dimension of the cylindrical shell is ; u , v , w This represents the displacement of any point on the surface of the shell. Indicates a corner. Indicates bending moment, Indicates shear force. Indicates axial internal force. Represents equivalent torque 。 4. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 2, characterized in that, when n An axisymmetric defect is located in the cylindrical shell. n The compatibility conditions and fixed boundary conditions for each end are as follows: or Take cylindrical coordinate system as ,in, r For the radial direction of the cylindrical shell, It is a cylindrical shell with a circumferential orientation. x For the axial direction of the cylindrical shell, , The superscripts for X and Y indicate sequential numbering of multiple regions; when n =1, the position of the dividing line is ,when n =2, the dividing line is located at... The axial dimension of the cylindrical shell is ; u , v , w Indicates displacement. Indicates a corner. Indicates bending moment, Indicates shear force. Indicates axial internal force. It represents the equivalent torque.

5. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 1, characterized in that, The intermediate coefficients are denoted as: , , , , , , , ; ， ； They are 8 relatively independent constants; It is an equation The root: In the above equations , , This indicates the dimensionless buckling critical load.

6. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 1, characterized in that, Based on the buckling mode equivalence assumption, compatibility condition, fixed boundary condition, and expressions for the buckling modes in each region, the method for obtaining the buckling modes in each region is as follows: When an axisymmetric defect is located at a non-end of a cylindrical shell, the cylindrical shell structure is divided into three regions by two dividing lines, including one region containing the defect and two regions without defects. The compatibility condition and the fixed boundary condition are as follows: Take cylindrical coordinate system as ,in, r For the radial direction of the cylindrical shell, It is a cylindrical shell with a circumferential orientation. x For the axial direction of the cylindrical shell, , The superscripts for X and Y indicate sequential numbering of multiple regions; the positions of the two dividing lines are respectively... The axial dimension of the cylindrical shell is ; u , v , w This represents the displacement of any point on the surface of the shell. Indicates a corner. Indicates bending moment, Indicates shear force. Indicates axial internal force. Indicates equivalent torque; Expressions for buckling modes in each region , and Substituting into the above formula, after simplification, we obtain a system of linear equations containing 24 equations. ,in, Vector of undetermined constants , The corresponding coefficient matrix ，k =1,2,…,24; Determine the vector of undetermined constants in the system of equations and critical buckling load ; vector of undetermined constants and critical buckling load Once confirmed, , and To determine the value.

7. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 6, characterized in that, Determine the vector of undetermined constants in the system of equations and critical buckling load The method is as follows: The necessary and sufficient condition for a cylindrical shell structure to have non-zero buckling modes during buckling is: ; According to the equation Determine the dimensionless critical buckling load containing the defect region. ; Will Substitute the value into the system of equations The vector of undetermined constants is obtained. .

8. The buckling analysis method for a functionally graded material cylindrical shell structure with axisymmetric defects according to claim 1, characterized in that, The defect includes circumferential stiffening.

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