Linear buckling calculation method for multi-layer fiber composite material cylindrical shell

Abstract

The invention discloses a linear buckling calculation method for a multi-layer fiber composite material cylindrical shell. The method comprises the following steps: constructing a displacement expression and a transverse stress expression of a double-helix mode for a multi-layer fiber composite material cylindrical shell structure; solving a balance equation of the three-dimensional elastic composite cylindrical shell under the action of hydrostatic pressure, and obtaining an expression of prestress in three directions; constructing a three-dimensional deformation differential equation taking displacement and transverse stress as unknown quantities; based on a displacement expression and a transverse stress expression of a double-helix mode, constructing a state space equation set taking displacement and transverse stress as state vectors; and constructing the relationship between the state vectors of the inner surface layer and the outer surface layer of the multi-layer fiber composite material cylindrical shell structure, solving the buckling modes of the multi-layer fiber composite material cylindrical shell, and obtaining the buckling load in each buckling mode. According to the method, the overall buckling load of the underwater composite material thick-wall structure can be accurately estimated.

Description

Technical Field

[0001] The present application relates to the technical field of composite materials, and more specifically, to a linear buckling calculation method for a multilayer fiber composite cylindrical shell. Background Art

[0002] In the field of submersibles or submarines, the fiber-reinforced composite materials can achieve the same load-bearing capacity and effective load at a lower weight and cost by utilizing their high specific strength, high specific stiffness, seawater corrosion resistance, and strong designability. In particular, the advantages of multi-layer fiber-wound composite pressure-resistant structures are more obvious. However, the anisotropy and multi-layer form of composite materials make the buckling load-bearing characteristics of composite thick-walled cylindrical shells more complicated. Local stress excess and internal defects in composite structures are very likely to cause a serious decrease in the load-bearing capacity of the entire shell, which puts higher requirements on the accurate solution and prediction of the buckling characteristics of deep-water composite pressure-resistant structures.

[0003] However, the existing methods cannot accurately estimate the complex stress field inside the thick-walled composite structure, and thus cannot effectively estimate the bearing capacity of the entire shell. There are two main limitations: 1. The eccentric effect of the fiber laying angle makes it difficult to accurately solve the differential equation of the three-dimensional deformation of the composite multilayer structure; 2. The membrane stress of the cylindrical shell based on the two-dimensional shell theory cannot consider the defects of the prestress in the thickness direction and the non-uniform distribution of the prestress between layers. An accurate solution method for the buckling characteristics of deep-water composite pressure-resistant structures is urgently needed to solve the above problems. Summary of the invention

[0004] In response to at least one defect or improvement need in the prior art, the present application provides a linear buckling calculation method for a multi-layer fiber composite cylindrical shell, which can solve at least one of the problems existing in the above-mentioned prior art.

[0005] To achieve the above object, according to a first aspect of the present application, a linear buckling calculation method for a multilayer fiber composite cylindrical shell is provided, the method comprising:

[0006] Construct displacement expressions and transverse stress expressions of the double helix mode for the multilayer fiber composite cylindrical shell structure;

[0007] Solve the three-dimensional elastic equilibrium equation of composite cylindrical shell under hydrostatic pressure and obtain the prestress expression;

[0008] Construct three-dimensional deformation differential equations with displacement and lateral stress as unknown quantities;

[0009] For each composite material layup, a state space equation set with displacement and transverse stress as state vectors is constructed based on the displacement expression and transverse stress expression of the double helix mode;

[0010] The relationship between the state vectors of the inner and outer surfaces of the multilayer fiber composite cylindrical shell structure is constructed. Combined with the known boundary conditions of the inner and outer surfaces, the buckling mode of the multilayer fiber composite cylindrical shell is solved and the buckling load under each buckling mode is obtained.

[0011] Furthermore, in the above-mentioned multi-layer fiber composite column shell linear buckling calculation method, the multi-layer fiber composite materials in the multi-layer fiber composite column shell structure include oblique laying, orthogonal laying, unidirectional 0° laying, unidirectional 90° laying, symmetrical laying and asymmetrical laying.

[0012] Furthermore, in the above-mentioned linear buckling calculation method of the multilayer fiber composite cylindrical shell, the displacement expression and the transverse stress expression of the double helix mode are constructed, specifically including:

[0013] In the cylindrical coordinate system, the displacement expression of the double helix mode is:

[0014]

[0015] Where (u, v, w) represent the displacements in three directions, λ1+λ2=2mπ / L is the constraint condition of the correlation coefficient, m is the axial half-wave number, n is the annular wave number, and L is the length of the cylindrical shell;

[0016] According to the three-dimensional elastic theory and the state space method, for each layer of fiber composite material, the transverse stress expression of the double helix mode is:

[0017]

[0018] where σ r is the normal stress, τ xr and τ rθ is the shear stress, r represents the thickness direction, θ represents the hoop direction, and x represents the axial direction.

[0019] Furthermore, the linear buckling calculation method of the multilayer fiber composite cylindrical shell mentioned above, the solving of the three-dimensional elastic deformation differential equation of the composite cylindrical shell under hydrostatic pressure to obtain the expression of the transverse prestress specifically includes:

[0020] Under hydrostatic pressure, the normal stress generated by the fiber composite cylindrical shell structure includes the initial stress σ in the thickness direction. r 0 , initial hoop stress σ θ 0 and the initial axial stress σ x 0 In the cylindrical coordinate system, there are equilibrium equations for the initial stress in the thickness direction, the initial stress in the hoop direction, and the initial stress in the axial direction:

[0021]

[0022] The stress and displacement have the following relationship

[0023]

[0024] Under the actual hydrostatic pressure, the stress generated in the fiber composite cylindrical shell structure is gradient along the thickness direction and evenly distributed along the circumferential and axial directions.

[0025]

[0026] τ xr 0 =τ xθ 0 =τ rθ 0 =0

[0027] Get the expression for initial stress

[0028]

[0029] When the ply angle is not zero,

[0030]

[0031] Taking the limits of the above four expressions, we get σ x 0 , σ r 0 and σ θ 0 After solving the expression, the normal stress in three directions is obtained as

[0032]

[0033] The hydrostatic pressure on the shell surface only exists in the normal direction, that is, the hydrostatic pressure p ext , the pressure inside the shell is zero,

[0034]

[0035] The coefficients of the transverse stress expression are obtained as

[0036]

[0037] Furthermore, the above-mentioned linear buckling calculation method of the multilayer fiber composite cylindrical shell, the construction of a three-dimensional deformation differential equation with displacement and lateral stress as unknown quantities, specifically includes:

[0038] Combining the material constitutive relation and the displacement-strain relation in the cylindrical coordinate system, the relationship between the in-plane stress and displacement is obtained as follows:

[0039]

[0040] Combining the equilibrium equations of initial stress in the thickness direction, initial stress in the hoop direction, and initial stress in the axial direction, the three-dimensional deformation differential equation containing displacement terms and transverse stress terms is obtained.

[0041]

[0042] Based on the relationship between transverse stress and displacement and The expression is

[0043]

[0044] The three-dimensional deformation differential equation with displacement and lateral stress as unknowns is obtained as

[0045]

[0046] The coefficient p i for

[0047]

[0048] Furthermore, the linear buckling calculation method of the multilayer fiber composite cylindrical shell mentioned above, wherein the state space equations with displacement and transverse stress as state vectors are constructed, specifically includes:

[0049] Get the radial derivatives of normal stress and shear stress

[0050]

[0051]

[0052] Decouple the partial differential equations of the normal stress and shear stress derivatives with respect to the radial direction to obtain the spatial state equation group

[0053]

[0054] Among them, F j =[U,V,W,X,Y,Z] T is the state vector of the jth layer, Q j is a sixth-order coefficient matrix.

[0055] Furthermore, the above-mentioned linear buckling calculation method of the multilayer fiber composite cylindrical shell realizes the solution of the buckling mode of the multilayer fiber composite cylindrical shell and obtains the buckling load under each buckling mode, which specifically includes:

[0056] Solve the spatial state equations and obtain the general solution:

[0057]

[0058] in, is the state vector of the outer side of the jth layer, is the state vector inside the jth layer;

[0059] Constructing the relationship between the state variables of the inner and outer shell layers

[0060]

[0061] Among them, F + p is the outermost state variable, is the innermost state variable, T is the transfer matrix, and the expression is Where p is the total number of plies;

[0062] Combined with the free surface boundary conditions of the inner and outer layers, the relationship between the state variables of the inner and outer layers is solved to obtain the buckling load under each buckling mode.

[0063] In general, the above technical solutions conceived by the present application can achieve the following beneficial effects compared with the prior art:

[0064] The present application provides a linear buckling calculation method for a multi-layer fiber composite cylindrical shell. By constructing displacement expressions and transverse stress expressions of a double helix mode, a state space equation group with displacement and transverse stress as state vectors is constructed for each composite layer according to the three-dimensional elastic deformation theory. Combined with the known boundary conditions of the inner and outer surfaces, a high-precision solution of the buckling mode of the entire shell is achieved. The buckling mode of a composite cylindrical shell laid at any angle can be solved, and the influence of axial, thickness and annular prestress is taken into account. It complies with the prestress characteristics of the shell under hydrostatic pressure, and can accurately estimate the stress field inside the composite material to ensure the safety of a submersible or submarine sailing underwater. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0066] Figure 1 A schematic flow chart of a linear buckling calculation method for a multilayer fiber composite cylindrical shell provided in an embodiment of the present application;

[0067] Figure 2 A schematic diagram of the fiber direction of a composite material structure laid at any angle provided in an embodiment of the present application;

[0068] Figure 3 A schematic diagram of the off-axis effect provided in an embodiment of the present application;

[0069] Figure 4 Displacement cloud diagram of the double helix mode displacement expression and the traditional Fourier series expansion displacement expression provided in the embodiments of the present application. DETAILED DESCRIPTION

[0070] In order to make the purpose, technical solutions and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not intended to limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0071] The terms "including" and "having" and any variations thereof in the specification and claims of the present application and the above drawings are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device comprising a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units that are not listed, or may optionally include other steps or units that are inherent to these processes, methods, products or devices.

[0072] The existing methods for accurately solving the buckling characteristics of deepwater composite pressure-resistant structures have the following problems:

[0073] (1) The eccentric effect of the fiber laying angle makes it difficult to accurately solve the differential equation of the deformation of the composite multilayer structure. The eccentric effect of the fiber laying angle is one of the main manifestations of the anisotropy of fiber-reinforced composite materials, that is, the fiber laying direction is not orthogonal to the cylindrical coordinate system, that is, non-zero or non-90° laying. For non-orthogonal laying of fiber layers, the material constitutive relationship in the cylindrical coordinate system can be expressed as:

[0074]

[0075] The above formula can be expressed as σ = Cε, where the elastic matrix C contains 13 independent elastic constants. The internal force-deformation relationship derived based on the two-dimensional shell theory is as follows:

[0076]

[0077] Among them, the stiffness matrix consists of the in-plane stiffness submatrix A, the bending-stretching coupling stiffness submatrix B and the bending stiffness submatrix D. For orthotropic materials, functionally gradient materials and isotropic materials, there are C 16 =C 26 =C 36 =C 45=0, A 16 =A 26 =A 45 =D 16 =D 26 = 0. For multilayer shells with symmetrical layers, there is also a coupled stiffness element B 16 =B 26 =0, which simplifies the derived deformation partial differential equation a lot. The traditional Fourier series expansion combined with the separation of variables method to represent the displacement can decouple the equation and thus achieve the solution, as shown in the following formula.

[0078]

[0079] However, for fiber multilayer cylindrical shells with eccentric effect at any angle, there is a C 16 , C 26 , C 36 , C 45 is not zero, which makes the eccentric stiffness term A in the stiffness matrix non-zero 16 , A 26 , A 45 , D 16 , D 26 This results in that when solving the deformation problem of composite materials laid at any angle, it is difficult to decouple the deformation equation by using the traditional Fourier series expansion combined with the separation of variables method to represent the displacement, which makes it difficult to accurately solve the deformation differential equation of the composite multilayer structure. When using the three-dimensional elastic theory combined with the state space method to solve the buckling characteristics of the composite multilayer thick shell, C 16 , C 26 , C 36 , C 45 Being non-zero also makes it difficult to decouple the state-space equations.

[0080] In view of the difficulty in solving the deformation differential equations of composite multilayer structures caused by the eccentric effect of fiber laying angle, the current research on the buckling characteristics of cylindrical shells is mainly focused on orthotropic materials, functional gradient materials and homogeneous isotropic material shells. There are only 9 independent elastic constants in the material constitutive relationship, and there is no problem in dealing with the eccentric effect. For special positive and negative staggered wound composite cylindrical shells, or when the lateral load is much larger than the in-plane load, the eccentric stiffness element A 16 , A 26 , A 45 , D 16 , D 26It is relatively small and can be treated as zero, which also avoids the problem of dealing with eccentric effects. This greatly simplifies the difficulty of solving the equations and can obtain relatively satisfactory accuracy. However, in actual engineering structures, the fiber laying angles are diverse and not all orthogonal. This treatment method is not applicable to the buckling problem of composite cylindrical shells laid at arbitrary angles or short shells dominated by axial forces. Some scholars have tried to abandon the traditional Fourier series expansion and instead express the displacement with orthogonal polynomials, but due to the rounding error of the computer, the numerical stability of high-order polynomials is not as good as that of the Fourier series form.

[0081] (2) The membrane stress of cylindrical shells based on two-dimensional shell theory cannot take into account the defects of non-uniform distribution of prestress in the thickness direction and prestress between layers. For the buckling problem of cylindrical shell structures under hydrostatic pressure, the stress inside the shell is not zero along the axial, thickness and circumferential directions, and is not a constant value along the coordinate axis, but a variable quantity. For the buckling problem of thick shells, this change gradient is even greater. However, the two-dimensional shell theory uses first-order or high-order functions to assume the lateral displacement field, which is not enough to describe the vibration problem of multi-layer structures with large changes in material properties along the lateral direction. It cannot take into account the continuity of lateral stress and the prestress that changes along the thickness direction. It is difficult to apply it to the buckling characteristics of thick composite shells under hydrostatic pressure.

[0082] The membrane stress of cylindrical shells in two-dimensional shell theory cannot consider the problem of non-uniform distribution of prestress in the thickness direction and prestress between layers. At present, when researchers use two-dimensional shell theory to study the buckling problem of thin shells, the influence of hydrostatic pressure is calculated according to the axial membrane force. Hoop membrane stress This method is accurate enough for studying the buckling of thin shells, but for thick shells, the prestress distribution in three directions under hydrostatic pressure is more uneven, and the applicability of the membrane stress treatment method becomes worse. When using three-dimensional elastic theory to study the buckling of composite cylindrical shells, scholars' research mainly focuses on the buckling of the entire shell under axial uniform load, and does not consider the influence of thickness direction and circumferential prestress, which is obviously not in line with the prestress characteristics of the shell under hydrostatic pressure.

[0083] In view of the above problems, this application provides a linear buckling calculation method for a multilayer fiber composite cylindrical shell. Figure 1 A schematic flow chart of a linear buckling calculation method for a multilayer fiber composite cylindrical shell provided in an embodiment of the present application includes:

[0084] 101. Construct the displacement expression and lateral stress expression of the double helix mode for the multilayer fiber composite cylindrical shell structure.

[0085] Specifically, Figure 2As shown in the figure, the fiber direction and geometric coordinate system of the composite material structure laid at any angle are neither coincident nor orthogonal. Under the action of external force, the structure has coupling of deformation modes in the x-direction and y-direction, that is, the plate direction, axial direction and circumferential direction, which is defined as the "eccentric effect", as shown in the figure. Figure 3 As shown in the figure, when the ply angle is 0°, the buckling deformation direction of the rectangular plate is consistent or orthogonal to the geometric coordinate system. When the ply angle is 45°, the buckling deformation direction of the rectangular plate presents a certain angle, and there is a coupling effect.

[0086] In order to accurately describe the coupling of deformation modes in different coordinate directions caused by the oblique ply pattern of the fiber composite cylindrical shell structure, this application adopts the double helix mode displacement expression, that is, the double Fourier series expansion to represent the displacement. In the cylindrical coordinate system, the displacement expression of the double helix mode is

[0087]

[0088] Where (u, v, w) represent the displacements in three directions, λ1+λ2=2mπ / L is the constraint condition of the correlation coefficient, m is the axial half-wave number, n is the annular wave number, and L is the length of the cylindrical shell;

[0089] According to the three-dimensional elastic theory and state space method, for each layer of fiber composite material, the three-directional displacement and three transverse stresses are used as state variables, and the transverse normal stress and shear stress can also be written in the following form according to the double helix mode:

[0090]

[0091] where σ r is the normal stress, τ xr and τ rθ is the shear stress, r represents the thickness direction, θ represents the hoop direction, and x represents the axial direction.

[0092] Using a double helix pattern to represent displacement and lateral stress can effectively handle the "eccentric effect" caused by skew paving. Take "0≤x≤6, 0≤θ≤1.5π, m=n=1, λ1 / λ2=1 / 9" as an example. Figure 4 As shown, by comparing the double helix mode displacement expression (right figure) with the traditional Fourier series expansion displacement expression (left figure), it can be seen that the displacement cloud map drawn based on the double helix mode displacement expression has the conditions and characteristics to describe the axial and circumferential coupled deformation of the skew-laid cylindrical shell.

[0093] Compared with the traditional displacement form of Fourier series combined with separation of variables method, this method can decouple the three-dimensional deformation differential equation and describe the deformation coupling in different coordinate directions.

[0094] 102. Solve the expression for the prestress caused by hydrostatic pressure and obtain the coefficients of the expression for the lateral stress.

[0095] Specifically, under the action of hydrostatic pressure, the cylindrical shell will not produce shear membrane stress, but only normal stress. The normal stress generated by the fiber composite cylindrical shell structure includes the initial stress in the thickness direction. Initial hoop stress and the initial stress in the axial direction In order to accurately describe the real stress distribution of the composite cylindrical shell structure in a hydrostatic pressure environment, the embodiment of the present application is based on the three-dimensional elastic theory. For each ply of the composite material, the constitutive relationship and strain-displacement relationship of the material in linear elastic mechanics are combined to derive the equilibrium equations of the initial stress in the thickness direction, the initial stress in the hoop direction, and the initial stress in the axial direction in the cylindrical coordinate system:

[0096]

[0097] The initial stress in the thickness direction That is, the axial membrane stress, the symbol is positive for tension and negative for pressure, and the initial stress in the hoop is The symbol is that the external pressure is negative, the internal pressure is positive, and the initial stress in the axial direction The base hoop stress of the film is greater than 0 for tensile stress and less than 0 for compressive stress.

[0098] The stress and displacement have the following relationship

[0099]

[0100] Under the actual hydrostatic pressure, the stress generated in the fiber composite cylindrical shell structure is gradient along the thickness direction and evenly distributed along the circumferential and axial directions.

[0101]

[0102] τ xr 0 =τ xθ 0 =τ rθ 0 =0

[0103] Get the expression for initial stress

[0104]

[0105] When the ply angle is not zero,

[0106]

[0107] Under hydrostatic pressure, the initial stresses of an axisymmetric shell satisfy the following equilibrium equations:

[0108]

[0109] Combining the above two equations, we can obtain an equation that only contains the lateral initial displacement:

[0110]

[0111] make

[0112]

[0113] Then the above equation containing only the initial lateral displacement can be simplified to

[0114]

[0115] Where r is a number greater than zero, r can be written as r = e t Substituting into the above formula, the variable coefficient second-order differential equation is transformed into a constant coefficient second-order differential equation

[0116]

[0117] The characteristic equation is

[0118]

[0119] The characteristic root is

[0120]

[0121] From the expressions of B1, B2, and B3, we can see that k1 and k2 are related to material parameters and ply angles. For common materials, these two roots are real roots, and the roots of the equation are

[0122]

[0123] After solving, the normal stresses in three directions are obtained as follows:

[0124]

[0125] Among them, C1 and C2 are still unknowns, and these two coefficients can be solved according to the initial boundary conditions of the inner and outer surfaces of the shell. When the cylindrical shell is placed in it, it is assumed that water is incompressible and that water only produces normal pressure on the shell surface, that is, the hydrostatic pressure p ext , the pressure inside the shell is 0,

[0126]

[0127] The coefficients of the transverse stress expression are obtained as

[0128]

[0129] The prestress derived here is more realistic than traditional membrane stress, and can solve the problem that membrane stress cannot consider the stress in the thickness direction of the cylindrical shell and the non-uniform distribution of interlayer stress.

[0130] 103. Construct a three-dimensional deformation differential equation with displacement and lateral stress as unknown quantities.

[0131] Specifically, according to the three-dimensional elastic deformation theory, only three-directional displacements (u, v, w) and three transverse stresses (σ r ,τ xr ,τ rθ ) is a three-dimensional deformation differential equation with unknown variables.

[0132] In order to construct a three-dimensional deformation differential equation with displacement and transverse stress as unknowns, other unknowns in the equilibrium equation are expressed as displacement and transverse stress. Combining the material constitutive relation and the displacement-strain relation in the cylindrical coordinate system, the relationship between the in-plane stress and the displacement is obtained as follows:

[0133]

[0134] Combining the equilibrium equations of initial stress in the thickness direction, initial stress in the hoop direction, and initial stress in the axial direction, the three-dimensional deformation differential equation containing displacement terms and transverse stress terms is obtained.

[0135]

[0136] Based on the relationship between transverse stress and displacement and The expression is

[0137]

[0138] The three-dimensional deformation differential equation with displacement and lateral stress as unknowns is obtained as

[0139]

[0140] The coefficient p i for

[0141]

[0142] 104. For each composite material ply, a state space equation group with displacement and transverse stress as state vectors is constructed based on the displacement expression and transverse stress expression of the double helix mode.

[0143] Specifically, obtain the derivatives of normal stress and shear stress with respect to the radial direction

[0144]

[0145]

[0146] In order to decouple the above three formulas, the partial differential equations of the radial derivatives of normal stress and shear stress are decoupled to obtain the spatial state equation group:

[0147]

[0148] Among them, F j =[U,V,W,X,Y,Z] T is the state vector of the jth layer, Q j is a sixth-order coefficient matrix.

[0149] 105. Construct the relationship between the state vectors of the inner and outer surfaces of the multilayer fiber composite cylindrical shell structure, combine the known boundary conditions of the inner and outer surfaces, solve the buckling mode of the multilayer fiber composite cylindrical shell, and obtain the buckling load under each buckling mode.

[0150] Specifically, the spatial state equation is a first-order homogeneous linear differential equation about the matrix. Solving the spatial state equation system, we obtain its general solution as

[0151]

[0152] in, is the state vector of the outer side of the jth layer, is the state vector on the inner side of the jth layer, and the corresponding spatial state equation and solution can be obtained for each layer of composite material layup.

[0153] In order to solve the critical buckling mode of the whole shell, the transfer matrix method is used to construct the relationship between the state variables of the inner and outer shell layers.

[0154]

[0155] Among them, F + p is the outermost state variable, is the innermost state variable, T is the transfer matrix, and the expression is Where p is the total number of plies; combined with the free boundary conditions of the inner and outer surface layers, that is, the stress of the inner surface layer is zero, the positive stress of the outer surface layer is the hydrostatic pressure, and the shear stress is zero, the transfer matrix is ​​simplified and solved to obtain the buckling load under each buckling mode, and the minimum value is the critical buckling load.

[0156] A linear buckling calculation method for a multi-layer fiber composite cylindrical shell provided in an embodiment of the present application constructs a displacement expression and a transverse stress expression of a double helix mode. According to the three-dimensional elastic deformation theory, a state space equation group with displacement and transverse stress as state vectors is constructed for each composite layer. Combined with the known boundary conditions of the inner and outer surface layers, a high-precision solution of the buckling mode of the entire shell is achieved. The buckling mode of a composite cylindrical shell laid at any angle can be solved, and the influence of the thickness direction, axial direction and annular prestress is taken into account. It conforms to the prestress characteristics of the shell under the action of hydrostatic pressure, and can accurately estimate the critical load of the overall crushing of the composite thick-walled cylindrical shell, thereby ensuring the safety of underwater navigation of a submersible or submarine.

[0157] Optionally, an embodiment of the present application provides a method for calculating the linear buckling of a multi-layer fiber composite column shell, in which the multi-layer fiber composite materials in the multi-layer fiber composite column shell structure are one or a mixture of oblique laying, unidirectional 0° laying, unidirectional 90° laying, orthogonal laying, symmetrical laying and asymmetrical laying.

[0158] The following is a specific example of a method for calculating the linear buckling of a multilayer fiber composite cylindrical shell provided in an embodiment of the present application:

[0159] Taking a simply supported composite multilayer cylindrical shell as an example, the displacement and lateral stress expressions of the double helix mode are:

[0160]

[0161] Substituting the above two equations into the derivatives of displacement and transverse stress with respect to the thickness direction coordinate, canceling the trigonometric terms (only showing the sine and cosine of λx+nθ, changing n in the following equation to -n will give the coefficients of the trigonometric terms with sine and cosine of λx-nθ), we can get

[0162]

[0163] According to the above four formulas, the coefficient matrix Q of the state space equations of each layer can be obtained j , substituting into the relationship between the state variables of the inner and outer shell surfaces, the transfer matrix of the whole shell state vector can be obtained, and the buckling mode can be solved by combining with the simply supported boundary.

[0164] In order to verify the theoretical method proposed in this application, the thick composite cylindrical shell mentioned in the literature (Shen Kechun "Stability Analysis of Fiber-Wound Cylindrical Shell under Hydrostatic Pressure", Journal of Northwestern Polytechnical University, 2018) was used as an example for calculation and compared with the traditional method in the literature and the finite element calculation results. The results are shown in Table 1. The parameters of the cylindrical shell are: length L = 0.695m, radius R = 0.15m, thickness h = 7.8mm; material parameters are: E1 = 121×10 9MPa, E2=8.6×10 9 MPa, μ 12 =0.253, μ 23 =0.421, G 12 =3.35×10 9 MPa,G 23 =2.85×10 9 MPa. Among these three methods, the finite element method is the most mature and has the highest accuracy. As can be seen from the table, for three different ply sequences, when the buckling mode is m=1 and n=3, the maximum error between the method proposed in this application and the finite element method does not exceed 5%, while the maximum error of the traditional method in the literature reaches 28%.

[0165] Table 1 Comparison with the results of literature calculations

[0166]

[0167] The shell material parameters of the second verification example are consistent with those of the first example, with a thickness of 30 mm, a thickness-to-diameter ratio of 0.2 (a thick shell), and a length and radius consistent with those of the first example. The critical buckling load and buckling mode for different ply sequences were calculated, which are in good agreement with the results of the finite element method calculation, with a maximum error of no more than 5%, as shown in Table 2.

[0168] Table 2 Comparison with finite element results

[0169]

[0170] It can be seen from the implementation results of the present application in the above two examples that the present application can be used to calculate and predict the critical buckling load of composite multilayer cylindrical shells.

[0171] It should be noted that, for the aforementioned method embodiments, for the sake of simplicity, they are all expressed as a series of action combinations, but those skilled in the art should be aware that the present application is not limited by the described order of actions, because according to the present application, certain steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily required by the present application.

[0172] The above is only an exemplary embodiment of the present disclosure, and the scope of the present disclosure cannot be limited thereto. That is, any equivalent changes and modifications made according to the teachings of the present disclosure are still within the scope of the present disclosure. After considering the specification and practicing the disclosure here, those skilled in the art will easily think of the implementation scheme of the present disclosure. This application is intended to cover any modification, use or adaptation of the present disclosure, which follows the general principles of the present disclosure and includes common knowledge or customary technical means in the technical field not recorded in the present disclosure. The description and examples are regarded as exemplary only, and the scope and spirit of the present disclosure are defined by the claims.

[0173] The technical features of the above embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0174] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application shall be included in the scope of protection of the present application.

Claims

1. A linear buckling calculation method for a multilayer fiber composite cylindrical shell, characterized in that: include: Construct displacement expressions and transverse stress expressions of the double helix mode for the multilayer fiber composite cylindrical shell structure; Solving the equilibrium equation of the three-dimensional elastic composite cylindrical shell under hydrostatic pressure to obtain an expression for the transverse stress; Construct three-dimensional deformation differential equations with displacement and lateral stress as unknown quantities; For each composite material layup, a state space equation set with displacement and transverse stress as state vectors is constructed based on the displacement expression and transverse stress expression of the double helix mode; The relationship between the state vectors of the inner and outer surfaces of the multilayer fiber composite cylindrical shell structure is constructed. Combined with the known boundary conditions of the inner and outer surfaces, the buckling mode of the multilayer fiber composite cylindrical shell is solved and the buckling load under each buckling mode is obtained.

2. The linear buckling calculation method of a multilayer fiber composite cylindrical shell according to claim 1, characterized in that: The laying method of the multi-layer fiber composite material in the multi-layer fiber composite material column shell structure includes one or more combinations of oblique laying, orthogonal laying, unidirectional 0° laying, unidirectional 90° laying, symmetrical laying and asymmetrical laying.

3. The linear buckling calculation method of a multilayer fiber composite cylindrical shell according to claim 1, characterized in that: The displacement expression and lateral stress expression for constructing the double helix mode specifically include: In the cylindrical coordinate system, the displacement expression of the double helix mode is: Where (u, v, w) represent the displacements in three directions respectively, λ1+λ2=2mπ / L is the restriction condition of the correlation coefficient, m is the half-wave number of the cylindrical shell in the axial direction, n is the wave number in the circumferential direction, and L is the length of the cylindrical shell; According to the three-dimensional elastic theory and the state space method, for each layer of fiber composite material, the transverse stress expression of the double helix mode is: where σ r is the normal stress, τ xr and τ rθ is the shear stress, r represents the thickness direction, θ represents the hoop direction, and x represents the axial direction.

4. The linear buckling calculation method of a multilayer fiber composite cylindrical shell according to claim 3, characterized in that: The method of solving the three-dimensional elastic equilibrium equation of the composite cylindrical shell under hydrostatic pressure to obtain the expression of the transverse stress specifically includes: Under hydrostatic pressure, the normal stress generated by the fiber composite column shell structure includes initial stress in the thickness direction. Initial hoop stress and the initial stress in the axial direction In the cylindrical coordinate system, there are equilibrium equations for the initial stress in the thickness direction, the initial stress in the hoop direction, and the initial stress in the axial direction: The stress and displacement have the following relationship The stress generated by the fiber composite cylindrical shell structure under the actual hydrostatic pressure is gradient along the thickness direction and evenly distributed along the circumferential and axial directions. t xr 0 =t xθ 0 =t rθ 0 =0 Get the expression for initial stress When the ply angle is not zero, Taking the limits of the above four expressions, we get σ x 0 , σ r 0 and σ θ 0 After solving the expression, the normal stress in three directions is obtained as Water only produces normal pressure on the shell surface, that is, hydrostatic pressure p ext , the pressure inside the shell is zero, The coefficients of the transverse stress expression are obtained as 5. The linear buckling calculation method of a multilayer fiber composite cylindrical shell according to claim 4, characterized in that: The construction of the three-dimensional deformation differential equation with displacement and lateral stress as unknown quantities specifically includes: Combining the material constitutive relation and the displacement-strain relation in the cylindrical coordinate system, the relationship between the in-plane stress and displacement is obtained as follows: Combining the equilibrium equations of initial stress in the thickness direction, initial stress in the hoop direction, and initial stress in the axial direction, the three-dimensional deformation differential equation containing displacement terms and transverse stress terms is obtained. Based on the relationship between transverse stress and displacement and The expression is The three-dimensional deformation differential equation with displacement and lateral stress as unknowns is obtained as The coefficient p i for p 29 =p1p 14 +p4p 13 ;p 30 =p2p 14 +p5p 13 ;p 31 =p2p 13 +p4p 14 ;p 32 =p3p 13 +p5p 14 ; p 33 =p6p 14 +p5p 13 ;p 34 =p1p 15 +p4p 14 ;p 35 =p2p 15 +p5p 14 ;p 36 =p2p 14 +p4p 15 ;p 37 =p6p 13 +p4p 14 ; p 38 =p3p 14 +p5p 15 ;p 39 =p5p 14 +p6p 15 ;p 40 =p6p 14 +p4p 15 ;p 41 =p 31 +p 37 -p9;p 42 =p 30 +p 33 -p7; p 43 =p 36 +p 40 -p7;p 44 =p 35 +p 39 -p8;p 45 =p 13 +p 15 6. The linear buckling calculation method for a multilayer fiber composite cylindrical shell according to claim 5, characterized in that: The construction of a state space equation group with displacement and lateral stress as state vectors specifically includes: Obtain the derivatives of normal and shear stresses with respect to the thickness direction Substitute the displacement and transverse stress double helix mode expressions into the displacement, normal stress and shear stress derivative partial differential equations with respect to the thickness direction to decouple them and obtain the spatial state equation group: Among them, F j =[U,V,W,X,Y,Z] T is the state vector of the jth layer, Q j is a sixth-order coefficient matrix.

7. The linear buckling calculation method of a multilayer fiber composite cylindrical shell according to claim 6, characterized in that: The buckling modes of multilayer fiber composite cylindrical shells are solved and the buckling loads under each buckling mode are obtained, including: Solve the spatial state equations and obtain the general solution: in, is the state vector of the outer side of the jth layer, is the state vector inside the jth layer; Constructing the relationship between the state vectors of the inner and outer surfaces of a cylindrical shell in, is the outermost state variable, is the innermost state variable, T is the transfer matrix, and the expression is Where p is the total number of plies; The relationship between the inner and outer surface state vectors is solved to obtain the buckling load under each buckling mode.

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