Method for determining elastic-plastic buckling critical load of ring-ribbed cylindrical shell

Abstract

The application discloses a kind of ring rib cylindrical shell elastic-plastic buckling critical load determination method, comprising the following steps: S1, the size and material parameters of ring rib cylindrical shell are obtained, and the Hamilton control equation of ring rib cylindrical shell is constructed;S2, the eigen vector of Hamilton control equation of ring rib cylindrical shell is calculated by separation of variables method, and first-third eigen parameters are obtained;S3, generalized displacement expression is constructed according to first-third eigen parameters;S4, generalized displacement expression is substituted into fixed boundary condition, and the elastic-plastic buckling critical load of ring rib cylindrical shell under fixed boundary condition is obtained.The application can break through the limit of semi-inverse method, does not need to assume the form of solution in advance, can be applied to the design of ring rib cylindrical shell, the application can efficiently and accurately obtain the elastic-plastic critical load of ring rib cylindrical shell, compared with traditional method, significantly expand the application range, can obtain the critical load of ring rib cylindrical shell under different boundary conditions.

Description

Technical Field

[0001] This invention belongs to the technical field of submarine structural main load-bearing component design, specifically relating to a method for determining the critical load of elastic-plastic buckling of a ring-ribbed cylindrical shell. Background Technology

[0002] Submarines, with their excellent stealth capabilities, play a crucial role in naval warfare. The ring-ribbed cylindrical hull is a primary load-bearing component, determining the submarine's diving depth and ensuring its safe operation. Therefore, the structural design of the ring-ribbed cylindrical hull is extremely important. Under hydrostatic pressure, the ring-ribbed cylindrical hull may experience strength failure or instability. However, with the use of high-strength materials, strength failure often does not occur. Therefore, predicting the elastoplastic critical load of the ring-ribbed cylindrical hull is of paramount importance.

[0003] Current standard methods are applicable to pressure-resistant structures made of high-strength steel with moderate diving depth, but their applicability to newer, lightweight, high-load-bearing pressure-resistant structures with greater diving depth (such as titanium alloy pressure-resistant structures) is poor. Regarding the theoretical calculation methods for the elastoplastic instability of ring-ribbed cylindrical shells, the traditional energy method first assumes an empirical deflection function, then substitutes the deflection expression into the energy functional to obtain the critical load through variational principles. However, for fixed-boundary structures, the displacement trial function is difficult to assume, making the traditional energy method difficult to apply. Since the actual boundary condition of ring-ribbed cylindrical shells in submarines is a fixed-boundary structure, a new method for analyzing the elastoplastic stability of ring-ribbed cylindrical shells under fixed-boundary conditions is urgently needed. Summary of the Invention

[0004] To address the aforementioned shortcomings in the existing technology, the present invention provides a method for determining the critical load of elastic-plastic buckling of a ring-ribbed cylindrical shell, which solves the problem that the traditional energy method is difficult to apply to the elastic-plastic stability analysis of submarine ring-ribbed cylindrical shells.

[0005] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: a method for determining the critical elastic-plastic buckling load of a ring-ribbed cylindrical shell, comprising the following steps: S1. Obtain the dimensions and material parameters of the ring-ribbed cylindrical shell, and construct the Hamiltonian governing equations for the ring-ribbed cylindrical shell; S2. Calculate the eigenvectors of the Hamiltonian control equations for the ring-ribbed cylindrical shell using the method of separation of variables, and obtain the first to third eigenparameters. S3. Construct a generalized displacement expression based on the first to third eigenparameters; S4. Substituting the generalized displacement expression into the fixed boundary condition, we obtain the critical load for elastic-plastic buckling of the ring-ribbed cylindrical shell under the fixed boundary condition.

[0006] Further: S1 includes the following sub-steps: S11. Obtain the dimensions of the ring-ribbed cylindrical shell and construct the buckling control equation; S12. The buckling control equation is introduced into the Hamiltonian system to obtain the Hamiltonian control equation for the ring-ribbed cylindrical shell.

[0007] Furthermore: the dimensions of the ring-ribbed cylindrical shell include the radius of the ring-ribbed cylindrical shell. R Shell thickness h rib web thickness b Rib wing plate width d Spacing between ribs l Material parameters include the tangential modulus of the material. Compared to Poisson ; The buckling control equation is specifically as follows:

[0008]

[0009]

[0010]

[0011]

[0012] In the formula, Along the axial direction of the cylindrical shell, Along the circumference of the cylindrical shell, u For cylindrical shell edge Displacement in direction, v For cylindrical shell edge Displacement in direction, w Let be the displacement of the cylindrical shell along the z-direction, where z is the thickness direction. The sign of the partial derivative. b 11 , b 12 , b 21 , b 22 and b 66 All are elastic coefficients. For along Surface force in the direction, For along Surface force in the direction, For the offset force, For along Bending moment in the direction, For along Bending moment in the direction, and All are torque. For along Shear force in the direction, For along Shear force in the direction, For along Pre-buckling stress in the direction, For along The pre-buckling stress in the direction.

[0013] Furthermore: In S12, the expression for the Hamiltonian governing equation of the annular ribbed cylindrical shell is specifically as follows:

[0014] In the formula, Z is the state vector, and T is the transpose symbol. θ As an intermediate parameter, and , The Hamiltonian matrix is ​​expressed as follows:

[0015] In the formula, F is the first submatrix, G is the second submatrix, and Q is the third submatrix, and its specific expression is as follows:

[0016]

[0017]

[0018] In the formula, The tangential modulus of the material. To take into account the moment of inertia of the shell plates, Let be a constant determined by the elastic coefficient, and its expression is: .

[0019] Furthermore: S2 specifically refers to: According to the separation of variables rule, the state vector Substituting this into the Hamiltonian governing equations for the ring-ribbed cylindrical shell, we obtain the intermediate equations, the specific expression of which is:

[0020] The eigenvectors are obtained from the intermediate equations. The specific expression is:

[0021] In the formula, For eigenvalues, The first eigenvalue, For the second eigenvalue, b n The third intrinsic parameter is expressed as follows:

[0022]

[0023]

[0024] In the formula, n It is an ordinal number, and .

[0025] Furthermore, in S3, the generalized displacement expression is specifically as follows:

[0026] In the formula, These are undetermined coefficients, determined by the boundary conditions at both ends.

[0027] Furthermore: In S4, the expression for the fixed-support boundary condition is specifically as follows:

[0028] In the formula, L is the length of the shell plate.

[0029] The beneficial effects of this invention are as follows: (1) This invention provides a method for determining the critical load of elastic-plastic buckling of a ring-ribbed cylindrical shell. It can overcome the limitations of the semi-inverse method, does not require prior assumption of the form of the solution, and can start from the control equation, import the control equation into the Hamiltonian system, and use methods such as separation of variables and symplectic eigenvalue expansion to solve the problem. Some new analytical solutions to the elastic-plastic buckling problem of the ring-ribbed cylindrical shell can be obtained and applied to the design of the ring-ribbed cylindrical shell.

[0030] (2) This invention provides a new theoretical method for solving the elastic-plastic buckling problem of ring-ribbed cylindrical shells, which can efficiently and accurately obtain the elastic-plastic critical load of ring-ribbed cylindrical shells. Compared with traditional methods, it significantly expands the application range and can obtain the critical load of ring-ribbed cylindrical shells under different boundary conditions.

[0031] (3) The present invention has high calculation accuracy and fast calculation efficiency, and is easy for engineers to use. It can provide a theoretical basis for the refined design of ring-ribbed cylindrical shell structures and guide engineers in design. Attached Figure Description

[0032] Figure 1 This is a flowchart of a method for determining the critical elastoplastic buckling load of a ring-ribbed cylindrical shell according to the present invention.

[0033] Figure 2 This is a schematic diagram of the cylindrical shell of the present invention.

[0034] Figure 3 This is a modal diagram of the elastic-plastic instability of the ring-ribbed cylindrical shell of the present invention. Detailed Implementation

[0035] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0036] Example 1: like Figure 1 As shown, in one embodiment of the present invention, a method for determining the critical load of elastic-plastic buckling of a ring-ribbed cylindrical shell includes the following steps: S1. Obtain the dimensions and material parameters of the ring-ribbed cylindrical shell, and construct the Hamiltonian governing equations for the ring-ribbed cylindrical shell; S2. Calculate the eigenvectors of the Hamiltonian control equations for the ring-ribbed cylindrical shell using the method of separation of variables, and obtain the first to third eigenparameters. S3. Construct a generalized displacement expression based on the first to third eigenparameters; S4. Substituting the generalized displacement expression into the fixed boundary condition, we obtain the critical load for elastic-plastic buckling of the ring-ribbed cylindrical shell under the fixed boundary condition.

[0037] S1 includes the following steps: S11. Obtain the dimensions of the ring-ribbed cylindrical shell and construct the buckling control equation; S12. The buckling control equation is introduced into the Hamiltonian system to obtain the Hamiltonian control equation for the ring-ribbed cylindrical shell.

[0038] In S11, the dimensions of the annular ribbed cylindrical shell include the radius of the annular ribbed cylindrical shell. R Shell thickness h Spacing between ribs l Material parameters include the tangential modulus of the material. Compared to Poisson ; The buckling control equation is specifically as follows:

[0039]

[0040]

[0041]

[0042]

[0043] In the formula, Along the axial direction of the cylindrical shell, Along the circumference of the cylindrical shell, u For cylindrical shell edge Displacement in direction, v For cylindrical shell edge Displacement in direction, w Let be the displacement of the cylindrical shell along the z-direction, where z is the thickness direction. The sign of the partial derivative. b 11 , b 12 , b 21 , b 22 and b 66 All are elastic coefficients. For along Surface force in the direction, For along Surface force in the direction, For the offset force, For along Bending moment in the direction, For along Bending moment in the direction, and All are torque. For along Shear force in the direction, For along Shear force in the direction, For along Pre-buckling stress in the direction, For along The pre-buckling stress in the direction.

[0044] In S12, the Hamiltonian governing equation for the ring-ribbed cylindrical shell is specifically expressed as follows:

[0045] In the formula, Z is the state vector, and T is the transpose symbol. θ As an intermediate parameter, and , The Hamiltonian matrix is ​​expressed as follows:

[0046] In the formula, F is the first submatrix, G is the second submatrix, and Q is the third submatrix, and its specific expression is as follows:

[0047]

[0048]

[0049] In the formula, The tangential modulus of the material. To take into account the moment of inertia of the shell plates, Let be a constant determined by the elastic coefficient, and its expression is: .

[0050] Under the total elasticity theory, the expressions for the elastic modulus and elastic coefficient are:

[0051] In the formula, The secant modulus of the material. For stress intensity, its expression is: ,in, For along Stress in the direction, and , For along Stress in the direction, and , g is the correction factor for the ring rib, and , F This represents the cross-sectional area of ​​the rib.

[0052] The Ramberg-Osgood model is used to represent the stress-strain relationship of ductile materials, and the tangential and secant moduli of the materials are further determined. ε The specific expression is:

[0053] In the formula, c These are material parameters, determined by the material properties.

[0054] Specifically, S2 is: According to the separation of variables rule, the state vector Substituting this into the Hamiltonian governing equations for the ring-ribbed cylindrical shell, we obtain the intermediate equations, the specific expression of which is:

[0055] The eigenvectors are obtained from the intermediate equations. The specific expression is:

[0056] In the formula, For eigenvalues, The first eigenvalue, For the second eigenvalue, b nThe third intrinsic parameter is expressed as follows:

[0057]

[0058]

[0059] In the formula, n It is an ordinal number, and .

[0060] In S3, the generalized displacement expression is specifically as follows:

[0061] In the formula, These are undetermined coefficients, determined by the boundary conditions at both ends.

[0062] In S4, the expression for the fixed boundary condition is specifically as follows:

[0063] In the formula, L is the length of the shell plate.

[0064] In this embodiment, the generalized displacement expression is substituted into the fixed boundary conditions to obtain a system of linear equations. If we set the determinant of the coefficient matrix of the linear equation system to 0, we can obtain the critical load for elastic-plastic buckling of the ring-ribbed cylindrical shell.

[0065] Example 2: This embodiment describes a specific implementation process provided in Embodiment 1.

[0066] like Figure 2 As shown, determine the dimensions, material properties, and rib type of the ring-ribbed cylindrical shell. Figure 1 As shown, the elastic modulus of the ring-ribbed cylindrical shell E =11500MPa, Poisson's ratio The length, thickness, and radius are as follows: L =200mm, h =1.5mm, R =100mm, number of ribs is N =9, intercostal spacing is l =20mm. Rib height and thickness are as follows: H =10mm, b =4mm, ribs are internal ribs. In the Ramberg-Osgood model: , By deriving the formula according to the method described in the invention, a generalized displacement expression is obtained. Substituting the above parameters into the displacement expression yields:

[0067] Substituting the generalized displacement expression into the fixed boundary conditions and solving the simultaneous equations, setting the determinant of the coefficient matrix to 0 determines the critical load. In this embodiment, the calculated critical load is P = 21.354 MPa. Substituting the critical load, the fundamental solution system is obtained, and further modal diagrams can be obtained, such as... Figure 3 As shown.

[0068] The beneficial effects of this invention are as follows: This invention provides a method for determining the critical load of elastic-plastic buckling of a ring-ribbed cylindrical shell, which can overcome the limitations of the semi-inverse method, does not require prior assumption of the form of the solution, and can start from the governing equation, import the governing equation into the Hamiltonian system, and use methods such as separation of variables and symplectic eigenvalue expansion to solve the problem, thus obtaining some new analytical solutions to the elastic-plastic buckling problem of the ring-ribbed cylindrical shell, which can be applied to the design of ring-ribbed cylindrical shells.

[0069] This invention provides a new theoretical method for solving the elastoplastic buckling problem of ring-ribbed cylindrical shells, which can efficiently and accurately obtain the elastoplastic critical load of ring-ribbed cylindrical shells. Compared with traditional methods, it significantly expands the application range and can obtain the critical load of ring-ribbed cylindrical shells under different boundary conditions.

[0070] This invention offers high computational accuracy and efficiency, making it easy for engineers to use. It provides a theoretical basis for the refined design of ring-ribbed cylindrical shell structures and guides engineers in their design process.

[0071] In the description of this invention, it should be understood that the terms "center," "thickness," "upper," "lower," "horizontal," "top," "bottom," "inner," "outer," and "radial," etc., indicating orientation or positional relationships based on the orientation or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying the relative importance or the number of technical features implicitly specified. Therefore, a feature defined by "first," "second," and "third" may explicitly or implicitly include one or more of that feature.

Claims

1. A method for determining the critical load for elastoplastic buckling of a ring-ribbed cylindrical shell, characterized in that, Includes the following steps: S1. Obtain the dimensions and material parameters of the ring-ribbed cylindrical shell, and construct the Hamiltonian governing equations for the ring-ribbed cylindrical shell; S2. Calculate the eigenvectors of the Hamiltonian control equations for the ring-ribbed cylindrical shell using the method of separation of variables, and obtain the first to third eigenparameters. S3. Construct a generalized displacement expression based on the first to third eigenparameters; S4. Substituting the generalized displacement expression into the fixed boundary condition, we obtain the critical load for elastic-plastic buckling of the ring-ribbed cylindrical shell under the fixed boundary condition. S1 includes the following steps: S11. Obtain the dimensions of the ring-ribbed cylindrical shell and construct the buckling control equation; S12. The buckling control equation is introduced into the Hamiltonian system to obtain the Hamiltonian control equation for the ring-ribbed cylindrical shell; In S11, the dimensions of the annular ribbed cylindrical shell include the radius of the annular ribbed cylindrical shell. R Shell thickness h and the distance between the ribs l Material parameters include the tangential modulus of the material. Compared to Poisson ; The buckling control equation is specifically as follows: In the formula, Along the axial direction of the cylindrical shell, Along the circumference of the cylindrical shell, u For cylindrical shell edge Displacement in direction, v For cylindrical shell edge Displacement in direction, w Let be the displacement of the cylindrical shell along the z-direction, where z is the thickness direction. The sign of the partial derivative. b 11 , b 12 , b 21 , b 22 and b 66 All are elastic coefficients. For along Surface force in the direction, For along Surface force in the direction, For the offset force, For along Bending moment in the direction, For along Bending moment in the direction, and All are torque. For along Shear force in the direction, For along Shear force in the direction, For along Pre-buckling stress in the direction, For along The pre-buckling stress in the direction.

2. The method of determining the elastic-plastic buckling load of a ring-ribbed cylindrical shell according to claim 1, wherein In S12, the Hamiltonian governing equation for the ring-ribbed cylindrical shell is specifically expressed as follows: where Z is a state vector, and T is a transpose symbol, θ is an intermediate parameter, and , is a Hamiltonian matrix, which is expressed as: In the formula, F is the first submatrix, G is the second submatrix, and Q is the third submatrix. Its specific expression is as follows: wherein is the tangential modulus of the material, is the moment of inertia of the shell accounting for the inertia of the shell, is a constant determined by the elastic coefficient, expressed as: .

3. The method according to claim 2, wherein Specifically, S2 is: According to the separation of variables method, the state vector is expressed as and substituted into the Hamiltonian governing equation of the ring-stiffened cylindrical shell, to obtain the intermediate equation, which is specifically expressed as The eigenvectors are obtained from the intermediate equations. The specific expression is: wherein is an eigenvalue, is a first eigenparameter, is a second eigenparameter, b n is a third eigenparameter, and is expressed by the following formula: wherein n n is an ordinal number, and .

4. The method for determining a buckling load of a ring-ribbed cylindrical shell according to claim 3, wherein In S3, the generalized displacement expression is specifically as follows: wherein are undetermined coefficients determined by the boundary conditions at both ends.

5. The method for determining the elastic-plastic buckling load of a ring-ribbed cylindrical shell according to claim 4, characterized in that, In S4, the expression for the fixed boundary condition is specifically as follows: In the formula, L Ls is the length of the shell plate.

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