Method for calculating stress and deformation of thin-walled cylinder under self-weight action

Abstract

The application discloses a calculation method of stress and deformation of a thin-walled cylinder under self-weight of a roller support, and specific steps include the following: S1, a cantilever model is used to calculate the bending moment and deflection of the cylinder in the longitudinal direction, and the optimal position of the roller support in the longitudinal direction is determined according to the maximum bending moment minimization and the maximum deflection minimization; S2, a half-circular curved beam model is established by using symmetry, and the bending moment and displacement of the cylinder in the transverse direction are calculated by using the Timoshenko beam theory; S3, the maximum stress of the cylinder in the transverse direction is calculated according to the bending moment, and the optimal half-enclosure angle required by the strength is determined; and S4, the ellipticity and root mean square roundness of the curved beam are calculated according to the displacement, and the optimal half-enclosure angle required by the deformation is determined. The application can accurately predict the maximum stress and deformation shape of the thin-walled cylinder, and can determine that the optimal enclosure angles of the roller support under the strength requirement and the deformation requirement are 67.4 DEG and 90 DEG respectively, and therefore, the optimal enclosure angle in engineering is recommended to be between 67.4 DEG and 90 DEG.

Description

Technical field:

[0001] The invention belongs to the field of marine engineering, and in particular relates to a method for calculating stress and deformation of a thin-walled cylinder supported by a roller frame under the action of its own weight. Background technology:

[0002] Large, thin-walled cylinders, crucial components of large-scale equipment and pressure vessels in the marine, chemical, and energy sectors, often require roller supports during manufacturing, transportation, and assembly. Rollers, a simple, compact, low-cost, and technologically mature auxiliary equipment for large-tonnage placement, facilitate the rotation, handling, and docking of cylinders, significantly improving construction efficiency. Due to the weight of the cylinder, changes in the roller's position and wrap angle can affect the cylinder's stress distribution and deformation.

[0003] If the roller frame wrap angle is selected according to the JB / T9187-1999 "Welding Roller Frame" standard, the roller frame wrap angle range should be between 45° and 110°. Currently, a 60° wrap angle is commonly used in practical projects. However, there is currently no suitable verification and calculation method for the specific selection of roller frame position and wrap angle. Therefore, analyzing the impact of different wrap angles on the cylinder is crucial for selecting the roller frame wrap angle. Summary of the invention:

[0004] The purpose of the present invention is to overcome the above shortcomings and provide a method for calculating the stress and deformation of a thin-walled cylinder supported by a roller frame under the action of its own weight.

[0005] The purpose of the present invention is achieved through the following technical solution: A method for calculating the stress and deformation of a thin-walled cylinder supported by a roller frame under the action of its own weight, the specific steps comprising:

[0006] S1. The longitudinal bending moment and deflection of the cylinder are calculated using the cantilever beam model. The optimal longitudinal position of the roller frame is determined by minimizing the maximum bending moment and maximum deflection of the beam.

[0007] S2. A semicircular curved beam model is established using symmetry in the transverse direction, and the bending moment and displacement in the transverse plane of the cylinder are calculated using the Timoshenko beam theory.

[0008] S3. Calculate the maximum stress in the cylinder surface based on the in-plane bending moment and determine the optimal half-wrapped angle required for strength;

[0009] S4. Calculate the ellipticity and root mean square roundness of the curved beam based on the displacement and determine the optimal half-angle required for deformation.

[0010] A further improvement of the present invention is that step S1 specifically includes:

[0011] S11. Establish an outrigger beam model and calculate the longitudinal bending moment and deflection of the cylinder to determine the optimal position of the roller frame in the outrigger beam. The calculation formula is:

[0012]

[0013] L is the longitudinal length of the cylinder, q is the weight of the cylinder distributed uniformly, x0 is the distance from the roller frame to the ends of the cylinder, M is the bending moment of the cantilever beam, and w is the deflection of the cantilever beam;

[0014] S12. According to formula (1) and formula (2), the maximum positive bending moment occurs in the span of the cantilever beam, and the bending moment is M L / 2 =qL 2 / 8-qLx0 / 2, the maximum negative bending moment occurs at the support point, and the value is M B =M A =qx0 2 / 2, according to the minimum optimization requirement of maximum bending moment, when the maximum positive bending moment value is equal to the maximum negative bending moment value, that is, M L / 2 =|M A |=|M B |, determine the optimal position of the support point under strength requirements as the distance from the end The maximum bending moment in the beam is The longitudinal bending moment is minimized, the bending normal stress is minimized, and it is the best strength design position;

[0015] S13. According to the deflection calculation formula, the maximum deformation deflection occurs at the mid-span of the cantilever beam or at both ends of the cantilever beam, and the deflections are w L / 2 =5q(L-2x) 4 / 384EI-qx 2 (L-2x) 2 / 16EI and w D =w C =-qx 4 / 8EI-qx 3 (L-2x) / 4EI+qx(L-2x) 3 / 24EI, in order to ensure the optimization of the maximum deflection, the values ​​at the point of maximum deflection must be equal, that is, w L / 2 =w D =w C , determine the optimal position of the support point under the deflection requirement is the distance x from the end w =0.233L, where the deflection is the smallest and is the optimal deflection deformation design position;

[0016] The longitudinal model determines the optimal position of the support point under the requirements of strength and deformation, and the distance from the end is and x w=0.233L, so the best range in the project is between 0.207L and 0.233L.

[0017] A further improvement of the present invention is that step S2 specifically includes:

[0018] S21. Based on the symmetry of the structure and load, the Timoshenko beam theory is used in the transverse direction of the cylinder to establish a half-semicircular curved beam model to calculate the in-plane bending moment and displacement of the cylinder.

[0019] The axial force, shear force and bending moment of the curved beam are:

[0020]

[0021] The various terms in the formula are the axial force, shear force and bending moment caused by gravity, support force, symmetrical pressure and symmetrical bending moment. The analytical expressions are:

[0022]

[0023] in: M G are the axial force, shear force, and bending moment caused by gravity within the cylinder surface; M P are the axial force, shear force and bending moment caused by the support force in the cylinder surface;

[0024] M X are the axial force, shear force and bending moment caused by the symmetrical pressure in the cylinder surface; are the axial force, shear force, and bending moment caused by the symmetrical bending moment in the cylinder surface, θ is the angle corresponding to the desired deformation, t is the thickness of the cylinder section steel plate, R is the outer radius of the cylinder, γ is the specific gravity of the steel, E is the elastic modulus of the steel, A is the cross-sectional area of ​​the curved beam, I is the moment of inertia, and G is the shear modulus;

[0025] S22. The curved beam structure is quadratically indeterminate. The boundary conditions determined by symmetry are:

[0026]

[0027] S23. Use the unit force method to calculate the symmetrical pressure F X And the symmetrical bending moment M0 is calculated as follows:

[0028] In formula (10), the expressions of the internal forces generated by the unit force and unit bending moment at the lower end are as follows:

[0029]

[0030] in: They represent the axial force, shear force and bending moment of the curved beam under the action of axial unit force respectively; They represent the axial force, shear force and bending moment of the curved beam under unit bending moment respectively;

[0031] S24. Calculate the symmetrical pressure F according to formula (9) and formula (10). X And the expression of the symmetrical bending moment M0 is as follows:

[0032]

[0033] The symmetrical pressure F X Substitute the values ​​of axial force, shear force and bending moment M0 into formulas (4), (5), (6) and (7), and then substitute the obtained values ​​into formula (3) to solve the distribution of axial force, shear force and bending moment of the cylinder in the plane.

[0034] A further improvement of the present invention is that step S3 specifically includes calculating an analytical formula for the bending normal stress at any position of the curved beam, and the calculation formula is:

[0035]

[0036] Where: θ is the bending normal stress at any position of the curved beam, W = t 2 / 6 is the bending section modulus of the curved beam. In formula (14), the point where the in-plane bending moment of the cylinder is the maximum is the contact point between the roller frame and the cylinder, and its maximum normal bending stress is also here. Further, the half wrap angle of the roller frame is optimized according to the strength requirement, and θ = α is substituted into M and σ. θ In the equation, take the derivative of α and the calculation formula is as follows:

[0037]

[0038] According to formula (16), when the half angle α M When ≈33.7°, the bending moment at the contact point between the cylinder and the roller frame is the smallest, that is, the hoop stress in the cylinder surface is the smallest under this half wrap angle.

[0039] A further improvement of the present invention is that step S4 includes applying a unit radial force at any position and calculating the radial displacement according to the force method, and the expression is as follows:

[0040] When 0≤δ≤α, the radial displacement equation is:

[0041]

[0042] When α≤δ≤π, the radial displacement equation is:

[0043]

[0044] The internal force expression of the curved beam under the radial unit force is:

[0045]

[0046] Where: δ is the angle corresponding to the internal force of the curved beam; according to the calculation formula (19), the radial displacement u is calculated r ;

[0047] The ellipticity and root mean square roundness of the curved beam at different half-wrapped angles are calculated based on the radial displacement. The calculation formula is:

[0048]

[0049] Where: ε is the ellipticity of the curved beam at different half-wrapped angles; e is the root mean square roundness of the curved beam at different half-wrapped angles; D max is the maximum deformation diameter of the curved beam; D min is the minimum deformation diameter of the curved beam;

[0050] According to the transverse model, the optimal wrap angle range of the roller frame is between 67.4° and 90° under the requirements of strength and deformation.

[0051] Compared with the prior art, the present invention has the following advantages:

[0052] The present invention uses an extended beam model to calculate the longitudinal bending moment and deflection of the cylinder, and determines the optimal longitudinal position of the roller frame based on minimizing the maximum bending moment and maximum deflection of the beam. Transversely, a semicircular curved beam model is established using symmetry, and the Timoshenko beam theory is used to calculate the bending moment and displacement within the transverse plane of the cylinder. The maximum stress within the cylinder surface is calculated based on the in-plane bending moment, and the optimal half-angle of support required for strength is determined. The ellipticity and root mean square roundness of the curved beam are calculated based on the displacement, and the optimal half-angle of support required for deformation is determined. The present invention accurately predicts the maximum stress and deformed shape of a thin-walled cylinder, and can determine that the optimal angle of support for the roller frame under strength and deformation requirements is 67.4° and 90°, respectively. Therefore, the optimal angle of support recommended in engineering is between 67.4° and 90°. Description of the drawings:

[0053] Figure 1 These are longitudinal and transverse model diagrams of the thin-walled cylinder supported by a roller frame in the present invention.

[0054] Figure 2 This is a diagram of the transversely symmetrical semicircular curved beam model of the cylinder in the present invention.

[0055] Figure 3 Schematic diagram of the variation of the in-plane bending moment of the thin-walled cylinder with different angles in the present invention.

[0056] Figure 4The figures are the deformation diagrams of the cylinder before and after deformation at different half-wrapped angles obtained by the method of the present invention and Ansys simulation.

[0057] Figure 5 Schematic diagram of the variation of the ovality and RMS roundness of the cylinder with the half wrap angle of the present invention. Specific implementation method:

[0058] In order to deepen the understanding of the present invention, the present invention will be further described in detail below with reference to embodiments and drawings. The embodiments are only used to explain the present invention and do not constitute a limitation on the scope of protection of the present invention.

[0059] The calculation method of stress and deformation of thin-walled cylinder supported by roller frame under the action of deadweight includes the following steps:

[0060] S1. The longitudinal bending moment and deflection of the cylinder are calculated using the cantilever beam model. The optimal longitudinal position of the roller frame is determined by minimizing the maximum bending moment and maximum deflection of the beam.

[0061] S2. A semicircular curved beam model is established using symmetry in the transverse direction, and the bending moment and displacement in the transverse plane of the cylinder are calculated using the Timoshenko beam theory.

[0062] S3. Calculate the maximum stress in the cylinder surface based on the in-plane bending moment and determine the optimal half-wrapped angle required for strength;

[0063] S4. Calculate the ellipticity and root mean square roundness of the curved beam based on the displacement and determine the optimal half-angle required for deformation.

[0064] The step S1 specifically includes:

[0065] S11. Build an outrigger beam model, such as Figure 1 As shown, the longitudinal bending moment and deflection of the cylinder are calculated to determine the optimal position of the roller frame in the outrigger beam. The calculation formula is:

[0066] (2)

[0067] L is the longitudinal length of the cylinder, q is the weight of the cylinder distributed uniformly, x0 is the distance from the roller frame to the ends of the cylinder, M is the bending moment of the cantilever beam, and w is the deflection of the cantilever beam;

[0068] S12. According to formula (1) and formula (2), the maximum positive bending moment occurs in the span of the cantilever beam, and the bending moment is M L2 =qL 2 / 8-qLx0 / 2, the maximum negative bending moment occurs at the support point (A / B section), and the value is M B =M A =qx0 2 / 2, according to the minimum optimization requirement of maximum bending moment, when the maximum positive bending moment value is equal to the maximum negative bending moment value, that is, M L / 2 =|M A |=|M B |, determine the optimal position of the support point under strength requirements as the distance from the end The maximum bending moment in the beam is The longitudinal bending moment is minimized, the bending normal stress is minimized, and it is the best strength design position;

[0069] S13. According to the deflection calculation formula, the maximum deformation deflection occurs at the mid-span of the cantilever beam or at both ends of the cantilever beam (C and D), and the deflections are w L / 2 =5q(L-2x) 4 / 384EI-qx 2 (L-2x) 2 / 16EI and w D =w C =-qx 4 / 8EI-qx 3 (L-2x) / 4EI+qx(L-2x) 3 / 24EI, in order to ensure the optimization of the maximum deflection, the values ​​at the point of maximum deflection must be equal, that is, w L / 2 =w D =w C , determine the optimal position of the support point under the deflection requirement is the distance x from the end w =0.233L, where the deflection is the smallest and is the optimal deflection deformation design position;

[0070] The longitudinal model determines the optimal position of the support point under the requirements of strength and deformation, and the distance from the end is and x w =0.233L, so the best range in the project is between 0.207L and 0.233L.

[0071] The step S2 specifically includes:

[0072] S21, based on the symmetry of structure and load, e.g. Figure 2 As shown, the Timoshenko beam theory is used in the transverse direction of the cylinder to establish a curved beam model in the shape of half a semicircle to calculate the in-plane bending moment and displacement of the cylinder;

[0073] The axial force, shear force and bending moment of the curved beam are:

[0074]

[0075] The various terms in the formula are the axial force, shear force and bending moment caused by gravity, support force, symmetrical pressure and symmetrical bending moment. The analytical expressions are:

[0076]

[0077]

[0078] in: M G are the axial force, shear force, and bending moment caused by gravity within the cylinder surface; M P are the axial force, shear force and bending moment caused by the support force in the cylinder surface;

[0079] M X are the axial force, shear force and bending moment caused by the symmetrical pressure in the cylinder surface; are the axial force, shear force, and bending moment caused by the symmetrical bending moment in the cylinder surface, θ is the angle corresponding to the desired deformation, t is the thickness of the cylinder section steel plate, R is the outer radius of the cylinder, γ is the specific gravity of the steel, E is the elastic modulus of the steel, A is the cross-sectional area of ​​the curved beam, I is the moment of inertia, and G is the shear modulus;

[0080] S22. The curved beam structure is quadratically indeterminate. The boundary conditions determined by symmetry are:

[0081]

[0082] S23. Use the unit force method to calculate the symmetrical pressure F X And the symmetrical bending moment M0 is calculated as follows:

[0083]

[0084] In formula (10), the expressions of the internal forces generated by the unit force and unit bending moment at the lower end are as follows:

[0085]

[0086] in: They represent the axial force, shear force and bending moment of the curved beam under the action of axial unit force respectively; They represent the axial force, shear force and bending moment of the curved beam under unit bending moment respectively;

[0087] S24. Calculate the symmetrical pressure F according to formula (9) and formula (10). X And the expression of the symmetrical bending moment M0 is as follows:

[0088]

[0089] The symmetrical pressure F XSubstitute the values ​​of axial force, shear force and bending moment M0 into formulas (4), (5), (6) and (7), and then substitute the obtained values ​​into formula (3) to solve the distribution of axial force, shear force and bending moment of the cylinder in the plane.

[0090] Step S3 specifically includes calculating the analytical formula of the bending normal stress at any position of the curved beam, and the calculation formula is:

[0091]

[0092] Where: θ is the bending normal stress at any position of the curved beam, W = t 2 / 6 is the bending section modulus of the curved beam. In formula (14), the point where the in-plane bending moment of the cylinder is the maximum is the contact point between the roller frame and the cylinder, and its maximum normal bending stress is also here. Further, the half wrap angle of the roller frame is optimized according to the strength requirement, and θ = α is substituted into M and σ. θ In the equation, take the derivative of α and the calculation formula is as follows:

[0093]

[0094] According to formula (16), when the half angle α M When ≈33.7°, the bending moment at the contact point between the cylinder and the roller frame is the smallest, that is, the hoop stress in the cylinder surface is the smallest under this half wrap angle.

[0095] The step S4 includes applying a unit radial force at any position and calculating the radial displacement according to the force method. The expression is as follows:

[0096] When 0≤δ≤α, the radial displacement equation is:

[0097]

[0098] When α≤δ≤π, the radial displacement equation is:

[0099]

[0100] The internal force expression of the curved beam under the radial unit force is:

[0101]

[0102] Where: δ is the angle corresponding to the internal force of the curved beam; according to the calculation formula (19), the radial displacement u is calculated r ;

[0103] The ellipticity and root mean square roundness of the curved beam at different half-wrapped angles are calculated based on the radial displacement. The calculation formula is:

[0104]

[0105] Where: ε is the ellipticity of the curved beam at different half-wrapped angles; e is the root mean square roundness of the curved beam at different half-wrapped angles; D max is the maximum deformation diameter of the curved beam; D min is the minimum deformation diameter of the curved beam;

[0106] In summary, the longitudinal model determines the optimal positions of the support points under the requirements of strength and deformation, respectively, the distance from the end is and x w =0.233L, so the optimal range recommended for engineering is between 0.207L and 0.233L. Transverse examples demonstrate that this method can accurately predict the maximum stress and deformed shape of thin-walled cylinders. The optimal wrap angles of the roller frame, meeting strength and deformation requirements, are determined to be 67.4° and 90°, respectively. Therefore, the optimal wrap angle recommended for engineering is between 67.4° and 90°.

[0107] Now analyze as Figure 1 The stress and deformation of the thin-walled cylinder supported by the roller stand are shown. The geometric parameters of the thin-walled cylinder are: L = 10m, R = 4.95m, t = 0.1m; material parameters: ρ = 7800×10 3 kg / m 3 , E = 200 GPa, υ = 0.3; the acceleration of gravity is g = 9.8 N / kg, and the shear coefficient of the Timoshenko beam is k = 5 / 6.

[0108] The optimal position of the roller frame on the cylinder under the strength and deformation requirements is determined according to the symmetrical cantilever beam model in the longitudinal direction. The optimal position of the roller frame to the cylinder end under the strength and deformation requirements is calculated by formulas (1) and (2) and is x M ≈2.07m and x w ≈2.23m.

[0109] First, according to the bending moment equation (3-14), the position of the maximum bending moment (dangerous section) is determined to be the support point, and compared with the finite element results to avoid the influence of contact stress, and the circumferential stress of the inner wall of the cylinder support point is calculated according to the bending normal stress formula (15). This point is in a unidirectional stress state, and its value is equal to the Mise stress. Then, according to the radial displacement formula (17-18), the radial deformation displacement of each point of the thin-walled cylinder is calculated, and the ovality and root mean square roundness are further calculated. Table 1 shows the Mise stress (close to the maximum Mise stress of the structure) and the ovality and root mean square roundness of the inner wall of the support point of the cylinder calculated by theory and simulation under different half-wrapped angles. As can be seen from the table, the Mise stress and ovality given by the theoretical solution and the simulation results are very consistent, verifying the accuracy of the method of the present invention.

[0110] Table 1 Maximum mise stress and deformation of the cylinder under different roller frame half angles

[0111]

[0112]

[0113] According to the principle of strength optimization, the maximum bending moment is minimized, and the optimal half-wrapped angle is determined to be α = 33.7° using formula (16). For the three cases where the half-wrapped angle is α = 30°, α = 33.7°, and α = 35°, Figure 3 The variation of the in-plane bending moment of the thin-walled cylinder with different angles θ is given. It can be seen that the bending moment at the support point is the maximum bending moment of the cylinder, and the maximum bending moment is the smallest when α = 33.7°, which verifies the optimal half-wrapped angle under the strength requirement determined by formula (16).

[0114] Figure 4 The deformation diagrams of the cylinder before and after deformation at different half-wrapped angles obtained by the method of the present invention and Ansys simulation are given. Comparative analysis shows that the theoretical deformation and the simulated deformation are highly consistent, further verifying the correctness of the method of the present invention. As the half-wrapped angle increases, the shape of the thin-walled cylinder after deformation gradually changes from a flat ellipse to an inverted pear. In order to further analyze the influence of the half-wrapped angle on the deformation of the cylinder, Figure 5 The variation of the cylinder ovality and root mean square roundness with the half wrap angle is given. As the half wrap angle α increases from 20°, the roundness and root mean square roundness decrease slowly. When the half wrap angle reaches 45°, the roundness and root mean square roundness increase rapidly. When the half wrap angle is equal to 45°, the ovality and root mean square roundness are both minimum, the ovality is close to 0, and the root mean square roundness is 8.167mm. At this time, the deformed shape is an inverted pear shape (such as Figure 3 In Figure d), RMS roundness is a better measure of deformation than ovality. Therefore, when prioritizing deformation optimization, a half-wrap angle of 45° (i.e., a wrap angle of 90°) minimizes cylinder deformation. Taking all factors into consideration, combining longitudinal strength optimization with deflection optimization, the optimal range of the distance between the roller support point and the end is between 0.207L and 0.223L. Combining transverse strength optimization with deformation optimization, the cylinder wrap angle should be between 67.4° and 90°.

[0115] Those skilled in the art will appreciate that the present invention is not limited to the foregoing embodiments. The foregoing embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. The calculation method of stress and deformation of thin-walled cylinder supported by roller frame under the action of deadweight is characterized by: The specific steps include: S1. The longitudinal bending moment and deflection of the cylinder are calculated using the cantilever beam model. The optimal longitudinal position of the roller frame is determined by minimizing the maximum bending moment and maximum deflection of the beam. S2. A semicircular curved beam model is established using symmetry in the transverse direction, and the bending moment and displacement in the transverse plane of the cylinder are calculated using the Timoshenko beam theory. S3. Calculate the maximum stress in the cylinder surface based on the in-plane bending moment and determine the optimal half-wrapped angle required for strength; S4. Calculate the ellipticity and root mean square roundness of the curved beam based on the displacement, and determine the optimal half-angle required for deformation; The step S1 specifically includes: S11. Establish an outrigger beam model and calculate the longitudinal bending moment and deflection of the cylinder to determine the optimal position of the roller frame in the outrigger beam. The calculation formula is: L is the longitudinal length of the cylinder, q is the weight of the cylinder distributed uniformly, x0 is the distance from the roller frame to the ends of the cylinder, M is the bending moment of the cantilever beam, and w is the deflection of the cantilever beam; S12. According to formula (1) and formula (2), the maximum positive bending moment occurs in the span of the cantilever beam, and the bending moment is M L / 2 =qL 2 / 8-qLx0 / 2, the maximum negative bending moment occurs at the support point, and the value is M B =M A =qx0 2 / 2, according to the minimum optimization requirement of maximum bending moment, when the maximum positive bending moment value is equal to the maximum negative bending moment value, that is, M L / 2 =|M A |=|M B |, determine the optimal position of the support point under strength requirements as the distance from the end The maximum bending moment in the beam is The longitudinal bending moment is minimized, the bending normal stress is minimized, and it is the best strength design position; S13. According to the deflection calculation formula, the maximum deformation deflection occurs at the mid-span of the cantilever beam or at both ends of the cantilever beam, and the deflections are w L / 2 =5q(L-2x) 4 / 384EI-qx 2 (L-2x) 2 / 16EI and w D =w C =-qx 4 / 8EI-qx 3 (L-2x) / 4EI+qx(L-2x) 3 / 24EI, in order to ensure the optimization of the maximum deflection, the values ​​at the point of maximum deflection must be equal, that is, w L / 2 =w D =w C , determine the optimal position of the support point under the deflection requirement is the distance x from the end w =0.233L, where the deflection is the smallest and is the optimal deflection deformation design position; The longitudinal model determines the optimal position of the support point under the requirements of strength and deformation, and the distance from the end is and x w =0.233L, so the best range in the project is between 0.207L and 0.233L; The step S2 specifically includes: S21. Based on the symmetry of the structure and load, the Timoshenko beam theory is used in the transverse direction of the cylinder to establish a half-semicircular curved beam model to calculate the in-plane bending moment and displacement of the cylinder. The axial force, shear force and bending moment of the curved beam are: The various terms in the formula are the axial force, shear force and bending moment caused by gravity, support force, symmetrical pressure and symmetrical bending moment. The analytical expressions are: in: M G are the axial force, shear force, and bending moment caused by gravity within the cylinder surface; M P are the axial force, shear force and bending moment caused by the support force in the cylinder surface; M X are the axial force, shear force and bending moment caused by the symmetrical pressure in the cylinder surface; are the axial force, shear force, and bending moment caused by the symmetrical bending moment in the cylinder surface, θ is the angle corresponding to the desired deformation, t is the thickness of the cylinder section steel plate, R is the outer radius of the cylinder, γ is the specific gravity of the steel, E is the elastic modulus of the steel, A is the cross-sectional area of ​​the curved beam, I is the moment of inertia, and G is the shear modulus; S22. The curved beam structure is quadratically indeterminate. The boundary conditions determined by symmetry are: S23, using the unit force method, apply axial unit force and axial unit bending moment Acting on the lower end of the statically determinate curved beam, calculate the symmetrical pressure F X And the symmetrical bending moment M0 is calculated as follows: In formula (10), the expressions of the internal forces generated by the unit force and unit bending moment at the lower end are as follows: in: They represent the axial force, shear force and bending moment of the curved beam under the action of axial unit force respectively; They represent the axial force, shear force and bending moment of the curved beam under unit bending moment respectively; S24. Calculate the symmetrical pressure F according to formula (9) and formula (10). X And the expression of the symmetrical bending moment M0 is as follows: The symmetrical pressure F X Substitute the values ​​of axial force, shear force and bending moment M0 into formulas (4), (5), (6) and (7), and then substitute the obtained values ​​into formula (3) to solve the distribution of axial force, shear force and bending moment of the cylinder in the plane.

2. The method for calculating stress and deformation of a thin-walled cylinder supported by a roller frame under its own weight according to claim 1 is characterized by: The step S3 specifically includes calculating an analytical formula for the bending normal stress at any position of the curved beam, and the calculation formula is: Where: θ is the bending normal stress at any position of the curved beam, W = t 2 / 6 is the bending section modulus of the curved beam. In formula (14), the point where the in-plane bending moment of the cylinder is the maximum is the contact point between the roller frame and the cylinder, and its maximum normal bending stress is also here. Further, the half wrap angle of the roller frame is optimized according to the strength requirement, and θ = α is substituted into M and σ. θ In the equation, take the derivative of α and the calculation formula is as follows: According to formula (16), when the half angle α M When ≈33.7°, the bending moment at the contact point between the cylinder and the roller frame is the smallest, that is, the hoop stress in the cylinder surface is the smallest under this half wrap angle.

3. The method for calculating the stress and deformation of a thin-walled cylinder supported by a roller frame under its own weight according to claim 1 is characterized by: The step S4 includes applying a unit radial force at any position and calculating the radial displacement according to the force method. The expression is as follows: When 0≤δ≤α, the radial displacement equation is: When α≤δ≤π, the radial displacement equation is: The internal force expression of the curved beam under the radial unit force is: Where: δ is the angle corresponding to the internal force of the curved beam; according to the calculation formula (19), the radial displacement u is calculated r ; The ellipticity and root mean square roundness of the curved beam at different half-wrapped angles are calculated based on the radial displacement. The calculation formula is: Where: ε is the ellipticity of the curved beam at different half-wrapped angles; e is the root mean square roundness of the curved beam at different half-wrapped angles; D max is the maximum deformation diameter of the curved beam; D min is the minimum deformation diameter of the curved beam; According to the transverse model, the optimal wrap angle range of the roller frame is between 67.4° and 90° under the requirements of strength and deformation.

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