Symplectic Superposition-Based Analytical Solutions for Buckling of Non-Lévy-Type Orthotropic Cylindrical Shells
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摘要: 该文基于笔者提出的辛叠加方法得到了经典解法难以直接获得的典型非Lévy型正交各向异性开口圆柱壳屈曲问题的解析解. 首先，基于Donnell薄壳理论建立了正交各向异性开口圆柱壳屈曲问题的Hamilton体系控制方程，然后将非Lévy型边界下的原问题拆分为两个子问题，在Hamilton体系下利用分离变量和辛本征展开等数学手段对子问题进行求解，最后基于原问题边界条件，通过子问题解的叠加求得原问题的解析解. 数值算例表明，辛叠加解析解与有限元数值解结果吻合良好. 同时，定量研究了长度和厚度等参数对屈曲载荷的影响. 相比于半逆解法等传统解析方法，辛叠加方法基于严格的数学推导，无需假定解的形式，可以获得更多类似问题的解析解.Abstract: Based on the symplectic superposition method (SSM) pioneered by the authors, the buckling problem of typical non-Lévy-type orthotropic cylindrical shells was solved analytically, which is difficult to handle with conventional analytical methods. The Hamiltonian system-based governing equations for buckling of orthotropic cylindrical shells were firstly established based on Donnell's shell theory. The original problem under non-Lévy-type boundary conditions was then divided into 2 subproblems, and each subproblem was solved with the mathematical techniques incorporating separation of variables and symplectic eigen expansion within the Hamiltonian framework. The analytical solution of the original problem was finally given through the superposition of the sub-solutions to satisfy the boundary conditions of the original problem. The numerical examples under consideration show that, the SSM-based analytical solutions are in good agreement with the finite element results. In addition, the effects of parameters including the length and the thickness on the critical buckling loads were quantitatively studied. Compared with the conventional analytical methods such as the semi-inverse method, the SSM works based on rigorous mathematical derivation without any assumption of the solution forms, and can obtain reliable analytical solutions to more similar issues.
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Key words:
- orthotropic /
- cylindrical shell /
- buckling /
- symplectic superposition method /
- analytical solution
1) (我刊编委李锐来稿) -
图 2 开口圆柱壳屈曲的辛叠加示意图
Figure 2. Symplectic superposition for buckling of a cylindrical panel
图 5 不同厚度下，开口圆柱壳临界屈曲载荷随厚度的变化曲线随曲率的变化曲线
Figure 5. Curvature-dependent critical buckling loads on cylindrical shells with different lengths on cylindrical shells with different thicknesses
表 1 a=b=R=1 m时，开口圆柱壳前十阶屈曲载荷的收敛性研究(单位：kN/m)
Table 1. Convergence of the 1st 10 buckling loads on the cylindrical shell with a=b=R=1 m (unit: kN/m)
δ/m number of series terms mode 1 2 3 4 5 6 7 8 9 10 1 10 1 220.2 1 392.5 1 501.8 1 531.1 1 830.1 1 887.1 1 986.8 2 049.8 2 201.7 2 286.1 20 1 220.2 1 392.5 1 501.8 1 531.1 1 830.1 1 887.1 1 986.7 2 049.8 2 201.7 2 286.0 30 1 220.2 1 392.5 1 501.8 1 531.1 1 830.1 1 887.1 1 986.7 2 049.8 2 201.7 2 286.0 0.05 10 285.61 300.36 307.11 322.16 367.86 374.32 383.09 385.92 406.60 430.85 20 285.61 300.35 307.08 322.16 367.86 374.30 383.07 385.91 406.58 430.84 30 285.61 300.35 307.08 322.16 367.86 374.30 383.07 385.91 406.58 430.84 
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表 2 b=1 m，δ=0.01 m时，开口圆柱壳的前十阶屈曲载荷(单位：kN/m)
Table 2. The 1st 10 buckling loads on the cylindrical shell with b=1 m and δ=0.01 m (unit: kN/m)
a/m R/m method mode 1 2 3 4 5 6 7 8 9 10 1 1 FEM 1 218.3 1 366.8 1 473.7 1 510.5 1 789.6 1 869.8 1 988.4 1 990.6 2 129.8 2 209.1 present 1 220.2 1 392.5 1 501.8 1 531.1 1 830.1 1 887.1 1 986.7 2 049.8 2 201.7 2 286.0 2 FEM 757.89 831.37 1 091.2 1 117.9 1 165.4 1 240.7 1 646.8 1 792.0 1 826.1 1 852.6 present 765.18 835.57 1 114.0 1 138.6 1 182.1 1 268.5 1 685.3 1 862.5 1 899.7 1 911.4 10 FEM 460.16 607.93 876.33 955.33 1 093.4 1 164.2 1 601.8 1 649.9 1 729.0 1 809.0 present 465.05 614.49 897.36 978.51 1 108.2 1 191.8 1 639.5 1 720.4 1 802.8 1 867.2 2 1 FEM 1 074.7 1 089.4 1 140.3 1 169.6 1 249.8 1 363.0 1 377.4 1 472.8 1 569.4 1 580.9 present 1 091.5 1 093.6 1 154.4 1 185.4 1 259.3 1 373.8 1 401.7 1 500.8 1 596.8 1 609.9 2 FEM 580.60 630.27 667.79 766.72 838.42 842.57 902.04 903.58 935.23 965.62 present 584.52 635.65 668.25 767.03 843.13 850.34 913.12 914.70 947.30 975.79 10 FEM 172.86 262.98 413.54 460.57 468.95 663.31 721.07 752.43 815.28 852.77 present 173.45 264.58 416.49 465.46 473.21 674.36 727.81 765.29 824.88 863.90
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表 3 b=1 m，δ=0.005 m时，开口圆柱壳的前十阶屈曲载荷(单位：kN/m)
Table 3. The 1st 10 buckling loads on the cylindrical shell with b=1 m and δ=0.005 m (unit: kN/m)
a/m R/m method mode 1 2 3 4 5 6 7 8 9 10 1 1 FEM 284.75 299.42 306.74 320.64 366.45 372.05 382.91 384.16 407.24 429.56 present 285.61 300.35 307.08 322.16 367.86 374.30 383.07 385.91 406.58 430.84 2 FEM 154.95 174.48 188.02 192.21 228.72 237.54 252.14 255.64 274.23 284.70 present 154.94 175.10 188.71 192.72 229.59 238.04 251.97 257.08 276.15 286.74 10 FEM 63.880 79.045 115.44 123.84 139.42 149.47 204.84 215.42 224.94 232.71 present 64.008 79.224 116.00 124.47 139.74 150.21 205.71 217.29 226.92 234.12 2 1 FEM 267.91 270.50 272.42 274.90 291.75 300.12 306.17 309.58 313.35 334.04 present 268.73 270.68 273.02 275.64 292.06 301.30 306.66 311.87 314.53 335.13 2 FEM 138.49 138.74 144.98 148.91 159.28 172.95 175.71 187.93 199.89 201.64 present 138.79 138.92 145.27 149.24 159.52 173.18 176.28 188.62 200.38 202.34 10 FEM 29.704 41.073 54.651 61.639 63.996 88.736 92.837 97.687 104.21 109.03 present 29.727 41.116 54.736 61.757 64.127 89.029 93.008 98.038 104.43 109.30
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