Secondary Buckling Analysis of Thin Rectangular Plates Based on the Wavelet Galerkin Method
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摘要:
通过经典的弹性矩形薄板，研究了小波Galerkin法(WGM)在非线性屈曲问题数值求解方面的应用。首先，介绍了基于小波Galerkin法的von Kármán方程离散格式，然后提出了离散方程Jacobi矩阵和Hesse矩阵的一个简便计算方法，并讨论了基于小波离散格式的特征方程法、扩展方程法和伪弧长法等非线性屈曲分析方法。其次，较为详细地分析了弹性矩形薄板的二次屈曲平衡路径以及长宽比、边界条件和双向压缩对波形跳跃的影响。数值结果表明，小波Galerkin法在求解矩形板屈曲临界载荷时仍然有良好的收敛性，所获结果与稳定性实验、二次摄动法和非线性有限单元法的结果也非常一致，而结合不同分岔计算方法的可行性，更使其可为典型板壳的复杂非线性稳定性问题提供一种高效的空间离散方法。
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关键词:
- 小波Galerkin法 /
- 矩形薄板 /
- 二次屈曲 /
- 波形跳跃
Abstract:Application of the wavelet Galerkin method (WGM) to numerical solution of nonlinear buckling problems was studied with classical elastic thin rectangular plates. First, the discretized scheme of the von Kármán equation were introduced, then a simple calculation approach to the Jacobian and Hessian matrices based on the WGM was proposed, and the wavelet discretized scheme-based eigenvalue equation method, the extended equation method and the pseudo arc-length method for nonlinear buckling analysis were discussed. Second, the secondary post-buckling equilibrium paths of elastic thin rectangular plates and the effects of aspect ratios, boundary conditions and bi-directional compression on the mode jumping behaviors, were discussed in detail. Numerical results show that, the WGM possesses good convergence for solving buckling loads on rectangular plates, and the obtained equilibrium paths are in good agreement with those from the stability experiments, the 2-step perturbation method and the nonlinear finite element method. Given the feasibility of combination with different bifurcation computation methods, the WGM makes an efficient spatial discretization method for complex nonlinear stability problems of typical plates and shells.
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Key words:
- wavelet Galerkin method /
- rectangular thin plate /
- secondary buckling /
- mode jumping
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图 1 四边简支板屈曲载荷的收敛阶数
Figure 1. Convergence orders of buckling loads for 4-edge simply supported plates
图 2 方形薄板单向压缩的后屈曲路径：(a) 四边简支；(b) 四边固支
Figure 2. Post-buckling paths of square plates under uniaxial compression: (a) 4-edge simply supported; (b) 4-edge fixed
图 3 四边固支方形薄板的二次屈曲系数
Figure 3. Secondary buckling coefficients of 4-edge fixed square plates
图 4 四边简支薄板二次屈曲平衡路径：(a) β=1.0；(b) β=2.0
Figure 4. Secondary buckling paths of 4-edge simply supported plates: (a) β=1.0; (b) β=2.0
图 5 四边固支薄板二次屈曲平衡路径：(a) β=1.0；(b) β=2.0
Figure 5. Secondary buckling paths of 4-edge fixed plates: (a) β=1.0; (b) β=2.0
图 6 方形薄板（β=1.0）后屈曲平衡曲面：(a) λ=110；(b) λ=159.991
Figure 6. Equilibrium surfaces of square plates （β=1.0） on postbuckling paths: (a) λ=110; (b) λ=159.991
图 7 矩形薄板（β=2.0）后屈曲平衡曲面：(a) λ=350；(b) λ=1811.765
Figure 7. Equilibrium surfaces of rectangular plates (β=2.0) on postbuckling paths: (a) λ=350; (b) λ=1811.765
图 8 加载比例k=0.2四边固支方板双向压缩的二次屈曲路径
Figure 8. Secondary buckling paths of a 4-edge fixed square plate with load ratio k=0.2
表 1 单向受压弹性矩形薄板的二次屈曲系数
Table 1. Secondary buckling coefficients of rectangular plates under uniaxial compression
β 4-edge simply supported 4-edge fixed k12 k22 k12 k22 1.0 174.518 7.715 16.210 12.767 1.5 21.693 5.292 24.259 13.066 2.0 11.139 5.351 45.893 13.297 2.5 57.023 4.302 10.315 8.099 3.0 55.059 4.694 52.947 9.087 
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