Exact Solutions for Axisymmetric Bending of Laminated Cylindrical Shells With General Boundary Conditions
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摘要: 复合材料层合圆柱壳因其材料与结构的优异特性，在化工、海洋和航空航天工程等领域中得到了广泛使用，该结构承载时在层间和边界附近的局部力学响应较为复杂，并会影响整体结构的工作性能. 状态空间法是求解层合结构精确解的一种有效方法，在处理非简支边界条件时通常需要借助数值方法. 该文基于层合圆柱壳的状态空间框架，将非简支端的边界位移函数也作为状态变量引入状态方程，建立了严格满足边界条件的齐次状态方程；然后，利用层合渐近技术将状态矩阵的系数常数化，得到了任意子层的状态传递关系，结合层间的连续性条件推出了层合圆柱壳的状态传递关系；最后，通过Fourier级数引入内外表面的载荷条件，得到了结构轴对称弯曲问题的精确解. 算例表明，所得解与有限元解吻合得很好，能够给出层合圆柱壳应力和位移沿轴向和径向的精确分布规律. 然后，分析了不同边界条件和铺层形式对静力响应的影响. 另外，通过固支端与自由端及其附近的位移与应力分布规律说明了两种约束产生的端部效应.Abstract: Due to the excellent properties of materials and structures, composite laminated cylindrical shells are widely used in such key fields as chemical, marine and aerospace engineering. However, the local mechanical responses near the interfaces and boundaries are complex and affect the performances of the structures. As an effective method to obtain the exact solutions of the laminated structures, the state space method needs the numerical simulation to deal with the non-simply supported boundaries. Based on the state space framework for laminated cylindrical shells, the boundary displacement functions at the non-simply supported ends were introduced into the state equations as state variables, and homogeneous state equations were established to strictly satisfy the boundary conditions. Then, the variable coefficients in the state equations were converted to constants with the lamination approximate method, and the transfer relations of the mechanical quantities along the thickness of the laminated cylindrical shell were established. Finally, the loading conditions on the surfaces of the cylindrical shells were introduced with the Fourier series, and the exact solutions to the axisymmetric bending problems were obtained. The examples show that, the present solutions are consistent with the finite element ones, and give the exact distributions of the stresses and displacements along the axial and radial directions of the laminated cylindrical shells. In addition, the displacement and stress distributions near the clamped and free ends help illustrate the end effects of the two constraints.
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图 1 层合圆柱壳及其层合渐近模型
Figure 1. The laminated cylindrical shell and its approximate sub-layers
图 2 层合圆柱壳位移和应力沿轴向x的分布
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 2. Displacement and stress distributions along the x direction of the laminated cylindrical shells
图 3 层合圆柱壳跨中位移和应力沿径向r的分布
Figure 3. Displacement and stress distributions along the r direction of the laminated cylindrical shells
图 4 CC层合圆柱壳自由端附近位移和应力沿径向r的分布
Figure 4. Displacement and stress distributions along the r direction of the CC laminated cylindrical shells
图 5 CF层合圆柱壳位移和应力沿径向r的分布
Figure 5. Displacement and stress distributions along the r direction of the CF laminated cylindrical shells
表 1 层合圆柱壳特定点的位移和应力
Table 1. Displacements and stresses of laminated cylindrical shells
$\frac{r-r_{\mathrm{a}}}{h}$ CC CS CF FEM present FEM present FEM present u
CC: x=l/4
CS/CF: x=l0 0.349 7 0.339 2 -11.156 6 -11.047 4 0.647 9 0.573 2 1/3 0.258 9 0.258 2 6.081 5 5.996 0 0.470 7 0.473 6 0.5 0.071 8 0.071 4 0.672 5 0.670 5 0.038 5 -0.113 1 2/3 -0.108 8 -0.109 1 0.200 3 0.199 7 -0.195 8 -0.268 8 1 -0.183 1 -0.183 4 8.762 5 8.759 5 -0.297 6 -0.301 5 w
x=l/20 1.209 4 1.209 2 1.210 1 1.209 8 1.209 0 1.208 5 1/3 1.070 3 1.070 4 1.071 0 1.071 0 1.069 9 1.069 8 0.5 1.017 7 1.017 7 1.018 4 1.018 4 1.017 2 1.017 3 2/3 0.993 0 0.993 0 0.993 7 0.993 7 0.992 5 0.992 4 1 0.975 7 0.975 7 0.976 24 0.976 3 0.975 2 0.975 4 10σr
x=l/20.1 -9.694 9 -9.765 5 -9.695 0 -9.760 4 -9.694 8 -9.748 8 1/3 -9.048 7 -9.056 8 -9.050 2 -9.056 7 -9.054 5 -9.027 7 0.5 -4.389 3 -4.399 3 -4.389 4 -4.399 5 -4.388 3 -4.385 4 2/3 -0.289 8 -0.290 2 -0.288 6 -0.290 8 -0.286 4 -0.291 2 0.9 -0.082 9 -0.077 9 -0.082 9 -0.080 3 -0.082 9 -0.051 8 τrx
x=00.1 3.028 4 2.942 8 3.028 8 3.065 0 3.028 6 3.062 3 1/3 1.546 4 1.039 2 1.546 4 1.039 3 1.546 4 1.039 5 0.5 0.831 7 0.839 9 0.831 7 0.840 0 0.831 7 0.839 6 2/3 1.221 3 0.872 2 1.221 3 0.872 3 1.221 2 0.873 1 0.9 1.475 3 1.547 7 1.475 4 1.496 1 1.475 0 1.496 1 100σx
x=l/20 -7.383 2 -7.834 1 -7.958 8 -8.180 8 -5.213 2 -6.430 8 1/3- -7.417 2 -8.012 8 -7.417 2 -7.894 3 -6.105 6 -6.780 0 1/3+ -3.671 0 -3.713 2 -3.367 1 -3.704 9 -3.622 1 -3.655 3 0.5 0.632 6 0.619 5 0.654 6 0.641 2 0.543 9 0.639 6 2/3- 4.533 6 4.555 2 4.586 4 4.592 4 4.534 4 4.528 6 2/3+ 4.668 4 5.356 7 5.885 6 6.223 4 4.754 4 5.142 1 1 4.910 0 5.132 5 6.370 8 6.484 0 4.850 4 4.955 1 σθ
x=l/20 0.167 7 0.168 7 0.167 8 0.168 7 0.167 8 0.168 5 1/3- 0.129 9 0.129 9 0.130 0 0.130 0 0.130 0 0.130 2 1/3+ 5.432 4 5.436 4 5.436 0 5.440 0 5.430 4 5.433 9 0.5 5.042 8 5.046 0 5.046 4 5.049 6 5.040 4 5.044 1 2/3- 4.810 4 4.813 5 4.814 0 4.816 8 4.808 0 4.810 2 2/3+ 0.189 5 0.189 7 0.189 8 0.189 9 0.189 4 0.189 5 1 0.177 7 0.177 6 0.178 0 0.177 9 0.177 6 0.177 6 
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