薄梁板壳的数值计算涉及关于挠度的4阶微分方程,其困难在于构造C₁连续的近似函数;同时,由于薄曲梁和曲壳控制方程的复杂性,通常用直梁或平板单元近似地模拟曲梁或曲壳,容易产生几何误差进而带来力学分析上的误差。前期研究采用独立覆盖流形法实现了基于厚梁板壳假设的精确几何曲梁和曲壳分析,本文在此基础上讨论了这种新型流形法的分区级数解的C₁连续性,完成了基于Euler-Bernoulli梁理论和Kirchhoff-Love板壳理论的精确几何薄曲梁和曲壳分析,并解决了几何公式推导复杂的问题。详细给出了薄曲梁的计算公式,简述了薄曲壳的计算过程,将前期文献中的算例在薄梁板壳假设下重新计算,验证了方法的有效性,相比厚梁板壳假设可节省约30%的自由度。研究成果同时展示了应用独立覆盖流形法求解4阶微分方程的潜力。

The numerical calculation of thin beam, plate and shell involves the fourth-order differential equation about deflection whose difficulty lies in constructing approximation functions with C₁ continuity. In the meantime, due to the complexity of the governing equation, the thin curved beam and curved shell are usually simulated approximately by using straight beam or flat plate elements, which is prone to generate geometric errors and then brings errors in mechanical analysis. In our previous study, manifold method based on independent covers is used to analyze curved beam and shell with exact geometry based on the assumption of thick beam and shell. On this basis, the C₁ continuity of the piecewise-defined series solutions of the new manifold method is discussed. The thin curved beam and shell with exact geometry is analyzed based on Euler-Bernoulli beam theory and Kirchhoff-Love shell theory, and the complexity of derivation of geometric formula is overcome. The calculation formula of thin curved beam is given in detail, and the process of thin curved shell is briefly described. The examples in previous study are recalculated under the assumption of thin beam, plate and shell, which verifies the effectiveness of the proposed method. Compared with the assumption of thick beam, plate and shell, the method saves about 30% of the degree of freedom. Meanwhile, the research demonstrates the potential of solving the fourth-order differential equations by applying manifold method based on independent covers.