Improvement of Classical Shell Theory and Its Application to Numerical Analysis of Stability for Shells

在固体非线性弹性一般理论中可以证实，如果外力具有势能，失稳只能以静力失稳的形式发生，而绝不会发生动力失稳。这意味着线性化的壳体稳定性方程在数学上是自伴的。由于基本理论最初是固体形式的，因此，用任何方法得到的壳体稳定性方程都必然是自伴的。但一些壳体稳定性理论方程，如。Vlassoy型方程的稳定性与自伴性是不一致的，构造与上述理论相一致的基本方程，特别是稳定性方程的自伴性是很重要的。在我们的研究中，壳的基本方程在以下假设下被简化。1)相对于中间表面的法向轴的旋转可以忽略不计。2)由于壳体足够薄，因此假定壳体承受的载荷分布在壳体的中表面上，在上述两种假定下，可以推导出三种类型的方程，如：考虑横向剪切变形的Sanders型和Vlassoy型。然后分别利用这类方程，采用Galerkin法和有限差分法结合Newton-Raphson法对圆柱壳和球壳在非保守外力作用下进行了数值分析。结果表明，有限差分法是构造用于平衡状态分析和稳定性分析的刚度矩阵的有效方法。采用Calerkin法、有限差分法、Newton - Raphson法、Newmark B法和稳定性分析新方法，在颤振域中求出了复杂状态下的极限环。

It can be confirmed in the general theory of nonlinear elasticity of solid that the lost of stability can take place only in the form of static instability and the dynamic instability by no means take place, if the external forces have a potential. This means that the linearized equation for the stability of shell is self-adjoint mathematically. As the basic theory is originally in the form of solid, the stability equation for the shell obtained from any kinds of methodology should be self - adjoint definitely. But some equations of theories for stability of shells, eg. Vlassoy type equations, have inconsistency with a self-adjointness.It is important to compose the basic equations consistent with the theory mentioned above, especially the self- adjoint property of the stability equations. In our study, the basic equations of shells are simplified under the following assumptions. 1) Rotations to the normal axis of the middle surface may be neglected. 2) The shell is sufficiently thin, therefore the shell is assumed to be loaded with loads distributed over the middle surface.Under the two assumptions mentioned above, three types of equations, eg. the type considering the transverse shear deformation, Sanders type and Vlassoy type, are expressed. And then using these types equations respectively, numerical analysis of cylindrical shells and spherical shells under the non-conservative external forces are performed by using Galerkin method and finite difference method together with Newton - Raphson method. Consequently it was shown that the finite difference method is very effective to make up the stiffness matrix for the analysis of equilibrium state and the stability analysis. Furthermore the limit cycle described in complicated state were found out in the flutter domain by the usual method of analysis by using Calerkin method, finite difference method together with Newton - Raphson method, Newmark b method and the new method of stability analysis.