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

在固体非线性弹性的一般理论中可以证实，如果外力有势，稳定性的丧失只能以静失稳的形式发生，而动失稳则绝不发生。这意味着壳层稳定性的线性化方程在数学上是自伴随的。由于基础理论本来是固体形式的，所以用任何一种方法得到的壳的稳定性方程都必须是明确的自伴随方程。但是关于壳层稳定性的一些理论方程，例如。Vlassoy型方程，具有不一致的自伴随性。建立与上述理论相一致的基本方程，特别是稳定性方程的自伴随性，具有重要的意义。在我们的研究中，壳的基本方程在以下假设下进行了简化。1)中面法向轴的旋转可以忽略。2)壳足够薄，因此假定壳的荷载分布在中表面。在上述两个假设下，有三种类型的方程，例如：考虑横向剪切变形的类型为Sanders型和Vlassoy型。然后分别采用Galerkin法、有限差分法和Newton - Raphson法对非保守外力作用下的圆柱壳和球壳进行了数值分析。结果表明，有限差分法是构造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、有限差分法与Newton - Raphson法、Newmark b法及新的稳定性分析方法，在颤振域内找到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.