Abstract

The plate problem has been investigated for centuries while exact solutions for anisotropic plates are still hard to obtain. In this paper, we aim to get exact solutions for rectangular anisotropic plates with four clamped edges through the state space method. A state space equation for the anisotropic elasticity is derived from the linear elasticity theory for the first time. The Fourier series in exponential form are adopted in the current work. This can transform the transfer matrix with differential operators in state space into a constant matrix. After superposition and differential treatments, the equations for the boundary conditions of four clamped edges are then combined with the state space equation for the anisotropic elasticity, forming a new compound state space which represents the 3 D anisotropic plate problem to be solved. By use of the state space method, a solvable linear equation system under the compound state space is formed. Example solution cases for both orthotropic and monoclinic plates with different thicknesses are provided. For comparison purpose, numerical results from the finite element method are also given to indicate the reliability and accuracy of the current state space method.

摘要

板块问题已经研究了几个世纪，而各向异性板块的精确解仍然难以获得。在本文中，我们旨在通过状态空间方法获得具有四个夹紧边缘的矩形各向异性板的精确解。首次从线弹性理论推导出各向异性弹性的状态空间方程。目前的工作采用指数形式的傅里叶级数。这可以将状态空间中带有微分算子的传递矩阵转换为常数矩阵。经过叠加和微分处理，将四个夹紧边缘的边界条件方程与各向异性弹性状态空间方程结合起来，形成一个新的复合状态空间，代表待求解的3D各向异性板问题。利用状态空间方法，形成了复合状态空间下的可解线性方程组。提供了不同厚度的正交各向异性板和单斜板的示例解决方案。为了进行比较，还给出了有限元法的数值结果，以表明当前状态空间方法的可靠性和准确性。