关键词: 从平面弹性力学的基本方程出发, 利用Fourier积分变换等数学手段, 推导出了单层平面问题的刚度矩阵, 然后按有限元法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程和Fourier积分逆变换, 得到在任意静荷载作用下多层弹性平面问题的精确解。由于刚度矩阵不含有正指数项, 计算时不会出现溢出现象, 从而克服了传递矩阵法的缺点。推导过程中摒弃了应力函数的选择, 使得问题的求解更加合理化, 同时也为进一步研究此类问题如温度场、动力学等奠定了理论基础。计算实例证明了推导结果的准确性。

Abstract: Exact solution of multilayer elastic plane in a rectangular coordinate system is obtained on the basis of Fourier integral transformation and stiffness matrix method. Stiffness matrix for monolayer derived firstly based on the fundamental elasticity equations and some mathematic methods such as Fourier integral transformation. And then the stiffness matrix is established for multilayer elastic plane, using the finite element concepts in which layers are completely contacted. Therefore, exact solution of multilayer elastic plane is obtained from the solution of algebra equation formed by stiffness matrix and inverse Fourier integral transformation. Due to the element of matrix is not included positive exponential function, the calculation is not overflowed. The shortages of transfer matrix method are overcome. This method is clear in concept, and the corresponding formulas are given not only simple but also convenient for application. More important thing is the method can be used to solve other problems for multilayered elastic plane, such as thermo field and dynamics. A numerical example is presented to prove the correction of calculated results.

Key words: multilayered elastic plane, stiffness matrix, Fourier integral transform