基于Mathematica的单元刚度矩阵显式解析解推导

Explicit Analytical Solutions for Element Stiffness Matrix Based on Mathematica

借助Mathematica给出了平面四边形单元刚度矩阵的两种显式解析形式.第一种形式采用直接积分方法,所得的刚度矩阵解析形式比较复杂;第二种结果基于高斯积分方法,所得的刚度矩阵解析形式相对简单.解析解的推导过程中对复杂的常数项部分用简单的假设变量代替,如对Jacobi行列式所做的变量代换,刚度矩阵的积分形式得到简化.最后,通过具体算例验证了两种显式解析形式进行单元刚度矩阵显式求解的有效性和实用性.

The paper proposes two explicit solutions for plane quadrilateral stiffness matrix based on mathematica.The first solution is produced by direct integration with a complicated form.The second solution is obtained from Gauss integration with a relatively simple form.During the generation of the analytical form,the complex constant item is replaced by simple hypothetical variables,such as the variable replacement of the determination of Jacobi matrix.The integrated form of stiffness matrix is simplified.Finally,the effectiveness and practicability of the two explicit solutions are verified by an example.