欧拉-伯努利梁单元刚度矩阵推导

Derivation of Element Stiffness Matrix of Euler-Bernoulli Beam Element

为明确欧拉-伯努利梁单元刚度矩阵推导过程,用统一的坐标系统和参数正方向规定,包括荷载、位移、内力和应力等,对等截面梁详细介绍了不同受力模式包括伸缩、扭转和弯曲问题的拉格朗日形函数和厄米特形函数,系统给出了刚度矩阵推导所需的基本方程包括几何方程、本构方程和平衡方程,并最终根据虚功原理得到单元刚度矩阵。这不仅有助于对单元刚度矩阵以及推导过程的理解,还可为参阅有限元专业书籍提供帮助。

The derivation process of element stiffness matrix of Euler-Bernoulli beam element was presented in detail in this paper. The derivation was based on the uniform coordinate and positive direction regulations of loads, displacements, internal forces and stresses. Following the detailed description of shape functions for tension, torsion and bending deformation, the basic equations including the geometric equations, physical equations and equilibrium equations were provided as well. Then the element stiffness matrix was deduced according to the principle of minimum potential energy. The complete and systematic derivation process is very helpful for understanding of element stiffness matrix.