铁摩辛柯梁单元刚度矩阵推导

Derivation of element stiffness matrix of Timoshenko beam element

为明确ANSYS中基于铁摩辛柯梁理论的Beam188单元的单刚矩阵,以形函数为基础,根据最小势能原理系统给出了铁摩辛柯梁单元的刚度矩阵推导过程,尤其将常规2节点单元的刚度矩阵扩展至3节点单元,包括等截面梁和变截面梁,并给出了各自的单刚矩阵理论表达式。以矩形、圆形、箱形和圆环四种截面为例,分别给出单刚矩阵理论值与ANSYS中Beam188单元刚度矩阵进行对比。结果表明,3节点等/变截面梁的单刚矩阵与理论推导所得结果一致,但2节点梁的单刚矩阵与理论推导所得结果存在偏差,且此偏差仅表现在与弯曲相关的元素中,并且随着单元长度的减小,两者的偏差也越来越小。另外,对比时应注意,ANSYS所得截面参数如剪切系数和极惯性矩与理论值存在较大偏差。

The study is initiated for the element stiffness matrix of Timoshenko beam element named Beam188 in the software ANSYS. Based on the shape functions and the principle of minimum potential energy, the derivation process and expressions of element stiffness matrix are provided both for the 2-node and 3-node elements, and both for constant-section beam and tapered-section beam. Then four types of sections including rectangular, circular, box and tube are chosen as examples, and their element stiffness matrixes got from theoretical derivation and from Beam188 in ANSYS are compared. Results show that the element stiffness matrixes of Beam188 are identical to the theoretical results for 3-node element, no matter constant or tapered-section. For 2-node element, however, the element stiffness matrixes of Beam188 show some difference on the elements which are related to the bending deformation, and the difference would be reduced with the reduction of element length. Moreover, the section properties got from ANSYS would be different from the theoretical value, especially for the shear factor andattention should be paid on.