一维有限单元的节点位移精度

研究高次杆单元和梁单元的节点位移精度问题.首先求出一端固支均匀杆和悬臂梁在任意次多项式形式分布载荷作用下的位移精确解,然后用二次杆单元、五次欧拉梁单元和三次铁木辛柯梁单元求得了节点位移.通过比较有限元解与精确解以及利用静力凝聚方法,发现一次以上杆单元、三次以上欧拉梁单元以及三次以上铁木辛柯梁单元都可以给出精确的端点位移.

NODAL DISPLACEMENT ACCURACY OF ONE-DIMENSIONAL FINITE ELEMENTS

One-dimensional (1D) components as rod, shaft and beam are widely used in engineering structures. This paper investigates the nodal displacement accuracy of 1D high-order rod element and beam elements subjected to arbitrary polynomial distributed loads. First, the exact solutions are derived for the uniform fixed-free rod and the Euler beam as well as the Timoshenko beam,then the free-end nodal displacements are obtained by using a second-order rod element, a fifth-order Euler beam element and a third-order Timoshenko beam element, respectively. By comparing the FEM results with the exact solutions, it is shown that accurate nodal displacements can be obtained by using linear or high-order rod elements, and cubic or high-order Euler and Timoshenko beam elements. Moreover, by taking the second-order rod element as an example, the present results and conclusions are validated by using the static condensation method.