用高阶单元分析平面应力结构的自由振动

Higher Order Element for Free Vibration Analysis of Plane Stress Structures

基于YKCheung带面内转角的平面应力单元 ,提出了一种用于剪力墙结构自由振动分析的 32～ 5 6自由度的高阶单元 ,单元的每个节点采用u ,u/x和v ,v/x为自由度 ,在墙体水平方向采用 3次Hermite函数 ,铅垂方向采用了 3～ 6次Lagrange多项式 ,对开孔墙体则提出了按剪切刚度等效和质量密度等效的连续化方法 .为了克服高阶元列式复杂的困难 ,采用下标求和形式的表达式通过标准有限元过程获得了单元刚度矩阵和质量矩阵 ,再用迭代法计算结构自振频率 给出了 5个算例 ,验证了高阶元所具有的高精度 ,并发现了剪力墙结构前3阶振型中当高宽比H/B≤ 10时 ,第 3阶振型为竖向振动 ,而当H/B≤ 2时 ,第 2 。

Based on the Cheung's beam-type element with rotational DOF, this work developed a higher order element of 32 56 DOF for vibration analysis of shear wall structures. The element employed u, &partu/&part.r, v and &partv/&part.r as DOF for each node, and 3-order Hermite polynomial and up to 6-order Lagrange polynomial as interpolation function for displacement modelling. For wall with openings, a continuum element was adopted on the principle of equivalence of shear stiffness and mass density. In order to simplify the formulation of higher order element, all expressions were formulated by summation of subscript and the element stiffness matrix and mass matrix were obtained by standard FEM procedure. Vibration frequence of structure was computed by iteration method. Five examples given in this paper verified the excellent accuracy of higher order element and discovered that among the first 3 vibration modes of shear wall structures, the 3rd vibration mode at the ratio of height to width H/B&le10 and the 2nd and 3rd vibration mode at H/B&le2 are all vertical vibrations of wall.