平面梁結構在位移負荷作用下之幾何非線性分析

Abstract

本文提出一個共旋轉全拉格蘭日有限元素推導法及數值程序以探討平面尤拉梁在位移負荷作用下的幾何非線性行為。 本文中的梁元素有兩個節點，每一節點具有三個自由度。節點座標、增量位移與增量旋轉、以及結構的平衡方程式都是定義在一組固定的總體座標系統上；而梁元素在應變與元素剛度矩陣，則是在當前元素位置上建立的元素座標上定義。並在當前元素座標上以虛功原理配合完全非線性梁理論的一致線性化有系統地推導元素節點內力。 本文提出一個基於牛頓－拉福森法的增量迭代法以求解受位移負荷之結構的非線性平衡方程式。為通過平衡路徑中之極限點及轉彎點，本文修改Crisfield的定弧長控制法[6]與Fried的正交軌線法[15]，使其能夠應用在受到一個負荷參數的位移負荷作用之結構分析。本文中提出以二次曲線內插法決定增量位移向量弧長，以求得挫屈負荷；並使用逆冪法計算挫屈模態；為了探討結構挫屈後的行為，本文採用位移擾動法，配合增量迭代法追蹤次要平衡路徑。 本文以例題來驗證本文所提之數值方法的準確性及有效性，並探討不同的平面梁結構受到位移負荷作用時的幾何非線性行為。

A co-rotational total Lagrangian finite element formulation and a numerical procedure for the geometrically nonlinear analysis of planar Euler beam under displacement loading are presented. The nodal coordinates, displacements, rotations, and the equilibrium equations of the structure are defined in a fixed global set of coordinates. The beam element has two nodes with three degrees of freedom per node. The kinematics of beam element is defined in terms of element coordinates, which are constructed at the current configuration of the beam element. The element internal nodal forces are systematically derived by consistent linearization of the fully geometrically non-linear beam theory using the virtual work principle in the current element coordinates. A incremental-iterative method based on the Newton-Raphson method is proposed for solving nonlinear equibrium equations with displacement loading. In order to pass limit points and turning points, Crisfield's constant arc length method [6] and Fried's orthogonal trajectory method [15], which are suitable for one parameter force loading, are adapted so that they are suitable for one parameter displacement loading. A parabolci interpolation method of the are length is proposed to find the buckling load. An inverse power method is used to calculate the buckling mode. A perturbation displacement method is used to find the secondary equilibrium path. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and investigate the geometrically nonlinear behavior of planar beam structures under displacement loading.