不對稱斷面開口薄壁梁之幾何非線性動態分析

Abstract

本研究的主要目的是以共旋轉全拉格朗日推導法推導一不對稱斷面之開口薄壁梁元素，探討三維梁之非線性動態反應。本文中推導的梁元素有兩個節點，各節點有七個自由度，元素節點定於梁元素兩端點斷面之剪心，並以剪心軸當元素的參考軸。本研究在元素當前的變形位置建立元素座標，並在該座標系統描述元素的運動，將元素座標視為一固定的局部座標，所以梁元素在元素座標系統的速度及加速度為絕對速度及絕對加速度。為了描述元素斷面的方位，本研究在元素斷面的形心建立一剛接於其上的元素斷面座標，並使用旋轉向量來描述元素節點斷面座標的旋轉，但用三個旋轉參數來描述元素座標與元素斷面座標間的變形旋轉，故推導梁元素時需要兩組節點參數。 本文利用非線性梁理論、D’Alembert原理和虛功原理及一致性二階線性化在當前的元素座標上推導梁元素的節點變形力、節點慣性力。本文保留元素的節點變形力至節點參數之二次項，保留元素的節點慣性力中所有節點參數對時間微分的項，但不考慮節點參數與節點參數對時間微分項之耦合項。元素的剛度矩陣是由元素的節點變形力對節點參數的微分求得，元素的慣性矩陣是由元素的節點慣性力對節點參數之時間導數(time derivatives)的微分求得。 本文基於Newmark直接積分法和Newton-Raphson迭代法來求解非線性運動方程式。本研究以文獻上的例題，驗證本研究之方法的效率及準確性。

A corotational total Lagrangian (CRTL) finite element formulation for the geometrically nonlinear dynamic analysis of asymmetric thin-walled beam with large rotations but small strain is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear center of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. For the purpose of treating arbitrarily large rotation of node in space, the orientation of the node is described by a base coordinate system rigidly tied to each node of the discretized structure, and a nodal rotation vector is used to represent the finite rotation of the base coordinate system. The values of nodal rotation vectors are reset to zero at current configuration, thus, the values of the first and second time derivative of the nodal rotation vector are equal the values of the spatial nodal angular velocity and acceleration. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the element. The current element coordinate system is regarded as an inertial local coordinate system, not a moving coordinate system. Thus, the first and the second time derivative of the position vector defined in the element coordinates are the absolute velocity and absolute acceleration. Three rotation parameters referred to the current element coordinates are defined to determine the orientation of element cross section. The deformation of the beam element is determined by the displacements of the shear center axis and the rotations of element cross section. The element deformation nodal forces and inertia nodal forces are systematically derived by the d'Alembert principle, the virtual work principle and consistent second order linearization in the current element coordinates. The element stiffness matrix may be obtained by differentiating the element deformation nodal forces with respect to the element nodal parameters, and the element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the element nodal parameters, and their first and second time derivatives. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear equations of motion. The standard Newmark method is applied to the incremental displacement and rotational vectors, and their time derivatives. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.