雙對稱斷面薄壁梁之幾何非線性動態分析

Abstract

本文的主要目的是以共旋轉有限元素法推導一個三維梁元素，並探討三維雙對稱薄壁梁的非線性動態反應。本文提出一個使用顯積分法解含旋轉自由度之非線性運動方程式的數值程序，並比較使用顯積分法及隱積分法求解非線性運動方程式之效率及準確性。 本文使用共旋轉有限元素法推導一個兩個節點，每個節點有七個自由度的梁元素，並在梁元素當前的變形位置上建立元素座標，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。本研究使用旋轉向量來描述元素節點的旋轉，用三個旋轉參數來描述元素的變形旋轉，故推導梁元素時需要兩組節點參數，此兩組節點參數間的轉換矩陣為節點旋轉參數的函數。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至節點參數之零次項及由兩組節點參數的轉換矩陣引進的節點參數一次項。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及節點參數對時間之微分的全微分求得。 本研究用文獻上的例題說明本研究推導之梁元素的正確性及本文提出之顯積分法的數值程序的可行性。本研究並以數值例題探討慣性力及慣性矩陣中之轉換矩陣及還有慣性矩陣中不同的項對非線性動態分析之效率及準確性的影響。本研究也以數值例題比較使用顯積分法的中央差分法與使用的隱積分法的Newmark積分法並配合Newton-Raphson增量迭代法之效率及準確性。

A co-rotational finite element formulation for the geometrically nonlinear dynamic analysis of doubly symmetric thin-walled beam with large rotations but small strain is presented. A numerical procedure based on the central difference method is proposed for the solution of the nonlinear equations of motion with rotation degrees of freedom. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the centroid of the end cross sections of the beam element and the centroid axis is chosen to be the reference axis. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the beam element. The deformation of the beam element is determined by the displacements of the centroid axis and the rotations of element cross section. Three rotation parameters are defined to describe the relative orientation between the element cross section coordinate system rigidly tied to the unwrapped cross section and the current element coordinate system. A rotation vector is used to represent the finite rotation of a base coordinate system rigidly to each node of the discretized structure. The relation among the variation of the three rotation parameters, three infinitesimal rotations about the axes of the current element coordinate system, and the rotation vector is derived. Three set of nodal parameters are employed here for the determination of the displacement fields of the element and for the assembly of the structural equations. 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, and an incremental method based on the central difference method are employed here for the solution of the nonlinear equations of motion. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. The possibility to simplify the tangent inertia matrix is investigated. The accuracy and efficiency of the Newmark method and the central difference method are investigated and compared.