雙對稱斷面開口薄壁梁之幾何非線性動態分析 的一致性共旋轉推導法

Abstract

本研究以一致性共旋轉法推導一個雙對稱斷面開口薄壁尤拉梁元素，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十四個自由度。本研究用元素節點對固定座標的位移及旋轉向量更新其位置及其斷面的方位，並在梁元素節點當前的位置及斷面方位建立一個固定元素座標及與其重合的移動元素座標。本研究用三個旋轉參數來描述元素斷面的方位與元素座標的關係。本研究利用在當前固定元素座標的元素節點位移、旋轉向量及其擾動量、速度、加速度、角速度、角加速度，推導出元素節點在當前固定元素座標的擾動位移和擾動旋轉向量與元素節點旋轉參數之擾動量的關係、擾動後之移動元素座標與當前固定元素座標的關係、移動元素座標的角速度及角加速度、元素節點的變形參數對時間的一次及二次微分。本研究利用虛功原理和D’Alembert原理，以及完整的幾何非線性梁理論的一致性二階線性化在當前的固定元素座標推導元素節點變形力及慣性力。本研究由元素節點變形內力的改變與擾動位移的關係推導梁元素的切線剛度矩陣。本研究將元素節點慣性力表示成元素節點之絕對速度、加速度、角速度、角加速度的函數，所以元素的質量矩陣與陀螺效應之關係矩陣分別由元素的節點慣性力對元素之節點加速度向量的微分及元素之節點速度向量的微分求得。本研究發現以一致性共旋轉推導法與共旋轉全拉格朗日推導法得到的節點慣性力有些差異，但這些差異會在元素增加時會趨近於零。 本研究以數值例題說明本研究之梁元素的準確性，並探討本文之一致性共旋轉法與文獻上共旋轉全拉格朗日法在動態分析時之差異性。另外本研究比較Newmark積分法與中央差分法在不同例題之效率及準確性。

A consistent co-rotational (CCR) finite element formulation for geometrically nonlinear dynamic analysis of doubly symmetric 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 centroids of the end cross sections of the beam element and the centroid axis is chosen to be the reference axis. A rotation vector is used to represent the finite rotation of coordinate systems rigidly tied to each node of the discretized structure. The incremental nodal displacement vectors and rotation vectors in global coordinates are used to update the node locations and orientation of the element. The deformations of the beam element are described in a current moving element coordinate system constructed at the current node locations and orientation of the beam element. 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. The element equations are derived in a fixed current element coordinates which are coincident with the current moving element coordinates. The perturbed displacements and spatial rotation, velocity and acceleration, angular velocity and angular acceleration of the current moving element coordinates, and the variation of the element nodal rotation parameters corresponding to the perturbation of element nodal displacement vectors and rotation vectors in the current fixed element coordinates are consistently determined and expressed in terms of the current element nodal displacements and rotation parameters, nodal velocities and accelerations, and nodal angular velocities and angular accelerations. The element deformation and inertia nodal forces are derived using the virtual work principle, the d’Alembert principle, and the consistent second order linearization of the fully geometrically nonlinear beam theory. In element deformation nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered. In the element inertia nodal forces, the terms up to the second order of time derivatives of deformation parameters are retained. However, the coupling between rotation parameters and their time derivatives are not considered in the element inertia nodal forces. In this study, the element inertia nodal forces are expressed in terms of element nodal velocities and accelerations, and nodal angular velocities and accelerations. Thus, the element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the element nodal velocities and accelerations. There is a slight difference between the present element inertia nodal forces and that derived using corotational total Lagrangian (CRTL) formulation. However, the effect of the difference may be negligible with the decrease of element size. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed for the solution of nonlinear equations of motion. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. It is found that the difference between the dynamic responses obtained using CCR formulation and CRTL formulation is negligible for all examples studied.