雙對稱開口薄壁Timoshenko梁之非線性動態分析

Abstract

本研究的主要目的是以共旋轉法推導一個三維Timoshenko梁元素，探討三維雙對稱開口薄壁Timoshenko梁的非線性動態反應。該梁元素的節點變形力及切線剛度矩陣之推導採用一致性共旋轉法，但該梁元素的節點慣性力及慣性矩陣之推導採用共旋轉全拉格朗日推導法。 本文中推導的梁元素有兩個節點，每個節點有7個自由度，元素節點定在梁元素端點之斷面形心，並以形心軸當作梁元素變形的參考軸。本研究用節點旋轉向量更新元素節點斷面的方位，在梁元素當前的變形位置上建立元素座標，並在其上定義三個旋轉參數來描述元素斷面的方位與元素座標的關係及元素的變形。梁元素的節點變形力、節點慣性力是利用非線性梁理論、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，並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對時間的微分到二次項，但不考慮節點參數與其微分的耦合項。本文由元素節點變形內力的改變量與擾動節點位移的關係推導梁元素的切線剛度矩陣，元素的慣性矩陣是由元素的節點慣性力對節點參數對時間之微分的微分求得。 本研究利用Newmark積分法搭配Newton-Raphson增量迭代法來解非線性運動方程式，以數值例題來驗證本文所提出之Timoshenko梁元素之效率及正確性，並探討Timoshenko梁的非線性動態反應。

A co-rotational finite element formulation for the nonlinear dynamic analysis of bisymmetric thin-walled Timoshenko beams is presented. The element deformation nodal force and tangent stiffness matrix are derived by consistent co-rotational formulation. The element inertia nodal force and inertia matrix are derived by co-rotational total Lagrangian formulation. 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 axis of centroid 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 the current element coordinate system constructed at the current configuration 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 exact kinematics of the Timoshenko beam is considered. The element deformation nodal forces and inertia nodal forces are derived using the nonlinear beam theory, d’Alembert principle, virtual work principle, and consistent second degree linearization at the current coordinate of the beam element. The terms up to the second order of spatial derivatives of deformation parameters are retained in the element deformation nodal forces, and the terms up to the second order of time derivatives of deformation parameters are retained in the element inertia nodal forces. However, the coupling between deformation parameters and their time derivatives are not considered in the element inertia nodal forces. The element tangent stiffness matrix is derived using the relations between the variation of the element nodal displacement vectors and rotation vectors and the corresponding variation of element nodal forces. The element inertia matrices may be obtained by differentiating the element inertia nodal forces with respect to the first and second time derivatives of the element nodal parameters. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson 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.