雙對稱開口薄壁梁元素之一致性共旋轉推導法及其在挫屈分析的應用

Abstract

本研究的主要目的是以一致致性共旋轉法提出一個推導雙對稱開口薄壁梁元素節點內力及切線剛度矩陣的方法。 本研究用虛功原理推導梁元素節點內力時，元素節點內力所作的虛功是在元素受虛位移擾動前的元素座標上推導，但元素應力所作的虛功是在元素受虛位移擾動後的元素座標上推導，即將元素座標建立在元素受虛位移擾動後的位置，並在其上定義元素的變形及推導虛應變。本研究推導的元素節點內力能滿足靜力的平衡。本研究由元素節點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。因本研究在推導元素的節點內力時，扣除了虛位移中剛體運動的部分，而元素的節點內力與元素一起剛體運動，所以不能僅由元素節點內力對節點參數微分求得，還要考慮元素節點內力在剛體運動時因方向改變造成的元素節點內力的改變。本文解非線性平衡方程式的數值計算方法是基於牛頓－拉福森(Newton-Raphson)法配合弧長控制(arc length control)法的增量迭代法。本研究中以系統切線剛度矩陣之行列式值為零當作挫屈準則，利用弧長的二分法求得挫屈負荷，為了測試本研究提出的方法的有效性及準確性，本研究以不同的例題，探討其幾何非線性行為及挫屈負荷並與文獻的結果比較。

A consistent procedure is proposed to derive the element internal nodal force and tangent stiffness matrix for doubly symmetric thin-walled beam element with open section using the virtual work principle combined with co-rotational total Lagrangian formulation. When the virtual displacement method is used to derive the element nodal force, the external virtual work done by the element internal nodal force and the virtual nodal displacement are defined in the current element coordinates, which are regarded as fixed coordinates. However, the internal virtual work done by the element stress and the virtual strain corresponding to the virtual nodal displacement are defined in an element coordinates which are constructed at the disturbed configuration of the beam element corresponding to virtual nodal displacement. Note that the rigid body motion part in the virtual displacement is eliminated in the derivation of the internal virtual work. The tangent stiffness matrix is derived from the increment of the element nodal force corresponding to an infinitesimal incremental displacement. The increment of the element nodal force comprises the direction change of the element internal nodal force corresponding to the rigid body motion part in the infinitesimal incremental displacement and the increment of the element nodal force corresponding to the deformation part in the infinitesimal incremental displacement. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the buckling load of doubly symmetric thin-walled beams.