雙對稱變斷面薄壁梁受軸力及扭矩作用之幾何非線性分析

Abstract

本研究採用文獻[1]中6節點退化薄壁元素的觀念，用共旋轉法提出一6節點20個自由度的退化薄壁元素以分析雙對稱I型變斷面薄壁梁的幾何非線性行為。本文中之元素僅考慮軸向位移與軸向旋轉造成的側向位移，每個元素有4個元素節點與2個斷面節點，且元素的變形皆在建立於元素當前的變形位置的元素座標系統中描述。本文中利用大位移理論的二階一致線性化，考慮了元素節點力軸向與扭轉變形之間的偶合作用。 本文解非線性平衡方程式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。 本文中的例題探討不同變斷面I型薄壁梁受不同軸向負載下的扭轉挫屈負荷與挫屈後的行為，並驗證文獻上變斷面梁之扭轉挫屈負荷詭論的正確性，此詭論為當梁翼板材料減少時其扭轉挫屈負荷反而增加；或當梁翼板材料增加時其扭轉挫屈負荷反而降低。本文中亦探討了變斷面梁受軸向力與扭矩同時作用下的幾何非線性行為。

A six node degenerate element is proposed based on the concept of Ref. [1] by using consistent co-rotational finite element formulation for the geometric nonlinear analysis of doubly symmetric thin-walled I beam with slow varying flange. Only the axial displacement and axial rotation are considered for the element developed here. The kinematics of the element is governed by two sectional nodes and four true element nodes. The deformations of the element are described in the current element coordinate system, which is constructed at the current configuration of the element. In element nodal forces, all coupling between twisting and stretching deformations of the element is considered by consistent second-order linearization of the large displacement theory. 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. Numerical examples are studied to verify the accuracy of the present method and investigate the torsional buckling load and post-buckling behavior of thin-walled I beams with different variable sections subjected to different axial loads. The geometric nonlinear behavior of thin-walled I beams subjected to axial load and axial torque simultaneously are also studied.