薄壁開口梁之幾何非線性挫屈及挫屈後行為研究

Abstract

本論文以共旋轉全拉格蘭日法推導一個二節點十四個自由度的薄壁開口梁元素，並將其應用在梁結構的幾何非線性分析，非線性挫屈及挫屈後分析。 本論文中元素節點定在剪心，取剪心軸當作梁元素變形的參考軸，元素變形是以未變形前梁元素的幾何形狀當作參考，並且在當前梁元素變形的位置上建立元素座標來描述元素變形。元素節點內力是藉由虛功原理及完全非線性梁理論的一致二階線性化系統地推導。軸向扭轉率的三階項是所有三階項中的支配項，而且是反映梁受到純扭矩時產生非線性行為重要的內力項，因此在元素節點內力中必須予以考慮。 本論文採用基於牛頓-拉福森法配合定弧長控制法的增量迭帶法解非線性平衡方程式。對梁結構的非線性挫屈及挫屈後分析僅考慮非陀螺保守系統。本論文提出一個弧長的拋物線內差法來求非線性挫屈負荷，對應的挫屈模態可化成廣義的特徵值問題並利用逆冪法求解。為了從主要平衡路徑進入次要平衡路徑，將所求得的挫屈模態放大至適當倍數當作擾動位移，然後在非線性挫屈負荷的平衡位置上加入此擾動位移。 本論文以文獻上取得的實驗結果以及非線性殼元素所得到的數值結果來驗證本論文所推導的非線性薄壁開口梁元素的準確性。本論文也與文獻上其他梁元素所得到的數值結果做了比較。本論文並以數值例題分析了斷面形狀、斷面尺寸、梁的細長比、翹曲的邊界條件以及負荷點的作用位置對結構行為的影響。

Studies on the geometrically nonlinear behavior and nonlinear buckling analysis of thin-walled open-section beams have been relatively rare. A two-node displacement-based thin-walled open-section beam element with seven degrees of freedom per node is developed by using corotational total Lagrangian formulation for the geometrically nonlinear, nonlinear buckling and postbuckling analysis of thin-walled open-section beams. In this thesis, element nodes are chosen to be the shear centers of end sections of the element. The shear center axis is employed as the reference axis of the beam element. The element deformations are referred to the initial undeformed geometry of the beam element and described in element coordinates which are constructed at the current configuration of the beam element. The internal nodal forces are systematically derived by using virtual work principal and a consistent second-order linearization of the fully geometrically nonlinear beam theory based on the exact kinematics of Euler beam to consider the coupling among bending, twisting and stretching deformations for the beam element. The third-order term of twist rate is the dominant term for the third-order terms and may be a very important term to reflect the nonlinear behavior of the beam subjected to a pure torque. Hence, the third-order term of twist rate is also considered in the element nodal forces. 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. For the buckling and postbuckling analysis of the structural system, only the nongyroscopic conservative system is considered in this thesis. A parabolic interpolation method of the arc length is proposed to find the nonlinear buckling load. An inverse power method for the solution of the generalized eigenvalue problem is used to find the corresponding buckling mode. In order to gain access to the secondary path from the primary path, at the bifurcation point a perturbation displacement proportional to the first buckling mode is added. To verify the accuracy of the present finite element formulation, numerical examples are studied and compared with published experimental results and numerical results obtained by nonlinear shell elements available in the literature. Comparisons between the present numerical results and those obtained by other beam elements or other methods available in the literature are also given in this thesis. Case studies are performed to investigate the effects of section geometry, slenderness ratio, warping boundary conditions, and location of loading point on the elastic buckling load and postbuckling behavior of the thin-walled beam structures.