三維挫屈梁之非線性分析

Abstract

本研究使用共旋轉全拉格蘭日有限元素法分析空間中一個兩端固定之細長梁，先在一端受固定的扭轉角，再施加端點軸向強制位移後的挫屈負荷及挫屈後行為。 本文採用由蕭國模博士與林文一博士所提出的一致性共旋轉全拉格蘭日有限元素法[18]來推導梁元素，應變採用以Green strain以及Engineering strain來進行分析，以完整非線性梁理論之二階一致線性化(consistent second order linearzaton)推導含撓曲、扭曲及軸向變形間耦合效應梁元素。 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。以系統切線剛度矩陣之行列式值為零做為挫屈準則。 數值例題將探討空間中不同細長比之細長梁，在受到固定轉角及軸向壓縮之挫屈負荷及挫屈後的軸向反力，比較以Green strain以及Engineering strain來分析的差異，並考慮梁之自重以及梁之初始不完美的影響。

The buckling and postbuckling behavior of clamped spatial rods subjected to a prescribed end axial rotation first, and then under compressive axial displacements is investigated using the corotational finite element method. The consistent co-rotational finite element formulation for three-dimensional Euler beam presented by Hsiao and Lin [18] is employed here. Both the Green strain and engineering strain are used for the measure of strain here. All coupling among bending, twisting, and stretching deformations for beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam 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 presented to investigate the effect of the prescribed end axial rotation and slenderness ratio on the initial, buckling, and postbuckling end axial reaction force of spatial rods under end compressive axial displacements. The results obtained using the Green strain and the engineering strain, are compared. The effect of self-weight and initial imperfection is investigated also.