三維固端梁之第二次挫屈分析

Abstract

本研究所稱的固端梁是一端固定，另一端僅側向固定的細長直梁，若將此固端梁施加軸向壓縮位移負荷，當壓縮量到達某一臨界值時，此固端梁會發生尤拉挫屈(Euler buckling)，本文中稱其為第一次挫屈，如繼續加以軸向壓縮，則該梁會側向拱起，本文將這個挫屈後的固端梁稱為挫屈梁。若將此挫屈梁繼續施加軸向壓縮位移，則其主要平衡路徑，即原固端梁的次要平衡路徑，為一穩定的平衡路徑。當軸向壓縮位移到達某一臨界值時，該挫屈梁會發生一面外(Out of plane)的側向－扭轉挫屈，本文稱其為第二次挫屈(Secondary buckling)。 本文用共旋轉有限元素法探討不同斷面之固端梁受端點軸向位移負荷的挫屈及挫屈後的行為，還有探討兩端固定之挫屈梁，中心點受側向集中位移負荷之非線性挫屈及挫屈後的行為。 本文在當前的元素座標上，以尤拉梁正確的變形機制、工程應變、虛功原理與有限元素法、非線性梁理論的一致線性化（consistent linearization），推導一個十四個自由度的梁元素。本研究採用基於牛頓法及定弧長法的增量迭代法求解非線性平衡方程式，以系統的切線剛度矩陣的行列式值是否為零作為挫屈的判斷準則。 本文中用不同的數值例題比較以工程應變和Green strain 推導之梁元 I 素對不同問題的影響，再以不同的數值例題探討不同斷面之固端梁的第二次挫屈、挫屈後的行為及影響固端梁第二次挫屈位移之因素，最後再探討挫屈梁中心點受側向集中位移負荷之非線性挫屈及挫屈後的行為。

The originally straight fixed-end beam is compressed axially at one end. When the axial compression is greater than a critical value, the beam will buckle in the lateral direction which has the smallest second moment of area of the beam cross section. This buckling is the so called Euler buckling and referred to as the first buckling here. The fixed-end beam after the first buckling is stable and referred to as the buckled beam. If the buckled beam is further compressed axially, it may buckle out of plane. This buckling is referred to as the secondary buckling here. A consistent co-rotational finite element method is used to investigate the geometric nonlinear buckling and postbuckling behavior of the fixed-end beam subjected to the axial compression and the buckled beam with fixed axial compression subjected to a mid-span lateral displacement. The beam element developed here has two nodes with seven degrees of freedom per node. The deformations of the beam element are described using the exact kinematics of the Euler beam in the current element coordinate system constructed at the current configuration of the beam element. The element nodal forces are derived using the virtual work principle, consistent second-order linearization of the fully geometrically non-linear beam theory, and engineering strains and stresses. III IV 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 compare the results obtained by the beam elements derived using the engineering strain and the Green strain, respectively. The effects of different rigidity ratios of the beam cross section on the second dimensionless buckling displacement of the fixed-end beam subjected to the axial compression are investigated through numerical examples. The buckling modes corresponding to the first two critical points and the postbuckling behavior of the buckled beam with different fixed axial compression subjected to a mid-span lateral displacement are also investigated.