雙對稱開口薄壁Timoshenko梁之非線性分析

Abstract

本研究的主要目的是以一致性共旋轉法探討雙對稱開口薄壁Timoshenko梁的幾何非線性挫屈及挫屈後的分析。 本文中推導的梁元素有兩個節點，每個節點有7個自由度。本文中將元素節點定在斷面剪心，並取剪心軸當作梁元素變形的參考軸。本研究在當前梁元素變形的位置上建立元素座標，並在其上描述元素的變形。為了準確描述Timoshenko元素的變形，本研究利用虛功原理並考慮剪力修正係數推導元素節點內力，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。線性剛度矩陣包含在元素切線剛度矩陣裡面。 本文求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。 為了驗證本研究提出的方法的準確性與有效性，本研究以不同數值例題探討剪應變對雙對稱開口薄壁梁之負荷—位移曲線及挫屈負荷的影響，並與Euler–Bernoulli梁的結果做比較。

A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of bisymmetric thin-walled Timoshenko beams is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the centroid of the end cross-sections of the beam element and the axis of centroid is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The exact kinematics of the Timoshenko beam is considered. The element nodal forces are derived using the virtual work principle with the consideration of the shear correction factor. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. A consistent second-order linearization of the element nodal forces is used here. Thus, all coupling among bending, shearing, twisting, and stretching deformations of the beam element is retained. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix. 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 studied and compared with the results obtained by using Euler beam element to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of the shear deformation on the loading–deflection curves and buckling load of the bisymmetric thin-walled beams.