不對稱開口薄壁梁元素之一致性共旋轉推導法及其在挫屈分析的應用

Abstract

本研究的主要目的是以一致性共旋轉法推導一個不對稱開口薄壁梁元素，並探討其節點內力及切線剛度矩陣中一些高次項對結構之非線性行為及挫屈負荷的影響。 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。本研究利用虛功原理，完整的幾何非線性梁理論的一致性二次線性化推導元素節點內力，本研究的節點內力除了保留全部的二次項外還保留了軸向扭轉率的三階項。 本研究用虛功原理推導梁元素節點內力時，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，所以不能僅由元素節點內力對節點參數微分求得切線剛度矩陣，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。 本文中採用牛頓－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，以系統切線剛度矩陣之行列式值為零當作挫屈準則，本研究以數值例題分析梁結構的幾何非線性行為及挫屈負荷，並與文獻比較，以說明本研究提出的方法的正確性。因隨著元素數目的增加，梁元素之長度，剪心軸側向位移的一次微分，及扭轉角會趨近於零，所以元素節點內力及剛度矩陣中含這些量之項亦會趨近於零，故本研究亦以數值例題探討這些項對元素之收斂性分析的影響。

A consistent procedure is proposed to derive the element internal nodal force and tangent stiffness matrix for asymmetric thin-walled beam element with open section using the virtual work principle combined with co-rotational total Lagrangian formulation. The effects of some higher order terms of element node forces and tangent stiffness matrix on the buckling load and postbuckling behavior of beam structures. In this thesis, a two-node element with seven degrees of freedom per node is developed. The element nodes are chosen to be the shear center of end section of the element. The shear center axis is employed as the reference axis of the beam element. The element deformation are referred to the undeformed geometry of the beam element and described in element coordinates which are constructed at the current configuration of the beam element. The element internal nodal forces are systematically derived by using virtual work principal and a consistent second-order linearization of the fully geometrically nonlinear beam theory. In this study, the terms up to the second order of rotation parameters and their spatial derivatives and the third-order term of twist rate are retained in the element nodal forces. When the virtual displacement method is used to derive the element nodal force, the external virtual work done by the element internal nodal force and the virtual nodal displacement are defined in the current element coordinates, which are regarded as fixed coordinates. However, the internal virtual work done by the element stress and the virtual strain corresponding to the virtual nodal displacement are defined in an element coordinates which are constructed at the disturbed configuration of the beam element corresponding to virtual nodal displacement. Note that the rigid body motion part in the virtual displacement is eliminated in the derivation of the internal virtual work. The tangent stiffness matrix is derived from the increment of the element nodal force corresponding to an infinitesimal incremental displacement. The increment of the element nodal force comprises the direction change of the element internal nodal force corresponding to the rigid body motion part in the infinitesimal incremental displacement and the increment of the element nodal force corresponding to the deformation part in the infinitesimal incremental displacement. 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. To verify the accuracy of present formulation, numerical examples are studied and compared with published experimental results and numerical results for geometrically nonlinear behavior and nonlinear buckling load. With the increase of element number, the length of beam element, the twist angle and slopes of the beam axis will approach to zero. Thus, The corresponding terms in element internal nodal forces and element stiffness matrices will approach to zero too. To investigate the effects of these terms on the convergent solution of buckling load and deflections for beam structures, convergence tests are carried out for different numerical examples.