平面三角形薄殼元素之共旋轉推導法

Abstract

本研究主要目的是以共旋轉全拉格朗日推導法(Co-rotational total Lagrangian formulation)、von Karman板正確的變形機制、一致性二階線性化(Consistent second order linearization)及虛功原理，推導一個具面內旋轉自由度平面三角殼元素，並將其應用在薄殼結構的幾何非線性及挫屈分析。 本文中推導的平面三角殼元素有3節點、每個節點有9個自由度，元素的節點自由度為節點位移向量的3個分量、節點旋轉向量的3個分量及節點平面應變的3個分量。本研究將元素的三個節點建立在板的中平面上，在三個節點的當前位置建立一個元素座標，並用中平面變形後的位移及法向量描述殼元素的變形。本研究利用極分解定理(Polar decomposition theorem)將殼中平面的變形梯度(Deformation gradient)分解成一個旋轉矩陣(Rotation matrix)和一個伸縮矩陣(Stretch matrix)的乘積，並用一剛接在元素中平面的座標系統的旋轉表示變形梯度中的剛體旋轉。本研究用三個旋轉參數來描述該中平面座標系統的旋轉。 本文採用牛頓-拉福森(Newton-Raphson)法和弧長控制(Arc-length control)法的增量疊代法來解結構的非線性平衡方程式。本研究分析文獻上常見的殼結構基準問題，並與文獻上的線性解、非線性解、挫屈負荷比較。本研究探討元素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。

A facet triangular shell element with drilling degree of freedom is developed by using a corotational total Lagrangian formulation for the geometrically nonlinear analysis of thin shell structure with large rotation but small strain. The element developed has three nodes with nine degrees of freedom per node. The element nodes are chosen to be located at the mid-plane of the plate element. The deformations of the shell element are described in a current element coordinate system constructed at the current configuration of the shell element. The element nodal forces are derived using the virtual work principle, the exact kinematics of the von Karman plate, and the consistent second order linearization. The element tangent stiffness matrix may be obtained by differencing the element nodal force with respect to nodal parameters. The deformation of the shell element is determined by the displacements of the mid-plane and the rotations of a mid-plane coordinate system associated with each point of the mid-plane relative to the current element coordinate system. The origin of the mid-plane coordinates is rigidly tied to mid-plane. Three rotation parameters are defined to describe the rotation of the mid-plane coordinate system. For convenience, two set of nodal parameters are employed to determine the displacement fields of the element. The first set of nodal parameters is chosen to be three nodal displacements, three nodal rotation parameters, and three strains. The second set of nodal parameters is chosen to be three nodal displacement and six nodal values of the first spatial derivative of displacements. To determine the relationship between these two sets of nodal parameters, the deformation gradient at each element node is decomposed into the product of a proper orthogonal matrix and a right stretch matrix by using the polar decomposition theorem, and the rotation matrix corresponding to the rotation of the mid-plane coordinate system relative to the current element coordinate system is regarded as the proper orthogonal matrix. Two sets of nodal parameters are used for the assembly of the structural equations. 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 determinant of the structure is used as the criterion of the buckling state. Benchmark problems for linear and geometric nonlinear analysis of shells given in the literature are studied to demonstrate the accuracy and efficiency of the proposed shell element. The effect of the first order terms of the transformation matrix between the variation of the two sets of nodal parameters on the equilibrium path and buckling load are also investigated.