高階幾何非線性平面三角形薄殼元素之研究

Nonlinear analysis of the high order triangular thin flat shell element

黃楚璋(Chu-Chang Huang)

指導教授 ： 蕭國模

國立交通大學/工學院/機械工程系所/博士(2018年)

摘要

本研究之主要目的是以共旋轉全拉格朗日推導法(Corotational total Lagrangian formulation)提出一個具面內旋轉自由度之高階平面三角殼元素，並將其應用在薄殼結構的幾何非線性及挫屈分析。
本研究推導的平面三角殼元素有三個在元素中平面的節點，每個節點有九個自由度。在三個節點當前的位置建立一個元素座標，並在當前的元素座標上並用中平面的位移及其變形後之法向量的方位描述殼元素的變形。為描述變形後之法向量的方位，本研究定義了三個旋轉參數。為了方便推導，本研究採用兩組節點自由度，一組為三個位移分量、三個旋轉向量的分量、三個應變，用來組合殼結構之系統節點內力及剛度矩陣；另一組為三個位移分量、六個位移分量的一次微分，用來決定元素內部的位移場。為決定兩組節點自由度的關係矩陣，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。並將該旋轉矩陣視為剛接在元素中平面之座標系統的旋轉矩陣。
本研究用虛功原理、Kirchhoff板正確的變形機制及一致性二階線性化(Consistent second order linearization)，推導元素節點內力，用元素節點內力對節點自由度的微分推導元素切線剛度矩陣。
本研究採用基於弧長法和牛頓-拉福森法的增量迭代法解非線性平衡方程式，以結構切線剛度矩陣之行列式值為零偵測挫屈點。本研究分析各種文獻上常見的殼結構基準問題，並與文獻上的非線性解、挫屈負荷比較，以測試本研究之殼元素的性能，並探討元素節點內力及切線剛度矩陣之高次項對平衡路徑及挫屈負荷的影響。

關鍵字

有限原素法 ； 平面殼元素 ； 非線性 ； 挫屈負荷

並列摘要

The purpose of this research is to develop a high order facet triangular shell element with drilling degree of freedom by using a corotational total Lagrangian formulation for the geometrically nonlinear analysis and buckling analysis of thin shell structure with large rotation but small strain.
The element developed here has three nodes with nine degrees of freedom per node. The element nodes are chosen to be located at the mid-surface of the 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 deformation of the shell element is determined by the displacements of the mid-surface and the orientation of normal of the deformed mid-surface. Three rotation parameters are defined to describe the orientation of normal of the mid-surface. For convenience, two set of nodal parameters are employed. The first set of nodal parameters is used for the assembly of the structural equations. They are chosen to be three components of nodal displacement vector, three components of nodal rotation vector, and three strain components. The second set of nodal parameters is used to determine the element displacement fields. They are chosen to be three components of nodal displacement vector and six nodal values of the first spatial derivative of displacement components. 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 (rotation matrix) and a right stretch matrix by using the polar decomposition theorem. The rotation matrix is regarded as the orientation matrix of a mid-surface coordinate system rigidly attached to the mid-surface. The element nodal forces are derived using the virtual work principle, the exact kinematics of the Kirchhoff plate, and the consistent second order linearization. The element tangent stiffness matrix is obtained by differencing the element nodal force with respect to nodal parameters.
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 geometric nonlinear analysis and buckling analysis of shells given in the literature are studied to investigate the effect of the high-order terms of the element internal nodal force and stiffness matrix on the equilibrium path and buckling load.

並列關鍵字

finite element ； thin flat shell element ； Nonlinear analysis

參考文獻

[8] 林寬政, “平面三角形殼元素之改善研究”, 交通大學機械工程學系碩士論文, 台灣, 新竹, 2010.

[9] 盧致群,“高階平面三角形殼元素之研究”, 交通大學機械工程學系碩士論文, 台灣, 新竹, 2011.

[1] K. J. Bathe, L. W. Ho, “A simple and effective element for analysis of general shell structures”, Computer and Structure, vol. 13, pp. 673–681, 1981.

[2] C. Pacoste, “Co-rotational flat facet triangular element for shell instability”, Computer Methods in Applied Mechanics and Engineering, vol. 156, pp. 75–110, 1998.

[3] P. Khosravi, R. Ganesan, R. Sedaghati, “An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures”, Computer and Structure, vol. 86, pp. 850–858, 2008.