高階平面三角形殼元素之研究

Abstract

本文主要目的為探討一個高階平面三角形殼元素在薄殼結構之幾何非線性分析的精確性。本文以共旋轉(co-rotational formulation)有限元素法及增量迭代法來探討薄殼的幾何非線性行為。本文將採用文獻上一個具旋轉自由度的三角形平面元素與一個 連續的高階三角形板元素疊加成一個3節點的高階平面三角殼元素，元素的節點自由度為3個位移、3個旋轉、3個平面應變及3個側向位移二次微分。本文以殼結構之切線剛度矩陣的行列式值來偵測平衡路徑上的分歧點及極限點。 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並以數值例題測試該高階平面三角形殼元素的性能。

A new high order flat triangular shell element for the geometrically nonlinear analysis are presented. In this paper, co-rotational finite element formulation and incremental-iterative method are employed. The new shell element is the superposition of a triangular membrane element with drilling degree of freedom and a continuous high order triangular plate element. The element has 3 nodes and 12 degrees of freedom per node. The degrees of freedom at each node are 3 translations, 3 rotations, 3 membrane strains and 3 second derivative of lateral displacement. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion to detect the buckling state. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is used for solving nonlinear equilibrium equations. Several numerical examples are used to demonstrate the performance of the shell element.