平面三角形殼元素之改善研究

Abstract

本文主要目的為改善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)有限元素法來探討薄殼的幾何非線性行為。本文採用文獻上具旋轉自由度的三角形平面元素與三角形板元素疊加成一個3節點的平面三角殼元素，該元素的節點自由度為3個位移、3個旋轉、及3個平面應變。本文中提出一個決定三角殼元素之節點變形參數的方法。本文以共旋轉total Lagrangian法推導平面三角殼元素幾何剛度矩陣，並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。本研究以文獻上的數值例題探討本研究採用之平面三角殼元素的性能，來說明本文提出的方法的正確性及功效，同時探討不同的元素幾何剛度矩陣對平衡迭代和偵測平衡路徑上分歧點及挫屈模態的影響。

The objective of this paper is to improve the accuracy and efficiency of the flat triangular shell element for the geometric nonlinear analysis. In this paper, the co-rotational finite element formulation is employed. The 3-node triangular shell element employed here is the superposition of the triangular membrane element with drilling degrees of freedom and triangular plate element proposed in the literature. The element has nine degrees of freedom per node: three translations, three rotations, and three membrane strains. A motion process to determine the element deformation nodal rotations is proposed. A co-rotational total Lagrangian formulation is used to derive the geometric stiffness matrix of the triangular shell element. 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 employed for the solution of nonlinear equilibrium equations. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of different geometric stiffness matrices derived using different approximations on the convergence rate of equilibrium iteration and the value of the buckling load are also investigated through numerical examples.