具旋轉自由度之三角平面元素的共旋轉推導法

Abstract

本研究提出一個具旋轉自由度之三角平面元素的共旋轉推導法，本方法可以用來分析具大旋轉但小應變之二維平面的幾何非線性問題。本研究採用的元素為3節點18個自由度的QST (quadratic strain triangle) 元素，該元素的節點參數為兩個節點位移分量、一個節點旋轉及三個節點應變分量，節點應變是定義於一個剛接在該節點的基礎座標上。元素的變形及方程式都是定義於一個建立在當前元素變形位置的元素座標上。本研究提出一個元素的運動過程，以決定元素在當前變形位置及當前元素座標上的節點參數值。 本研究以文獻上常用的線性例題探討本研究採用之元素的性能，同時探討不同的邊界條件及負荷型態對於線性解之收斂速率及精確度的影響。 本研究採用基於牛頓-拉福森(Newton-Raphson)法和弧長控制(arc-length control)法的增量迭代法來解非線性平衡方程式。本研究以數值例題說明本文提出的方法的正確性及功效。

A co-rotational formulation of triangular plane element with drilling degrees of freedom is presented for the nonlinear analysis of two-dimensional plane problems with large rotations but small strains. The element employed here to implement the present method is a 3-node quadratic strain triangle (QST) element. The element nodal parameters are chosen to be two components of nodal displacement vector, one nodal drilling rotation, and three nodal strain components. The nodal strains are defined in a nodal base coordinates which is rigidly tied to the corresponding node. Element deformations and element equations are defined in terms of element coordinates which are constructed at the current configuration of the element. A motion process of the element is proposed to determine the values of the element nodal parameters corresponding to the current deformation of the element in the current element coordinates. Benchmark examples in the literature for linear analysis are studied to investigate the performance of the employed element. The effects of boundary conditions and load types on the convergence rate and accuracy of solution are also investigated. Numerical results show that the performance of the QST element employed here is excellent. An incremental-iterative method based on Newton-Raphson method and arc-length control method is used for solving nonlinear equilibrium equations. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method.