以迭代法進行平面曲梁之線性與非線性分析

An Iterative Method For Linear and Nonlinear Analysis of Planar Curved Beams

簡秀伊(Hsiu-Yi Chien)

指導教授 ： 楊永斌

國立臺灣大學/工學院/土木工程學研究所/碩士(2006年)

https://doi.org/10.6342/NTU.2006.00595

摘要

本文旨在探討以迭代法改善有限元素法求解之精度，以曲梁為例，分成線性與幾何非線性分析兩個部分加以研究。
一般以有限元素法分析平面曲梁線性問題時，將曲梁切割成許多元素來表示。然而這些元素因為受到諧和條件及形狀函數選擇的限制，無法完全滿足曲梁的各項條件。因此在分析時，常會造成很大的誤差。本文將引用由前人所提出的一套迭代流程，來改善分析結果之精準度。由前人之研究可知，在迭代流程中，預測子(predictor)只影響迭代的次數或收斂的速度，而校正子(corrector)卻會影響計算結果的精準度。為驗證此一道理，本文將採用品質並不完美的曲梁元素之幾何勁度矩陣作為預測子，並以準確的校正子，即準確之力與位移關係式，透過迭代流程以消除分析結果之誤差。
另一方面，針對幾何非線性分析部分，前人已經運用剛體運動法則推導出直梁元素之幾何勁度矩陣，本文將進一步延伸應用剛體法則於平面曲梁元素之幾何勁度矩陣的推導，並以外顯式表示之。對於剛體而言，其幾何勁度矩陣可視為模擬元素在受力狀態下進行剛體位移的行為，與材料係數及斷面特性無關聯，故當元素兩端具有相同剛體模態時，不論它是直梁或曲梁元素，都具有相同的幾何勁度矩陣。根據此一原理，我們可以將直梁元素之幾何勁度矩陣，經座標系統轉換後，而得到曲梁元素之幾何勁度矩陣。此一矩陣的可靠性將在實例中加以驗證。

關鍵字

迭代法 ； 幾何非線性 ； 幾何勁度矩陣

並列摘要

In this study, the concept of iterations will be adopted to improve the accuary of finite element solutions. We shall take the planar curved beam as an example. Both linear and geometrically nonlinear analyses will be discussed.
In analyzing the linear behavior of planer curved beams by the finite element method, the beam is usually discretized into a number of elements. However, owing to the constraints of compatibility conditions and selection of shape functions, some errors can still occur for the linear problems. In this study, theiterative procedure proposed by previous researchers will be employed to improve the accuracy of the solution obtained.
According to previous researches, in an iterative procedure, the predictor affects only the number of iterations or the speed of convergence, while the corrector determines solely the accuracy of the iterative solution. To verify such a concept, stiffness matrices with some defects will be used in this study as the predictor, while accurate force-displacement relations will be used as the corrector. It will be demonstrated thatthrough the iterative procedure, the accuracy of the solution can be significantly improved..
For the nonlinear part, it is realized that the rigid body rule was successfully applied to derivation of the geometric stiffness matrix for the planer straight beam element. Such a procedure will be followed in this study to derive the geometric stiffness matrix for the planer curved beam problem, which will be presented in explicit form.
As for a rigid body, the geometric stiffness matrix can be regard as the ability of the element with initial forces in undergoing the rigid motion. For this reason, the geometric stiffness matrix is only related to the shape of the element, but not the properties of the materials or cross sections. Thus, the geometric stiffness matrices for the straight and curved beams will be the same, as long as they have identical nodal degrees of freedom. It follows that the geometric stiffness matrix for a curved beam can derived from that for the corresponding straight beam merely by transformation of the coordinates from the Cartesian system to the curvilinear system. The capability of such an element will be verified in the numerical studies.

並列關鍵字

Iterative Method ； Curved Beam ； Geometric Stiffness Matrix

參考文獻

[26]林詩勃(2005)，簡易非線性三角板元素，國立台灣大學土木工程學研究

[27]陳中偉(2005)，以迭代法修正梁及三角形元素之缺陷，國立台灣大學土木工程學研究所碩士論文，指導教授:楊永斌。

[1]Batoz, J.L., and Dhatt, G., “Incremental displacement alogorithms for nonlinear problems,” Int. J. for Numer. Meth. in Eng. 1979; 14:1262-1266

[2]Cook, R. D., Concepts and Applications of Finite Element Analysis, 2nd ed., John Wiley, New York, 1981.

[3]Chang, J. T., “Derivation of the higher-order stiffness matrix of a space frame element,” Finite Elements in Analysis and Design 2004; 41(1): 15-30