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

• Main Authors: Hsiu-Yi Chien, 簡秀伊
• Other Authors: 楊永斌
• Format: Others
• Language: zh-TW
• Published: 2006
• Online Access: http://ndltd.ncl.edu.tw/handle/87630742415586356050

碩士 === 國立臺灣大學 === 土木工程學研究所 === 94 === 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.