以合理的外在勁度矩陣進行結構幾何非線性分析

Abstract

有關於結構的幾何非線性分析，文獻中一般利用元素的自然變形來求解元素內力，即對元素軸向位移進行修正，得到元素在變形過程時軸向的實際伸長量，此法於梁式結構的分析中可以得到很好的結果，但是對於其他自然變形於有限元素法中不易定義之結構型式，此法則較不容易應用；此外，在過去文獻中，也可以外在勁度矩陣的觀念將剛體運動效應排除，但是此一概念在計算上並不如自然變形法來得精確，也未被廣泛使用，因此，本研究的目的在於推導合理之外在勁度矩陣以改善此法，增加其可行性，使得此一概念在計算元素內力時能更加完善。在本研究中利用文獻中對於二維梁及三維梁所定義的各種變形模態，以及幾何勁度矩陣的性質，推導外在勁度矩陣，並針對三維梁元素於空間中旋轉所引致的彎矩效應作修正，以外在勁度矩陣概念建立內力校正式，在經由實例分析過後，可以驗證本研究所建立的內力校正式在分析非線性結構的可行性，並滿足幾何非線性分析的需求。

A geometrically nonlinear analysis can be basically decomposed into two phases: the predictor phase and corrector phase. How to calculate the member forces for each incremental step in the corrector phase is a critical issue, as it controls the accuracy of the solution. In many previous works, the member forces are obtained by a well-known method based on the natural deformations. In calculating the natural deformations, the axial displacement of each element is modified as an approximation of the real axial extensions. This method is effective in the analysis of structures with beam elements, but it cannot be readily applied to other types of elements for which the natural deformations are difficult to define. On the other hand, in some previous studies, the concept of external stiffness matrix has also been used to eliminate the influence of rigid body motions in calculating the member forces. This concept has the advantage that the natural displacements need not be directly computed, but in comparison with the former method, it is not accurate enough, which may lead to slow convergence for some problems. Therefore, the objective of this thesis is to improve the external stiffness matrix method and to extend its practicability by deriving physically qualified external stiffness matrices.Based on various natural and rigid body modes presented previously for the two-dimensional and three-dimensional beams, along with the properties of the external stiffness matrix, this thesis will first derive qualified external stiffness matrices, which will then be modified through incorporation of the moments induced by the rotation of the beam in the three-dimensional space. Next, the formula for calculating the member forces incorporating the concept of external stiffness matrix will be derived. Finally, through the numerical verifications, it is demonstrated that the member force formula presented herein is feasible and can be generally used in the geometrically nonlinear analysis of structures.