| 摘要: | 結構分析之首要目的為決定結構體系在已知載重下所對應構件
中之應力、應變以及位移等資料。在計算力學體系出現之前，大多將
結構劃分為簡單之桿、梁、板、殼等不同類型之構件，透過材料力學
及結構力學推導其受力後之力學模型，並針對複雜之力學問題進行簡
化以便於計算。
隨著電腦計算機之出現，眾多學者結合力學與數值運算而發展出
了計算力學，使得力學理論於應用上更為方便且可處理之問題更為廣
泛。其中以虛功法所推導之非線性有限元素雖可以得到可靠且精確之
分析模擬，但推導過程往往過於複雜，且傳統有限元係以矩陣模式做
為計算之基礎，當結構體系過於龐大且受力模式過於複雜時，往往需
消耗過多計算時間處理矩陣運算，且矩陣運算有時受限於其數學上求
解之奇異狀況而導致數值運算之問題。
本研究採用一簡單之幾何非線性處理之程序，亦即搭配共轉座標
演算法，藉由建立非耦合型態之運動方程式並搭配隱式Newmark-β
法以求解控制方程式，且推導一系列之二維固體元素，進行數值分
析，探討其於高度非線性分析上之能力。;The main idea of structural analysis is to determine the stress, strain and
displacement of the system. Before the computational mechanics had
been developed, several analysis were conducted by treating the
structures as a series of bar, beam, plate and shell members and derived
the simplified governing equations which were based on the mechanics of
material and structural mechanics.
Since the computers have been developed, mechanics finally can combine
with the numerical analysis, the computational mechanics started to be
widely applied and had a great effect on convenience and solving difficult
problems. Among several numerical method, the nonlinear finite element,
which is based on the principle of virtual work, can give more accuracy in
the analysis, but sometimes the derivation is too complicated, also,
traditional finite element method is of coupled matrix formulation, this
phenomenon lead to a difficult point on solving the simultaneous
equations if the structural system is huge enough or the mechanical
behavior is complex, when a system contains the above problems the
matrix calculation will waste too much time or even can make the matrix
solution singular.
In order to solve the problems of traditional finite element, in this study
we consider an easy way to treat the geometric nonlinearity, that is,
corotational formulation, also, constructing the uncoupled-type equations
of motion and solving them by the use of implicit Newmark-β method,
and deriving nonlinear plane solid elements as well as testing their ability
at highly geometrically nonlinear analysis. |
