二維固體有限元素之幾何非線性動力分析

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張顥(Hao Chang) 查詢紙本館藏

畢業系所

土木工程學系

論文名稱

二維固體有限元素之幾何非線性動力分析

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 計算力學
★ 幾何非線性
★ 共轉座標演算法
★ 非耦合型態
★ 隱式Newmark-β 法
★ 二維固體元素

關鍵字(英)

★ computational mechanics
★ geometric nonlinearity
★ corotational formulation
★ uncoupled
★ implicit Newmark-β method
★ plane solid element

論文目次

摘要……………………………………………………..……………….I
Abstract………….……………………………………………………….II
誌謝…………………………………………………...……………….III
目錄..…………………………………………………..…..…………..IV
表目錄………………………………………………..……….……VIII
圖目錄………………………………………………..…….………IX
第一章緒論………………………………………...…..……………….1
1.1 研究動機與目的………………………………..………………1
1.2 文獻回顧……………………………………..…………………2
1.2.1 向量式有限元素法………………….………………….2
1.2.2 幾何非線性有限元素法……………..…………….4
1.3 論文架構…….…………………………………………………7
第二章幾何非線性有限元素法…………………….……………..8
2.1 平衡方程式與虛功原理…………………………..…………8
2.2 應變與應力分析…………………………………..………..11
2.2.1 參考構形…………………………………..……...…..11
2.2.2 變形梯度…………………………………..……...…..12
2.2.3 應變與應力………………………………….…...…..15
2.2.4 應力與應變描述之空間轉換………………….…..19
2.3 幾何非線性有限元推演法…………….……………………24
2.3.1 更新式增量拉格朗日演算法.………………….…25
2.3.2 更新式增量共轉座標演算法.………………….…29
第三章二維共轉固體有限元之推導…………………………...…39
3.1 離散模型與控制方程式………………….….………………39
3.2 共轉座標與剛體旋轉……………...………….………………41
3.2.1 共轉座標之取………………………………………..41
3.2.2 剛體旋轉計算………………………………..……..42
3.3 固體元素之節點應力與內力…………………………..……..46
3.3.1 三節點等應元…………………………………..…….47
3.3.2 四節點等參數元……………………………………...56
3.4 固體元素之節點外力…………………………………..……..61
3.4.1 三節點等應變元…………………………...…..……..61
3.4.2 四節點等參數元………...............................…..……..63
3.5 固體元素之節點質量………....................................…..……..64
第四章採用隱式直接積分法之有限元素……………………..……..74
4.1 隱式 Newmark-直接積分計算程序…………….………74
4.2 二維固體元素勁度比例阻尼力計算……………………….83
第五章數值算例分析與驗證…………………………………………90
5.1 元素之剛體轉動檢驗…………………………………………90
5.1.1 三節點等應變元之剛體旋轉測試…………………90
5.1.2 四節點等參數元之剛體旋轉測試…………………91
5.2 大變形與大變位之靜力分析…………………….………...…91
5.2.1 柔性懸臂梁受靜態集中載重分析…….………...…92
5.2.2 柔性懸臂梁受靜態均佈載重分析…….………...…93
5.2.3 方形框架之大變位分析….…………………...…...…95
5.2.4 無約束直梁端點受均佈剪應力之運動...……........…96
5.3 結構失穩行為分析…………………………………………....97
5.3.1 底端固接頂端自由之理想柱挫屈分析……...……....98
5.3.2 兩端鉸接之理想柱挫屈分析……...………………....99
5.4 動力分析…………………………………………………......101
5.4.1 兩端固定之繩索大變位振動分析……………......101
5.4.2 單點支承梁受自重之單擺運動………………......102
5.4.3 雷利阻尼之動力歷時分析驗證………………......103
第六章結論與未來展望…………………………………...….…..…153
6.1 結論……………………………………………………...…153
6.2 未來展望……………………………………………………..156
參考文獻………………………………………………..…….……….157
附圖………..………………………………………………….……….162

參考文獻

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指導教授

李姿瑩(Tzu-Ying Lee)

審核日期

2016-1-27

推文