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系統識別號 U0026-2606201119334500
論文名稱(中文) 圓柱座標系統下彈性力學問題之狀態空間解析法
論文名稱(英文) State Space Approach to Some Problems of Elasticity in Cylindrical Coordinates
校院名稱 成功大學
系所名稱(中) 土木工程學系碩博士班
系所名稱(英) Department of Civil Engineering
學年度 99
學期 2
出版年 100
研究生(中文) 曾維德
研究生(英文) Wei-Der Tseng
電子信箱 wdtseng@mail.njtc.edu.tw
學號 n6894101
學位類別 博士
語文別 中文
論文頁數 114頁
口試委員 指導教授-譚建國
口試委員-葉超雄
召集委員-吳光鐘
口試委員-洪宏基
口試委員-王仲宇
口試委員-宋見春
口試委員-陳東陽
中文關鍵字 異向彈性力學  懸臂圓柱  曲樑  圓柱正向性  特徵函數展開  Hamiltonian  狀態空間  辛正交性 
英文關鍵字 Anisotropic elasticity  Circular cylinders  Curved beams  Cylindrical orthotropy  Eigenfunction expansion  Hamiltonian  State space  Symplectic orthogonality 
學科別分類
中文摘要 異向彈性力學問題之解析通常係儘可能消去基本方程式之應力或位移分量,以簡化問題之控制方程式,不可避免的是,所得之控制方程式階數升高。廣受重視者殆為Lekhnitskii與Stroh解析模式,傳統解析模式所能求解之問題僅限於廣義平面問題,難以推廣至三維問題。本文發展一異向彈性力學問題之Hamiltonian狀態空間解析架構,將異向彈性力學之基本方程式,以位移向量及其共軛之應力向量為狀態向量表示,藉由分析力學中之Hamiltonian變分及Legendre變換,建構出一明確、易處理之狀態方程式及輸出方程式。本狀態空間解析模式所內含之特徵系統具備Hamiltonian特性,得以藉分離變數及特徵函數展開解析三維問題。為闡述如何應用本解析架構,文中列舉若干傳統方法未能處理之二維及三維問題,包括三維懸臂圓柱受拉、扭、彎作用,及二維圓柱正向性曲樑及圓拱之精確解析,並評估各問題中之端部效應,廣義平面應變解、樑彎曲基本理論解於相關問題以及異向性材料之適用性。
英文摘要 Problems of anisotropic elasticity are conventionally formulated on the basis of Lekhnitskii's formalism or Stroh's formalism, which are intended for problems of plane deformations of anisotropic elastic bodies. To eliminate the unknown variables from the basic equations as many as possible, the conventional approach for problems of the cylindrical coordinate system leads to higher-order partial differential equations with variable coefficients. Extension of the classical formalisms to problems of coupled fields is difficult. In this work,a state space formalism for linear elasticity of cylindrically anisotropic materials is developed by taking the displacement vector and the stress vectors as the state variables. By means of Hamiltonian formulation and the Legendre transformation, the basic equations of anisotropic elasticity in cylindrical coordinates are formulated into the state space framework in which the state equation, the output equation, and the boundary conditions are expressed neatly in terms of the state vector that comprises the displacement vector and the associated conjugate stress vector as the dual variables. Hamiltonian symplecticity of the formalism are examined at length, which provide an essential basis for developing a solution approach using separation of variables and eigenfunction expansion. For illustration,3-D exact analysis of extension, torsion and bending of a circular cantilever and 2-D exact analysis of cylindrically orthotropic curved beams and arches subjected to inplane loads are studied. The fixed-end effects and applicability of the solutions of generalized plane strains and the elementary theory of bending of beams are evaluated.
論文目次 摘要 I
Abstract II
誌謝 III
目錄 IV
表目錄 VII
圖目錄 VIII
符號表 XI
第一章 緒論 1
1 研究動機 1
2 論文大綱 2
第二章 基本理論 3
1 基本方程式 3
2 邊界條件及主要方向之選取 6
3 Lagrangian及Hamiltonian解析方式 10
4 狀態方程式 12
4.1 以z為特定座標之狀態方程式 12
4.2 以r為特定座標之狀態方程式 13
4.3 以θ為特定座標之狀態方程式 14
5 解析方法 16
5.1 函數形式 16
5.2 問題之分解 18
6 特徵系統之Hamiltonian特性 20
6.1 系統矩陣之特性 20
6.2 非重根特徵值對應之特徵向量之辛正交 23
6.3 重根特徵值對應之特徵向量之辛正交 25
6.4 辛正交之應用及隱含之物理意義 27
第三章 曲樑及圓拱問題 31
1 問題陳述 31
1.1 狀態方程式與邊界條件 31
1.2 傳統解析方式 34
2 狀態空間解析 36
2.1 特別解 36
2.2 狀態方程式之解 41
2.3 應用辛正交決定係數 45
3 結果與討論 47
第四章 有限長圓柱之軸對稱問題 57
1 問題陳述 57
1.1 狀態方程式與邊界條件 57
1.2 傳統解析方式 59
2 狀態空間解析 61
2.1 狀態方程式之解 61
2.2 應用辛正交決定係數 67
3 結果與討論 68
第五章 有限長圓柱之三維問題 78
1 問題陳述 78
1.1 狀態方程式與邊界條件 78
1.2 傳統解析方式 81
2 狀態空間解析 82
2.1 狀態方程式之解 82
2.2 應用辛正交決定係數 90
3 結果與討論 92
第六章 結論與後續研究 103
參考文獻 105
附錄A 109
附錄B 112
自述 113
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