臺灣博碩士論文加值系統

本文利用一階剪應變形的假設與虛功原理推導平版大變形理論，而以移動最小二乘法進行數值模擬，並引入Quasi-Hermite Type Formulation將變位及合應力進行聯結，使方程式簡化且降低變量之微分階數，以避免直接處理複雜之平衡方程式，求解其偏微分方程。數值求解時，利用Newton-Raphson Method將非線性方程式轉換成線性之方程組並加以迭代計算，求解大變形後之位置，進而分析其力學行為。本文數值算例有三，其一為四邊簡支平版受雙正弦荷重，用以驗證理論於線性部分之數值解是否與線性理論解析解相近似，另二算例為懸臂平版受純彎矩作用及在不同支承條件下受均佈荷重作用之分析。利用本文之數值解與理論之解析解比較，用以驗證本文理論與數值方法之正確性，討論大變形下之力學行為。

The study emphasizes on utilizing Large Deformation Theory derived from the first-order shear deformation assumption and the principle of virtual work. Furthermore, we adopt the moving least squares method for numerical simulation. In order to avoid complex equilibrium equations resulted from direct solution, a Quasi-Hermite Type Formulation is introduced in the stress-displacement relations to simplify the equation and reduce the orders of differentiation of variable. The nonlinear equations will be calculated based on the Newton-Raphson Method. The study demonstrates three numerical examples. First is a simply supported plate under the sinusoidally-distributed load, compared with the linear part of the numerical solution of the present method with analytic solution to verify accuracy of the numerical method; the other is the plate subjected to pure bending moment, and the third is by adopting the plate under one uniform load while different boundary conditions are applied. By comparing the data obtained from theoretic assumptions and results from the simulations, the theory adopted in the study has high accuracy and is proven to perform the structural behavior under significant deformation.

摘要 I
Abstract II
誌謝 VIII
目錄 X
圖目錄 XIII
符號說明 XV
第一章 緒論 1
1.1 研究動機 1
1.2 文獻回顧 3
1.3 分析方法 7
1.4 本文架構 8
第二章 平版大變形理論 9
2.1 平版座標系統 9
2.2 平版變形前中平面特性 9
2.3 變形後平版之中平面特性描述 10
2.3.1 變形後中平面之基本性質 10
2.3.2 平版變形前後中平面基底向量之關係 11
2.4 平版大變形應變分析 14
2.4.1 平版變形之幾何描述 14
2.4.2 Lagrangian應變 16
2.5 平版大變形之合應力與中平面應變關係 18
2.6 平衡方程式 19
2.6.1 虛功原理 19
2.6.2 第二種Piola-Kirchhoff 應力張量與Cauchy應力張量 21
2.6.3 斷面合應力與合彎矩 22
2.6.4 合應力之平衡方程式 23
2.6.5 邊界條件方程式 25
第三章 非線性方程式解析之數值方法 27
3.1 修正量計算方程 28
3.2 平衡方程修正量關係式 30
3.2.1 斷面合力平衡方程式 30
3.2.2 斷面彎矩平衡方程式 30
3.2.3 轉角限制方程式 30
3.3 邊界條件修正量關係式 31
3.3.1 法線方向合力邊界條件 31
3.3.2 切線方向合力邊界條件 31
3.3.3 an方向合力邊界條件 31
3.3.4 彎矩邊界法線方向分量 31
3.3.5 彎矩邊界切線方向分量 32
第四章 移動最小二乘法理論推導 33
4.1 移動最小二乘法 33
4.2 Quasi-Hermite Type Formulation 38
4.3 鄰近點與權重函數之選取 41
第五章 數值算例 42
5.1. 四邊簡支矩形版受雙正弦載重 42
5.1.1. 求解變位之解析解 42
5.1.2. 驗證解析解並討論非線性行為 43
5.1.3. 收斂率比較 44
5.2. 均向性平版受單向純彎矩 45
5.2.1. 均向性平版受單向純彎矩之理論分析 45
5.2.2. 數值分析 46
5.3. 正方形平版受均佈荷重作用 47
5.3.1. Case1：四邊皆為固定端受均佈載重 47
5.3.2. Case2：四邊皆為鉸支承受均佈載重 47
5.3.3. Case3：兩邊固端兩邊鉸支受均佈荷重 48
5.4. 兩端為自由端之矩形版受均佈荷重 49
5.4.1. Case1：在x=0與x=Lx為固定端 49
5.4.2. Case2：在x=0與x=Lx為鉸支承 49
5.4.3. Case3：在x=0為固定端，x=Lx為鉸支承 49
第六章 結論 51
參考文獻 52
圖 54

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