在本論文中，主要探討圓形板及環形板軸對稱振動分析在位移邊界條件之外滿足應力邊界條件的影響。採用由傳統位移變化原理(conventional displacesment–type variational principle)及Reissner’s principle所組合成的一種新的軸對稱有限元素(Axisymmetric finite element)加以分析。藉由這種列式，應力會如同位移一樣是主要的變數。而應力和位移邊界條件也可以容易並正確的加以使用。 藉由此種方法得到一些典型圓形板及環形板的軸對稱振動頻率，並且和位移形式的軸對稱有限元素分析相互比較。研究結果發現，傳統的有限元素分析雖然沒有符合應力邊界條件，卻仍然可以得到足夠準確的圓形板及環形板振動頻率。

In the present study, effect of satisfying stress boundary conditions, in addition to displacement boundary conditions, in the axisymmetric vibration analysis of circular and annular plates is investigated. A new axisymmetric finite element, which is based on a combination of the conventional displacement-type variational principle and the Reissner’s principle, is proposed. With this formulation, stresses, like displacements, are primary variables, and both displacement and stress boundary conditions can be easily and exactly imposed. Axisymmetric vibration frequencies of some typical circular and annular plates are then obtained with the present approach and are compared with those by the displacement-type axisymmetric finite element. It is found that the conventional finite element, though not satisfying stress boundary conditions, can still obtain sufficiently accurate vibration frequencies of circular and annular plates.

摘要 …………………………………………………… 2
目錄 …………………………………………………… 4
表目錄 ……………………………………………… 6
圖目錄 ……………………………………………… 7

第一章 緒論 ……………………………………… 8
1-1 前言 ……………………………………… 8
1-2 文獻回顧 ………………………………… 8
1-2-1 板的自由振動解析解方法 …………………… 9
1-2-2 古典板理論 …………………………………… 9
1-2-3 Mindlin板理論 ……………………………… 10
1-2-4 較高階剪力變形理論 ……………………… 11
1-2-5 板的自由振動數值解方法 …………………… 11
1-2-6 三維彈性力學有限元素法 ………………… 12

第二章 滿足應力邊界條件對圓形板及環形板軸對稱振動分析的影響 …………………………………………… 14
2-1 前言 …………………………………………… 14
2-2 理論推導 ……………………………………… 14

第三章 問題解析 …………………………………… 23
3-1 問題描述 …………………………………… 23

第四章 結果與討論 ………………………………… 29
4-1 收斂試驗 ……………………………………… 29
4-2 文獻比較與結果之討論 ……………………… 30

第五章 結論與建議 ………………………………… 46
5-1 結論 ………………………………………… 46
5-2 建議 ………………………………………… 46

參考文獻 …………………………………………… 47
附 錄 …………………………………………… 53

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