本文擬以平面應變有限元素法分析樑結構之自由振動行為。對於具有相當寬度之樑，以彈性力學為基礎，推導其位移型平面應變有限元素列式。並且不加入其它的假設及簡化，針對所有的位移邊界條件都能夠如實加入，而與傳統僅根據中性面行為的樑理論分析有所差異，因此本文所採用的分析方法相較於傳統樑理論的分析，應該能夠更準確的求取樑結構之自然振動頻率及振動模式。
本文所求得之數值分析結果將分別與尤拉樑理論、Timoshenko樑理論及高階樑理論做比較，以了解本文所採用的方法與其他傳統樑理論間的差異，並且探討幾何形狀及邊界條件對於整體樑結構之自然振動頻率的影響。

In the present study , the free vibration of beam structures is analyzed by a plane strain finite element method . The displacement-type plane strain finite element formulation is based on elasticity and has no any other assumptions and simplifications except that the beam is of moderate depth . Also, all the displacement boundary conditions can be imposed exactly . These are the advantages that beam theories of conventional approach, which simulate beams with neutral plane behaviors, do not have. Therefore the present analyses should be able to obtain natural frequency and mode shape more accurately than conventional methods.
The numerical values of natural frequencies of the present approach will be compared with previously published results of the Euler-Bernoulli beam theory and Timoshenko beam theory ,
and further with the high order beam theory to reveal their diff -erences .The effects of geometry ratio and displacement boundary condition on natural frequency of beams are also discussed.

目錄

目錄 ………………………………………………………………...i
圖目錄 …………………………………………………………….iii
表目錄 …………………………………………………………….iv
摘要 ………………………………………...……………………..vi
第一章 緒論 ……..………………………………………………..1
1-1 前言 ……………………………………………………….1
1-2 文獻回顧 ………………………………………………….2
1-2-1 尤拉樑理論 …………………………………..…..3
1-2-2 Timoshenko樑理論 …………………………….5
1-2-3 高階剪切變形理論 ……………………………..7
1-2-4 三維彈性力學分析法 ………….………….……..8
1-2-5 有限元素法 ……………………………………..9
第二章 樑之平面應變有限元素挫曲分析 ………..……………11
2-1 前言 ...….…………………………………………….....11 2-2 理論推導 ……..………………………………………...11

第三章 問題解析 ………………………………………………..19
3-1 前言 …….……………………………………………..19 3-2 問題描述 ……………………………………………...19
(1) 材料性質 ……………………………………………....19
(2) 幾何比例 ……………………………………………....20
(3) 邊界條件 ……………………………………………....20
(4) 元素選取 ……………………………………………....20
(5) 自然振動頻率之無因次化 ……………………………21
第四章 結果與討論 ……………………………………………..23
4-1 前言 …………………………………………………...23
4-2 收斂試驗 ……………………………………………...23
4-3 文獻與本文結果之比較與討論 ……………………...25
第五章 結論與建議 ……………………………………………..67
5-1 結論 …………………………………………………...67
5-2 建議 …………………………………………………...68
參考文獻 …………………………………………………………69
附錄A ………………………………………………………..…...74
附錄B ………………………………………………………….…75

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