一般分析樑結構大多以Euler樑理論為基礎，其忽略了剪力對樑結構的影響，因為大部分樑結構的跨深比較大，所產生的剪力效應相對的小；當分析跨深比較小的樑結構，或是具有較明顯的剪力效應結構時，剪力效應就必須考慮，於是本文應用了Timoshenko樑理論來分析跨深比較小的樑結構。
　　向量式有限元素法為新穎的數值計算方法，不需求解矩陣方程組及疊代計算，大量減少了數值分析中所必須面對的困難，本文以向量式有限元素法來推導Timoshenko樑元素以分析短跨距深樑結構物。由早先樑理論的文獻可知，Euler樑理論忽略了剪力效應，因此我們將向量式有限元素法的基礎理論改為有加入剪力效應的Timoshenko樑理論，探討向量式有限元素法是否能夠應用Timoshenko樑理論來分析結構物。
　　本文以向量式有限元素法應用Timoshenko樑理論的數值方法，分析的結構有深樑、簡支樑、懸臂樑及門型剛架。簡支樑、懸臂樑及門型剛架的靜力分析驗證本文理論及程式計算的正確性；深樑的靜力分析與Ahmed(1996)利用有限差分法的數值做比對；並以簡支樑、懸臂樑及門型剛架的動力分析，探討改變跨深比及外力條件的分析結果。

In this study, a vector form intrinsic finite element (VFIFE) is derived and applied to study both the static and dynamic responses of deep short beams under dynamic loadings. It is already known that the application of classical beam theory known as Euler’s beam theory to beams with large ratio of D/L (depth/span larger than 1/4), a short-deep beam, may not necessarily obtain satisfactory results for the stress analysis of the beam. One of the main presumptions from the classical Euler’s beam theory is that the plane of the cross-section remains plane and normal to the neutral axis of the beam after deformation. This presumption is no more true when the beam subject to loadings is a short-deep beam because the bending stress is no longer a dominant stress while the other secondary effects may have more severe influences on the mechanical behavior of the beam. This study by utilizing the vector form intrinsic finite element method (VFIFE) to derive a new element for the Timoshenko beam provides an alternative method for the analysis of a short-deep beam, particularly, subject to dynamic loadings. By taking the advantage of the VFIFE that is a time-saving scheme for the dynamic analysis, the element of Timoshenko-beam is derived along with the dynamic solution procedure. The motions in transverse direction and the rotation at each node of the beam are calculated and presented into figures. The results from numerical analysis are also verified with theoretical solution (exact analytical solution) and further compared to the results obtained from traditional finite element method.

總目錄 i
圖目錄 iii
表目錄 viii
第一章 緒論 1
1.1 文獻回顧與研究動機 1
1.2 向量式有限元研究背景 4
1.3 本文內容 5
第二章 向量式有限元基本理論與Timoshenko樑理論 9
2.1 向量式有限元素法的離散化 9
2.2 平面剛架元節點變形及座標系統(deformation coordination system) 10
2.3 平面剛架元的增量內力計算 13
2.3.1 Timoshenko樑理論 13
2.3.2 Timoshenko樑理論應用於向量有限元的增量內力計算 14
2.4 平面剛架元質點運動方程式的差分式 19
2.5 顯式時間積分 21
2.6 計算流程 23
第三章 結果與討論 27
3.1 向量有限元靜力分析 27
3.1.1 向量有限元基本驗證 27
3.1.2 深樑分析 29
3.2 動力分析 29
3.2.1 理想階梯負載(Ideal Step Loading) 30
3.2.2 脈衝負載(Impulse Loading) 31
3.2.3 諧合負載(Harmonic Loading) 32
3.2.4 地震力(Earthquake Loading) 33
第四章 結論與建議 84
4.1 結論 84
4.2建議 85
附錄 A 86
參考文獻 90

1.A. V. Murthy, “ Vibrations of short beams ”, American Institute of Aeronautics and Astronautics Journal, Vol. 8, pp. 34-38, 1970.
2.A. Tesslor and S. B. Dong , “On a hierarchy of conforming Timoshenko beam elements”, Computer and Structures, Vol. 14, pp. 335-344, 1981.
3.Anil K. Chopa, Dynamics of structures Theory And Applications To Earthquake Engineering, 2nd Edition,2001.
4.D. L. Thomas, J. M. Wilson and R. R. Wilson, “ Timoshenko beam finite elements “, Journal of Sound and Vibration, Vol. 31(3), pp.315-330, 1973.
5.G. R. Cowper, “ The shear coefficient in Timoshenko’s beam theory ”, ASME, Journal of Applied Mechanics, Vol. 33, pp.335-340, 1966.
6.J.G. and L.E. Goodman,”Natural frequencies of continuous beams of uniform span length”,Journal of Applied Mechanics,Vol. 18,pp. 217-218,1951.
7.J.N. Goodier,” On the problems of the beam and plate in the theory of elasticity”,Transactions of the Royal Society of Canada,Vol. 32,pp. 65-88,1938.
8.J. Thomas and B. A. H. Abbas, “ Finite element model for dynamic analysis of Timoshenko beam ”, Journal of Sound and Vibration, Vol.41(3), pp. 291-299, 1975.
9.J. N. Reddy, An Introduction To The Finite Element Method, 3rd Edition McGRAW-Hill, 2006.
10.J. N. Reddy ,”On Locking-Free Shear Deformable Beam Finite Elements”, Computer Methods In Applied Mechanics and Engineering, Vol. 149 , pp. 113 -132,1997.
11.K. K. Kapur, “ Vibration of a Timoshenko beam using a finite element approach ”, Journal of the Acoustical Society of America , Vol. 40, pp. 1058-1063, 1966.
12.L.E. Goodman,”Flexural vibration in uniform beams according to the Timoshenko theory”,Journal of Applied Mechanics,Vol. 21,pp. 202-204,1954.
13.N. G. Stephen and M. Levinson, “ A second order beam theory ”,Journal of Sound and Vibration, Vol. 67(3), pp. 293-305, 1979.
14.R. E. Nickell and G. A. Secor, “Convergence of consistently derived Timoshenko beam finite elements”, International Journal for Numerical Method in Engineering, Vol. 5, pp. 243-245, 1972.
15.S.P. Timoshenko,”On the correction for shear of the differential equation for transverse vibrations of prismatic bars”, Philosophical Magazines ,Vol. 41 ,744 -746,1921.
16.S.P. Timoshenko,”On the transverse vibration of bar of uniform cross-section”, Philosophical Magazines,Vol. 43,pp. 125-131,1922.
17.S. B. Dong and J. A. Wolf, “Effect of transverse shear deformation on vibrations of planar structures composed of beam-type elements”, Journal of the Acoustical Society of America, Vol. 53, pp. 120-127, 1973.
18.Shih, C., Wang, Y. K., Ting, E. C., “Fundamentals of a vector form intrinsic finite element: Part III. Convected material frame and examples,” Journal of Mechanics, Vol. 20, No. 2, pp. 133-143, (2004).
19.S. Reaz Ahmed, A. B. M. Idris and Md. Wahhaj Uddin, ”Numerical solution of both ends fixed deep beams”, Computers and Structures, Vol. 61,No 1 ,pp. 21-29, 1996.
20.T. J. R. Hughes ,R. L. Taylor and W. Kanoknukulchoii, “A simple and efficient plate element for bending”, International Journal for Numerical Method in Engineering, Vol. 11 ,pp. 1529-1943, 1977.
21.Ting, E. C., Shih, C., Wang, Y. K., “Fundamentals of a vector form intrinsic finite element: Part I. basic procedure and a plane frame element,” Journal of Mechanics, Vol. 20, No. 2, pp. 113-122, (2004a).
22.Ting, E. C., Shih, C., Wang, Y. K., “Fundamentals of a vector form intrinsic finite element: Part II. plane solid elements,” Journal of Mechanics, Vol. 20, No. 2, pp. 123-132, (2004b).
23.W. Carnegie, J. Thomas and E. Dokumuci, “ An improved method of matrix displacement analysis in vibration problems ”, Aeronaut. Quart. , Vol. 20, pp. 321-332, 1969.
24.王仁佐，”向量式結構運動分析”，國立中央大學土木工程系博士論文，2005。
25.曾國瑋，”向量式有限元於剛架式海域結構物之動力分析”，國立中山大學海洋環境及工程學系碩士論文，2007。
26.丁承先、王仲宇、吳東岳、王仁佐、莊清鏘，”運動解析與向量式有限元”，2007。