數值積分表示微分元素法（DQEM）為一種分析連體力學問題的數值方法。此數值分析法除了能有系統地編成電腦程式外，因其具有較高的耦合特性，且考慮所有的基本條件，故使用較少的離散點就能得到收斂，可有效地求得精確解，大幅降低計算機的運算量。
數值積分有限元素法是將欲分析的結構物分割成有限個元素，再利用數值積分表示微分法的技巧，對定義於各個元素內的統御微分或偏微分方程式，兩個相鄰元素相連接的相鄰邊界上之轉接條件式及領域邊界上之邊界條件式，做數值的離散化。
本篇論文主要是針對I型樑的振動問題做研究，以期望能減少結構物的變形量，並加以分析其振動頻率對結構物的影響。往後的研究方向，就朝向更具體實際的複雜結構物做更好的驗證。

The differential quadrature element method (DQEM) is a numerical analysis method for analyzing continuum mechanics problems. The numerical procedure of this method can systematically implemented into a computer program. The coupling of solutions at discrete points is strong. In addition, all fundamental relations are considered in constructing the overall discrete points, and accurate results can be obtained by using less arithmetic operations which can reduce the computer CPU time required.
Like FEM, in using DQEM to solve a problem the domain is separated into many elements. The DQ discretization is carried out on an element-basis. The discretized governing differential or partial differential equations defined on the elements, transition condition on inter-element boundaries and boundary conditions are assembled to
obtain an overall algebraic system.
In this paper, the DQEM analysis model of nonprismatic I beams resting on elastic foundations is developed, and the related computer problem is implemented. Sample problems of static deformation and free vibration are analyzed. They prove that the developed DQEM analysis
model is excellent.

【1】 Bellman,R.and Casti,J.,”Differential quadrature and long term integration,”Journal of Mathematical Analysis and
Applications,Vol. 34, pp.235-238（1971）.
【2】 Bellman, R., Kashef, B. G., and Casti, J.”Differential quadrature： a technique for the rapid solation of non-linear partial differential eruations,”Journal of Computational Physics,Vol. 10, pp. 40-52
（1972.）.
【3】 Bert, C. W., Jang, S. K., and Striz, A. G., “Two new approximate methods for analyzing free vibration of structural components,”
AIAA Journal, Vol.26, No. 5,pp.612-618（1988）.
【4】 Bert, C. W., Wang, X., and Striz, A. G., “Differential quadrature for static and free vibration analysis of anisotropic plates by differebtial quadrature method ： a semi-analytical approach ,”
Journal of Sound Vibration, Vol. 30, pp. 1737-1744（1993）.
【5】 Bert, C. W. and Malik, M., “Free vibration analysis of tapered rectangular platrs by differential quadrature method ： a semi-analytical approach ,” Journal of Sound Vibration, Vol. 190,
No. 1, pp. 41-63（1996）.
【6】 Bhat, R. B., Laura, P. A. A., Gutierrez, R. G., Cortinez, V. H., and Sanzi, V. H., “Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness,” Journal of Sound Vibration, Vol. 138,
pp. 205-219（1990）.
【7】 Civan, F. and Sliepcevich, C. M., “Differentail quadrature for multidimensional problems,” Journal of Mathematical Analysis
and Applications, Vol. 101, pp. 423-443（1984）.
【8】 ”I-DEAS Master Series 4.0,” SDRC（1996）.
【9】 Jang, S. K., Bert, C. W., and Striz, A. G.,”Application of differrntail quadrature to static analysis of structural components,” International Journal for Numerical Methods in Engineering, Vol. 28, pp. 561-577（1989）.
【10】Kukreti, A. R., Farsa, J., and Bert, C. W.,”Differential quadrature and rayleigh-ritz methods to determine the fundamental frequencises of simply supported rectangular plates with linerly varying thickness,” Journal of Sound Vibration, Vol.189, No.1, pp
103-122（1996）.
【11】Malik, M. and Bert, C. W., “Differential quadrature solation for steady state incompressible and compressible lubrication problems,” ASME Journal of Tribology, Vol. 116, pp. 296-302
（1994）.
【12】林育南,“數值積分表示微分元素法的研究”,國立成功大學造
船暨船舶工程學系碩士論文,（1995）.
【13】黃志偉,“數值積分表示微分元素法剪變形變斷面樑分析模
式”, 國立成功大學造船暨船舶工程學系碩士論文,（1997）.
【14】謝明祺, ‘數值積分表示微分元素法具彈性基座樑分析模式’,國
立成功大學造船暨船舶工程學系碩士論文,（1997）.
【15】Sung K.Jang, Charles W Bert and Alfred G. Striz,”Application of Differential Quadrature to Staric Analysis of Structral Components”, International Journal for Numerical Methods in
Engineering, Vol. 28, pp. 561-577（1989）.
【16】J.O. Mingle, “The Method of Differential Quadrature for Transient
Nonlinear Diffusion”,J. Math. Anal., Vol. 60, pp. 569-599（1977）.
【17】Herbert Reismann and Peter S. Pawlik, “Elasticity Theory and
Applications”, Chp 6, pp. 217-221.
【18】Gere and Timoshenko,”Mechanics of Meterials”, PWS-KENT,
Third Edition.
【19】J.B.CARR, The Effect of Shear Flexibility and Rotatoty Inertia on the Nateral Frequencies of Uniform Beams,The aeronautical
Quarterly ,Vol. 21,pp. 79-90（1970）.
【20】C.N.Chen,”A differential Quadrature Element Method”, the third U.S.National congress on Computional Mechanics, Dalls, USA,（1995）.
【21】C.N. Chen, “The-Dimentional Truss Model of the Differential Quadrature Element Method”, Submitted to Computer Methods in
Applied Mechanics and Enginessring.
【22】C.N.Chen,”The-Dimensional Frame Model of the Differential Quadrature Element Method”, Computers and Structures, in press.
【23】G.R.Cowper,”The Shear Coefficient in Timoshenko’s
Beam,”Journal of Applied Mechanics,pp. 335-340（1966）.