本篇論文主要探討遵守古典Kirchhoff理論之圓板具非線性邊界的靜態問題。在文中使用Shifting Method處理非線性邊界，再以Maple作圖，最後以實際的工程係數討論其最終結果。最後我們可以發現對於此類非線性邊界的靜態問題只需利用Shifting Method就可得出解。此外本文也有探討對於不同個數的邊界假設之下之結果，探討其撓度行為。另外本文也提供具非線性彈簧底座之圓板正確解，不僅完整的將過程表現出，更提供簡單的方法判別微擾參數的極限方法，讓解法更完整。

In this paper, we have studied static deflection of circular plate (Kirchhoff plate) with nonlinear elastic boundary conditions. The paper will be exhibited completed solution to two problems. The first one will show the circular plate’s both ends with nonlinear boundary condition. The first step is to simplify governing equation and boundary condition by non-dimensional process. Then using Shifting Method solve static plate deflection problem with nonlinear boundary. And finally, we will use Maple program to plot 3D-diagram. The second one will show the circular plate’s with nonlinear elastic foundation. In the problem, we’ll use variation parameter method to solve particular solution. Finally, we even show the easy way to get the critical e value to fit the perturbation method. In the conclusion, these problems with nonlinear boundary condition can be solved by Shifting Function Method. We can get completely exact solution.

參考文獻
[1]S.Timoshenko, and S.,Woinowsky-Krieger, “Theory of Plates and Shells,”2nd.,McGraw,New York,1959.
[2]R.P. Agarwall, G. Akriv, “Boundary value problems occurring in plate deflection,” Journal of Computation and Applied Mathmatics, Vol. 8, No. 3, pp. 145-154, 1982.
[3]J. Rashidinia , R. Jalilian ,“Non-polynomial spline for solution of boundary-value problems in plate deflection theory,” International Journal of Computer Mathematics,Vol. 84, No. 10, 1483-1494,2007.
[4]A. Hasanov, “Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate ,”International Journal of Non-Linear Mechanics, pp.711-721,2007.
[5]W. Z. Chien, K. Y. Yeh, “ON THE LARGE DEFLECTION OF RECTANGULAR PLATE ,” International Congress of Applied Mechanics,Bruxelles, pp.387-394.
[6]A. Cabada, R.L. Pouso , “Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundaryconditions,” Nonlinear Analysis, Vol. 42 , pp.1377 -1396.
[7]J. Ehme, P.W. Eloe, J. Henderson , “Upper and lower solution methods for fully nonlinearboundary value problems,” Journal of Differential Equations, Vol.180, pp. 51–64,2002.
[8]A. Bhimaraddi, “Large amplitude vibrations of imperfect antisymmetric,” Journal of Sound and Vibration, Vol. 162, No. 3, pp. 457-470, 1993.
[9]J.J. Nieto, R. Rodríguez-López , “Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions,” International Journal of Computer Mathematics,Vol. 40, pp. 433-442,2000.
[10]S.Lopez, “Geometrically Nonlinear Analysis of Plates and Cylindrical Shells by a Predictor-Corrector Method,” Computer and Structures, Vol. 79, pp. 1405-1414, 2001.
[11]O. Civalek, “Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation,” Journal of Sound and Vibration, Vol. 294, pp. 966-980, 2006.
[12]J. G. Eisley, “Nonlinear vibration of beams and rectangular plates,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 15, No. 2, pp. 167-175, 2005.
[13]George Junkin II, R. T. Davis, “General non-linear plate theory applied to a circular plate with large deflections,” International Journal of Non-Linear Mechanics. Vol. 7, pp. 503-526,1972.
[14]M. K. Singha, R. Daripa, “Nonlinear vibration of symmetrically laminated composite skewplates by finite element method,” International Journal of Non-Linear Mechanics, Vol. 42, pp. 1144-1152, 2007.
[15]G. Pohit, K. N. Saha, D. Misra, S. Ghosal, “Nonlinear free vibration analysis of square plates with various boundary conditions,” Journal of Sound and Vibration, Vol. 287, pp. 1031-1044, 2005.
[16]O. Civalek, A. Yavas, “Large Deflection Static Analysis of Rectangular Plates On Two Parameter Elastic Foundations,” International Journal of Science & Technology, Vol. 1, No. 1, pp. 43-50, 2006.
[17]J. H. Huang, T. L. Wu, K. K. Huang, “Nonlinear Static And Dynamic Analysis of Functionally Graded Plates,” International Journal of Applied Mechanics and Engineering, Vol. 11, No. 3, pp. 679-698, 2006.
[18]G. R. Fernandes, W. S. Venturini, “Non-linear boundary element analysis of floor slabs reinforced with rectangular beams,” Engineering Analysis with Boundary Elements ,Vol. 31, pp.721–737,2007.
[19]R. Sircar, “Fundamental Frequency of Vibration of a Rectangular Plate on a Nonlinear Elastic Foundation,” Indian Journal pure apply Mathe, Vol.11, No.2, pp. 252-255, 1980.
[20]T. Horibe, N. Asano, “Large Deflection Analysis of Rectangular Plates on Two-parameter Elastic Foundation by the Boundary Strip Method,” JSME International Journal, Series A, Vol. 44, No. 4, 2001.
[21]S. Y. Lee, S. M. Lin, “Dynamic analysis of nonuniform beams with time-dependent elastic boundary conditions,” ASME J. Applied Mechanics, Vol. 63, pp. 474-478, 1996
[22]S. Y. Lee, W. R. Wang, T. Y. Chen, “A general approach on the mechanical analysis of non-uniform beams with non-homogeneous elastic boundary conditions,” ASME J. Vibration and Acoustics Vol. 120, pp. 164-169, 1998.
[23]S. Y. Lee, Y. H. Kuo, “Exact solutions for the analysis of general elastically restrained non-uniform beams,” ASME Journal of Applied Mechanics, Vol. 59, pp. S205-S212, 1992.
[24]S. M. Lin, “The closed-form solution for the forced vibration of non-uniform plates with distributed time-dependent boundary conditions,” Journal of Sound and Vibration, Vol. 232, No.3, pp. 493-590, 1999.
[25]王何維,邊界條件為非線性之板的靜態分析,國立成功大學機械工程學系碩士論文(2009)