本論文旨在利用柴比雪夫共位法求解異向性積層板殼問題的控制微分方程式。此方法是針對異向積層板殼，其具有任意疊層形式、不同的邊界條件及負荷等的異向性積層板殼問題求得結果。而上述的問題是無法簡單地採用納維爾及李維的方法來獲得解答。
柴比雪夫多項式具有正交及快速收歛的特性。他們和高斯－羅貝多共位點被用來逼近本論文中所提及的一些問題。同時，這些用本論文之方法所得到的解是一組方程式，其結果之可應用性比其他直接獲得數據的方法大得多。此外，柴比雪夫多項式可以經由簡單的數學代換，從原有的函數範圍[-1,1]變換成任意的範圍。一般而言，大部份關於異向性積層平板的研究都是針對於矩形平板的問題而來。然而，經由柴比雪夫多項式的優點，不但此類問題可以解決，而且亦可求解非矩形平板問題。
最後，在範例的章節中，附上數個範例，同時列出應用本論文的方法所產生的位移、應力合力及力矩合力的結果。將這些結果與有限元素法（NASTRAN）的數值結果相比較，以彰顯應用本方法求解的精密度。

The purpose of this work is to solve governing differential equations of laminated anisotropic plates and shells by using the Chebyshev collocation method. This method yields these results those can not be accomplished easily by both Navier’s and Levy’s methods in the case of any kind of stacking sequence in composite laminates with the variety of boundary conditions subjected to any type of loading.
The Chebyshev polynomials have the characteristics of orthogonality and fast convergence. They and Gauss-Lobatto collocation points can be utilized to approximate the solution of these problems in this paper. Meanwhile, these results obtained by the method are presented as some mathematical functions that they are more applicable than some sets of data obtained by other methods. On the other hand, by simply mathematical transformation, it is easy to modify the range of Chebyshev polynomials from the interval [-1,1] into any intervals. In general, the research on laminated anisotropic plates is almost focused on the case of rectangular plate. It is difficult to handle the laminated anisotropic plate problems with the non-rectangular borders by traditional methods. However, through the merits of Chebyshev polynomials, such problems can be overcome as stated in this paper.
Finally, some cases in the chapter of examples are illustrated to highlight the displacements, stress resultants and moment resultants of our proposed work. The preciseness is also found in comparison with numerical results by using finite element method incorporated with the software of NASTRAN.

Acknowledgments………………………………………………………i
Contents…………………………………………………………………ii
Abstract (Traditional Chinese) ……………………..…………………vi
Abstract (English) ……………………………………………………viii
List of Tables………………………………….…………………………x
List of Figures……………………………………………………….…xii
List of Symbols………………………………………………………xv

Chapter 1 Introduction……..…………………………………………1
1.1 Background………...……………………………………………1
1.1.1 Plates and Shells…………………………………..……1
1.1.2 Chebyshev Polynomials………………………………..3
1.2 Organization………………………..……………………………5
Chapter 2 Chebyshev Polynomials…………………………………7
Chapter 3 Formulation for Laminated Anisotropic Plate and Shell……………………………………………………..…10
3.1 Differential Geometry of Shells……………………………..…10
3.2 Laminated Composite Materials……………………………….15
3.3 Governing Equations for Plates………………………………..19
3.4 Governing Equations for Cylindrical Shells….……………..…20
Chapter 4 Boundary Conditions…...………………………………23
Chapter 5 Chebyshev Collocation Method…...………….…………26
5.1 Chebyshev Collocation Method for Plate and Shell…….…..…26
5.2 Chebyshev Collocation Method for Non-Rectangular Plate…..31
Chapter 6 Examples………………………………………………….40
6.1 Examples for Rectangular Plates………………………………41
6.1.1 Case 1: Four-Layered Cross-Ply Simply Supported Composite Plate………………………………………41
6.1.2 Case 2: Two-Layered Angle-Ply All Clamped Composite Plate………………………………………46
6.1.3 Case 3: Four-Layered Cross-Ply Part Free and Part Clamped Composite Plat……………………………..48
6.1.4 Case 4: 16-Layered Quasi-Isotropic Simply Supported Composite Plate………………………………………50
6.2 Examples for Non-Rectangular Plates…………………………50
6.2.1 Case 5: Four-Layered Cross-Ply All Clamped Plate with Non-Rectangular Boundaries……...…………………50
6.2.2 Case 6: Four-Layered Cross-Ply all Clamped Plate with Curved Boundaries……...……………………………53
6.3 Examples for Cylindrical Shells………………………….……54
6.3.1 Case 7: Four-Layered Cross-Ply Clamped Composite Cylindrical Shell……………………………………...55
6.3.2 Case 8: Four-Layered Cross-Ply Clamped and Simply Supported Composite Cylindrical Shell……...………57
Chapter 7 Discussion……………...………………………………….59
Chapter 8 Conclusion and Future Work………………...………….62
8.1 Conclusion…………………………..…………………………62
8.2 Future Work……………………………………………………62
References……………………………………………………………..121
Appendix A……………………………………………………………126
A.1 Full Lists of Equation (3.28)………………………………….126
Appendix B……………………………………………………………129
B.1 Full Lists of Equation (3.35)………………………………….129
B.2 Full Lists of Equation (3.36)………………………………….129
Appendix C……………………………………………………………129
C.1 Full Lists of Equation (3.43)………………………………….129
C.2 Full Lists of Equation (3.44)………………………………….132
Appendix D……………………………………………………………129
D.1 Full Lists of Equation (5.10a)……..………………………….134
D.2 Full Lists of Equation (5.11a)……….…………………….….134
D.3 Full Lists of Equation (5.12a)……….…………………….….135
D.4 Full Lists of Equation (5.12f)……….…………………….….135
D.5 Full Lists of Equation (5.12g)……….…………………….….135
D.6 Full Lists of Equation (5.12h)……….…………………….….136
D.7 Full Lists of Equation (5.12i)……….…………………….…..136
Appendix E……………………………………………………………129
E.1 Full Lists of Equation (5.20a)……..………………………….137
E.2 Full Lists of Equation (5.21a)……….…………………….….137
E.3 Full Lists of Equation (5.22a)……….…………………….….138
E.4 Full Lists of Equation (5.22f)……….…………………….….139
E.5 Full Lists of Equation (5.22g)……….…………………….….140
E.6 Full Lists of Equation (5.22h)……….…………………….….141
E.7 Full Lists of Equation (5.22i)……….…………………….…..142
Appendix F……………………………………………………………143
F.1 Full Lists of Equation (6.4a)………………………………….143
F.2 Full Lists of Equation (6.4b)………………………………….143
F.3 Full Lists of Equation (6.5c)………………………………….144
F.4 Full Lists of Equation (6.5d)………………………………….144
F.5 Full Lists of Equation (6.6e)………………………………….145
F.6 Full Lists of Equation (6.6f)………………………………….146
F.7 Full Lists of Equation (6.6g)………………………………….146
F.8 Full Lists of Equation (6.6h)………………………………….146
Vita…………………………….........…………………………………147

[1] Novozhilov, V. V., Translated by Lowe, P. G., Edited by Radok, J. R. M., The Theory of Thin Shells, Groningen: P. Noordhoff, 1959.
[2] Vlasov, V. Z., General Theory of Shells and Its Application to Engineering, Moscow-Leningrad, 1949, NASA Technical Translation NASA-TT-F-99, 1964.
[3] Kraus, H., Thin Elastic Shells: An Introduction to the Theoretical Foundations and the Analysis of Their Static and Dynamic Behavior, John Wiley and Sons, Inc., 1972.
[4] Ambartsumyan, S. A., Theory of Anisotropic Shells, Moscow, 1961, English Translation, NASA-TT-F118, 1964.
[5] Sokolnikoff, I. S., Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua, John Wiley and Sons, Inc., 1964.
[6] Flügge, W., Stress in Shells, Springer, Berlin, 1967.
[7] Lekhnitskii, S. G., Anisotropic Plates, Gordon and Breach Science Publishers, New York, 1986
[8] Ambartsumyan, S. A., Theory of Anisotropic Plates, Technomic Publishing Company Inc., 1970.
[9] Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Company Inc., 1987.
[10] Whitney, J. M. and Riley, M. B., “Elastic Properties of Fiber Reinforced Composite Materials,” AIAA Journal, Vol. 4, p. 1537, 1966.
[11] Nelson, R. B. and Lorch, D. R., “A refined Theory for Laminated Orthotropic Plates,” ASME Journal of Applied Mechanics, Vol. 41, p. 171, 1974.
[12] Reissner, E. and Stavsky, Y., “Bending and Stretching of Certain Type of Heterogeneous Aeolotropic Elastic Plates,” ASME Journal of Applied Mechanics, Vol. 28, pp. 402-408, 1961.
[13] Dong, S. B., Pister, K. S. and Taylor, R. L., “On the Theory of Laminated Anisotropic Shells and Plates,” Journal of the Aerospace Science, Vol. 29, pp. 969-975, 1962.
[14] Whitney, J. M. and Leissa, A. W., “Analysis of a Simply Supported Laminated Anisotropic Rectangular Plate,” AIAA Journal, Vol. 8, No. 1, pp. 28-33, 1969.
[15] Sun, C. T. and Whitney, J. M., “Theory for Dynamic Response of Laminated Plates,” AIAA Journal, Vol. 11, No. 2, pp. 178-184, 1973.
[16] Whitney, J. M. and Sun, C.-T., “A Refined Theory for Laminated Anisotropic, Cylindrical Shells,” ASME Journal of Applied Mechanics, Vol. 41, No. 2, pp. 471-476, 1974.
[17] Green, A. E. and Naghdi, P. M., “A Theory of Laminated Composite Plates,” IMA Journal of Applied Mathematics, Vol. 29, p. 1, 1982.
[18] Chaudhuri, R. A., “Arbitrarily Laminated, Anisotropic Cylindrical Shell Under Internal Pressure,” AIAA Journal, Vol. 24, No. 11, pp. 1851-1858, 1986.
[19] Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, 1944.
[20] Askey, R., Orthogonal Polynomials and Special Functions, Philadelphia, Pennsylvania, 1975.
[21] Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, Science Publishers, Inc., 1978.
[22] Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1977.
[23] Mund, E. H., “A Short Survey on Preconditioning Techniques in Spectral Calculations,” Applied Numerical Mathematics, Vol. 33, Issue 1-4, pp. 61-70, 2000.
[24] Shen, J., Wang, F. and Xu, J. C., “A Finite-Element Multigrid Preconditioner for Chebyshev-Collocation Methods,” Applied Numerical Mathematics, Vol. 33, Issue 1-4, pp. 471-477, 2000.
[25] Clenshaw, C. W., “The Numerical Solution of Linear Differential Equations in Chebyshev Series,” Proceedings of the Cambridge Philological Society, Vol. 53, pp. 134-149, 1957.
[26] Fox, L, “Chebyshev Methods for Ordinary Differential Equation,” The Computer Journal, Vol. 4, pp. 318-331, 1962.
[27] Lanczos, C., Applied Analysis, Prentice-Hall, Englewood cliffs, New Jersey, 1664.
[28] Wright, K., “Chebyshev Collocation Methods for Ordinary Differential Equation,” The Computer Journal, Vol. 6, pp. 358-365, 1963/1964.
[29] Schultz, W. W., Lee, N. Y., and Boyd, J. P., “Chebyshev Pseudospectral Method of Viscous Flows with Corner Singularities,” Journal of Scientific Computing, Vol. 4, No. 1, pp. 1-24, 1989.
[30] Karageorghis, A, “A Note on the Satisfaction of the Boundary Conditions for Chebyshev Collocation Methods in Rectangular Domains,” Journal of Scientific Computing, Vol. 6, No. 1, pp. 21-26, 1991.
[31] Boyd, J. P., Chebyshev & Fourier Spectral Methods, Springer-Verlag, 1989.
[32] Lu, Y. Y., “A Chebyshev Collocation Method for Computing the Eigenvalues of the Laplacian,” International Journal for Numerical Method in Engineering, Vol. 38, pp. 231-243, 1995.
[33] Kjellmert, B., “A Chebyshev Collocation Multidomain Method to Solve the Reissner-Mindlin Equations for the Transient Response of an Anisotropic Plate Subjected to Impact,” International Journal for Numerical Method in Engineering, Vol. 40, pp. 3689-3702, 1997.
[34] Ma, H. P. and Sun, W. W., “A Legendre-Petrov-Galerkin and Chebyshev Collocation Method for 3rd-Order Differential- Equations,” SIAM Journal on Numerical Analysis, Vol. 38, Issue 5, pp. 1425-1438, 2000.
[35] Botella, O. and Peyret, R., “Computing Singular Solutions of the Navier-Stokes Equations with the Chebyshev-Collocation Method,” International Journal for Numerical Methods in Fluids, Vol. 36, Issue 2, pp 125-163, 2001.
[36] Fox, L. and Parker, I. B., Chebyshev Polynomials in Numerical Analysis, Oxford University Press, 1968.
[37] Rivlin, T. J., The Chebyshev Polynomials, John Wiley and Sons, Inc., New York, 1974.
[38] Jones, R. M., Mechanics of Composite Materials, Scripta Book Company, 1975.
[39] Christensen, R. M., Mechanics of Composite Materials, Krieger Publishing Company, Florida, 1979.
[40] Tsai, S. W., and Hahn, H. T., Introduction to Composite Materials, Technomic Publishing Company, Inc., 1980.
[41] Halpin, J. C., Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., 1984.
[42] Vinson, J. R., Sierakowski, R. L., The behavior of Structures Composed of Composite Materials, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1986.
[43] Chawla, K. K., Composite Materials Science and Engineering, Springer-Verlag, New York, 1987.
[44] Agarwal, B. D. and Broutman, L. J., Analysis and Performance of Fiber Composites, Second Edition, John Wiley and Sons, Inc., New York, 1990.
[45] Gibson, R. F., Principles of Composite Material Mechanics, McGraw-Hill, Inc., New York, 1994.
[46] Timoshenko, S. and Woinowsky-Krieger, S, Theory of Plates and Shells, Second Edition, McGraw-Hill, Inc., 1959.
[47] England, A. H., Complex Variable Methods in Elasticity, John Wiley and Sons, Inc., 1971.
[48] Green, A. E. and Adkins, J. E., Large Elastic Deformations, Oxford University Press, 1970.
[49] Clenshaw, C. W., and Norton, H. J., “The Solution of Nonlinear Ordinary Differential Equations in Chebyshev Series,” The Computer Journal, Vol. 6, pp. 88-92, 1963/1964.
[50] Norton, H. J., “The Iterative Solution of Non-Linear Ordinary Differential Equations on Chebyshev Series,” The Computer Journal, Vol. 7, pp. 76-85, 1964.