本文在三維線性彈性力學架構下，依據改良三維漸近解析理論並藉由微擾方法分析複合圓柱層殼之幾何非線性行為。文中考慮幾何非線性效應之三維彈性力學方程式包括Green-Lagrange的應變與位移關係式、以second Piola-Kirchhoff應力張量表示之平衡方程式及單斜晶體材料遵循之廣義虎克定律。依據改良三維漸近解析理論，重新整理三維非線性分析相關之基本方程式，再經過適當的無因次化、漸近展開、連續積分及將橫向剪力變形的影響提至首階等推衍程序後，求得具遞迴特性之各階問題相關控制方程式，文中顯示von Karman一階剪力變形理論（FSDT）即為三維非線性理論之首階近似理論。在首階及高階問題的控制方程式中，其線性項之微分運算子均相同，非線性項中含有未知變數的各項則以具規則性的形式呈現，而其他的非齊性項則可藉由較低階問題解經計算求得。所以，本漸近理論可由系統化的求解方式循序漸近的分析求得圓柱複合層殼的三維非線性分析解。

　　Within the framework of the three-dimensional (3D) nonlinear elasticity, a refined asymptotic theory is developed for the nonlinear analysis of laminated circular cylindrical shells. In the present formulation, the basic equations including the nonlinear relations between the finite strains (Green strains) and displacements, the nonlinear equilibrium equations in terms of the Kirchhoff stress components and the generalized Hooke’s law for a monoclinic elastic material are considered. After using proper nondimensionalization, asymptotic expansion, successive integration and then bringing the effects of transverse shear deformation into the leading-order level, we obtain recursive sets of the governing equations for various orders. It is shown that the von Karman-type first-order shear deformation theory (FSDT) is derived as a first-order approximation to the 3D nonlinear theory. The differential operators in the linear terms of governing equations for the leading order problem remain identical to those for the higher-order problems. The nonlinear terms related to the unknowns of the current order appear in a regular pattern and the other nonhomogeneous terms can be calculated by the lower-order solutions. It is also illustrated that the nonlinear analysis of laminated circular cylindrical shells can be made in a hierarchic and consistent way.

Bellman, R. and Casti, J., Differential quadrature and long-term integration, J. Math. Anal. Appl. 34, 235–238, 1971.
Bellman, R. B., Kashef, G. and Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comp. Physics 10, 40–52, 1972.
Bert, C. W., Wang X. and Striz, A. G., Differential quadrature for static and free vibration analysis of anisotropic plates, Int. J. Solid Struct. 30, 1737-1744, 1993.
Bert, C. W. and Malik, M., Differential quadrature method in computational mechanics: A Review, Appl. Mech. Rev. 49, 1–27, 1996.
Bhaskar, K. and Varadan, T. K., Exact elasticity solution for laminated anisotropic cylindrical shells, J. Appl. Mech. 60, 41-47, 1993.
Chia, C. Y., Nonlinear analysis of plates, McGraw-Hill, New York, 1980.
Chia, C. Y., Geometrically nonlinear behavior of composite plates: a review. Appl. Mech. Review 41, 439-451, 1988.
Dennis, S.T., A Galerkin solution to geometrically nonlinear laminated shallow shell equations. Comput. & Struct. 63, 859-874, 1997.
Du, H., Lim, M. K. and Lin, R. M., Application of generalized differential quadrature method to structure problems, Int. J. Numer. Method Eng. 37, 1881-1896, 1994.
Feng, Y. and Bert, C. W., Application of the quadrure method to flexural vibration analysis of geometrically non-linear beam, Nonlinear Dynamics 3, 13-18, 1992.
Fung, Y. C., Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
Icardi, U., The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise models, Compos. Struct. 29, 349-364, 1994.

Lekhnitskii, S. G., Theory of elasticity of an anisotropic elastic body, Holden-Day, San Franciso, 1963.
Librescu, L., Refined geometrically nonlinear theories of anisotropic laminated shells, Quarterly Appl. Math. XLV, 1-22, 1987.
Love, A. E. H., A treatise on the mathematical theory of elasticity, Dover, New York, 1944.
Moussaoui, F. and Benamar, R., Nonlinear vibrations of shell-type structures: a review with bibliography, J. Sound Vibr. 255, 161-184, 2002.
Nayfeh, A. H., Introduction to perturbation techniques, Wiley, New York, 1981.
Novozhilov, V. V., Foundations of the nonlinear theory of elasticity, Graylock Press, New York, 1953.
Reddy, J.N., Chandrashekhara, K., Nonlinear analysis of laminated shells including transverse shear strains, AIAA J. 23, 440-441, 1985.
Ren, J. G., Analysis of laminated circular cylindrical shells under axisymmetric loading, Compos. Struct. 30, 271-280, 1995.
Saada, A. S., Elasticity: theory and applications, Pergamon Press, New York, 1974.
Sathyamoorthy, M., Nonlinear vibration analysis of plates: a review and survey of current developments, Appl. Mech. Review 40, 1553-1561, 1987.
Shu, C. and Richard, B. E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 15, 791-798, 1992.
Striz, A. G., Chen, W. and Bert, C. W., Static analysis of structures by the quadrature element meyhod（QEM）, Int. J. Solid Struct. 31, 2807-2818, 1994.
Striz, A. G., Wang X. and Bert, C. W., Harmonic differential method and applications to structural components, Acta Mechanica 111, 85-94, 1995.
Sun, C.T. and Chin, H., On large deflection effects in unsymmetric cross-ply composite laminates, J. Compos. Mater, 22, 1045-1059, 1988.
Tan, H. F., Tian, Z. H. and Du, X. W., A new geometrical nonlinear laminated theory for large deformation analysis, Int. J. Solids Struct. 37, 2577-2589, 2000.
Toorani, M. H. and Lakis, A. A., General equations of anisotropic plates and shells including transverse share deformations, rotary inertia and initial curvature effects, J. Sound Vibr. 237, 561-615, 2000.
Turvey, G.J., Large deflection cylindrical bending analysis of cross-ply laminated strips, J. Mech. Engng. Sci. 23, 21-29, 1981.
Washizu, K., Variational methods in elasticity & plasticity, Pergamon Press, New York, 1974.
Wu, C. P. and Chen, C. W., Elastic buckling of multilayered anisotropic conical shells, J. Aero. Engng. ASCE 14, 29-36, 2001.
Wu, C. P. and Chiu, S. J., Thermoelastic buckling of laminated composite conical shells. J. Therm. Stresses 24, 881-901, 2001.
Wu, C. P. and Chiu, S. J., Thermally induced dynamic instability of laminated composite conical shells, Int. J. Solids Struct. 39, 3001-3021, 2002.
Wu, C. P. and Hung, Y. C., Asymptotic theory of laminated circular conical shells, Int. J. Engng. Sci. 37, 977-1005, 2001.
Wu, C. P., Hung, Y. C. and Lo, J. Y., A refined asymptotic theory of laminated circular conical shells, Euro. J. Mech. A/Solids 21, 281-300, 2002.
Wu, C. P., Tarn, J. Q. and Chen, P. Y., Refined asymptotic theory of doubly curved laminated shells, J. Engng. Mech. ASCE 123, 1238-1246, 1997.
Wu, C. P. and Hung Y. C., Asymptotic theory of laminated circular conical shells, Int. J. Engng. Sci. 37, 977-1005, 1999.
Wu, C. P., Tarn, J. Q. and Chi, S. M., Three-dimensional analysis of doubly curved laminated shells, J. Engng. Mech. ASCE 122, 391-401, 1996a.

Wu, C. P., Tarn, J. Q. and Chi, S. M., An asymptotic theory for dynamic response of doubly curved laminated shells, Int. J. Solids Struct. 33, 3813_3841, 1996b.
Wu, C. P., Tarn, J. Q. and Tang, S. C., A refined asymptotic theory for dynamic analysis of doubly curved laminated shells, Int. J. Solids Struct. 35, 1953-1979, 1998.
Wu, C. P. and Wu, C. H., Asymptotic differential quadrature solutions for the free vibration of laminated conical shells, Comput. Mech. 25, 346-357, 2000.
Xu, C. S. and Chia, C. Y., Non-linear analysis of unsymmetrically laminated moderately thick shallow spherical shells, Int. J. Nonlinear Mech. 29, 247-259, 1994.
Xu, C. S., Xia, Z. Q. and Chia, C. Y., Nonlinear theory and vibration analysis of laminated truncated thick conical shells, Int. J. Nonlinear Mech. 31, 139-154, 1996.
Yuan, F. G., Yang, W. and Kim, H., Analysis of axisymmetrically-loaded filament wound composite cylindrical shells, Comp. Struct. 50, 115-130, 2000.
Zienkiewicz, O. C., The finite element method, McGraw-Hill, London, UK, 1976.