Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis

ABSTRACT

This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present formulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to investtigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behavior of laminated shells of revolution.

Cite this paper

F. Tornabene and A. Ceruti, "Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis,"*World Journal of Mechanics*, Vol. 3 No. 1, 2013, pp. 1-25. doi: 10.4236/wjm.2013.31001.

F. Tornabene and A. Ceruti, "Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis,"

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[2] W. Flügge, “Stresses in Shells,” Springer-Verlag, Berlin, 1960. doi:10.1007/978-3-662-01028-0

[3] A. L. Gol’denveizer, “Theory of Elastic Thin Shells,” Pergamon Press, Oxford, 1961.

[4] V. V. Novozhilov, “Thin Shell Theory,” P. Noordhoff, Groningen, 1964.

[5] V. Z. Vlasov, “General Theory of Shells and Its Application in Engineering,” NASA-TT-F-99, 1964.

[6] S. A. Ambartusumyan, “Theory of Anisotropic Shells,” NASA-TT-F-118, 1964.

[7] H. Kraus, “Thin Elastic Shells,” John Wiley & Sons, Hoboken, 1967.

[8] A. W. Leissa, “Vibration of Plates,” NASA-SP-160, 1969.

[9] A. W. Leissa, “Vibration of Shells,” NASA-SP-288, 1973.

[10] ?. Marku?, “The Mechanics of Vibrations of Cylindrical Shells,” Elsevier, Amsterdam, 1988.

[11] E. Ventsel and T. Krauthammer, “Thin Plates and Shells,” Marcel Dekker, New York, 2001. doi:10.1201/9780203908723

[12] W. Soedel, “Vibrations of Shells and Plates,” Marcel Dekker, New York, 2004.

[13] E. Reissner, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” Journal of Applied Mechanics, Vol. 12, 1945, pp. 66-77.

[14] P. L. Gould, “Finite Element Analysis of Shells of Revolution,” Pitman Publishing, New York, 1984.

[15] P. L. Gould, “Analysis of Plates and Shells,” PrenticeHall, Upper Saddle River, 1999.

[16] M. S. Qatu, “Accurate Theory for Laminated Composite Deep Thick Shells,” International Journal of Solids and Structures, Vol. 36, No. 19, 1999, pp. 2917-2941. doi:10.1016/S0020-76839800134-6

[17] M. S. Qatu, “Vibration of Laminated Shells and Plates,” Elsevier, Amsterdam, 2004.

[18] M. H. Toorani and A. A. Lakis, “General Equations of Anisotropic Plates and Shells Including Transverse Shear Deformations, Rotary Inertia and Initial Curvature Effects,” Journal of Sound and Vibration, Vol. 237, No. 4, 2000, pp. 561-615. doi:10.1006/jsvi.2000.3073

[19] M. H. Toorani and A. A. Lakis, “Free Vibration of NonUniform Composite Cylindrical Shells,” Nuclear Engineering and Design, Vol. 237, No. 17, 2006, pp. 17481758. doi:10.1016/j.nucengdes.2006.01.004

[20] J. N. Reddy, “Mechanics of Laminated Composites Plates and Shells,” CRC Press, New York, 2003.

[21] A. Messina, “Free Vibrations of Multilayered Doubly Curved Shells Based on a Mixed Variational Approach and Global Piecewise-Smooth Functions,” International Journal of Solids and Structures, Vol. 40, No. 12, 2003, pp. 3069-3088. doi:10.1016/S0020-76830300115-X

[22] C. P. Wu and C. Y. Lee, “Differential Quadrature Solution for the Free Vibration Analysis of Laminated Conical Shells with Variable Stiffness,” International Journal of Mechanical Sciences, Vol. 43, No. 8, 2001, pp. 18531869. doi:10.1016/S0020-74030100010-8

[23] A. Ceruti, A. Liverani and G. Caligiana, “Fairing with Neighbourhood LOD Filtering to Upgrade Interactively B-Spline into Class-A Curve,” International Journal on Interactive Design and Manufacturing, 2012. http://link.springer.com/article/10.1007%2Fs12008-012-0181-9

[24] G. Farin, “Curves and Surfaces for Computer Aided Geometric Design,” Academic Press, Waltham, 1990.

[25] L. Piegl and W. Tiller, “The NURBS Book,” Springer, Berlin, 1997. doi:10.1007/978-3-642-59223-2

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