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 — Oak Academic Publishing
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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
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.
S. Timoshenko and S. Woinowsky-Krieger, “Theory of Plates and Shells,” McGraw-Hill, New York, 1959.
W. Flügge, “Stresses in Shells,” Springer-Verlag, Berlin, 1960. doi:10.1007/978-3-662-01028-0
A. L. Gol’denveizer, “Theory of Elastic Thin Shells,” Pergamon Press, Oxford, 1961.
V. V. Novozhilov, “Thin Shell Theory,” P. Noordhoff, Groningen, 1964.
V. Z. Vlasov, “General Theory of Shells and Its Application in Engineering,” NASA-TT-F-99, 1964.
S. A. Ambartusumyan, “Theory of Anisotropic Shells,” NASA-TT-F-118, 1964.
H. Kraus, “Thin Elastic Shells,” John Wiley & Sons, Hoboken, 1967.
A. W. Leissa, “Vibration of Plates,” NASA-SP-160, 1969.
A. W. Leissa, “Vibration of Shells,” NASA-SP-288, 1973.
?. Marku?, “The Mechanics of Vibrations of Cylindrical Shells,” Elsevier, Amsterdam, 1988.
E. Ventsel and T. Krauthammer, “Thin Plates and Shells,” Marcel Dekker, New York, 2001. doi:10.1201/9780203908723
W. Soedel, “Vibrations of Shells and Plates,” Marcel Dekker, New York, 2004.
E. Reissner, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” Journal of Applied Mechanics, Vol. 12, 1945, pp. 66-77.
P. L. Gould, “Finite Element Analysis of Shells of Revolution,” Pitman Publishing, New York, 1984.
P. L. Gould, “Analysis of Plates and Shells,” PrenticeHall, Upper Saddle River, 1999.
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
M. S. Qatu, “Vibration of Laminated Shells and Plates,” Elsevier, Amsterdam, 2004.
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
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
J. N. Reddy, “Mechanics of Laminated Composites Plates and Shells,” CRC Press, New York, 2003.
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
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
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
G. Farin, “Curves and Surfaces for Computer Aided Geometric Design,” Academic Press, Waltham, 1990.
L. Piegl and W. Tiller, “The NURBS Book,” Springer, Berlin, 1997. doi:10.1007/978-3-642-59223-2
C. Shu, “Differential Quadrature and Its Application in Engineering,” Springer, Berlin, 2000. doi:10.1007/978-1-4471-0407-0
D. N. Paliwal, R. K. Pandey and T. Nath, “Free Vibration of Circular Cylindrical Shell on Winkler and Pasternak Foundations,” International Journal of Pressure Vessel and Piping, Vol. 69, No. 1, 1996, pp. 79-89. doi:10.1016/0308-01619500010-0
?. Civalek, “Geometrically Nonlinear Dynamic Analysis of Doubly Curved Isotropic Shells Resting on Elastin Foundation by a Combination of Harmonic Differential Quadrature-Finite Difference Methods,” International Journal of Pressure Vessel and Piping, Vol. 82, No. 6, 2005, pp. 753-761. doi:10.1016/j.ijpvp.2004.12.003
G. B. Golovko, P. Z. Lugovoi and V. F. Meish, “Solution of Axisymmetric Dynamic Problems for Cylindrical Shells on an Elastic Foundation,” International Applied Mechanics, Vol. 43, No. 12, 2007, pp. 785-793. doi:10.1007/s10778-008-0006-5
A. H. Sofiyev, “The Buckling of FGM Truncated Conical Shells Subjected to Axial Compressive Load and Resting on Winkler-Pasternak Foundations,” International Journal of Pressure Vessel and Piping, Vol. 87, No. 12, 2010, pp. 753-761. doi:10.1016/j.ijpvp.2010.08.012
A. H. Sofiyev, “Buckling Analysis of FGM Circular Shells under Combined Loads and Resting on Pasternak Type Elastic Foundations,” Mechanics Research Communications, Vol. 37, No. 6, 2010, pp. 539-544. doi:10.1016/j.mechrescom.2010.07.019
C. Bert and M. Malik, “Differential Quadrature Method in Computational Mechanics,” Applied Mechanics Reviews, Vol. 49, No. 1, 1996, pp. 1-27. doi:10.1115/1.3101882
K. M. Liew, J. B. Han and Z. M. Xiao, “Differential Quadrature Method for Thick Symmetric Cross-Ply Laminates with First-Order Shear Flexibility,” International Journal of Solids and Structures, Vol. 33, No. 18, 1996, pp. 2647-2658. doi:10.1016/0020-76839500174-3
L. Hua and K. Y. Lam, “Frequency Characteristics of a Thin Rotating Cylindrical Shell Using the Generalized Differential Quadrature Method,” International Journal of Mechanical Sciences, Vol. 40, No. 5, 1998, pp. 443459. doi:10.1016/S0020-74039700057-X
K. M. Liew and T. M. Teo, “Modeling via Differential Quadrature Method: Three-Dimensional Solutions for Rectangular Plates,” Computer Methods in Applied Mechanics and Engineering, Vol. 159, No. 3, 1998, pp. 369381. doi:10.1016/S0045-78259700279-X
T. Y. Ng, H. Li, K. Y. Lam and C. T. Loy, “Parametric Instability of Conical Shells by the Generalized Differential Quadrature Method,” International Journal for Numerical Methods in Engineering, Vol. 44, No. 6, 1999, pp. 819-837. doi:10.1002/SICI1097-02071999022844:6 3.0.CO;2-0
J.-B. Han and K. M. Liew, “Static Analysis of Mindlin Plates: The Differential Quadrature Element Method DQEM,” Computer Methods in Applied Mechanics and Engineering, Vol. 177, No. 1-2, 1999, pp. 51-75. doi:10.1016/S0045-78259900371-0
F.-L. Liu and K. M. Liew, “Differential Quadrature Element Method: A New Approach for Free Vibration of Polar Mindlin Plates Having Discontinuities,” Computer Methods in Applied Mechanics and Engineering, Vol. 179, No. 3-4, 1999, pp. 407-423. doi:10.1016/S0045-78259900049-3
K. M. Liew and F.-L. Liu, “Differential Quadrature Method for Vibration Analysis of Shear Deformable Annular Sector Plates,” Journal of Sound and Vibration, Vol. 230, No. 2, 2000, pp. 335-356. doi:10.1006/jsvi.1999.2623
L. Hua and K. Y. Lam, “Orthotropic Influence on Frequency Characteristics of Rotating Composite Laminated Conical Shell by the Generalized Differential Quadrature Method,” International Journal of Solids and Structures, Vol. 38, No. 22, 2001, pp. 3995-4015. doi:10.1016/S0020-76830000272-9
G. Karami and P. Malekzadeh, “A New Differential Quadrature Methodology for Beam Analysis and the Associated Differential Quadrature Element Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 32, 2002, pp. 3509-3526. doi:10.1016/S0045-78250200289-X
K. M. Liew, T. Y. Ng and J. Z. Zhang, “Differential Quadrature-Layerwise Modeling Technique for Three Dimensional Analysis of Cross-Ply Laminated Plates of Various Edge Supports,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 35, 2002, pp. 38113832. doi:10.1016/S0045-78250200309-2
T. Y. Wu, Y. Y. Wang and G. R. Liu, “Free Vibration Analysis of Circular Plates Using Generalized Differential Quadrature Rule,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 46, 2002, pp. 5365-5380. doi:10.1016/S0045-78250200463-2
T. C. Fung, “Stability and Accuracy of Differential Quadrature Method in Solving Dynamic Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 13-14, 2002, pp. 1311-1331. doi:10.1016/S0045-78250100324-3
K. M. Liew and Y. Q. Huang, “Bending and Buckling of Thick Symmetric Rectangular Laminates Using the Moving Least-Squares Differential Quadrature Method,” International Journal of Mechanical Sciences, Vol. 45, No. 1, 2003, pp. 95-114. doi:10.1016/S0020-74030300037-7
K. M. Liew, Y. Q. Huang and J. N. Reddy, “Moving Least Squares Differential Quadrature Method and Its Applications to the Analysis of Shear Deformable Plates,” International Journal for Numerical Methods in Engineering, Vol. 56, No. 15, 2003, pp. 2332-2351. doi:10.1002/nme.646
K. M. Liew, Y. Q. Huang and J. N. Reddy, “Vibration Analysis of Symmetrically Laminated Plates Based on FSDT Using the Moving Least Squares Differential Quadrature Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 12, 2003, pp. 22032222. doi:10.1016/S0045-78250300238-X
T. Y. Wu, Y. Y. Wang and G. R. Liu, “A Generalized Differential Quadrature Rule for Bending Analyses of Cylindrical Barrel Shells,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 13-14, 2003, pp. 1629-1647. doi:10.1016/S0045-78250200650-3
Y. Q. Huang and Q. S. Li, “Bending and Buckling Analysis of Antisymmetric Laminates Using the Moving Least Square Differential Quadrature Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 33-35, 2004, pp. 3471-3492. doi:10.1016/j.cma.2003.12.039
X. Wang and Y. Wang, “Free Vibration Analyses of Thin Sector Plates by the New Version of Differential Quadrature Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 36, 2004, pp. 3957-3971. doi:10.1016/j.cma.2004.02.010
P. Malekzadeh, G. Karami and M. Farid, “A Semi-Analytical DQEM for Free Vibration Analysis of Thick Plates with Two Opposite Edges Simply Supported,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 45, 2004, pp. 4781-4796. doi:10.1016/j.cma.2004.05.005
?. Civalek, “Geometrically Nonlinear Dynamic Analysis of Doubly Curved Isotropic Shells Resting on Elastic Foundation by a Combination of HDQ-FD Methods,” International Journal Pressure Vessels and Piping, Vol. 82, No. 6, 2005, pp. 470-479. doi:10.1016/j.ijpvp.2004.12.003
E. Viola and F. Tornabene, “Vibration Analysis of Damaged Circular Arches with Varying Cross-Section,” Structural Integrity & Durability, Vol. 1, 2005, pp. 155-169. doi:10.3970/sdhm.2005.001.155
Z. Zong, K. Y. Lam and Y. Y. Zhang, “A Multidomain Differential Quadrature Approach to Plane Elastic Problems with Material Discontinuity,” Mathematical and Computer Modelling, Vol. 41, No. 4-5, 2005, pp. 539-553. doi:10.1016/j.mcm.2003.11.009
E. Viola and F. Tornabene, “Vibration Analysis of Conical Shell Structures Using GDQ Method,” Far East Journal of Applied Mathematics, Vol. 25, No. 1, 2006, pp. 2339. http://www.pphmj.com/abstract/2166.htm
?. Civalek, “Linear Vibration Analysis of Isotropic Conical Shells by Discrete Singular Convolution DSC,” International Journal of Structural Engineering and Mechanics, Vol. 25, No. 1, 2007, pp. 127-130.
F. Tornabene, “Modellazione e Soluzione di Strutture a Guscio in Materiale Anisotropo Modelling and Solution of Shell Structures made of Anisotropic Materials,” Ph.D. Thesis, DISTART Department, University of Bologna, Bologna, 2007.
F. Tornabene and E. Viola, “Vibration Analysis of Spherical Structural Elements Using the GDQ Method,” Computers & Mathematics with Applications, Vol. 53, No. 10, 2007, pp. 1538-1560. doi:10.1016/j.camwa.2006.03.039
E. Viola, M. Dilena and F. Tornabene, “Analytical and Numerical Results for Vibration Analysis of Multi-Stepped and Multi-Damaged Circular Arches,” Journal of Sound and Vibration, Vol. 299, No. 1-2, 2007, pp. 143163. doi:10.1016/j.jsv.2006.07.001
X. Wang, “Nonlinear Stability Analysis of Thin Doubly Curved Orthotropic Shallow Shells by the Differential Quadrature Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 17-20, 2007, pp. 2242-2251. doi:10.1016/j.cma.2006.11.009
X. Wang, X. Wang and X. Shi, “Accurate Buckling Loads of Thin Rectangular Plates under Parabolic Edge Compressions by the Differential Quadrature Method,” International Journal of Mechanical Sciences, Vol. 49, No. 4, 2007, pp. 447-453. doi:10.1016/j.ijmecsci.2006.09.004
A. Marzani, F. Tornabene and E. Viola, “Nonconservative Stability Problems via Generalized Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 315, No. 1-2, 2008, pp. 176-196. doi:10.1016/j.jsv.2008.01.056
F. Tornabene and E. Viola, “2-D Solution for Free Vibrations of Parabolic Shells Using Generalized Differential Quadrature Method,” European Journal of Mechanics— A/Solids, Vol. 27, No. 6, 2008, pp. 1001-1025. doi:10.1016/j.euromechsol.2007.12.007
A. Alibeigloo and R. Modoliat, “Static Analysis of CrossPly Laminated Plates with Integrated Surface Piezoelectric Layers Using Differential Quadrature,” Composite Structures, Vol. 88, No. 3, 2009, pp. 342-353. doi:10.1016/j.compstruct.2008.04.018
F. Tornabene, “Vibration Analysis of Functionally Graded Conical, Cylindrical and Annular Shell Structures with a Four-Parameter Power-Law Distribution,” Computer Methods in Applied Mechanics and Engineering, Vol. 198, No. 37-40, 2009, pp. 2911-2935. doi:10.1016/j.cma.2009.04.011
F. Tornabene and E. Viola, “Free Vibrations of Four-Parameter Functionally Graded Parabolic Panels and Shell of Revolution,” European Journal of Mechanics—A/Solids, Vol. 28, No. 5, 2009, pp. 991-1013. doi:10.1016/j.euromechsol.2009.04.005
F. Tornabene and E. Viola, “Free Vibration Analysis of Functionally Graded Panels and Shells of Revolution,” Meccanica, Vol. 44, No. 3, 2009, pp. 255-281. doi:10.1007/s11012-008-9167-x
F. Tornabene, E. Viola and D. J. Inman, “2-D Differential Quadrature Solution for Vibration Analysis of Functionally Graded Conical, Cylindrical and Annular Shell Structures,” Journal of Sound and Vibration, Vol. 328, No. 3, 2009, pp. 259-290. doi:10.1016/j.jsv.2009.07.031
E. Viola and F. Tornabene, “Free Vibrations of Three Parameter Functionally Graded Parabolic Panels of Revolution,” Mechanics Research Communications, Vol. 36, No. 5, 2009, pp. 587-594. doi:10.1016/j.mechrescom.2009.02.001
L. Yang and S. Zhifei, “Free Vibration of a Functionally Graded Piezoelectric Beam via State-Space Based Differential Quadrature,” Composite Structures, Vol. 87, No. 3, 2009, pp. 257-264. doi:10.1016/j.compstruct.2008.01.012
F. Tornabene, A. Marzani, E. Viola and I. Elishakoff, “Critical Flow Speeds of Pipes Conveying Fluid by the Generalized Differential Quadrature Method,” Advances in Theoretical and Applied Mechanics, Vol. 3, No. 3, 2010, pp. 121-138. http://m-hikari.com/atam/atam2010/atam1-4-2010/
A. Alibeigloo and V. Nouri, “Static Analysis of Functionally Graded Cylindrical Shell with Piezoelectric Layers Using Differential Quadrature Method,” Composite Structures, Vol. 92, No. 8, 2010, pp. 1775-1785. doi:10.1016/j.compstruct.2010.02.004
A. Andakhshideh, S. Maleki and M. M. Aghdam, “NonLinear Bending Analysis of Laminated Sector Plates Using Generalized Differential Quadrature,” Composite Structures, Vol. 92, No. 9, 2010, pp. 2258-2264. doi:10.1016/j.compstruct.2009.08.007
P. Malekzadeh and A. Alibeygi Beni, “Free Vibration of Functionally Graded Arbitrary Straight-Sided Quadrilateral Plates in Thermal Environment,” Composite Structures, Vol. 92, No. 11, 2010, pp. 2758-2767. doi:10.1016/j.compstruct.2010.04.011
O. Sepahi, M. R. Forouzan and P. Malekzadeh, “Large Deflection Analysis of Thermo-Mechanical Loaded Annular FGM Plates on Nonlinear Elastic Foundation via DQM,” Composite Structures, Vol. 92, No. 10, 2010, pp. 2369-2378. doi:10.1016/j.compstruct.2010.03.011
M. H. Yas and B. Sobhani Aragh, “Three-Dimensional Analysis for Thermoelastic Response of Functionally Graded Fiber Reinforced Cylindrical Panel,” Composite Structures, Vol. 92, No. 10, 2010, pp. 2391-2399. doi:10.1016/j.compstruct.2010.03.008
F. Tornabene, “Free Vibrations of Laminated Composite Doubly-Curved Shells and Panels of Revolution via the GDQ Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 200, No. 9, 2011, pp. 931-952. doi:10.1016/j.cma.2010.11.017
F. Tornabene, “2-D GDQ Solution for Free Vibrations of Anisotropic Doubly-Curved Shells and Panels of Revolution,” Composite Structures, Vol. 93, No. 7, 2011, pp. 1854-1876. doi:10.1016/j.compstruct.2011.02.006
F. Tornabene, “Free Vibrations of Anisotropic DoublyCurved Shells and Panels of Revolution with a Free-Form Meridian Resting on Winkler-Pasternak Elastic Foundations,” Composite Structures, Vol. 94, No. 1, 2011, pp. 186-206. doi:10.1016/j.compstruct.2011.07.002
F. Tornabene, A. Liverani and G. Caligiana, “FGM and Laminated Doubly Curved Shells and Panels of Revolution with a Free-Form Meridian: A 2-D GDQ Solution for Free Vibrations,” International Journal of Mechanical Sciences, Vol. 53, No. 6, 2011, pp. 446-470. doi:10.1016/j.ijmecsci.2011.03.007
F. Tornabene, A. Liverani and G. Caligiana, “Laminated Composite Rectangular and Annular Plates: A GDQ Solution for Static Analysis with a Posteriori Shear and Normal Stress Recovery,” Composites Part B: Engineering, Vol. 43, No. 4, 2012, pp. 1847-1872. doi:10.1016/j.compositesb.2012.01.065
F. Tornabene, A. Liverani and G. Caligiana, “General Anisotropic Doubly-Curved Shell Theory: A Differential Quadrature Solution for Free Vibrations of Shells and Panels of Revolution with a Free-Form Meridian,” Journal of Sound & Vibration, Vol. 331, No. 22, 2012, pp. 4848-4869. doi:10.1016/j.jsv.2012.05.036
F. Tornabene, “Meccanica delle Strutture a Guscio in Materiale Composito. Il Metodo Generalizzato di Quadratura Differenziale,” Esculapio, Bologna, 2012.
E. Viola, F. Tornabene and N. Fantuzzi, “General HigherOrder Shear Deformation Theories for the Free Vibration Analysis of Completely Doubly-Curved Laminated Shells and Panels,” Composite Structures, Vol. 95, No. 1, 2013, pp. 639-666. doi:10.1016/j.compstruct.2012.08.005
F. Tornabene and E. Viola, “Static Analysis of Functionally Graded Doubly-Curved Shells and Panels of Revolution,” Meccanica, 2012. doi:10.1007/s11012-012-9643-1
Sh. Hosseini-Hashemi, M. Fadaee and M. Es’haghi, “A Novel Approach for In-Plane/Out-of-Plane Frequency Analysis of Functionally Graded Circular/Annular Plates,” International Journal of Mechanical Sciences, Vol. 52, No. 8, 2010, pp. 1025-1035. doi:10.1016/j.ijmecsci.2010.04.009
X. Zhao and K. M. Liew, “Free Vibration Analysis of Functionally Graded Conical Shell Panels by a Meshless Method,” Composite Structures, Vol. 93, No. 2, 2011, pp. 649-664 doi:10.1016/j.compstruct.2010.08.014