FGM and Laminated Doubly-Curved and Degenerate Shells Resting on Nonlinear Elastic Foundations: A GDQ Solution for Static Analysis with a Posteriori Stress and Strain Recovery

Francesco Tornabene, J N Reddy

Abstract


This work focuses on the static analysis of functionally graded (FGM) and laminated doubly-curved shells and panels resting on nonlinear and linear elastic foundations using the Generalized Differential Quadrature (GDQ) method. The First-order Shear Deformation Theory (FSDT) for the aforementioned moderately thick structural elements is considered. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Several types of shell structures such as doubly-curved shells (elliptic and hyperbolic hyperboloids), singly-curved (spherical, cylindrical and conical shells), and degenerate panels (rectangular plates) are considered in this paper. The main contribution of this paper is the application of the differential geometry within GDQ method to solve doubly-curved FGM shells resting on nonlinear elastic foundations. The linear Winkler-Pasternak elastic foundation has been considered as a special case of the nonlinear elastic foundation proposed herein. The discretization of the differential system by means of the GDQ technique leads to a standard nonlinear problem, and the Newton-Raphson scheme is used to obtain the solution. Two different four-parameter power-law distributions are considered for the ceramic volume fraction of each lamina. In order to show the accuracy of this methodology, numerical comparisons between the present formulation and finite element solutions are presented. Very good agreement is observed. Finally, new results are presented to show effects of various parameters of the nonlinear elastic foundation on the behavior of functionally graded and laminated doubly-curved shells and panels.

Keywords


Static Analysis; Laminated Composite Doubly-Curved Shells and Panels; Nonlinear Elastic and Winkler-Pasternak Foundation; First-order Shear Deformation Theory; Generalized Differential Quadrature Method

Full Text:

PDF

References


S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959.

W. Flügge, Stresses in Shells, Springer-Verlag, 1960.

A.L. Gol’denveizer, Theory of Elastic Thin Shells, Pergamon Press, 1961.

V.V. Novozhilov, Thin Shell Theory, P. Noordhoff, 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, 1967.

A.W. Leissa, Vibration of Plates, NASA-SP-160, 1969.

A.W. Leissa, Vibration of Shells, NASA-SP-288, 1973.

Š. Markuš, The Mechanics of Vibrations of Cylindrical Shells, Elsevier, 1988.

E. Ventsel, T. Krauthammer, Thin Plates and Shells, Marcel Dekker, 2001.

W. Soedel, Vibrations of Shells and Plates, Marcel Dekker, 2004.

J.N. Reddy, Theory and Analysis of Plates and Shells, 2nd ed., CRC Press, Boca Raton, FL, 2007.

P.L. Gould, Finite Element Analysis of Shells of Revolution, Pitman Publishing, 1984.

P.L. Gould, Analysis of Plates and Shells, Prentice-Hall, 1999.

M.S. Qatu, Accurate Theory for Laminated Composite Deep Thick Shells, Int. J. Solids Struct. 36 (1999) 2917–2941.

M.S. Qatu, Vibration of Laminated Shells and Plates, Elsevier, 2004.

J.N. Reddy and C.F. Liu, “A higher-order shear deformation theory for laminated elastic shells,” Int. J. Engng. Sci., 23 (1985), 319–330.

M.H. Toorani, A.A. Lakis, General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects, J. Sound Vib. 237 (2000), 561–615.

J.N. Reddy, Mechanics of Laminated Composites Plates and Shells, 2nd ed., CRC Press, New York, 2004.

D.N. Paliwal, R.K. Pandey, T. Nath, Free vibration of circular cylindrical shell on Winkler and Pasternak foundations, Int. J. Press. Ves. Piping 69 (1996), 79–89.

O. Civalek, Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods, Int. J. Press. Ves. Piping 82 (2005), 753–761.

G.B. Golovko, P.Z. Lugovoi, V.F. Meish, Solution of axisymmetric dynamic problems for cylindrical shells on an elastic foundation, Int. Appl. Mech. 43 (2007), 785–793.

A.H. Sofiyev, The buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler-Pasternak foundations, Int. J. Press. Ves. Piping 87 (2010), 753–761.

A.H. Sofiyev, Buckling analysis of FGM circular shells under combined loads and resting on Pasternak type elastic foundations, Mech. Res. Commun. 37 (2010), 539–544.

A.H. Sofiyev, N. Kuruoglu, Natural frequency of laminated orthotropic shells with different boundary conditions and resting on the Pasternak type elastic foundation, Compos. Part B Eng. 42 (2011), 1562–1570.

H.-T. Thai, D.-H. Choi, A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation, Compos. Part B Eng. 43 (2012), 2335–2347.

Y. Kiani, M.R. Eslami, An exact solution for thermal buckling of annular FGM plates on an elastic medium, Compos. Part B Eng. 45 (2013), 101–110.

Ö. Civalek, Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches, Compos. Part B Eng. 50 (2013) 171–179.

L.F. Qiana, R.C. Batra, L.M. Chen, Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method, Compos. Part B Eng. 35 (2004), 685–697.

C.M.C. Roque, A.J.M. Ferreira, R.M.N. Jorge, Modelling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions, Compos. Part B Eng. 36 (2005), 559–572.

J.R. Xiao, D.F. Gilhooley, R.C. Batra, J.W. Gillespie Jr., M.A. McCarthy, Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method, Compos. Part B Eng. 39 (2008), 414–427.

E. Carrera, S. Brischetto, M. Cinefra, M. Soave, Effects of thickness stretching in functionally graded plates and shells, Compos. Part B Eng. 42 (2011), 123–133.

A.J.M. Ferreira, C.M.C. Roque, A.M.A. Neves, R.M.N. Jorge, C.M.M. Soares, K.M. Liew, Buckling and vibration analysis of isotropic and laminated plates by radial basis functions, Compos. Part B Eng. 42 (2011), 592–606.

G. Giunta, F. Biscani, S. Belouettar, E. Carrera, Hierarchical modelling of doubly curved laminated composite shells under distributed and localised loadings, Compos. Part B Eng. 42 (2011), 682–691.

A.J.M. Ferreira, E. Carrera, M. Cinefra, C.M.C. Roque, O. Polit, Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations, Compos. Part B Eng. 42 (2011), 1276–1284.

C. Shu, Differential Quadrature and Its Application in Engineering, Springer, Berlin, 2000.

C. Bert, M. Malik, Differential quadrature method in computational mechanics, Appl. Mech. Rev. 49 (1996), 1–27.

C. Shu, H. Du, Free vibration analysis of composites cylindrical shells by DQM, Compos. Part B Eng. 28B (1997), 267–274.

L. Hua, K.Y. Lam, Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method, Int. J. Mech. Sci. 40 (1998), 443–459.

K.M. Liew, T.M. Teo, Modeling via differential quadrature method: Three-dimensional solutions for rectangular plates, Comput. Methods Appl. Mech. Engrg. 159 (1998), 369–381.

J.-B. Han, K.M. Liew, Static analysis of Mindlin plates: The Differential Quadrature Element Method (DQEM), Comput. Methods Appl. Mech. Engrg. 177 (1999), 51–75.

K.Y. Lam, L. Hua, On free vibration of a rotating truncated circular orthotropic conical shell, Compos. Part B Eng. 30 (1999) 135–144.

S. Moradi, F. Taheri, Delamination buckling analysis of general laminated composite beams by differential quadrature method, Compos. Part B Eng. 30 (1999), 503–511.

K.M. Liew, F.-L. Liu, Differential quadrature method for vibration analysis of shear deformable annular sector plates, J. Sound Vib. 230 (2000), 335–356.

L. Hua, Influence of boundary conditions on the free vibrations of rotating truncated circular multi-layered conical shells, Compos. Part B Eng. 31 (2000), 265–275.

L. Hua, K.Y. Lam, Orthotropic influence on frequency characteristics of rotating composite laminated conical shell by the generalized differential quadrature method, Int. J. Solids Struct. 38 (2001), 3995–4015.

G. Karami, P. Malekzadeh, A new differential quadrature methodology for beam analysis and the associated differential quadrature element method, Comput. Methods Appl. Mech. Engrg. 191 (2002), 3509–3526.

K.M. Liew, T.Y. Ng, J.Z. Zhang, Differential quadrature-layerwise modeling technique for three dimensional analysis of cross-ply laminated plates of various edge supports, Comput. Methods Appl. Mech. Engrg. 191 (2002), 3811–3832.

K.M. Liew, Y.Q. Huang, Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method, Int. J. Mech. Sci. 45 (2003) 95–114.

K.M. Liew, Y.Q. Huang, J.N. Reddy, Moving least squares differential quadrature method and its applications to the analysis of shear deformable plates, Int. J. Numer. Methods Eng. 56 (2003) 2332–2351.

K.M. Liew, Y.Q. Huang, J.N. Reddy, Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method, Comput. Methods Appl. Mech. Engrg. 192 (2003) 2203–2222.

T.Y. Wu, Y.Y. Wang, G.R. Liu, A generalized differential quadrature rule for bending analyses of cylindrical barrel shells, Comput. Methods Appl. Mech. Engrg. 192 (2003), 1629–1647.

J. Yang, H.-S. Shen, Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions, Compos. Part B Eng. 34 (2003), 103–115.

Y.Q. Huang, Q.S. Li, Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method, Comput. Methods Appl. Mech. Engrg. 193 (2004), 3471–3492.

X. Wang, Y. Wang, Free vibration analyses of thin sector plates by the new version of differential quadrature method, Comput. Methods Appl. Mech. Engrg. 193 (2004), 3957–3971.

P. Malekzadeh, G. Karami, M. Farid, A semi-analytical DQEM for free vibration analysis of thick plates with two opposite edges simply supported, Comput. Methods Appl. Mech. Engrg. 193 (2004), 4781–4796.

Ö. Civalek, Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of HDQ-FD methods, Int. J. Press. Ves. Piping 82 (2005), 470–479.

E. Viola, F. Tornabene, Vibration analysis of damaged circular arches with varying cross-section, Struct. Integr. Durab. (SID-SDHM) 1 (2005), 155–169.

Z. Zong, K.Y. Lam, Y.Y. Zhang, A multidomain differential quadrature approach to plane elastic problems with material discontinuity, Mathematical and Computer Modelling 41 (2005), 539–553.

E. Viola, F. Tornabene, Vibration analysis of conical shell structures using GDQ method, Far East J. Appl. Math. 25 (2006), 23–39.

Ö. Civalek, Linear vibration analysis of isotropic conical shells by Discrete Singular Convolution (DSC), Int. J. Struct. Eng. Mech. 25 (2007), 127–130.

F. Tornabene, Modellazione e Soluzione di Strutture a Guscio in Materiale Anisotropo (Modelling and Solution of Shell Structures made of Anisotropic Materials), PhD Thesis, University of Bologna—DISTART Department, 2007.

F. Tornabene, E. Viola, Vibration analysis of spherical structural elements using the GDQ method, Comput. Math. Appl. 53 (2007), 1538–1560.

E. Viola, M. Dilena, F. Tornabene, Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches, J. Sound Vib. 299 (2007), 143–163.

X. Wang, Nonlinear stability analysis of thin doubly curved orthotropic shallow shells by the differential quadrature method, Comput. Methods Appl. Mech. Engrg. 196 (2007), 2242–2251.

X. Wang, X. Wang, X. Shi, Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method, Int. J. Mech. Sci. 49 (2007), 447–453.

A. Marzani, F. Tornabene, E. Viola, Nonconservative stability problems via generalized differential quadrature method, J. Sound Vib. 315 (2008), 176–196.

F. Tornabene, E. Viola, 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method, Eur. J. Mech. A-Solid 27 (2008), 1001–1025.

A. Alibeigloo, R. Modoliat, Static analysis of cross-ply laminated plates with integrated surface piezoelectric layers using differential quadrature, Compos. Struct. 88 (2009), 342–353.

F. Tornabene, Free vibration analysis of functionally graded conical, cylindrical and annular shell structures with a four-parameter power-law distribution, Comput. Methods Appl. Mech. Engrg. 198 (2009), 2911–2935.

F. Tornabene, E. Viola, Free vibrations of four-parameter functionally graded parabolic panels and shell of revolution, Eur. J. Mech. A-Solid 28 (2009), 991–1013.

F. Tornabene, E. Viola, Free vibration analysis of functionally graded panels and shells of revolution, Meccanica 44 (2009), 255–281.

F. Tornabene, E. Viola, D.J. Inman, 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical and annular shell structures, J. Sound Vib. 328 (2009) 259–290.

E. Viola, F. Tornabene, Free vibrations of three parameter functionally graded parabolic panels of revolution, Mech. Res. Commun. 36 (2009), 587–594.

L. Yang, S. Zhifei, Free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature, Compos. Struct. 87 (2009), 257–264.

F. Tornabene, A. Marzani, E. Viola, I. Elishakoff, Critical flow speeds of pipes conveying fluid by the generalized differential quadrature method, Adv. Theor. Appl. Mech. 3 (2010), 121–138.

A. Alibeigloo, V. Nouri, Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method, Compos. Struct. 92 (2010), 1775–1785.

A. Andakhshideh, S. Maleki, M.M. Aghdam, Non-linear bending analysis of laminated sector plates using generalized differential quadrature, Compos. Struct. 92 (2010), 2258–2264.

Sh. Hosseini-Hashemi, M. Fadaee, M. Es’haghi, A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates, Int. J. Mech. Sci. 52 (2010), 1025–1035.

P. Malekzadeh, A. Alibeygi Beni, Free vibration of functionally graded arbitrary straight-sided quadrilateral plates in thermal environment, Compos. Struct. 92 (2010), 2758–2767.

O. Sepahi, M.R. Forouzan, P. Malekzadeh, Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM, Compos. Struct. 92 (2010) 2369–2378.

M.H. Yas, B. Sobhani Aragh, Three-dimensional analysis for thermoelastic response of functionally graded fiber reinforced cylindrical panel, Compos. Struct. 92 (2010), 2391–2399.

F. Tornabene, Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method, Comput. Methods Appl. Mech. Engrg. 200 (2011), 931–952.

F. Tornabene, 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution, Compos. Struct. 93 (2011), 1854–1876.

F. Tornabene, Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler-Pasternak elastic foundations, Compos. Struct. 94 (2011), 186–206.

F. Tornabene, A. Liverani, 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, Int. J. Mech. Sci. 53 (2011), 446–470.

X. Zhao, K.M. Liew, Free vibration analysis of functionally graded conical shell panels by a meshless method, Compos. Struct. 93 (2011), 649–664.

F. Tornabene, A. Liverani, G. Caligiana, Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery, Compos. Part B Eng. 43 (2012), 1847–1872.

F. Tornabene, A. Liverani, G. Caligiana, Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using Generalized Differential Quadrature Method, Int. J. Mech. Sci. 61 (2012), 71–87.

F. Tornabene, A. Liverani, 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, J. Sound Vib. 331 (2012), 4848–4869.

F. Tornabene, Meccanica delle Strutture a Guscio in Materiale Composito. Il Metodo Generalizzato di Quadratura Differenziale, Esculapio, 2012.

F. Tornabene, 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 J. Mech. 3 (2013), 1–25.

F. Tornabene, A. Ceruti, Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method, Math. Probl. Eng., vol. 2013, Article ID 867079, 1–33.

F. Tornabene, E. Viola, Static analysis of functionally graded doubly-curved shells and panels of revolution, Meccanica 48 (2013), 901–930.

F. Tornabene, E. Viola, N. Fantuzzi, General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels, Compos. Struct. 104 (2013), 94–117.

F. Tornabene, N. Fantuzzi, E. Viola, A.J.M. Ferreira, Radial basis function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single layer formulation, Compos. Part B Eng. (2013), doi: http://dx.doi.org/10.1016/j.compositesb.2013.07.026.

F. Tornabene, N. Fantuzzi, E. Viola, J.N. Reddy, Winkler-Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels, Compos. Part B Eng. (2013), doi: http://dx.doi.org/10.1016/j.compositesb.2013.06.020.

E. Viola, F. Tornabene, N. Fantuzzi, General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels, Compos. Struct. 95 (2013), 639–666.

E. Viola, F. Tornabene, N. Fantuzzi, Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories, Compos. Struct. 101 (2013), 59–93.

E. Viola, F. Tornabene, N. Fantuzzi, Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape, Compos. Struct. (2013), doi: http://dx.doi.org/10.1016/j.compstruct.2013.07.034.

E. Viola, F. Tornabene, N. Fantuzzi, DiQuMASPAB Software, DICAM Department, Alma Mater Studiorum—University of Bologna (http://software.dicam.unibo.it/diqumaspab-project).


Refbacks

  • There are currently no refbacks.