Elsevier

Composite Structures

Volume 112, June 2014, Pages 44-65
Composite Structures

Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

https://doi.org/10.1016/j.compstruct.2014.01.039Get rights and content

Abstract

This study focuses on the static analysis of functionally graded conical shells and panels and extends a previous formulation by the first three authors. A 2D Unconstrained Third order Shear Deformation Theory (UTSDT) is used for the evaluation of tangential and normal stresses in moderately thick functionally graded truncated conical shells and panels subjected to meridian, circumferential and normal uniform loadings. To investigate the behavior of the functionally graded structures at issue, a four parameter power law function is considered. The initial curvature effect is discussed and the role of the parameters in the power law function is shown. The conical shell problem described in terms of seven partial differential equations is solved by using the generalized differential quadrature (GDQ) method. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. The stress recovery is worked out to reconstruct the correct distribution of transverse stress components. Accurate stress profiles for general loading combinations applied at the extreme surfaces are obtained. The influence of the semi vertex angle is pointed out.

Introduction

As it is well known, the classical bending and shear deformation theories have been developed for the analysis of composite structures. Structures with a ratio of thickness to representative dimensions equal to 1/20 or less are considered to be thin and the classical bending theory will be adopted, whereas structures with the ratio greater than 1/20 are studied by means of shear deformation theories. Reissner [1], [2] proposed the first order shear deformation plate and shell theories based on kinematics analysis. Mindlin [3] suggested a first order shear deformation plate theory that included rotary inertia terms for the free vibrations of plates. Because the first order shear deformation theories based on Reissner–Mindlin kinematics violated the zero shear stress condition on the top and bottom surfaces of the shell or plate, a shear correction factor was required to compensate for the error due to a constant shear strain assumption through the thickness. The Reissner–Mindlin theory has been applied to the analysis of a variety of structures. Whitney [4], [5] investigated the shear correction factors for orthotropic laminates under static loads and analyzed the effects of shear deformation on the bending of laminated plates. Whitney and Pagano [6] considered the shear deformation of heterogeneous anisotropic plates and Reissner [7] developed a consistent treatment of transverse shear deformations in laminated anisotropic plates. However, in order to obtain a better prediction of shear deformation and transverse normal strains in laminated structures, higher order theories are required. Over the years, several higher order shear deformation theories have been developed by different authors [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], prevalently with reference to the plate structure and cubic expansion of the displacement field. Lo et al. [9] wrote a theory of homogeneous plate deformation which accounts for the effects of transverse shear deformation, transverse normal strain and a non-linear distribution of the in plane displacements with respect to the thickness coordinate. Later, they extended their third order formulation to laminated plates [10]. Murthy [11] presented an improved transverse shear deformation theory for laminated anisotropic plates under bending. The displacement field was chosen so that the transverse shear stress vanished on the plate surfaces with the aim to remove the use of shear correction factor in computing shear stresses. Levinson [12] presented a refined theory for the static and dynamic analyses of isotropic plates by using different displacement field expressions. Reddy [13] pointed out that the equilibrium equations derived by Murthy [11] and Levinson [12] resulted to be inconsistent. He wrote a simple higher order theory for laminated composites plates with a consistent derivation of the displacement field and associated equilibrium equations. Later, Reddy and Liu [14] extended the higher order shear deformation theory to shells. Several layerwise models, which contain full three dimensional kinematics and constitutive relations have been developed too. Reddy [15] suggested a layerwise theory by giving an accurate description of the three dimensional displacement field which was expanded as a linear combination of the thickness coordinate and unknown functions of position of each layer. Di Sciuva [16], [17] formulated an improved shear deformation theory, the so called zig-zag theory. He considered a two dimensional theory by adopting a displacement field with piecewise linear variation of the membrane displacement and a constant value of the transverse displacement through the thickness. The fulfilment of the static and geometric continuity conditions was obtained and the influence of the distortion of the deformed normal segment was included. Reddy and Kim [18] pointed out that all the higher order theories are substantially expressed in the form of the displacement expansions used. The equilibrium equations can be derived in different ways. When the principle of virtual displacements is used, additional terms in the form of higher stress resultants are involved.

Bisegna and Sacco [19] derived a general procedure, based on the conjecture that plate theories can be derived from the three dimensional elasticity, by imposing suitable constraints on the stress and strain fields. They used the equilibrium equations to carry out the shear stress in the thickness of the plate. A layer wise laminate theory rationally deduced from the three dimensional elasticity was also presented [20]. Bischoff and Ramm [21] discussed the physical significance of higher order kinematic and static variables in a three dimensional shell formulation. Auricchio and Sacco [22] presented new mixed variational formulations for a first order shear deformation laminate theory and considered the out of plane stresses as primary variables of the problem. They determined the shear stress profile either by independent piecewise quadratic functions in the thickness or by satisfying the three dimensional equilibrium equations written in terms of mid-plane strains and curvatures. Carrera [23], [24] traced a critical overview about the capability of the Reissner’s mixed variational theorem (RMVT) to study multilayered plates and shells. He stated that the interface continuity of transverse shear stresses, as well as the zig-zag form of displacements in the thickness shell direction, are easily introduced by RMWT. Kulikov and Plotnikova [25] developed models for the analyses of multilayered Timoshenko–Mindlin-type shells for the analysis of composite shells, where the effect of transverse shear and transverse normal strains was included. They calculated the axial displacement, the vertical displacement, and the moments resultant by varying the geometric shell parameters. Carrera and Brischetto [26] extended the thickness locking mechanism to shell geometries by considering the thin shell theory, the first order shear deformation theory, higher order theories, mixed theories and layer wise theories. Their investigation confirmed that the thickness locking can be identified as a shell theory problem and has no relation with the numerical methods. Moreover, they observed that in order to avoid the thickness locking the shell theories would require at least a parabolic distribution of transverse displacement component. Matsunaga [27] determined the natural frequency, the buckling stress and the stress distribution of functionally graded shallow shells. He used the method of power series expansion of displacements components and derived the fundamental set of governing equations through the Hamilton’s principle. He proved that a 2D higher order deformation theory can predict accurately not only the natural frequencies and buckling stresses but also the through the thickness stress and displacement distributions. Cinefra et al. [28] proposed a variable kinematic shell model, based on Carrera’s unified formulation, to solve dynamic and static shell problems. They compared classical shell theories with the refined ones based on the Reissner mixed variational theorem. The distribution of the vibration modes and stress components in the thickness shell directions was also investigated. Carrera et al. [29] evaluated the effect of thickness stretching in functionally graded plate/shell structures in the thickness direction. Various FGM plates and shells with different geometries and material properties, as proposed by Zenkour [30] and Kashtalyan and Menshykova [31], were examined. They also confirmed the Koiter’s recommendation [32] which states that an increase in the order of expansion for in-plane displacements can result meaningless if the thickness stretching is discarded in the plate/shell theories (constant transverse displacement). Liew et al. [33] presented an overview on the development of element free or meshless methods in the analysis of composite structures. Recently, Asemi et al. [34] furnished an elastic solution of a two dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads, such as internal, external, and axial pressure. They applied the finite element method (FEM) by using Rayleigh–Ritz energy formulation. The influence of semi vertex angle of the cone and the power law exponents on different distributions of displacements and stresses was investigated. Numerical solutions for hollow cylindrical and truncated conical shells, with various loading and boundary conditions, were assessed. Aghdam et al. [35] conducted the bending analysis of moderately thick functionally graded conical panels, subjected to uniform and non-uniform distributed loadings. They applied the first order shear deformation theory and solved the governing equations by the extended Kantorovich method. The influence of the volume fraction exponent on the distribution of the normalized deflection and moment was underlined.

A lot of works deal with the dynamic response of revolution conical shells [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51]. For the sake of brevity only a few will be explicitly quoted [36], [37], [38], [39], [40], [41], [42], [43], [44]. Khatri and Asnani [36] conducted the vibration and damping analysis of multilayered conical shells. They wrote the governing equation of motion for axisymmetric and antisymmetric vibrations of a general multilayered conical shell consisting of an arbitrary number of orthotropic material layers. They applied the Galerkin method for finding the approximate solutions to the shell with various edge conditions. Lam et al. [37] used the generalized differential quadrature method as a numerical technique for the analysis of the free vibration of truncated conical panels. They considered clamped and simply supported isotropic truncated conical panels and studied the effect of the semi vertex angle on the frequency characteristics. Liew et al. [38] studied the free vibration of conical shells by using the classical thin shell theory. The effects of the semi vertex angle and the boundary conditions were dealt with. Li et al. [39] calculated the natural frequencies and the forced vibration responses of conical shells, using the Rayleigh–Ritz method. Sofiyev [40] analyzed the vibration and stability behavior of freely supported truncated and complete FGM conical shells subjected to external pressure. The material properties were assumed to vary continuously through the thickness of the conical shells, by following a simple power law. The dynamic behavior of functionally graded conical shells by means of the FSDT and the GDQ numerical technique was investigated in [41], [42]. A double form of the simple power law distributions was considered and the effect of the power exponent on the natural frequencies of the graded conical shells was shown. Different types of non-uniform grid point distributions were used. Zhao and Liew [43] developed the free vibration analysis of functionally graded conical shell panels by a meshless method. The accuracy of the proposed method was verified by executing convergence studies in terms of the number of nodes. The effects of the volume fraction, boundary condition, semi vertex angle, and length to thickness ratio on the frequency characteristics of the functionally graded conical shells were monitored. Recently, Tornabene et al. [44] handled functionally graded and laminated doubly curved shells and panels of revolution with a free-form meridian. They furnished a 2D GDQ solution for free vibration by using the first order shear deformation theory (FSDT) and including the curvature effect in the formulation.

Among the numerical works which dealt with the buckling of conical shell, the following ones are considered. Wu and Chiu [45] focused on the thermally induced dynamic instability of composite conical shells. Dulmir et al. [46] treated the axisymmetric static and dynamic buckling of composite truncated conical cap. Bhangale et al. [47] studied the linear thermoelastic buckling and free vibration behavior of functionally graded truncated conical shells. Sofiyev [48] analyzed the stability of functionally graded truncated conical shells under aperiodic impulsive loading. Naj et al. [49] examined the thermal and mechanical instability of functionally graded conical shells. Sofiyev [50] analyzed the stability behavior of freely supported FGM conical shells subjected to external pressure. Recently, he also characterized the influence of the initial imperfection on the linear buckling response of FGM truncated conical shells [51].

The aim of the present study is to extend the previous formulation by the first three authors [52] to the determination of accurate stress profiles for functionally conical shells and panels. As far as the static analysis of functionally graded conical panels and shells is concerned, shear deformation theories of various degrees have been applied. The kinematic model of the first order conceived by Reissner and Mindlin has been overcome by the ones of higher order theories, which lead to the accurate determination of the sliding strain. By fixing the Taylor’s expansion of displacement field at the third order and taking constant the transverse displacement, two third order shear deformation theories are recurrent in the literature background: the third order theories of constrained and unconstrained nature. The first one was originally formulated by Reddy [13], whereas the second one was firstly proposed by Leung [53] and is considered as an evolution of the FSDT. The need to constrain the resulting kinematic model in the Reddy’s formulation, by enforcing the null value of sliding strains on the boundary surfaces, has the proper advantage to make the boundary conditions satisfied. Differently, the Leung’s third order model does not introduce any constraint and it also allows to consider shearing loads on the boundary surfaces. In this paper, the authors reconsider the unconstrained third order theory and write it for functionally graded conical panels and shells. As in the previous work devoted to FGM cylindrical shells and panels [52], the authors combine the Leung’s theory with the stress recovery technique in order to conduct the static analysis under shear and normal constant loads at the extreme surfaces of graded conical shells or panels. The ceramic volume fraction follows a four parameter power exponent law. They start from the definition of a seven parameter displacement field, use the strain–displacement relations enriched by the initial curvature effect, the constitutive equations and the internal actions in terms of the displacement parameters. With the definition of the external transverse and shear uniform loads written in terms of the ones acting on the upper and lower surfaces, the principle of virtual displacements is applied to derive the indefinite equilibrium equations and the boundary conditions. The substitution of the internal actions in terms of generalized displacements in the indefinite equilibrium equation system leads to seven fundamental equations. The generalized differential quadrature method (GDQ) [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92] is applied in order to solve the fundamental system and obtain the solution in terms of the seven independent displacement parameters. Using the constitutive equations, the membrane meridian and circumferential stress responses along the thickness direction for different class of functionally graded materials are determined. With the in-plane stress components indirectly derived from the GDQ-solution [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92] of the fundamental system, the integration of the three dimensional indefinite equilibrium equations is carried out. In order to satisfy the boundary condition at extreme surfaces, the determined transverse shear or normal stress is refined as shown in the previous paper by the first three authors [52]. In this manner, the transverse and normal stresses are plotted throughout the thickness by means of the GDQ solution of the 3D indefinite equilibrium equations, for different types of functionally graded truncated conical panels and shells. The influences of the initial curvature effect, the semi vertex angle of the conical shell, the open angle of the conical panel, the thickness to radius ratio, the thickness to length ratio on the stress profiles are set forth. Moreover, the comparisons between the first and third order mechanical responses, the effect of the material coefficients, the difference between the mechanical behavior of cylindrical and truncated conical structures are carefully examined and depicted. Further publications related to other collocation methods used for the evaluation of the static and dynamic behavior of shell structures, as well as some other papers about stress recovery in bending of plates can be found in [93], [94], [95], [96], [97], [98], [99], [100], [101], [102].

Section snippets

Displacement field

A graded truncated conical shell is considered in the following. L0, R0, h denote the height, the parallel radius and the total thickness of the shell, respectively. The position of an arbitrary point P within the shell is located by the coordinates x(0  x  x0 = L0/cos α), s(0  s  s0 = ϑR0) upon the middle surface. The coordinate ζ is directed along the outward normal n and measured from the reference surface (−h/2  ζ  h/2), as shown in Fig. 1. The α-parameter is the angle of the semi-vertex of the cone

Stress recovery procedure

In order to discretize the derivatives in the governing Eq. (35), as well as the external boundary conditions and the compatibility conditions, the generalized differential quadrature method (GDQ) [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92] is used. In this study, the Chebyshev–Gauss–Lobatto (C–G–L) grid

Numerical investigation and stress profiles

In this numerical study the through the thickness stress distributions are furnished. Two shear deformation models are investigated: the generalized unconstrained third order (GUTSDT) and first order (GFSDT) shear deformation theories, which are enriched by the initial curvature effect. In order to define the correct profile of the transverse shear and normal stress profiles, the stress recovery is also proposed. The recovery procedure requires the knowledge of the membrane stress components.

Conclusions

The generalized third order shear deformation theory with the normal and shear stress recovery is extended to various types of functionally graded truncated conical panels and shells. By means of the GDQ method the shear τxn,τsn,τ¯xs and normal σ¯x,σ¯s,σn stress distributions are accurately determined along the thickness direction. By considering the present formulation it is possible to apply a general loading condition with the satisfaction of all the boundary conditions. It is shown how

Acknowledgements

This research was supported by the Italian Ministry for University and Scientific, Technological Research MIUR (40% and 60%). The research topic is one of the subjects of the Center of Study and Research for the Identification of Materials and Structures (CIMEST)-“M. Capurso” of the University of Bologna (Italy).

References (102)

  • E. Carrera et al.

    Effects of thickness stretching in functionally graded plates and shells

    Compos Part B: Eng

    (2011)
  • A.M. Zenkour

    Generalized shear deformation theory for bending analysis of functionally graded plates

    Appl Math Model

    (2006)
  • K.M. Liew et al.

    A review of meshless methods for laminated and functionally graded plates and shells

    Compos Struct

    (2011)
  • M.M. Aghdam et al.

    Bending analysis of moderately thick functionally graded conical panels

    Compos Struct

    (2011)
  • K.N. Khatri et al.

    Vibration and damping analysis of multilayered conical shells

    Compos Struct

    (1995)
  • K.Y. Lam et al.

    Generalized differential quadrature method for the free vibration of truncated conical panels

    J Sound Vib

    (2002)
  • K.M. Liew et al.

    Free vibration analysis of conical shells via the element-free kp-Ritz method

    J Sound Vib

    (2005)
  • F.M. Li et al.

    The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh–Ritz method

    Mech Res Commun

    (2009)
  • A.H. Sofiyev

    The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure

    Compos Struct

    (2009)
  • F. Tornabene et al.

    2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures

    J Sound Vib

    (2009)
  • X. Zhao et al.

    Free vibration analysis of functionally graded conical shell panels by a meshless method

    Compos Struct

    (2011)
  • F. Tornabene et al.

    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

    (2011)
  • C.P. Wu et al.

    Thermally induced dynamic instability of laminated composite conical shells

    Int J Solids Struct

    (2002)
  • R.K. Bhangale et al.

    Linear thermoelastic buckling and free vibration behavior of functionally graded truncated conical shells

    J Sound Vib

    (2006)
  • A.H. Sofiyev

    Thermoelastic stability of functionally graded truncated conical shells

    Compos Struct

    (2007)
  • R. Naj et al.

    Thermal and mechanical instability of functionally graded truncated conical shells

    Thin Wall Struct

    (2008)
  • A.H. Sofiyev

    The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure

    Compos Struct

    (2010)
  • A.H. Sofiyev

    Influence of the initial imperfection on the nonlinear buckling response of FGM truncated conical shells

    Int J Mech Sci

    (2011)
  • E. Viola et al.

    Numerical investigations of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

    Compos Struct

    (2012)
  • A.Y.T. Leung

    An Unconstrained third order plate theory

    Compos Struct

    (1991)
  • F. Civan et al.

    Differential quadrature for multi-dimensional problems

    J Math Anal Appl

    (1984)
  • E. Artioli et al.

    A differential quadrature method solution for shear-deformable shells of revolution

    Eng Struct

    (2005)
  • F. Tornabene et al.

    Vibration analysis of spherical structural elements using the GDQ method

    Comput Math Appl

    (2007)
  • E. Viola et al.

    Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches

    J Sound Vib

    (2007)
  • A. Marzani et al.

    Nonconservative stability problems via generalized differential quadrature method

    J Sound Vib

    (2008)
  • F. Tornabene et al.

    2-D solution for free vibration of parabolic shells using generalized differential quadrature method

    Eur J Mech A – Solids

    (2008)
  • F. Tornabene

    Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law function

    Comput Methods Appl Mech Eng

    (2009)
  • F. Tornabene et al.

    Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution

    Eur J Mech A – Solids

    (2009)
  • E. Viola et al.

    Free vibrations of three parameter functionally graded parabolic panels of revolution

    Mech Res Commun

    (2009)
  • F. Tornabene

    2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution

    Compos Struct

    (2011)
  • 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

    (2011)
  • F. Tornabene et al.

    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

    (2012)
  • F. Tornabene et al.

    Laminated composite rectangular and annular plates: a GDQ solution for static analysis with a posteriori shear and normal stress recovery

    Compos Part B: Eng

    (2012)
  • E. Viola et al.

    General higher order shear deformation theories for the vibration analysis of completely double-curved laminated shells and panels

    Compos Struct

    (2013)
  • F. Tornabene et al.

    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

    (2012)
  • E. Viola et al.

    Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories

    Compos Struct

    (2013)
  • F. Tornabene et al.

    General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels

    Compos Struct

    (2013)
  • E. Viola et al.

    Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape

    Compos Struct

    (2013)
  • F. Tornabene et al.

    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)
  • F. Tornabene et al.

    Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method

    Compos Struct

    (2014)
  • Cited by (127)

    View all citing articles on Scopus
    View full text