Vibration and buckling analysis of two-layered functionally graded cylindrical shell, considering the effects of transverse shear and rotary inertia H.A. Sepiani a, * , A. Rastgoo a , F. Ebrahimi a , A. Ghorbanpour Arani b a Department of Mechanical Engineering, School of Engineering, University of Tehran, Tehran, Iran b Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran article info Article history: Received 20 July 2009 Accepted 26 September 2009 Available online 1 October 2009 Keywords: Functionally graded material Vibration Buckling Stability Two-layered functionally graded cylindrical shell abstract This research investigates the free vibration and buckling of a two-layered cylindrical shell made of inner functionally graded (FG) and outer isotropic elastic layer, subjected to combined static and periodic axial forces. Material properties of functionally graded cylindrical shell are considered as temperature depen- dent and graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Theoretical formulations are presented based on two different methods of first-order shear deformation theory (FSDT) considering the transverse shear strains and the rotary iner- tias and the classical shell theory (CST). The results obtained show that the transverse shear and rotary inertias have considerable effect on the fundamental frequency of the FG cylindrical shell. The results for nondimensional natural frequency are in a close agreement with those in literature. It is inferred from the results that the geometry parameters and material composition of the shell have significant effect on the critical axial force, so that the minimum critical load is obtained for fully metal shell. Good agreement between theoretical and finite element results validates the approach. It is concluded that the presence of an additional elastic layer significantly increases the nondimensional natural frequency, the buckling resistance and hence the elastic stability in axial compression with respect to a FG hollow cylinder. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Functionally graded materials (FGMs) are now developed for general use as structural components in extremely high tempera- ture environments. Unlike fiber–matrix composites which have a mismatch of mechanical properties across an interface of two dis- crete materials bonded together and may result in debonding at high temperatures. FGMs have the advantage of being able to with- stand high temperature environments while maintaining their structural integrity, as presented at: the works done by the authors, in which the behavior of FGM plates combined with smart structures are investigated [1,2]. Formulation and theoretical analysis of the FGM plates and shells were presented by Reddy and Chin [3,4], Arciniega and Red- dy [5], Praveen et al. [6]. Shell structure made up of this composite material (FGM) is also one of the basic structural elements used in many engineering structures. Despite the evident importance in practical applications, investigations on the static and dynamic characteristics of FGM shell structures are still limited in number. Among those available, Pradhan [7], Gong [8] and Ng [9] inves- tigated the free vibration, elastic response and dynamic instability of FG cylindrical shells, respectively. Aforementioned studies, theoretical formulations were all based on classical shell theory, i.e., neglecting the effect of transverse shear strains. Employing the first-order shear deformation shell theory, Jafari [10] and Ravikiran [11] investigated both buckling and free vibration of functionally graded cylindrical shells and Bhangale [12] solved the free vibration problem for simply supported FG magneto-elec- tro-elastic shells. Shen [13] showed that in shell buckling, there is a boundary layer phenomenon where prebuckling and buckling dis- placement vary rapidly and performed the postbuckling analysis for imperfect, stiffened and multilayered cylindrical shells [14]. In some of above studies the material properties are assumed to be independent of the temperature. Cylindrical shells in engineering structures with large aspect ra- tios are typically stiffened against buckling by circumferential and longitudinal members, known as ring and stringers stiffeners, respectively. Accordingly, sandwich construction with lightweight additional layer materials has well known advantages of stiffness and strength to weight. These structures are widely used in appli- cations where weight consideration has great importance. Cylin- drical shell with an elastic layer under axial pressure is well known to be highly efficient in terms of combining high stiffness to weight. These structures consist of a fully condensed shell mate- rial supported by a low density external/internal layer. The forth author analyzed the elastic buckling of a thin cylindrical shell supported by an elastic core [15] showing that this structural 0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2009.09.052 * Corresponding author. E-mail address: sepiani@ut.ac.ir (H.A. Sepiani). Materials and Design 31 (2010) 1063–1069 Contents lists available at ScienceDirect Materials and Design journal homepage: www.elsevier.com/locate/matdes