The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells Zenon J.G.N. del Prado a,n , Ana Larissa D.P. Argenta a , Frederico M.A. da Silva a , Paulo B. Gonçalves b a Federal University of Goiás, School of Civil Engineering, Avenida Universitária, 1488, Setor Leste Universitário, 74605-200 Goiânia, GO, Brazil b Catholic University of Rio de Janeiro, Department of Civil Engineering, Rua Marquês de São Vicente, 225, Gávea, 22453-900 Rio de Janeiro, RJ, Brazil article info Article history: Received 14 December 2013 Received in revised form 26 March 2014 Accepted 27 March 2014 Keywords: Cylindrical shell Orthotropic material Lateral load Dynamic instability Non-linear vibration abstract The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell's non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Circular cylindrical shells subjected to various loading condi- tions are widely used in several engineering areas and industrial applications and their analysis has become an important research area in applied mechanics. These structures present large capacity to withstand both axial loads and lateral pressures; however, they may display complex dynamic behavior due mainly to geometric non-linearity and sensitivity to small imperfections. Although a large number of papers has been published on their non-linear behavior, only a small number of investigations is concerned with the analysis of orthotropic shells. However many natural or artificial materials present orthotropic properties and, as shown in this paper, optimal orthotropic materials can be developed to withstand a specific load. Also corrugated and densely stiffened materials can be described as an equivalent orthotropic materials, as shown by Briassoulis [1], Shen [2], Siad [3], Andrianov et al. [4] and Torkamani et al. [5]. One of the first works on the dynamic behavior of orthotropic cylindrical shells was published by Jain [6] who studied the free vibrations of orthotropic empty cylindrical shells and shells partially or completely filled with an incompressible and non- viscous fluid. Warburton and Soni [7] and Bradford and Dong [8] analyzed, respectively, the resonant response and lateral vibra- tions of orthotropic cylindrical shells. Chen et al. [9] and Chen and Ding [10] studied the free vibrations of both fluid-filled isotropic and orthotropic cylindrical shells, respectively. Using the Sanders–Koiter non-linear shell theory, Selmane and Lakis [11] studied the influence of geometric non-linearities associated with the shell and the fluid flow on the dynamics of empty and fluid- filled elastic thin orthotropic cylindrical shells. Del Prado et al. [12–15] and Argenta et al. [16], using Donnell's non-linear shallow shell theory, without considering the effect of shear deformation, studied the influence of the ratio of Young's modulus in the circumferential and axial direction as well as geometric relations, on the non-linear vibrations of a simply supported orthotropic cylindrical shell, sub- jected to axial and lateral time-dependent loads. Ip et al. [17], using Love's first-approximation shell theory, studied the free vibrations of fiber-reinforced composite cylindrical shells and the influence of the shell thickness on the flexural and stretching energy. An experimental and numerical analysis of a base-excited thin orthotropic cylindrical shell with a top mass was carried out by Mallon Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017 0020-7462/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ55 62 3209 6265. E-mail addresses: zenon@ufg.br (Z.J.G.N. del Prado), analarissaargenta@hotmail.com (A.L.D.P. Argenta), silvafma@ufg.br (F.M.A. da Silva), paulo@puc-rio.br (P.B. Gonçalves). Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎