Approximate analytical solutions for the nonlinear free vibrations of composite beams in buckling Samir A. Emam ⇑ Department of Mechanical Engineering, Faculty of Engineering, United Arab Emirates University, Al Ain, P.O. Box 17555, United Arab Emirates Department of Mechanical Design and Production, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt article info Article history: Available online 11 January 2013 Keywords: Nonlinear free vibration Composite beams Variational method Analytical solution abstract We present approximate analytical solutions for the nonlinear free vibrations of symmetrically or asym- metrically laminated composite beams in prebuckling and postbuckling. Simply supported and clamped– clamped boundary conditions are considered. Galerkin’s discretization is used to obtain the nonlinear ordinary differential equations governing the large-amplitude vibrations of composite beams in prebuck- ling and postbuckling, which are found to be of the same form. The variational method of He [20,21] is used to derive an approximate analytical solution for the nonlinear natural frequency and the nonlinear load–deflection relation. Results obtained by using the proposed analytical solution is compared with the finite element results available in the literature and a good agreement has been obtained. Numerical results to show the variation of the nonlinear natural frequency with the applied axial load for a variety of composite laminates are presented. The contribution of the amplitude of vibration on the nonlinear load–deflection response and the nonlinear natural frequency is found to be significant. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Fiber-reinforced laminated composite structures are used in many engineering applications due to their superior properties such as high specific strength and high specific stiffness. Structures that carry inplane loadings are subject to buckle. The critical buck- ling load defines the threshold at which the prebuckling equilib- rium position loses its stability. For beams, it is shown that the first buckling mode is the only stable equilibrium position, in the postbuckling state, and all higher order modes are unstable, Nayfeh and Emam [1]. This means that beams have a load-carrying capac- ity in postbuckling as well. Linear free vibration analysis is a basic element of understanding the dynamic response of a structure that is subjected to dynamic loadings. On the other hand, under severe dynamic loading conditions, structures may undergo large- amplitude vibrations. In this case, linear free vibration analysis will not be adequate and a nonlinear free vibration analysis becomes necessary. The nonlinear free vibrations of isotropic beams have received a considerable attention by many researchers [2–9]. On the other hand, a few studies have been reported on the nonlinear free vibrations of composite beams [10–14]. Gunda et al. [15] stud- ied large-amplitude vibrations of laminated composite beam with axially immovable ends with symmetric and asymmetric layup ori- entations by using the Rayleigh–Ritz (R–R) and finite element methods. Geometric nonlinearity of von-Karman type, which ac- counts for the midplane stretching, is considered. Results pre- sented in that study are valid as long as the beam in its prebuckling state. Baghani et al. [16] presented analytical expres- sions for large-amplitude free vibration and post-buckling analysis of unsymmetrically laminated composite beams on elastic founda- tion. Besides, the elastic foundation has cubic nonlinearity with shearing layer. The nonlinear governing equation is solved by employing the variational iteration method. They presented the ef- fects of different parameters on the ratio of nonlinear to linear nat- ural frequency and the post-buckling load–deflection relation. Their analysis was also valid in the prebuckling state. To the best of author’s knowledge, the nonlinear free vibrations of composite beams in the postbuckling state has not been ad- dressed yet, which was the motivation behind this study. The main objective of this study is to present an approximate analytical solu- tion for the nonlinear free vibrations of laminated composite beams in postbuckling using He’s variational principle. Simply sup- ported and clamped–clamped boundary conditions are used. The equation governing the large-amplitude vibrations of composite beams is a nonlinear integral partial-differential equation. A sin- gle-mode Galerkin discretization is used to reduce the governing equation into a nonlinear ordinary-differential equation. It is found out that the equations governing the nonlinear free vibrations of composite beams in prebuckling and postbuckling have a similar form. The model is validated by comparing present results with 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.12.044 ⇑ Tel.: +971 3 7135127. E-mail address: semam@uaeu.ac.ae Composite Structures 100 (2013) 186–194 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct