Analytical and finite element modeling of laminated composite and sandwich plates: An assessment of a new shear deformation theory for free vibration response Neeraj Grover, B.N. Singh n , D.K. Maiti Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721302, India article info Article history: Received 29 August 2012 Received in revised form 7 December 2012 Accepted 25 December 2012 Available online 8 January 2013 Keywords: Shear deformation theory Free vibration Composite plate Sandwich plate Navier solution Finite element abstract Recently developed inverse hyperbolic shear deformation theory by the authors is extended to analyze the free vibration response of laminated composite and sandwich plates. Euler–Lagrange equations are derived employing the principle of virtual work for the dynamic problem. A Navier type and finite element solutions are proposed to obtain the free vibration response of laminated composite and sandwich plates. A C 0 -continuous isoparametric biquadratic-quadrilateral serendipity element is used for the finite element solution of generalized higher order shear deformation theory so as to ensure its applicability to general laminates subjected to different combinations of boundary conditions. Higher modes of vibration are obtained for laminated cross-ply and angle-ply plates and efficiency of the theory is ensured by comparing the results with the existing results. It is observed that both analytical and finite element solutions with the present theory are capable for accurate prediction of the free vibration response. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The recent advances in the use of composite materials for the structural applications in aerospace, marine, and automotive industry due to their mechanical advantages of specific modulus and specific strength over conventional materials, improved impact and fatigue resistance, and design flexibility have moti- vated structural researchers to develop an accurate mathematical model in order to ensure the realistic behavior of the structure. The importance of transverse shear deformation for the analysis of laminated composite and sandwich plates is well established. The influence of transverse shear deformation is also significant for the analysis of vibration behavior. In the past few decades, various shear deformation theories have been proposed and implemented to evaluate the fundamental frequency and higher modes of vibration. Yang et al. [1] proposed a two-dimensional theory for heterogeneous plates including the effect of transverse shear deformation and rotary inertia and concluded that trans- verse shear deformations play a significant role for accurate prediction of dynamic response. Reddy [2] presented the finite element solution for free vibration of anti-symmetric angle ply plates by implementing transverse shear deformations proposed by Ref. [1]. However, it requires the use of shear correction factor. In order to consider the non-linear transverse shear stress distribution accurately and hence eliminate the requirement of shear correction factor, various higher order shear deformation theories (HSDT) have been proposed. Reddy and Phan [3] inves- tigated the natural frequencies and buckling loads of elastic plates based on well known higher order shear deformation theory proposed by Reddy [4] and hence need of shear correction factor vanishes. Authors concluded that the precise natural frequencies and vibration modes can be achieved by considering the accurate shear deformation in comparison to first order shear deformation theory (FSDT). Khdeir [5] proposed Levy type solutions in con- junction with state space approach for free vibration and buckling response using Reddy’s HSDT and obtained results for different combinations of boundary conditions. Kant and Mallikarjuna [6] presented a C 0 iso-parametric finite element to predict the vibration response of laminated composite and sandwich plates using 11 variable higher order shear deformation theory and five variable first order shear deformation theory proposed by Kant et al. [7]. Touratier [8] proposed a trigonometric shear deforma- tion theory which has been extensively used for static, buckling, and dynamic behavior of plates. A detailed description and review of the shear deformation plate theories can be found in Reissner [9], Noor and Burton [10], Reddy [11], Mallikarjuna and Kant [12], and Carrera [13]. Khdeir and Reddy [14] used second order shear deformation theory to analyze the free vibration response of laminated composite plates. Fares and Zenkour [15] investigated Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijmecsci International Journal of Mechanical Sciences 0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.12.010 n Corresponding author. Tel.: þ91 3222 283026; fax: þ91 3222 255303. E-mail addresses: bnsingh@aero.iitkgp.ernet.in, bnsingh9@rediffmail.com (B.N. Singh). International Journal of Mechanical Sciences 67 (2013) 89–99