Interlaminar stresses in thick rectangular laminated plates with arbitrary laminations and boundary conditions under transverse loads M. Tahani ⇑ , A. Andakhshideh Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran article info Article history: Available online 5 January 2012 Keywords: Composite laminated plates Edge effect Interlaminar stresses Three-dimensional multi-term extended Kantorovich method Transverse loading abstract In the present study, interlaminar stresses resulting from bending of thick rectangular laminated plates with arbitrary laminations and boundary conditions are analyzed analytically based on a three-dimensional multi-term extended Kantorovich method (3DMTEKM). Using the principle of minimum total potential energy, three systems of coupled ordinary differential equations with non- homogeneous boundary conditions are obtained. Then an iterative procedure is established to achieve analytical solution. The results obtained from this theory are compared with those of analytical solutions existing in the literature. It is found that the present results have excellent agreements with those obtained by layerwise theory. The results show that the multi-term EKM converges within only three terms of trial functions and the single-term EKM is not able to estimate the local interlaminar stresses near the boundaries of laminates. Finally, the power of the present approach in obtaining the interlam- inar stresses in thick rectangular laminated plates with general types of boundary conditions and lay- ups is examined. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Interlaminar stresses which arise at the edges of laminated composites play an important role in their analysis and design. They are a major source of concern in composite laminates because they can lead to delamination and failure of laminate at a much lower load than that predicted by in-plane failure criteria. Numerous investigators have proposed various analytical and numerical methods to examine the transverse stress behavior near the edges of laminates. However, because of inherent complexities involved in the problem, no exact solution is known for elasticity equations. Therefore, various approximate methods for determin- ing the interlaminar stresses are documented in the literature. These methods may, for convenience, be classified as either analyt- ical or numerical. Because of the exceptionally large number of pa- pers on the subject matter, only the pertinent pioneering works are referred here. The interested reader will find sufficient references to cover the literature in more depth in the review article by Kant and Swaminathan [1]. The first approximate solution of interlaminar shear stresses was proposed by Puppo and Evenson [2] based on a laminated model containing anisotropic layers separated by isotropic adhe- sive layers with interlaminar normal stress being neglected through the laminate. They showed that interlaminar shear stres- ses reach their maximum magnitudes at the free edges. Pipes and Pagano [3] subsequently obtained interlaminar stresses in a long symmetric laminate subjected to uniform axial strain. They utilized classical linear theory of elasticity to reveal the presence of significant stresses, both normal and shear, at the edges, be- tween plies of composite plates. They employed a finite difference technique to solve the coupled second-order partial differential governing equations in terms of displacements. Whitney [4] also evaluated free edge stresses in laminated composites using simple stress approximations in the form of products of exponential and trigonometric functions. Later, Pagano [5] examined the problem by extension of the higher-order plate theory. Tang [6] and Tang and Levy [7] presented an analytical method by means of a bound- ary-layer theory. In this case they were able to predict all three interlaminar stress components near free edges of a symmetric angle-ply composite laminate under uniaxial tension. Using perturbation techniques, Hsu and Herakovich [8] developed a zeroth-order solution for investigation of edge effects in symmetric angle-ply laminates. Wang and Choi [9,10] studied the free edge singularities by means of Lekhnitskii’s stress potential and the theory of anisotropic elasticity. Reissner [11–13] proposed a valuable mixed variational theo- rem for the approximate analysis of isotropic and anisotropic laminated elastic plates and shells. Reissner’s mixed variational theorem (RMVT) is a better description than classical variational formulation with only displacement variables. The classical varia- tional theories do not provide a priori interlaminar continuous 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.12.027 ⇑ Corresponding author. Tel./fax: +98 511 876 3304. E-mail address: mtahani@ferdowsi.um.ac.ir (M. Tahani). Composite Structures 94 (2012) 1793–1804 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct