JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 315 (2008) 318–342 Exact solution for linear buckling of rectangular Mindlin plates Shahrokh Hosseini-Hashemi a , Korosh Khorshidi a,Ã , Marco Amabili b a Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran b Dipartimento di Ingegneria Industriale, Universita` di Parma, Parco Area delle Scienze 181/A, Parma 43100, Italy Received 6 July 2007; received in revised form 31 January 2008; accepted 31 January 2008 Handling Editor: A.V. Metrikine Available online 14 March 2008 Abstract In the present paper, the Mindlin plate theory is used to study buckling of in-plane loaded isotropic rectangular plates with different boundary conditions. The novelty of the paper is that the analytical closed-form solution is developed without any use of approximation for a combination of six different boundary conditions; specifically, two opposite edges are simply supported and any of the other two edges can be simply supported, clamped or free. Monoaxial in-plane compressive loads on both directions are considered, as well as equal biaxial loads. The present analytical solution can be obtained with any required accuracy and can be used as benchmark. Dimensionless critical buckling loads and mode shapes are given for the six cases analyzed. The effect of boundary conditions, loading conditions, variations of aspect ratios and thickness ratios are examined and discussed in detail. Finally, based on comparison with previously published results, the accuracy of the results is shown. r 2008 Elsevier Ltd. All rights reserved. 1. Introduction Thick plates are important structural elements and are widely used in engineering applications. They can be analyzed by using the classical Kirchhoff thin plate theory, but, because the effects of transverse shear deformation are neglected, the deflections are underestimated and the natural frequencies and buckling loads are overestimated. In order to deal with thicker and laminated composite plates, the Mindlin theory of plates (first-order shear deformation theory) was introduced to take into account transverse shear strains. Five variables are used in this theory to describe the deformation: three displacements of the middle surface and two rotations. In case of flat plates (without geometric imperfections), the in-plane displacements are uncoupled from the transverse displacement and rotations. The Mindlin approach [1] does not satisfy the transverse shear boundary conditions at the top and bottom surfaces of the plate, since a constant shear angle through the thickness is assumed, and plane sections remain plane after deformation. As a consequence of this approximation, the Mindlin theory of plates requires shear correction factors for equilibrium considerations. For this reason, Reddy [1,2] has developed a nonlinear plate ARTICLE IN PRESS www.elsevier.com/locate/jsvi 0022-460X/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2008.01.059 Ã Corresponding author. Tel.: +98 21 7391 2912; fax: +98 21 7724 0488. E-mail address: k_khorshidi@iust.ac.ir (K. Khorshidi).