Buckling analysis of plates using the two variable refined plate theory Seung-Eock Kim a , Huu-Tai Thai a , Jaehong Lee b,Ã a Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong Kwangjin-ku, Seoul 143-747, Republic of Korea b Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Republic of Korea article info Article history: Received 16 January 2008 Received in revised form 17 July 2008 Accepted 5 August 2008 Available online 19 September 2008 Keywords: Refined plate theory Buckling analysis Isotropic plate Orthotropic plate Navier method abstract Buckling analysis of isotropic and orthotropic plates using the two variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results obtained by the present theory are compared with classical plate theory solutions, first-order shear deformable theory solutions, and available exact solutions in the literature. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order shear deformable theory. & 2008 Elsevier Ltd. All rights reserved. 1. Introduction The buckling of rectangular plates has been a subject of study in solid mechanics for more than a century. Many exact solutions for isotropic and orthotropic plates have been developed, most of them can be found in Timoshenko and Woinowsky-Krieger [1], Timoshenko and Gere [2], Bank and Jin [3], Kang and Leissa [4], Aydogdu and Ece [5], and Hwang and Lee [6]. In company with studies of buckling behavior of plate, many plate theories have been developed. The simplest one is the classical plate theory (CPT) which neglects the transverse normal and shear stresses. This theory is not appropriate for the thick and orthotropic plate with high degree of modulus ratio. In order to overcome this limitation, the shear deformable theory which takes account of transverse shear effects is recommended. The Reissner [7] and Mindlin [8] theories are known as the first-order shear deform- able theory (FSDT), and account for the transverse shear effects by the way of linear variation of in-plane displacements through the thickness. However, these models do not satisfy the zero traction boundary conditions on the top and bottom faces of the plate, and need to use the shear correction factor to satisfy the constitutive relations for transverse shear stresses and shear strains. For these reasons, many higher-order theories have been developed to improve in FSDT such as Levinson [9] and Reddy [10]. Shimpi and Patel [11] presented a two variable refined plate theory (RPT) for orthotropic plates. This theory which looks like higher-order theory uses only two unknown functions in order to derive two governing equations for orthotropic plates. The most interesting feature of this theory is that it does not require shear correction factor, and has strong similarities with the CPT in some aspects such as governing equation, boundary conditions and moment expressions. The accuracy of this theory has been demonstrated for static bending and free vibration behaviors of plates by Shimpi and Patel [11], therefore, it seems to be important to extend this theory to the static buckling behavior. In this paper, the two variable RPT developed by Shimpi and Patel [11] has been extended to the buckling behavior of orthotropic plate subjected to the in-plane loading. Using the Navier method, the closed-form solutions have been obtained. Numerical examples involving side-to-thickness ratio and mod- ulus ratio are presented to illustrate the accuracy of the present theory in predicting the critical buckling load of isotropic and orthotropic plates. Numerical results obtained by the present theory are compared with CPT solutions, FSDT solutions with different value of shear correction factor. 2. RPT for orthotropic plates 2.1. Basic assumptions of RPT Assumptions of the RPT are as follows: i. The displacements are small in comparison with the plate thickness h and, therefore, strains involved are infinitesimal. ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures 0263-8231/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.08.002 Ã Corresponding author. Tel.: +82 2 3408 3287; fax: +82 2 3408 4331. E-mail address: jhlee@sejong.ac.kr (J. Lee). Thin-Walled Structures 47 (2009) 455–462