Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 569356, 11 pages

http://dx.doi.org/10.1155/2015/569356

## On the Shear Buckling of Clamped Narrow Rectangular Orthotropic Plates

Department of Civil, Environmental and Natural Resources Engineering, Division of Structural and Construction Engineering-Timber Structures, Luleå University of Technology, 971 87 Luleå, Sweden

Received 19 October 2015; Accepted 29 October 2015

Academic Editor: Francesco Tornabene

Copyright © 2015 Seyed Rasoul Atashipour and Ulf Arne Girhammar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with stability analysis of clamped rectangular orthotropic thin plates subjected to uniformly distributed shear load around the edges. Due to the nature of this problem, it is impossible to present mathematically exact analytical solution for the governing differential equations. Consequently, all existing studies in the literature have been performed by means of different numerical approaches. Here, a closed-form approach is presented for simple and fast prediction of the critical buckling load of clamped narrow rectangular orthotropic thin plates. Next, a practical modification factor is proposed to extend the validity of the obtained results for a wide range of plate aspect ratios. To demonstrate the efficiency and reliability of the proposed closed-form formulas, an accurate computational code is developed based on the classical plate theory (CPT) by means of differential quadrature method (DQM) for comparison purposes. Moreover, several finite element (FE) simulations are performed via ANSYS software. It is shown that simplicity, high accuracy, and rapid prediction of the critical load for different values of the plate aspect ratio and for a wide range of effective geometric and mechanical parameters are the main advantages of the proposed closed-form formulas over other existing studies in the literature for the same problem.

#### 1. Introduction

The shear buckling analysis of clamped composite plates is of great importance in design of many types of engineering structures. Unlike the problem of normal buckling of plates, the shear buckling problem of plates is mathematically described by differential equations having a term with odd-order of derivatives with respect to each of the planar spatial coordinates. Therefore, their governing equations cannot be solved exactly. Such problems are almost always analysed and solved using different numerical approaches. Apart from the loading type, clamped boundary conditions at all plate edges make the problem more difficult for finding an exact analytical solution.

During the past decades, many investigators have studied the shear buckling problem of rectangular plates. One of the first efforts dealing with shear buckling analysis of clamped isotropic plates with finite dimensions can be attributed to Budiansky and Conner [1] using Lagrangian multiplier method. A useful review of the studies on the shear buckling of both isotropic and orthotropic plates was presented by Johns [2]. Shear buckling analysis of antisymmetric cross ply, simply supported rectangular plates was carried out by Hui [3] using Galerkin procedure. Kosteletos [4] studied shear buckling response of laminated composite rectangular plates with clamped edges using Galerkin method. Biggers and Pageau [5] computed shear buckling loads of both uniform and composite tailored plates using finite element method. Xiang et al. [6] employed pb-2 Rayleigh-Ritz approach to obtain critical shear loads of simply supported skew plates. Loughlan [7] studied the shear buckling of thin laminated composite plates and examined the effect of bend-twist coupling on their behaviour using a finite strip procedure. Lopatin and Korbut [8] utilized the finite difference method to investigate the shear buckling of thin clamped orthotropic plates. Shufirn and Eisenberger [9] analysed the buckling of thin plates under combined shear and normal compressive loads using the multiterm extended Kantorovich method. The shear buckling load of rectangular composite plates consisting of concentric rectangular layups was investigated by Papadopoulos and Kassapoglou [10] by means of a Rayleigh-Ritz approach. Wu et al. [11] calculated the critical shear buckling loads of rectangular plates by the extended spline collocation method (SCM). Uymaz and Aydogdu [12] carried out the shear buckling analysis of functionally graded plates for various boundary conditions based on the Ritz method. Shariyat and Asemi [13] performed a nonlinear elasticity-based analysis for the shear buckling of rectangular orthotropic functionally graded (FG) plates surrounded by elastic foundations using a cubic B-spline finite element approach.

Evidently, all the above-mentioned numerical studies have some deficiencies like convergence difficulties and being time-consuming compared to analytical and closed-form solutions. Therefore, it is not easy and time-efficient to predict the critical shear buckling loads and investigate the effect of various parameters by the use of numerical solution approaches. To the best of authors’ knowledge, no closed-form solution can be found in the literature for the shear buckling of composite rectangular plates with finite dimensions. To fill this apparent void, the present work is carried out to provide efficient and reliable explicit formulas for rapid prediction of the fundamental critical shear buckling loads of clamped orthotropic rectangular plates. The range of validity of the proposed closed-form formulas is extended by introducing a practical modification factor. Also, in order to demonstrate the efficiency and reliability of the proposed closed-form formulas, an accurate computational code is developed by means of differential quadrature method (DQM) for comparison purposes. Moreover, several finite element (FE) simulations are performed via ANSYS software.

This paper is only devoted to a principle study of the shear buckling behavior and, for illustration, is applied to a laminated veneer lumber (LVL) panel. Other failure modes, such as the shear strength, are not included in the analysis.

#### 2. Definition of the Problem and Governing Equations

Consider a clamped narrow rectangular orthotropic plate of length , width , and thickness , subjected to a uniformly distributed shear load per length (Figure 1). The coordinates system is shown in the figure. We employ the classical plate theory (CPT) of Kirchhoff to study the shear buckling of thin plates. The governing equation of CPT for the orthotropic plates is expressed aswhere is transverse displacement, and are stiffness coefficients of orthotropic materials and are defined as follows:in which and are modulus of elasticity of orthotropic material in and directions, respectively; is the in-plane shear modulus and are the Poisson’s ratios.