Engineering Structures 26 (2004) 1473–1482 www.elsevier.com/locate/engstruct Vibration of non-uniform thick plates on elastic foundation by diﬀerential quadrature method P. Malekzadeh a , G. Karami b, a Department of Mechanical Engineering, Persian Gulf University, Bushehr, Iran b Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Dolve 111, P.O. box 5285, Fargo, ND 58105, USA Received 4 March 2004; received in revised form 13 May 2004; accepted 18 May 2004 Abstract This paper presents a diﬀerential quadrature (DQ) solution for free vibration analysis of thick plates of continuously varying thickness on two-parameter elastic foundations. The formulations are based on the ﬁrst-order shear deformation theory taking into account the eﬀects of rotary inertia. The thickness of the plate may vary in one or two directions. The thickness variation might be assumed linear or non-linear. Diﬀerent types of boundary conditions, including free edges and corners, loaded edges with in-plane forces are formulated. The accuracy, convergence and versatility of the DQ procedure for the type of plate problems, with complicated governing diﬀerential equations and boundary conditions are examined and veriﬁed. # 2004 Elsevier Ltd. All rights reserved. Keywords: Vibration; Thick plate; Elastic foundation; DQM 1. Introduction Vibration problems of variable thickness plates on elastic foundations play an important role in many structural and foundation engineering problems. Exten- sive research has been carried out based on the thin plates theory [1]. It is well known fact that the classical plate theory over-predicts the eigenfrequencies at all modes in situations where shear deformations and rotary inertias become signiﬁcant. To improve the accuracy, use of theories that include the eﬀects of shear deformation and rotary inertia (Mindlin theory) is recommended especially for moderately thick plates [2]. Many researchers have carried out research on free vibration analysis of thick plates of variable thickness. Mikami and Yoshimura [3] applied the collocation method for the analysis of rectangular Mindlin plates with linearly varying thickness. Aksu and Al-Kaabi [4,5] presented a method based on a variational prin- ciple in conjunction with ﬁnite diﬀerence technique to calculate frequencies of rectangular Mindlin plates with linear and parabolic varying thicknesses. Mizusawa [6] used spline strip method to investigate the free vibration of Mindlin plates with linear variation of thickness. In afore-mentioned studies, only plates with two opposite edges simply supported and with a vary- ing thickness along one direction have been examined. Sakiyama and Huang [7] used Green function to obtain approximate solutions for natural frequencies of thin as well as moderately thick plates. Li [8] presented an analytical approach for free vibration of shear deformable beams in order to determine the natural frequency of elastically restrained non-uniform ﬂex- ural-shear plates under the action of in-plane load. Mizusawa and Kondo [9] used spline element method to investigate the free vibration analysis of moderately thick skew plates with varying thickness in one direc- tion. More recently, Cheung and Zhou [10] studied the free vibration of rectangular Mindlin plates with vari- able thickness using the Rayleigh–Ritz method. Cheung and Zhou assumed a continuous variation in thickness presented by a power function of the rec- tangular coordinates. A closed-form solution for the natural frequencies of simply supported Mindlin plates on Pasternak  Corresponding author. Tel.: +1-701-231-5859; fax: +1-701-231- 8913. E-mail address: g.karami@ndsu.nodak.edu (G. Karami). 0141-0296/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.05.008