Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening Liao-Liang Ke a,c , Jie Yang b, * , Sritawat Kitipornchai a a Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong b School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, P.O. Box 71, Bundoora, VIC 3083, Australia c Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China article info Article history: Available online 13 March 2009 Keywords: Functionally graded materials Timoshenko beam Open edge crack Geometric nonlinearity Ritz method Postbuckling abstract In this paper, the postbuckling response of beams made of functionally graded materials (FGMs) contain- ing an open edge crack is studied based on Timoshenko beam theory and von Kármán nonlinear kinemat- ics. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through thickness direction. Ritz method is employed to derive the nonlinear governing equations, which are then solved by using Newton–Raphson method to obtain the postbuckling load-end shortening curves and postbuckling deflection-end shortening curves. A detailed parametric study is conducted to study the influences of crack depth, crack location, material property gradient, and slenderness ratio on the postbuckling behavior of cracked FGM beams. It is found that both intact and cracked FGM beams exhibit similar postbuckling behavior under end shortening. Unlike isotropic homogeneous beams, bifurcation buckling does not occur for both intact and cracked FGM beams due to the presence of bending–extension coupling effect. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction It has long been known that crack defects cause local changes in structural stiffness hence the stability and dynamic characteristics of the structure as well. The effect of crack on the buckling and postbuckling behavior is an important topic in structural safety assessment and has received increasing research efforts. The buck- ling and postbuckling of cracked homogeneous structures have been studied extensively. Shaw and Huang [1] studied the effects of the crack length, boundary and loading conditions on the buck- ling behavior of cracked plates using the finite element method. Riks et al. [2] presented finite element buckling and postbuckling analyses of cracked plates loaded in tension. Estekanchi and Vafai [3] investigated the buckling of cylindrical shells with cracks of varying length and orientation. Based on the rotational spring model, Wang [4] obtained an analytical solution for the elastic sta- bility of an edge cracked beam subjected to a compressive follower force. Wang and Quek [5] proposed a repair technique using a pie- zoelectric patch that produce a local moment to counteract the loss of bending stiffness to restore the buckling load-carry capacity of a cracked column. The effect of the crack on the buckling capacity of the column is studied analytically. Brighenti [6] investigated the effect of crack lengths and locations on the buckling of cracked elastic rectangular plates under edge compression and/or tension. Alinia et al. [7] discussed the influence of central cracks on buck- ling and postbuckling behaviour of shear panels using the finite element analysis. It was shown that the length and angle of cracks may change the buckling behavior of shear panels. Recently, Skrinar [8] estimated the critical buckling load for slender transversely cracked beam-columns. Yazdchi and Anaraki [9] analyzed the load-carrying capacity of edge cracked columns with different boundary conditions and cross-sections subjected to concentric vertical loads. In both Refs. [8,9], the cracked section is modeled as a massless rotational spring. Functionally graded materials (FGMs) are inhomogeneous com- posites whose material properties vary gradually with respect to spatial coordinates. The material composition can be designed so as to improve the strength, toughness, high temperature with- standing ability, etc. to meet the desired structure performance. The buckling [10–16] and postbuckling [17–24] behaviors of FGM structures have been investigated extensively in the past 10 years. However, studies concerning the effect of crack defects on the stability of FGM structures are very limited. Yang and Chen [25] analytically discussed the influence of open edge cracks on the vibration and buckling of Euler–Bernoulli FGM beams with differ- ent boundary conditions. Ke et al. [26] considered the free vibra- tion and elastic buckling of cracked Timoshenko graded beams and obtained analytical solutions. These two studies are for linear buckling analysis only. To the best of authors’ knowledge, no pre- vious work has been done on the postbuckling analysis of cracked FGM structures. 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.03.003 * Corresponding author. Tel.: +61 3 99256169; fax: +61 3 99256092. E-mail address: j.yang@rmit.edu.au (J. Yang). Composite Structures 90 (2009) 152–160 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct