Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material

Mathematical Problems in Engineering, May 2013

Geometrically nonlinear static analysis of edge cracked cantilever Timoshenko beams composed of functionally graded material (FGM) subjected to a nonfollower transversal point load at the free end of the beam is studied with large displacements and large rotations. Material properties of the beam change in the height direction according to exponential distributions. The cracked beam is modeled as an assembly of two subbeams connected through a massless elastic rotational spring. In the study, the finite element of the beam is constructed by using the total Lagrangian Timoshenko beam element approximation. The nonlinear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. The convergence study is performed for various numbers of finite elements. In the study, the effects of the location of crack, the depth of the crack, and various material distributions on the nonlinear static response of the FGM beam are investigated in detail. Also, the difference between the geometrically linear and nonlinear analysis of edge cracked FGM beam is investigated in detail.

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Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material

Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material Şeref Doğuşcan Akbaş Şehit Muhtar Mah. Öğüt Sok, No:2/37, 34435 Beyoğlu, Istanbul, Turkey Received 11 December 2012; Revised 16 February 2013; Accepted 25 March 2013 Academic Editor: Salvatore Caddemi Copyright © 2013 Şeref Doğuşcan Akbaş. 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 Geometrically nonlinear static analysis of edge cracked cantilever Timoshenko beams composed of functionally graded material (FGM) subjected to a nonfollower transversal point load at the free end of the beam is studied with large displacements and large rotations. Material properties of the beam change in the height direction according to exponential distributions. The cracked beam is modeled as an assembly of two subbeams connected through a massless elastic rotational spring. In the study, the finite element of the beam is constructed by using the total Lagrangian Timoshenko beam element approximation. The nonlinear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. The convergence study is performed for various numbers of finite elements. In the study, the effects of the location of crack, the depth of the crack, and various material distributions on the nonlinear static response of the FGM beam are investigated in detail. Also, the difference between the geometrically linear and nonlinear analysis of edge cracked FGM beam is investigated in detail. 1. Introduction Functionally graded materials (FGMs) are special composites whose composition varies continuously as a function of position along thickness of a structure to achieve a required function. FGMs are generally made of a mixture of ceramic and metal to satisfy the demand of ultrahigh-temperature environment and to eliminate the interface problems. Typically, in an FGM, one face of a structural component is ceramic that can resist severe thermal loading and the other face is metal which has excellent structural strength. FGMs consisting of heat-resisting ceramic and fracture-resisting metal can improve the properties of thermal barrier systems because cracking and delamination, which are often observed in conventional layered composites, are reduced by proper smooth transition of material properties. The technology of FGMs was an original material fabrication technology proposed in Japan in 1984 by Sendai Group. Since the concept of FGMs has been introduced in 1980s, these new kinds of materials have been employed in many engineering application fields, such as aircrafts, space vehicles, defense industries, electronics, and biomedical sectors, to eliminate stress concentrations, to relax residual stresses, and to enhance bonding strength. Because of the wide material variations and applications of FGMs, it is important to study the responses of FGM structures to mechanical and other loadings. With the increased use of FGMs, understanding the mechanical behavior and safe performance of cracked FGM structures is very important. It is known that a crack in structure elements introduces a local flexibility, becomes more flexible and its dynamic, buckling, and static behaviors will be changed. Cracks cause local flexibility and changes in structural stiffness. In recent years, much more attention has been given to the nonlinear behavior of intact FGM beams. Rastgo et al. [1] investigated the instability of curved beams made of functionally graded material under thermal loading. Agarwal et al. [2] studied the geometrically nonlinear static and dynamic responses of functionally graded beams. Li et al. [3] investigated thermal postbuckling of functionally graded clamped-clamped Timoshenko beams subjected to transversely nonuniform temperature. Based on Kirchhoff’s assumption of straight normal line of beams and considering the effects of the axial elongation, the initial curvature and the stretching-bending coupling on the arch deformation, geometrically nonlinear governing equations of functionally graded arch subjected to mechanical, and thermal loads are derived by Song and Li [4]. Kang and Li [5] used large and small deformation theories to find the nonlinear solutions to a functionally graded material cantilever beam subjected to an end force. Kang and Li [6] analyzed the large deformation of a nonlinear cantilever functionally graded beam. Thermal postbuckling behavior of uniform slender FG beams is discussed independently using the classical Rayleigh-Ritz (RR) formulation and the versatile finite element analysis based on the von Karman nonlinear strain approximation by Anandrao et al. [7]. Kocatürk et al. [8] examined the full geometrically nonlinear static analysis of a cantilever Timoshenko (...truncated)


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Şeref Doğuşcan Akbaş. Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material, Mathematical Problems in Engineering, 2013, 2013, DOI: 10.1155/2013/871815