Free Vibration Characteristics Of Edge Cracked Functionally Graded Beams By Using Finite Element Method
|International Journal of Engineering Trends and Technology (IJETT)||
|© 2013 by IJETT Journal|
|Year of Publication : 2013|
|Authors : Dr. Şeref Doğuşcan AKBAŞ|
Dr. ?eref Do?u?can AKBA?. "Free Vibration Characteristics Of Edge Cracked Functionally Graded Beams By Using Finite Element Method". International Journal of Engineering Trends and Technology (IJETT). V4(10):4590-4597 Oct 2013. ISSN:2231-5381. www.ijettjournal.org. published by seventh sense research group.
This paper presents free vibration analysis of an edge cracked functionally graded cantilever beam. The differential equations of motion are obtained by using Hamilton’s principle. The considered problem is investigated within the Euler-Bernoulli beam theory by using finite element method. The cracked beam is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. Material properties of the beam change in the thickness direction according to exponential distributions. In order to establish the accuracy of the present formulation and results, the natural frequencies are obtained, and compared with the published results available in the literature. Good agreement is observed. In the study, the effects of the location of crack, the depth of the crack and different material distributions on the natural frequencies and the mode shapes of the cracked functionally graded beams are investigated in detail.
 A.D. Dimarogonas, “Vibration of Cracked Structures: A State of the Art Review,”Engineering Fracture Mechanics, vol. 55, no. 5, pp. 831–857, 1996.
 T. Yokoyama and M.C. Chen, “Vibration Analysis of Edge-Cracked Beams using a Line-Spring Model,” Engineering Fracture Mechanics, vol. 59, no. 3, pp. 403–409,1998.
 M. Kisa, J. Brandon, and M. Topcu, “Free Vibration Analysis of Cracked Beams by a Combination of Finite Elements and Component Mode Synthesis Methods,” Computers and Structures, vol. 67, no. 4, pp. 215–223, 1998.
 T.G. Chondros, A.D. Dimarogonas and J. Yao, “A continuous cracked beam vibration theory,” Journal of Sound and Vibration, vol. 215, no. 1, pp. 17–34, 1998.
 E.I. Shifrin and R. Ruotolo, “Natural frequencies of a beam with an arbitrary number of cracks,” Journal of Sound and Vibration, vol. 222, no. 3, pp. 409–23, 1999.
 M. K?sa and J.A. Brandon, “Free vibration analysis of multiple openedge cracked beams by component mode synthesis,” Structural Engineering Mechanics, vol. 10, no. 1, pp. 81-92, 2000.
 D.Y. Zheng and S.C. Fan, “Natural Frequency Changes of a Cracked Timoshenko Beam by Modified Fourier Series,” Journal of Sound and Vibration, vol. 246, no. 2, pp. 297–317, 2001.
 H.P. Lin, S.C. Chang and J.D. Wu, “Beam vibrations with an arbitrary number of cracks,” Journal of Sound and Vibration, vol. 258, no. 5, pp. 987–999, 2002.
 N.T. Khiem and T.V. Lien, “The dynamic stiffness matrix method in forced vibration analysis of multiple-cracked beam,” Journal of Sound and Vibration, vol. 254, no. 3, pp. 541–555, 2002.
 M.A. Mahmoud and M.A. Abou Zaid, “Dynamic response of a beam with a crack subject to a moving mass,” Journal of Sound and Vibration, vol. 256, no. 4, pp. 591–603, 2002.
 A. Nag, D. Roy Mahapatra, S. Gopalakrishnan, and T.S. Sankar, “A Spectral Finite Element with Embedded Delamination for Modeling of Wave Scattering in Composite Beams,” Composites Science and Technology, vol. 63, no. 15, pp. 2187–2200, 2003.
 D. Sreekanth Kumar, D. RoyMahapatra, and S. Gopalakrishnan, “A Spectral Finite Element for Wave Propagation and Structural Diagnostic Analysis in a Composite Beam with Transverse Cracks,” Finite Elements in Analysis and Design, vol. 40, no. 13-14, pp. 1729–1751, 2004.
 C. Bilello and L.A. Bergman, “Vibration of damaged beams under a moving mass: theory and experimental validation,” Journal of Sound and Vibration, vol. 274, no. 3-5, pp. 567–582, 2004.
 D.Y. Zheng and N.J. Kessissoglou, “Free vibration analysis of a cracked beam by finite element method,” Journal of Sound and Vibration, vol. 273, no. 3, pp. 457–475, 2004.
 M. K?sa, “Free Vibration analysis of a cantilever composite beam with multiple cracks.” Composites Science and Technology, vol. 64, no. 9, pp. 1391-1402, 2004.
 M.H. Hsu, “Vibration Analysis of Edge-Cracked Beam on Elastic Foundation with Axial Loading using the Differential Quadrature Method,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 1, pp. 1–17, 2005.
 B. Binici, “Vibration of beams with multiple open cracks subjected to axial force,” of Sound and Vibration, vol. 287, no. 1-2, pp. 277–295, 2005.
 S. Loutridis, E. Douka and L.J. “Hadjileontiadis, Forced vibration behaviour and crack detection of cracked beams using instantaneous frequency,” NDT&E International, vol., no. 5, pp. 411–419, 2005.
 H.P. Lin and S.C. Chang, “Forced responses of cracked cantilever beams subjected to a concentrated moving load,” International Journal of Mechanical Sciences, vol. 48, no. 12, pp. 1456–1463, 2006.
 J.A. Loya, L. Rubio and J. Fernández-Sáez, “Natural Frequencies for Bending Vibrations of Timoshenko Cracked Beams,” Journal of Sound and Vibration, vol. 290, no. 3-5, pp. 640–653, 2006.
 K. El Bikri, R. Benamar, and M.M. Bennouna, “Geometrically Non-Linear Free Vibrations of Clamped-Clamped Beams with an Edge Crack,” Computers & Structures. 84, no. 7, pp. 485–502, 2006.
 K. Aydin, “Vibratory Characteristics of Axially-Loaded Timoshenko Beams with Arbitrary Number of Cracks,” Journal of Vibration and Acoustics, vol. 129, no. 3, pp. 341–354, 2007.
 M. K?sa and M.A. Gürel, “Free vibration analysis of uniform and stepped cracked beams with circular cross sections,” International Journal of Engineering Science, vol. 45, no. 2-8, pp. 364–380, 2007.
 K. Aydin, “Vibratory characteristics of Euler-Bernoulli beams with an arbitrary number of cracks subjected to axial load.” Journal of Vibration and Control, vol. 14, no. 4, pp. 485-510, 2008.
 S. Caddemi and I. Caliò, “Exact closed-form solution for the vibration modes of the–Bernoulli beam with multiple open cracks,” Journal of Sound and Vibration, vol. 327, no. 3-5, pp. 473-489, 2009.
 M. Shafiei and N. Khaji, “Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load,” Acta Mechanica, vol. 221, no. 1-2, pp. 79–97, 2011.
 C. S. Wang and L. T. Lee, “Modified and Simplified Sectional Flexibility of a Cracked Beam,” Journal of Applied Mathematics, Hindawi Publishing Corporation, Vol. 2012, doi:10.1155/2012/543828, 2012.
 R. Sridhar, A. Chakraborty, and S. Gopalakrishnan, “Wave PropagationAnalysis in Anisotropic and Inhomogeneous Uncracked and Cracked Structures using Pseudospectral Finite Element Method,” International Journal of Solids and Structures, vol. 43, no. 16, pp. 4997–5031, 2006.
 V. Briman and L.W. Byrd, “Vibration of Damaged Cantilevered Beams Manufactured from Functionally Graded Materials,” AIAA Journal., vol. 45, no. 11, pp. 2747–2757, 2007.
 J. Yang, Y. Chen, Y. Xiang and X.L. Jia, “Free and Forced Vibration of Cracked Beams under an Axial Force and a Moving Load,” Journal of Sound and Vibration, vol. 312, no. 1-2, pp. 166–181, 2008.
 J. Yang and Y. Chen, “Free Vibration and Buckling Analyses of Functionally Graded Beams with Edge Cracks,” Composite Structures, vol. 83, no. 1, pp. 48–60, 2008.
 L. L. Ke, J. Yang, S. Kitipornchai and Yang Xiang, “Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials,” Mechanics of Advanced Materials and Structures, vol. 16, no. 6, pp. 488–502, 2009.
 Z. Yu and F. Chu, “Identification of crack in functionally graded material beams using the p- version of finite element method,”Journal of Sound and Vibration, vol. 325, no. 1-, pp. 69–84, 2009.
 L. L. Ke, J. Yang and S. Kitipornchai, “Postbuckling analysis of edge cracked graded Timoshenko beams under end shortening,” Composite Structures, vol. 90, no. 2, pp. 152–160, 2009.
 M.S. Matbuly, O. Ragb and M. Nassar, “Natural frequencies of a functionally graded cracked beam using the differential quadrature method.” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2307–16, 2009.
 H.Z. Ferezqi, M. Tahani and H.E. Toussi, “Analytical approach to free vibrations of cracked Timoshenko beams made of functionally graded materials.” Mechanics of Advanced Materials and Structures, vol. 17, no. 5, pp.353–65, 2010.
 T. Yan , S. Kitipornchai, J. Yang and X. Q. He, “Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load,” Composite Structures, vol. 93, no. 11, pp. 2992–3001, 2011.
 ?.D. Akba?, “Static analysis of a functionally graded beam with edge cracks on elastic foundation,” Proceedings of the 9 th International Fracture Conference, pp. 70-80 , ?stanbul, Turkey, 2011.
 T. Yan, J. Yang and S. Kitipornchai, “Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation,” Nonlinear Dynamics, vol. 67, no. 1, pp. 527–540, 2012.
 D. Wei, Y. Liu and Z. Xiang, “An analytical method for free vibration analysis of functionally graded beams with edge cracks”, Journal of Sound and Vibration, vol. 331, no. 7, pp. 1686–1700, 2012.
 ?.D. Akba?, “Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material,” Mathematical Problems in Engineering, Hindawi Publishing Corporation, Vol. 2013, doi: 10.1155/2013/871815, 2012.
 D. Broek, Elementary engineering fracture mechanics, Martinus Nijhoff Publishers, Dordrecht, 1986.
 F. Erdogan and B.H. Wu, “The Surface Crack Problem for a Plate with Functionally Graded Properties,” Journal of Applied Mechanics, vol. 64, no. 3, pp. 448–456, 1997.
Open edge crack, Free vibration, Functionally graded materials, Finite element analysis