Free Vibration Analysis of Functionally Graded Beams with Cracks

Document Type: Research Paper


1 Department of Production and Automation, Faculty of Mechanical Engineering, University of Prishtina “Hasan Prishtina”, 10000 Prishtina, Kosovo

2 Department of Mechanical Engineering, Faculty of Engineering, Nigde Omer Halisdemir University, 51245 Nigde, Turkey


This study introduces the free vibration analysis of multilayered symmetric sandwich Timoshenko beams, made of functionally graded materials with two edge cracked, using the finite element method and linear elastic fracture mechanic theory. The FG beam consists of 50 layers, located symmetrically to the neutral plane, whose material properties distribution change along the beam thickness, according to power and exponential laws. The constituent of each layer of the beam is different, but each layer is isotropic and homogeneous. Natural frequency values of a cantilever beam are calculated using a developed MATLAB code. There is good agreement between the present results and the published results from the literature. A detailed study is carried out to observe the effect of crack location, crack depth ratio, power law index and material distribution on the first four natural frequencies.


[1]    E. Demir, H. Çallioǧlu, and M. Sayer, Free vibration of symmetric FG sandwich Timoshenko beam with simply supported edges, Indian J. Eng. Mater. Sci., 20(6), 2013, 515–521.

[2]    J. Liu, Y. M. Shao, and W. D. Zhu, Free vibration analysis of a cantilever beam with a slant edge crack, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., 231(5), 2017, 823–843.

[3]    M. Attar, A. Karrech, and K. Regenauer-Lieb, Dynamic response of cracked Timoshenko beams on elastic foundations under moving harmonic loads, J. Vib. Control, 23(3), 2017, 432–457.

[4]    J. A. Loya, L. Rubio, and J. Fernández-Sáez, Natural frequencies for bending vibrations of Timoshenko cracked beams, J. Sound Vib., 290(3–5), 2006, 640–653.

[5]    D. Y. Zheng and N. J. Kessissoglou, Free vibration analysis of a cracked beam by finite element method, J. Sound Vib., 273(3), 2004, 457–475.

[6]    M. Kisa and J. Brandon, Free vibration analysis of multiple open-edge cracked beams by component mode synthesis, J. Sound Vib., 238(1), 2000, 1–18.

[7]    M. Kisa, J. Brandon, and M. Topcu, Free vibration analysis of cracked beams by a combination of finite elements and component mode synthesis methods, Comput. Struct., 67(4), 1998, 215–223.

[8]    M. H. H. Shen and C. Pierre, Free Vibrations of Beams With a Single-Edge Crack, J. Sound Vib., 170(2), 1994, 237–259, 1994.

[9]    J. Zeng, H. Ma, W. Zhang, and B. Wen, Dynamic characteristic analysis of cracked cantilever beams under different crack types, Eng. Fail. Anal., 74, 2017, 80–94.

[10]  M. Aydogdu and V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges, Mater. Des., 28(5), 2007, 1651–1656.

[11]  J. Yang and Y. Chen, Free vibration and buckling analyses of functionally graded beams with edge cracks, Compos. Struct., 83(1), 2008, 48–60.

[12]  A. E. Alshorbagy, M. A. Eltaher, and F. F. Mahmoud, Free vibration characteristics of a functionally graded beam by finite element method, Appl. Math. Model., 35(1), 2011, 412–425.

[13]  M. Şimşek, T. Kocatürk, and Ş. D. Akbaş, Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load, Compos. Struct., 94(8), 2012, 2358–2364.

[14]  Y. Yang, C. C. Lam, K. P. Kou, and V. P. Iu, Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method, Compos. Struct., 117(1), 2014, 32–39.

[15]  H. Su and J. R. Banerjee, Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams, Comput. Struct., 147, 2015, 107–116.

[16]  Y. Cunedioglu, Free vibration analysis of edge cracked symmetric functionally graded sandwich beams, Struct. Eng. Mech., 56(6), 2015, 1003–1020.

[17]  Y. Yilmaz and S. Evran, Free vibration analysis of axially layered functionally graded short beams using experimental and finite element methods, Sci. Eng. Compos. Mater., 23(4), 2016, 453–460.

[18]  W. R. Chen and H. Chang, Closed-Form Solutions for Free Vibration Frequencies of Functionally Graded Euler-Bernoulli Beams, Mech. Compos. Mater., 53(1), 2017, 79–98.

[19]  Y. F. Xing, Z. K. Wang, and T. F. Xu, Closed-form Analytical Solutions for Free Vibration of Rectangular Functionally Graded Thin Plates in Thermal Environment, Int. J. Appl. Mech., 10(3), 2018, 1850025.

[20]  D. Song, J. Shi, and Z. Liu, Vibration analysis of functionally graded plate with a moving mass, Appl. Math. Model.,  46, 2017, 141–160.

[21]  T. Van Lien, N. T. Duc, and N. T. Khiem, Free vibration analysis of multiple cracked functionally graded Timoshenko beams, Lat. Am. J. Solids Struct., 14(9), 2017, 1752–1766.

[22]  E. Demir, H. Çallioǧlu, and M. Sayer, Free vibration of symmetric FG sandwich Timoshenko beam with simply supported edges, Indian J. Eng. Mater. Sci., 20(6), 2013, 515–521.

[23]  R. F. Gibson, Principles of Composite Materials. New York: McGraw-Hill, 1994.

[24]  D. L. Logan, A first course in the finite element method, 4th ed. Toronto: Thomson, 2007.

[25]  Y. Cunedioglu and B. Beylergil, Free vibration analysis of laminated composite beam under room and high temperatures, Struct. Eng. Mech., 51(1), 2014, 111–130.

[26]  Ş. D. Akbaş, Free vibration characteristics of edge cracked functionally graded beams by using finite element method, Int. J. Eng. Trends Technol., 4(10), 2013, 4590–4597.

[27]  E. I. Shifrin and R. Ruotolo, Natural frequencies of a beam with an arbitrary number of cracks, J. Sound Vib., 222(3), 1999, 409–423.