Study of crack’s effect on the natural frequencies of bi-directional functionally graded beam

References

• Bardell, N. S. 1996. An engineering application of the h-p version of the finite element method to the static analysis of a Euler-bernoulli beam. Computers & Structures 59 (2):195–211. doi:10.1016/0045-7949(95)00252-9.
   Web of Science ®Google Scholar
• Bouzidi, I., A. Hadjoui, and A. Fellah. 2020. Dynamic analysis of functionally graded rotor-blade system using the classical version of the finite element method. Mechanics Based Design of Structures and Machines 49:1080–108. doi:10.1080/15397734.2019.1706558.
   Web of Science ®Google Scholar
• Chen, X. L, and K. M. Liew. 2004. Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads. Smart Materials and Structures 13 (6):1430–7. doi:10.1088/0964-1726/13/6/014.
   Web of Science ®Google Scholar
• Fellah, A., H. Abdelhamid, B. Brahim, and S. Ahmed. 2019. Study of the effect of an open transverse crack on the vibratory behavior of rotors using the h-p version of the finite element method. Journal of Solid Mechanics 11 (1):181–200. doi:10.22034/JSM.2019.664228.
   Google Scholar
• Gayen, D., D. Chakraborty, and Tiwari, R. 2017. Finite element analysis for a functionally graded rotating shaft with multiple breathing cracks. International Journal of Mechanical Sciences 134:411–23. doi:10.1016/j.ijmecsci.2017.10.027.
   Web of Science ®Google Scholar
• Karamanli, A. 2017. Static behaviour of two-directional functionally graded sandwich beams using various beam theories. New Trends in Mathematical Sciences 5:112–47. doi:10.20852/ntmsci.2017.161.
   Google Scholar
• Ke, L. L., J. Yang, S. Kitipornchai, and Y. Xiang. 2009. Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mechanics of Advanced Materials and Structures 16 (6):488–502. doi:10.1080/15376490902781175.
   Web of Science ®Google Scholar
• Li, L, and Y. Hu. 2017. Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory. Composite Structures 172:242–50. doi:10.1016/j.compstruct.2017.03.097.
   Web of Science ®Google Scholar
• Lien, T. V., N. T. Duc, and N. T. Khiem. 2017. Free Vibration Analysis of Multiple Cracked Functionally Graded Timoshenko Beams. Latin American Journal of Solids and Structures 14 (9):1752–66. doi:10.1590/1679-78253693.
   Web of Science ®Google Scholar
• Lü, C. F., W. Q. Chen, C. W. Xu, and Lim, R. Q. 2008. Semi-analytical elasticity solutions for bi-directional functionally graded beams. International Journal of Solids and Structures 45 (1):258–75. doi:10.1016/j.ijsolstr.2007.07.018.
   Web of Science ®Google Scholar
• Nejad, M. Z., A. Hadi, and A. Rastgoo. 2016. Buckling analysis of arbitrary twodirectional functionally graded Euler–Bernoulli nano beams based on nonlocal elasticity theory. International Journal of Engineering Science 103:1–10. doi:10.1016/j.ijengsci.2016.03.001.
   Web of Science ®Google Scholar
• Nguyen, D. H., Q. H. Nguyen, T. T. Tran, and V. T. Bui. 2017. Vibration of bidimensional functionally graded Timoshenko beams excited by a moving load. Acta Mechanica 228 (1):141–55. doi:10.1007/s00707-016-1705-3.
   Web of Science ®Google Scholar
• Panigrahi, B, and G. Pohit. 2018. Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique. Applied Mathematical Modelling 57:248–67. doi:10.1016/J.Apm.2018.01.022.
   Web of Science ®Google Scholar
• Piovan, M. T, and R. Sampaio. 2009. A study on the dynamics of rotating beams with functionally graded properties. Journal of Sound and Vibration 327 (1–2):134–43. doi:10.1016/j.jsv.2009.06.015.
   Web of Science ®Google Scholar
• Şimşek, M. 2015. Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Composite Structures 133:968–78. doi:10.1016/j.compstruct.2015.08.021.
   Web of Science ®Google Scholar
• Trinh, L. C., T. P. Vo, H. T. Thai, and T. K. Nguyen. 2018. Size-dependent vibration of bi-directional functionally graded micro beams with arbitrary boundary conditions. Composites Part B: Engineering 134:225–45. doi:10.1016/j.compositesb.2017.09.054.
   Web of Science ®Google Scholar
• Wang, Z. H., X. H. Wang, G. D. Xu, S. Cheng, and T. Zeng. 2016. Free vibration of two-directional functionally graded beams. Composite Structures 135:191–8. doi:10.1016/j.compstruct.2015.09.013.
   Web of Science ®Google Scholar
• Zhao, L., W. Q. Chen, and C. F. Lü. 2012. Symplectic elasticity for bi-directional functionally graded materials. Mechanics of Materials 54:32–42. doi:10.1016/j.mechmat.2012.06.001.
   Web of Science ®Google Scholar