Frequency Analysis of Cracked Porous Functionally Graded Beams on Elastic Foundation using Reddy Third Order Shear Deformation Theory

Document Type : Original Article

Authors

• Mohammad Amin Forghani
• yousef bazarganlari
• Parham Zahedinejhad
• Mohammad Javad Kazemzadeh Parsi

Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran

10.22067/jacsm.2021.68770.1007

Abstract

In this paper, Frequency analysis of cracked porous functionally graded beams which are resting on elastic foundation based on reddy third order shear deformation theory is studied. Mechanical property gradients defined in accordance with model of exponential law. Governing equations obtained with the aid of third order shear deformation theory and by considering elastic foundation effects within hamilton’s principle. Due to complicating closed form solution of equations, differential equations solved utilizing Generalized Differential Quadrature Method by considering various end conditions. In order to validate the results, comparisons are made with solutions which are available for other papers. This study shows that the difference between the results of this paper and the results of others is negligible. Finally the effects of geometrical parameters, exponential law indexes, crack location and depth, elastic foundation and porosity on natural frequencies of functionally graded beams are studied. The results of this paper and effects of these parameters can be used in the optimum design of functional graded beams and crack prediction, detection and monitoring techniques.

Keywords

• Frequency Analysis
• Exponential FGM
• Edge Crack
• Porous
• Winkler Pasternak
• Third Order Reddy Theory

Main Subjects

• Solid mechanics

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References

1. Yang, J., Chen, Y., Xiang, Y. and Jia, X.L., "Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load". Journal of Sound and Vibration , Vol. 312, pp. 166-181, (2008).
2. Yang, E.C., Zhao, X. and Li, Y.H., "Free vibration analysis for cracked FGM beams by means of a continuous beam model", Shock and Vibration, pp. 1-13, )2015(.
3. Wei, D., Liu, Y. and Xiang, Z., "An analytical method for free vibration analysis of functionally graded beams with edge cracks", Journal of Sound and Vibration, Vol. 331, pp. 1686-1700, (2012).
4. Erdogan, F., and B. H. Wu., "The surface crack problem for a plate with functionally graded properties.", pp. 449-456, (1997).
5. Shifrin, E.I. and Ruotolo, R., "Natural frequencies of a beam with an arbitrary number of cracks", Journal of Sound and vibration, Vol. 222, pp. 409-423, (1999).
6. Swamidas, A.S.J., Yang, X. and Seshadri, R., "Identification of cracking in beam structures using Timoshenko and Euler formulations", Journal of Engineering Mechanics, Vol.130, pp. 1297-1308, (2004).
7. Binici, B., "Vibration of beams with multiple open cracks subjected to axial force", Journal of Sound and Vibration, Vol. 287, pp. 277-295, (2005).
8. Lin, H.P. and Chang, S.C., "Forced responses of cracked cantilever beams subjected to a concentrated moving load", International journal of mechanical sciences, Vol. 48, pp. 1456-1463, (2006).
9. Hsu, M.H., "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, pp. 1-17, (2005).
10. Yang, J., Chen, Y., Xiang, Y. and Jia, X.L., "Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load", Journal of Sound and Vibration, Vol. 312, pp. 166-181, (2008).
11. Yang, J. and Chen, Y., "Free vibration and buckling analyses of functionally graded beams with edge cracks", Composite Structures, Vol. 83, pp. 48-60, (2008).
12. Matbuly, M.S., Ragb, O. and Nassar, M., "Natural frequencies of a functionally graded cracked beam using the differential quadrature method", Applied mathematics and computation, Vol. 215, pp. 2307-2316, (2009).
13. Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y., "Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials", Mechanics of Advanced Materials and Structures,16, pp. 488-502, (2009).
14. Ferezqi, H.Z., Tahani, M. and Toussi, H.E., "Analytical approach to free vibrations of cracked Timoshenko beams made of functionally graded materials", Mechanics of Advanced Materials and Structures, Vol. 17, 353-365, (2010).
15. Aydin, K., "Free vibration of functionally graded beams with arbitrary number of surface cracks", European Journal of Mechanics-A/Solids, Vol. 42, pp. 112-124, (2013).
16. Sherafatnia, K., Farrahi, G.H. and Faghidian, S.A., "Analytic approach to free vibration and buckling analysis of functionally graded beams with edge cracks using four engineering beam theories", International Journal of Engineering, 27, pp. 979-990, (2014).
17. Van Lien, T., Duc, N.T. and Khiem, N.T., "A New Form of Frequency Equation for Functionally Graded Timoshenko Beams with Arbitrary Number of Open Transverse Cracks", Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, Vol. 43, pp. 235-250, (2019).
18. Shabani, S. and Cunedioglu, Y., "Free vibration analysis of functionally graded beams with cracks", Journal of Applied and Computational Mechanics, Vol. 6, pp. 908-919, (2020).
19. رضایی م. و همکاران، " ارائه تابع اغتشاش ترک جدید برای آنالیز ارتعاشات عرضی تیر ترک‌دار" ، نشریه علوم کاربردی و محاسباتی در مکانیک، د. 25، ش. 2، ص 32-19، (1393)
20. علی جانی ع. و خمامی ابدی م.، " استخراج روابط صریح در تعیین فرکانس طبیعی تیر اویلر- برنولی دارای ترک روی بستر الاستیک با استفاده از روش رایلی" ، نشریه مهندسی مکانیک امیر کبیر، د. 52، ش. 5، ص 100-91، (1399).
21. Zahedinejad, P., "Free vibration analysis of functionally graded beams resting on elastic foundation in thermal environment", International Journal of Structural Stability and Dynamics, Vol.16, pp. 1550029-51, (2016).
22. Eisenberger, M., "Vibration frequencies for beams on variable one-and two-parameter elastic foundations", Journal of Sound and Vibration, 176, pp. 577-84, (1994).
23. Chen WQ, Lü CF, Bian ZG., "A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation", Applied Mathematical Modelling, Vol. 28, pp. 877-90, (2004).
24. Akbaş, Ş.D., "Free Vibration Analysis of Edge Cracked Functionally Graded Beams Resting on Winkler-Pasternak Foundation", International Journal of Engineering & Applied Sciences, Vol. 7, pp. 1-15, (2015).
25. Baghlani, A., Khayat, M. and Dehghan, S.M., "Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction", Applied Mathematical Modelling, Vol. 78, pp.550-575, (2020)
26. Yahia, S.A., Atmane, H.A., Houari, M.S.A. and Tounsi, A., "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Structural Engineering and Mechanics, Vol. 53, pp. 1143-1165, (2015).
27. Ebrahimi, F. and Mokhtari, M., "Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method", Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 37, pp. 1435-1444, (2015).
28. Avcar, M., "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel and Composite Structures, Vol. 30, pp. 603-615, (2019).
29. Ramteke, P.M. and Panda, S.K., "Free Vibrational Behaviour of Multi-Directional Porous Functionally Graded Structures", Arabian Journal for Science and Engineering, pp. 1-16, (2021).
30. Şimşek, M., "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nuclear Engineering and Design, 240, pp.  697-705, (2010).
31. Pradhan, K.K., Chakraverty, S., "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method", Composites Part B Engineering, Vol. 51, pp. 175-184, (2013).
32. Bert, CW., Malik, M., "Differential quadrature method in computational mechanics. a review", Applied mechanics reviews, 49, pp. 1-28, (1996).
33. Zhu, L.F., Ke, L.L., Zhu, X.Q., Xiang, Y. and Wang, Y.S., "Crack identification of functionally graded beams using continuous wavelet transform", Composite Structures, Vol. 210, pp. 473-485, (2019).
34. Erdogan, F., and B. H. Wu. "The surface crack problem for a plate with functionally graded properties." 449-456, (1997).
35. Aydin, K., "Free vibration of functionally graded beams with arbitrary number of surface cracks", European Journal of Mechanics-A/Solids, Vol. 42, pp. 112-124, (2013).
36. Wattanasakulpong, N. and Chaikittiratana, A., "Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method", Meccanica, 50, pp. 1331-1342, (2015).
37. Chen, D., Yang, J. and Kitipornchai, S., "Free and forced vibrations of shear deformable functionally graded porous beams", International journal of mechanical sciences, Vol. 108, pp. 14-22, (2016).
38. Bellman, R. and Casti, J.," Differential quadrature and long-term integration", Journal of Mathematical Analysis and Applications, Vol. 34, pp. 235-238, (1971).
39. Khiem, N.T. and Lien, T.V., "A simplified method for natural frequency analysis of a multiple cracked beam", Journal of sound and vibration, Vol. 245, pp. 737-751, (2001).