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http://dx.doi.org/10.12989/cac.2020.26.3.285

Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study  

AlSaid-Alwan, Hiyam Hazim Saeed (Graduate School of Natural and Applied Sciences, Suleyman Demirel University)
Avcar, Mehmet (Department of Civil Engineering, Suleyman Demirel University)
Publication Information
Computers and Concrete / v.26, no.3, 2020 , pp. 285-292 More about this Journal
Abstract
In engineering structures, to having the projected structure to serve all the engineering purposes, the theory to be used during the modeling stage is also of great importance. In the present work, an analytical solution of the free vibration of the beam composed of functionally graded materials (FGMs) is presented utilizing different beam theories. The comparison of supposed beam theory for free vibration of functionally graded (FG) beam is examined. For this aim, Euler-Bernoulli, Rayleigh, Shear, and Timoshenko beam theories are employed. The functionally graded material properties are assumed to vary continuously through the thickness direction of the beam with respect to the volume fraction of constituents. The governing equations of free vibration of FG beams are derived in the frameworks of four beam theories. Resulting equations are solved versus simply supported boundary conditions, analytically. To verify the results, comparisons are carried out with the available results. Parametrical studies are performed for discussing the effects of supposed beam theory, the variation of beam characteristics, and FGM properties on the free vibration of beams. In conclusion, it is found that the interaction between FGM properties and the supposed beam theory is of significance in terms of free vibration of the beams and that different beam theories need to be used depending on the characteristics of the beam in question.
Keywords
FGMs; free vibration; analytical solution; beam theories; comparative study;
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Times Cited By KSCI : 34  (Citation Analysis)
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1 Akgoz, B. and Civalek, O. (2017), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. B Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024.   DOI
2 Al Rjoub, Y.S. and Hamad, A.G. (2017), "Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method", KSCE J. Civil Eng., 21, 792-806. https://doi.org/10.1007/s12205-016-0149-6.   DOI
3 Alsaid-Alwan, H.H.S. (2017), "Free vibration analysis of functionally graded beam with different engineering theories", Master of Science Thesis, Suleyman Demirel University, Graduate School of Natural and Applied Sciences, Department of Civil Engineering, Isparta.
4 Anil, K.L., Panda, S.K., Sharma, N., Hirwani, C.K. and Topal, U. (2020), "Optimal fiber volume fraction prediction of layered composite using frequency constraints-A hybrid FEM approach", Comput. Concrete, 25(4), 303-310. https://doi.org/10.12989/cac.2020.25.4.303.   DOI
5 Avcar M. (2015), "Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam", Struct. Eng. Mech., 55, 871-884. https://doi.org/10.12989/sem.2015.55.4.871.   DOI
6 Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603.   DOI
7 Avcar, M. and Alsaid-Alwan, H. (2017a), "Free vibration of functionally graded Rayleigh beam", Int. J. Eng. Appl. Sci., 9, 127-137. http://dx.doi.org/10.24107/ijeas.322884.
8 Avcar, M. and Alsaid-Alwan, H. (2017b), "Free vibration analysis of functionally graded beams using different engineering theories", 4th International Conference on Computational and Experimental Science and Engineering (ICCESEN 2017), Antalya, Turkey.
9 Avcar, M. and Mohammed, W.K.M. (2018), "Free vibration of functionally graded beams resting on Winkler-Pasternak foundation", Arab. J. Geosci., 11(10), 232. https://doi.org/10.1007/s12517-018-3579-2.   DOI
10 Ayache, B., Bennai, R., Fahsi, B., Fourn, H., Atmane, H.A. and Tounsi, A. (2018), "Analysis of wave propagation and free vibration of functionally graded porous material beam with a novel four variable refined theory", Earthq. Struct., 15(4), 369-382. https://doi.org/10.12989/eas.2018.15.4.369.   DOI
11 Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 1, 409-423. https://doi.org/10.12989/scs.2015.18.2.409.
12 Aydogdu, M., Arda, M. and Filiz, S. (2018), "Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter", Adv. Nano Res., Int. J., 6(3), 257-278. http://dx.doi.org/10.12989/anr.2018.6.3.257.
13 Balubaid, M., Tounsi, A., Dakhel, B. and Mahmoud, S.R. (2019), "Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory", Comput. Concrete, 24(6), 579-586. http://dx.doi.org/10.12989/cac.2019.24.6.579.   DOI
14 Birman, V. and Byrd, L.W. (2007), "Modeling and analysis of functionally graded materials and structure", ASME Appl. Mech. Rev., 60(5), 195-216. https://doi.org/10.1115/1.2777164.   DOI
15 Chaabane, L.A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F.Z., Tounsi, A., Derras A., Bousahla A.A. and Tounsi, A. (2019), "Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation", Struct. Eng. Mech., 71(2), 185-196. http://dx.doi.org/10.12989/sem.2019.71.2.185.   DOI
16 Chakraverty, S. and Pradhan, K.K. (2016), Vibration Of Functionally Graded Beams and Plates, Academic Press.
17 Civalek, O. and Ozturk, B. (2010), "Free vibration analysis of tapered beam-column with pinned ends embedded in Winkler-Pasternak elastic foundation", Geomech. Eng., 2(1), 45-56. http://dx.doi.org/10.12989/gae.2010.2.1.045.   DOI
18 Dewangan, H.C., Panda, S.K. and Sharma, N. (2020a), "Experimental validation of role of cut-out parameters on modal responses of laminated composite-a coupled FE approach", Int. J. Appl. Mech., 2050068. https://doi.org/10.1142/S1758825120500684.
19 Ebrahimi, F., Barati, M.R. and Civalek, O (2020), "Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures", Eng. Comput., 36, 953-964. https://doi.org/10.1007/s00366-019-00742-z.   DOI
20 Dewangan, H.C., Sharma, N., Hirwani, C.K. and Panda, S.K. (2020b), "Numerical eigenfrequency and experimental verification of variable cutout (square/rectangular) borne layered glass/epoxy flat/curved panel structure", Mech. Bas. Des. Struct. Mach., 1-18. https://doi.org/10.1080/15397734.2020.1759432.
21 Hadji, L., Daouadji, T.H. and Bedia, E.A. (2015), "A refined exponential shear deformation theory for free vibration of FGM beam with porosities", Geomech. Eng.., 9(3), 361-372. https://doi.org/10.12989/gae.2015.9.3.361.   DOI
22 Hadji, L., Daouadji, T.H., Tounsi, A. and Bedia, E.A. (2014), "A higher order shear deformation theory for static and free vibration of FGM beam", Steel Compos. Struct., 16(5), 507-519. http://dx.doi.org/10.12989/scs.2014.16.5.507.   DOI
23 Han, M.S., Benaroya, H. and Wei, T. (1999), "Dynamics of transversely vibrating beams using four engineering theories", J. Sound Vib., 225, 935-988. https://doi.org/10.1006/jsvi.1999.2257.   DOI
24 Hirwani, C.K. and Panda, S.K. (2019), "Nonlinear thermal free vibration frequency analysis of delaminated shell panel using FEM", Compos. Struct., 224, 111011. https://doi.org/10.1016/j.compstruct.2019.111011.   DOI
25 Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A., Bedia, E.A. and Al-Osta, M.A. (2020), "A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: Bending and free vibration analysis", Comput. Concrete, 25(1), 37-57. http://dx.doi.org/10.12989/cac.2020.25.1.037.   DOI
26 Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B Eng., 28, 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9.   DOI
27 Kahya, V. and Turan, M. (2018), "Vibration and buckling of laminated beams by a multi-layer finite element model", Steel Compos. Struct., 28(4), 415-426. http://dx.doi.org/10.12989/scs.2018.28.4.415.   DOI
28 Kieback, B., Neubrand, A. and Riedel, H. (2003), "Processing techniques for functionally graded materials", Mater. Sci. Eng. A, 362, 81-106. https://doi.org/10.1016/S0921-5093(03)00578-1.   DOI
29 Koizumi, M. (1993), "The concept of FGM", Ceram. Tran. Funct. Grad. Mater., 34, 3-10. https://doi.org/10.1080/10426919508935030.
30 Li, S., Wan, Z. and Zhang, J. (2014), "Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories", Appl. Math. Mech., 35, 591-606. https://doi.org/10.1007/s10483-014-1815-6.   DOI
31 Mahamood, R.M., Akinlabi, E.T., Shukla, M. and Pityana, S. (2012), "Functionally graded material: an overview", Proceedings of the World Congress on Engineering, Vol III, WCE 2012, London, UK.
32 Mehar, K., Mishra, P.K. and Panda, S.K. (2020), "Numerical investigation of thermal frequency responses of graded hybrid smart nanocomposite (CNT-SMA-Epoxy) structure", Mech. Adv. Mater. Struct., 1-13. https://doi.org/10.1080/15376494.2020.1725193.
33 Nejadi, M.M. and Mohammadimehr, M. (2020), "Analysis of a functionally graded nanocomposite sandwich beam considering porosity distribution on variable elastic foundation using DQM: Buckling and vibration behaviors", Comput. Concrete, 25(3), 215-224. https://doi.org/10.12989/cac.2020.25.3.215.   DOI
34 Nguyen, T.K., Vo, T.P. and Thai, H.T. (2013), "Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory", Compos. B Eng., 55, 147-157. https://doi.org/10.1016/j.compositesb.2013.06.011.   DOI
35 Ramteke, P.M., Panda, S.K. and Sharma, N. (2019), "Effect of grading pattern and porosity on the eigen characteristics of porous functionally graded structure", Steel Compos. Struct., 33(6), 865-875 https://doi.org/10.12989/scs.2019.33.6.865.   DOI
36 Pandey, H.K., Hirwani, C.K., Sharma, N., Katariya, P.V., Dewangan, H.C. and Panda, S.K. (2019), "Effect of nano glass cenosphere filler on hybrid composite eigenfrequency responses-An FEM approach and experimental verification", Adv. Nano Res., 7(6), 419-429. https://doi.org/10.12989/anr.2019.7.6.419.   DOI
37 Patle, B.K., Hirwani, C.K., Singh, R.P. and Panda, S.K. (2018), "Eigenfrequency and deflection analysis of layered structure using uncertain elastic properties-a fuzzy finite element approach", Int. J. Approx. Reason., 98, 163-176. https://doi.org/10.1016/j.ijar.2018.04.013.   DOI
38 Rahmani, M.C., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Bedia, E.A., Mahmoud, S.R., Benrahou, K.H. and Tounsi, A. (2020), "Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory", Comput. Concrete, 25(3), 225-244. http://dx.doi.org/10.12989/cac.2020.25.3.225.   DOI
39 Rao, S.S. (2007), Vibration of Continuous Systems, Wiley, New York, USA.
40 Sahoo, S.S., Panda, S.K., Mahapatra, T.R. and Hirwani, C.K. (2019), "Numerical analysis of transient responses of delaminated layered structure using different mid-plane theories and experimental validation", Iran J. Sci. Technol. Tran. Mech. Eng., 43, 41-56. https://doi.org/10.1007/s40997-017-0111-3.   DOI
41 Sahouane, A., Hadji, L. and Bourada, M. (2019), "Numerical analysis for free vibration of functionally graded beams using an original HSDBT", Earthq. Struct., 17(1), 31-37. https://doi.org/10.12989/eas.2019.17.1.031.   DOI
42 Thai, H.T. and Vo, T.P. (2012), "Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories", Int. J. Mech. Sci., 62(1), 57-66. https://doi.org/10.1016/j.ijmecsci.2012.05.0143   DOI
43 Sahu, P., Sharma, N. and Panda, S.K. (2020), "Numerical prediction and experimental validation of free vibration responses of hybrid composite (Glass/Carbon/Kevlar) curved panel structure", Compos. Struct., 241, 112073. https://doi.org/10.1016/j.compstruct.2020.112073.   DOI
44 Shokravi, M. (2017), "Vibration analysis of silica nanoparticles-reinforced concrete beams considering agglomeration effects", Comput. Concrete, 19(3), 333-338. http://dx.doi.org/10.12989/cac.2017.19.3.333.   DOI
45 Suresh, S. and Mortensen, A. (1998), Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behavior of Graded Metals and Metal-Ceramic Composites, IOM Communications, London, UK.
46 Timoshenko, S.P. (1937), Vibration Problems in Engineering, D. Van Nostrand, Princeton, NJ, USA.
47 Wang, J.R., Liu, T.L. and Chen, D.W. (2007), "Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems with the effects of shear deformation and rotary inertia", Struct Eng. Mech., 26(1), 1-14. http://dx.doi.org/10.12989/sem.2007.26.1.001.   DOI
48 Yildirim, V. and Kiral, E. (2000), "Investigation of the rotary inertia and shear deformation effects on the out-of-plane bending and torsional natural frequencies of laminated beams", Compos. Struct., 49(3), 313-320. https://doi.org/10.1016/S0263-8223(00)00063-5.   DOI
49 Wang, X. and Li, S. (2016), "Free vibration analysis of functionally graded material beams based on Levinson beam theory", Appl. Math. Mech., 37, 861-878. https://doi.org/10.1007/s10483-016-2094-9.   DOI
50 Wattanasakulpong, N. and Ungbhakorn, V. (2012), "Free vibration analysis of functionally graded beams with general elastically end constraints by DTM", World J. Mech., 2, 297-310. https://doi.org/10.4236/wjm.2012.26036.   DOI
51 Civalek, O. and Kiracioglu, O. (2010), "Free vibration analysis of Timoshenko beams by DSC method", Int. J. Numer. Meth. Bio., 26(12), 1890-1898. https://doi.org/10.1002/cnm.1279.   DOI
52 Lee, J.W. and Lee, J.Y. (2019), "An exact transfer matrix method for coupled bending and bending vibrations of a twisted Timoshenko beam", Struct. Eng. Mech., 72(6), 797-807. http://dx.doi.org/10.12989/sem.2019.72.6.797.   DOI