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http://dx.doi.org/10.12989/scs.2018.29.3.363

Free vibration of tapered BFGM beams using an efficient shear deformable finite element model  

Nguyen, Dinh Kien (Institute of Mechanics, VAST)
Tran, Thi Thom (Institute of Mechanics, VAST)
Publication Information
Steel and Composite Structures / v.29, no.3, 2018 , pp. 363-377 More about this Journal
Abstract
An efficient and free of shear locking finite element model is developed and employed to study free vibration of tapered bidirectional functionally graded material (BFGM) beams. The beam material is assumed to be formed from four distinct constituent materials whose volume fraction continuously varies along the longitudinal and thickness directions by power-law functions. The finite element formulation based on the first-order shear deformation theory is derived by using hierarchical functions to interpolate the displacement field. In order to improve efficiency and accuracy of the formulation, the shear strain is constrained to constant and the exact variation of the cross-sectional profile is employed to compute the element stiffness and mass matrices. A comprehensive parametric study is carried out to highlight the influence of the material distribution, the taper and aspect ratios as well as the boundary conditions on the vibration characteristics. Numerical investigation reveals that the proposed model is efficient, and it is capable to evaluate the natural frequencies of BFGM beams by using a small number of the elements. It is also shown that the effect of the taper ratio on the fundamental frequency of the BFGM beams is significantly influenced by the boundary conditions. The present results are of benefit to optimum design of tapered FGM beam structures.
Keywords
bi-directional functionally graded material; tapered beams; first-order shear deformation theory; hierarchical functions; free vibration; finite element model;
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1 Cook, R.D., Malkus, D.S., Plesha, M.E. and Witt, R.J. (2002), Concepts and Applications of Finite Element Analysis, (4th Ed.), John Wiley & Sons, New York, NY, USA.
2 Frikha, A., Hajlaoui, A., Wali, M. and Dammak, F. (2016), "A new higher order $C^0$ mixed beam element for FGM beams analysis", Compos. Part B Eng., 106, 181-189.   DOI
3 Gan, B.S., Trinh, T.H., Le, T.H. and Nguyen, D.K. (2015), "Dynamic response of non-uniform Timoshenko beams made of axially FGM subjected to multiple moving point loads", Struct. Eng. Mech., Int. J., 53(5), 981-995.   DOI
4 Ghazaryan, D., Burlayenko, V.N., Avetisyan, A. and Bhaskar, A. (2017), "Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method", J. Eng. Math., DOI: 10.1007/s10665-017-9937-3
5 Hao, D. and Wei, C. (2016), "Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams", Compos. Struct., 141, 253-263.   DOI
6 Hein, H. and Feklistova, L. (2011), "Free vibrations of nonuniform and axially functionally graded beams using Haar wavelets", Eng. Struct., 33(12), 3696-3701.   DOI
7 Huang, Y. and Li, X.-F. (2010), "A new approach for free vibration of axially functionally graded beams with non-uniform crosssection", J. Sound Vib., 329(11), 2291-2303.   DOI
8 Huang, Y. and Li, X.-F. (2011), "Buckling analysis of nonuniform and axially graded columns with varying flexural rigidity", ASCE J. Eng. Mech., 137(1), 73-81.   DOI
9 Huang, Y., Yang, L.-E. and Luo, Q.-Z. (2013), "Free vibration of axially functionally graded Timoshenko beams with nonuniform cross-section", Compos. Part B Eng., 45(1), 1493-1498.   DOI
10 Trinh, L.C., Vo, P.T., Thai, H.T. and Nguyen, T.K. (2016), "An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads", Compos. Part B Eng., 100, 152-163.   DOI
11 Trinh, L.C., Vo, T.P., Thai, H-T. and Nguyen, T-K. (2018), "Sizedependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions", Compos. Part B Eng., 134, 225-245.   DOI
12 Wang, Z., Wang, X., Xu, G., Cheng, S. and Zeng, T. (2016), "Free vibration of two-directional functionally graded beams", Compos. Struct., 135, 191-198.   DOI
13 Wattanasakulpong, N., Prusty, B.G. and Kelly, D.W. (2011), "Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams", Int. J. Mech. Sci., 53(9), 734-743.   DOI
14 Zhao, Y., Huang, Y. and Guo, M. (2017), "A novel approach for free vibration of axially functionally graded beams with nonuniform cross-section based on Chebyshev polynomials theory", Compos. Struct., 168, 277-284.   DOI
15 Zienkiewicz, O.C. and Taylor, R.L. (1997), The Finite Element Method, Vol. 1: Basic Formulation an Linear Problems, (4th Ed.), Mc. Graw-Hill Book Company, London, UK.
16 Karamanli, A. (2017), "Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory", Compos. Struct., 174, 70-86.   DOI
17 Huynh, T.A., Lieu, X.Q. and Lee, J. (2017), "NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem", Compos. Struct., 160, 1178-1190.   DOI
18 Kadoli, R., Akhtar, K. and Ganesan, N. (2008), "Static analysis of functionally graded beams using higher order shear deformation theory", Appl. Math. Model., 32(12), 2509-2525.   DOI
19 Kahya, V. and Turan, M. (2017), "Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory", Compos. Part B Eng., 109, 108-115.   DOI
20 Kosmatka, J.B. (1995), "An improved two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams", Comput. Struct., 57(1), 141-149.   DOI
21 Lezgy-Nazargah, M. (2015), "Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach", Aerosp. Sci. Technol., 45, 154-164.   DOI
22 Li, X.-F. (2008), "A unified approach for analyzing static and dynamic behaviours of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound Vib., 318(4-5), 1210-1229.   DOI
23 Calim, F.F. (2016), "Transient analysis of axially functionally graded Timoshenko beams with variable cross-section", Compos. Part B, 98, 472-483.   DOI
24 Li, X.-F., Kang, Y.-A. and Wu, J.-X. (2013), "Exact frequency equations of free vibration of exponentially functionally graded beams", App. Acoust., 74(3), 413-420.   DOI
25 Li, L. and Zhang, D. (2015), "Dynamic analysis of rotating axially FG tapered beams based on a new rigid-flexible coupled dynamic model using the B-spline method", Compos. Struct., 124, 357-367.   DOI
26 Akgoz, B. and Civalek, O. (2013), "Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory", Compos. Struct., 98, 314-322.   DOI
27 Bambill, D.V., Rossit, C.A. and Felix, D.H. (2015), "Free vibrations of stepped axially functionally graded Timoshenko beams", Meccanica, 50(4), 1073-1087.   DOI
28 Birman, V. and Byrd, L.W. (2007), "Modeling and Analysis of functionally graded materials and structures", Appl. Mech. Rev., 60(5), 195-216.   DOI
29 Chakraborty, A., Gopalakrishnan, S. and Reddy, J.N. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539.   DOI
30 Lu, C.F., Chen, W.Q., Xu, R.Q. and Lim, C.W. (2008), "Semianalytical elasticity solutions for bi-directional functionally graded beams", Int. J. Solids Struct., 45, 258-275.   DOI
31 Mahi, A., Adda Bedia, E.A., Tounsi, A. and Mechab, I. (2010), "An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions", Compos. Struct., 92(8), 1877-1887.   DOI
32 Nguyen, D.K. (2013), "Large displacement response of tapered cantilever beams made of axially functionally graded material", Compos. Part B Eng., 55, 298-305.   DOI
33 Nguyen, D.K. and Gan, B.S. (2014), "Large deflections of tapered functionally graded beams subjected to end forces", Appl. Math. Model., 38(11-12), 3054-3066.   DOI
34 Nguyen, D.K. and Bui, V.T. (2017), "Dynamic analysis of functionally graded Timoshenko beams in thermal environment using a higher-order hierarchical beam element", Math. Prob. Eng. DOI: https://doi.org/10.1155/2017/7025750
35 Nguyen, D.K., Nguyen, Q.H., Tran, T.T. and Bui, V.T. (2017), "Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load", Acta Mech., 228, 141-155.   DOI
36 Nemat-Alla, M. and Noda, N. (2000), "Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading", Acta Mech., 144(3-4), 211-229.   DOI
37 Niknam, H., Fallah, A. and Aghdam, M.M. (2014), "Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading", Int. J. Non-Linear Mech., 65, 141-147.   DOI
38 Pydah, A. and Sabale, A. (2017), "Static analysis of bi-directional functionally graded curved beams", Compos. Struct., 160, 867-876.   DOI
39 Rajasekaran, S. and Tochaei, E.N. (2014), "Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order", Meccanica, 49(4), 995-1009.   DOI
40 Rajasekaran, S. (2013), "Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach", Meccanica, 48(5), 1053-1070.   DOI
41 Shahba, A. and Rajasekaran, S. (2012), "Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials", App. Math. Model., 36(7), 3094-3111.   DOI
42 Shahba, A., Attarnejad, R., Marvi, M.T. and Hajilar, S. (2011), "Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and nonclassical boundary conditions", Compos. Part B Eng., 42(4), 801-808.
43 Shafiei, N. and Kazemi, M. (2017), "Buckling analysis on the bidimensional functionally graded porous tapered nano-/microscale beams", Aerosp. Sci. Technol., 66, 1-11.   DOI
44 Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S. and Kazemi, M. (2017), "Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams", Comput. Method Appl. Mech. Eng., 322, 615-632.   DOI
45 Simsek, M. (2015), "Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions", Compos. Struct., 133, 968-997.   DOI
46 Tang, A.-Y., Wu, J.-X., Li, X.-F. and Lee, K.Y. (2014), "Exact frequency equations of free vibration of exponentially nonuniform functionally graded Timoshenko beams", Int. J. Mech. Sci., 89, 1-11.   DOI