A generalized finite element approach to the free vibration analysis of non-uniform axially functionally graded beam

Document Type : Article


Department of Mechanical Engineering, VSSUT Burla, Burla-768018, Sambalpur, Odisha, India


A generalized finite element approach, for the free vibration analysis of an axially functionally graded (AFG) beam, having non-uniform thickness, has been presented in the current analysis. The use of non-uniform beam element and the way of assembling the same, make the finite element model, a generalized one. The current approach can be used for beams of both uniform and non-uniform thickness, with any of the homogenous and inhomogeneous material variation. The governing equation for free vibration of beam has been derived considering Euler-Bernoulli beam theory and by using Euler-Lagrange's equation. The cross-section of the of the beam is decreasing along the length depending upon the exponential function considered for variation in thickness. The material inhomogeneity is as per the Power and Exponential law of material variation along the axial direction, taken from the literature. Mathematical modelling of geometric non-uniformity, material inhomogeneity and finite element analysis of the AFG beam, have been performed using MATLAB. The effect of geometric non-uniformity and material gradient parameters on the fundamental frequencies of vibration in different classical boundary conditions have been investigated. The efficacy of the current method has been ascertained by comparing the result of available literature.


[1]          Koizumi, M. “FGM activities in Japan”, Composites Part B: Engineering, 28(1), pp. 1-4 (1997).
[2]          Sobzak, J., Drenchev, L. “Metal based Functionally Graded Materials – Engineering and Modelling”, Bentham Science Publisher Ltd. pp. 1-24 (2009).  
[3]          Kapuria, M., Bhattacharya, M., Kumar, A. N. “Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation”, Composite Structures, 82(3), pp. 390-402 (2008).
[4]          Yang, J., Chen, Y. Free vibration and buckling analysis of functionally graded beams with edge cracks, Composite Structures, 83(1), pp. 48-60 (2008).
[5]          Alshorbagy, A.  E., Eltaher, M. A., Mahmoud, F. F. “Free vibration characteristics of functionally graded beam by finite element method”, Applied mathematical modelling, 35(1), pp. 412-425 (2011).
[6]          Simsek, M., Kocaturk, T. “Free and forced vibration of functionally graded beam subjected to a concentrated moving harmonic load”, composite structures, 90, pp. 465-473 (2009).
[7]          Pradhan, K. K., Chakraverty, S. “Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method”, Composites: Part B, 51, pp. 175-184 (2013).
[8]          Rao, R. S., Ganesan, N. “Dynamic response of non-uniform composite beams”, Journal of Sound and Vibration, 200(5), pp. 563-577 (1997).
[9]          Karami, G., Malekzadeh, P., Shahpari, S. A. “A DQEM for vibration of shear deformable non-uniform beams with general boundary conditions”, Engineering structures, 25, pp. 1169-1179 (2003). 
[10]          Aydogdu, M., Taskin, V. “Free vibration analysis of functionally graded beams with simply supported edges”, Material and Design, 28, pp. 1651-1656 (2007).
[11]          Nguyen, V. L., Quoc, T. H. “Bending and free vibration analysis of functionally graded plates using new eight unknown shear deformation theory by finite element method”, International Journal of advanced structural Engineering, 8, pp. 391-399 (2016).
[12]          Huang, Y., Li, X.  F. “A new approach for free vibration of axially graded beams with non-uniform cross-section”, Journal of Sound and vibration, 329, pp. 2291-2303 (2010).
[13]          Cao, D., Gao, Y., Zhang, W. “Free vibration of axially functionally graded beams using the asymptotic development method”, Engineering Structures, 173, pp. 442-448 (2018).
[14]          Ghayesh, M. H. “Non-linear vibration analysis of axially functionally graded shear-deformable tapered beams”, Applied mathematical modelling, 59, pp. 583-596 (2018).
[15]          Huang, Y., Wang, T., Zhao, Y. et al. “Effect of axially functionally graded material on whirling frequencies and critical speeds of spinning Timoshenko beam”, Composite structures, 192, pp. 355-367 (2018).
[16]          Salinic, S., Obradovic, A., Tomovic, A. “Free vibration analysis of axially functionally graded tapered, stepped and continuously segmented rods and beams”, Composites Part B, 150, pp. 135-143 (2018).
[17]          Zheng, S., Chen, D., Wang, H. “Size dependent nonlinear free vibration of axially functionally graded tapered microbeams using finite element method”, Thin-Walled Structures, 139, pp. 46-52 (2019).
[18]          Sahin, S., Karahan, E., Kilic, B., Ozdemir, O. “Finite element method for vibration analysis of Timoshenko beams”, 9th International Conference on Recent Advances in Space Technologies (RAST), Istanbul, Turkey, pp. 673-679 (2019).
[19]          Xie, K., Wang, Y., Fu, T. “Dynamic response of axially functionally graded beam with longitudinal-transverse coupling effect”, Aerospace Science and Technology, 85, pp. 85-95 (2019).
[20]          Sun, D. L., Li, X. F. “Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with non-uniform cross-section”, Mechanics based design of structures and machines, 47(1), pp. 102-120 (2019).
[21]          Hughes, T.J.R., Cottrell, J.A., Bzileves, Y. “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement”, Computer Methods in Applied Mechanics and Engineering, 194(39-41), pp. 4135-4195 (2005).
[22]          Nguyan, H.X., Nguyan, T., Wahab, M.A. et al. “A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory”, Computer Methods in Applied Mechanics and Engineering, 313, pp. 904-940 (2017).
[23]          Thanh, C.L., Ferreira, A.J.M., Wahab, M.A. “A refined size-dependent couple stress theory for laminated composite micro-plates using isogeometric analysis”, Thin-Walled Structures, 145, 106427 (2019).
[24]          Thanh, C.L., Tran, L.V., Bui, T.Q. et al. “Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates”, Composite Structures, 221, 110838 (2019).
[25]          Thanh, C.L., Van, P.P., Thai, C.H. et al. “Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory”, Composite Structures, 184(15), pp. 633-649 (2018).
[26]          Van, P.H., Tran, L.V., Ferreira, A.J.M. et al. “Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads”, Nonlinear Dynamics, 87, pp. 879-894 (2017).
[27]          Van, P.P., Thai, C.H., Xuan, H.N. et al. “Porosity-dependent nonlinear transient responses of functionally graded Nano-plates using iso-geometric analysis”, Composites Part B: Engineering, 164, pp. 215-225 (2019).
[28]          Thanh, C.L., Tran, L.V., Huu, T.V. et al.  “The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis”, Computer Methods in Applied Mechanics and Engineering, 350, pp. 337-361, (2019).
[29]          Nguyen, T.-T., Nguyen, N.-L., Lee, J. et al. “Analysis of non-uniform polygonal cross-sections for thin-walled functionally graded straight and curved beams”, Engineering structures, 226, 111366, (2021).
[30]          Nguyen, T.-T., Lee, J. “Interactive geometric interpretation and static analysis of thin-walled bi-directional functionally graded beams”, Composite Structures, 191, pp. 1-11 (2018).
[31]          Nguyen, T.T., Lee, J. “Flexural-torsional vibration and buckling of thin-walled bi-directional functionally graded beams”, Composites Part B, 154, pp. 351-362 (2018).
[32]          Rajasekaran, S., Khaniki, H. B. “Finite element static and dynamic analysis of axially functionally graded non-uniform small-scale beams based on nonlocal strain gradient theory”, Mechanics of Advanced Materials and Structures, 0, pp. 1-15 (2018).
[33]          Voigt, W. “Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper”. Annalen der Physik,  pp. 573-587 (1889).