Static bending analysis of two-directional functionally graded beam using simple Timoshenko beam elements



Two-directional FG beam, Static bending, Transverse deflection, Rotation


This article presents the static bending of two-directional functionally graded (FG) beam by using simple Timoshenko beam elements. The Matlab code developed based on the finite element formulation is validated by solving two-directional FG beam problems under distributed load and two boundary conditions. Numerical results which are in terms of maximum normalized transverse deflections are compared with the analytical solutions and the results from previous studies. Besides, the shapes of transverse deflection and rotation along the length of beams are also depicted in this article to provide specific views about the static behavior of proposed structure.


Download data is not yet available.


O. Carvalho, M. Buciumeanu, G. Miranda, S. Madeira, and F. S. Silva, "Development of a method to produce FGMs by controlling the reinforcement distribution," Materials & Design, vol. 92, pp. 233-239, 2016.

M. Chehel Amirani, S. M. R. Khalili, and N. Nemati, "Free vibration analysis of sandwich beam with FG core using the element free Galerkin method," Composite Structures, vol. 90, pp. 373-379, 2009.

S. P. Parida, P. C. Jena, and R. R. Dash, "FGM Beam analysis in Dynamical and Thermal surroundings using Finite Element Method," Materials Today: Proceedings, vol. 18, pp. 3676-3682, 2019.

S. Zghal, D. Ataoui, and F. Dammak, "Static bending analysis of beams made of functionally graded porous materials," Mechanics Based Design of Structures and Machines, pp. 1-18, 2020.

D. Wu, W. Gao, D. Hui, K. Gao, and K. Li, "Stochastic static analysis of Euler-Bernoulli type functionally graded structures," Composites Part B: Engineering, vol. 134, pp. 69-80, 2018.

H. L. Ton-That, H. Nguyen-Van, and T. Chau-Dinh, "A novel quadrilateral element for analysis of functionally graded porous plates/shells reinforced by graphene platelets," Archive of Applied Mechanics, vol. 91, pp. 2435-2466, 2021.

V. N. Burlayenko, H. Altenbach, T. Sadowski, S. D. Dimitrova, and A. Bhaskar, "Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements," Applied Mathematical Modelling, vol. 45, pp. 422-438, 2017.

M. S. Beg and M. Y. Yasin, "Bending, free and forced vibration of functionally graded deep curved beams in thermal environment using an efficient layerwise theory," Mechanics of Materials, vol. 159, p. 103919, 2021.

M. Iasiello, N. Bianco, W. K. S. Chiu, and V. Naso, "The effects of variable porosity and cell size on the thermal performance of functionally-graded foams," International Journal of Thermal Sciences, vol. 160, p. 106696, 2021.

M. ?im?ek, T. Kocatürk, and ?. D. Akba?, "Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory," Composite Structures, vol. 95, pp. 740-747, 2013.

H. L. Ton-That, "Plate Structural Analysis Based on a Double Interpolation Element with Arbitrary Meshing," Acta Mechanica et Automatica, vol. 15, pp. 91-99, 2021.

Vo-Duy Quang and T.-T. H. Lan, "Free vibration of simply supported steel I-girders with trapezoidal web corrugations," Reports in Mechanical Engineering, vol. 1, pp. 141-150, 2020.

I. Katili, T. Syahril, and A. M. Katili, "Static and free vibration analysis of FGM beam based on unified and integrated of Timoshenko’s theory," Composite Structures, vol. 242, p. 112130, 2020.

S.-R. Li, D.-F. Cao, and Z.-Q. Wan, "Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams," Applied Mathematical Modelling, vol. 37, pp. 7077-7085, 2013.

H. L. Ton-That, "The Linear and Nonlinear Bending Analyses of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Based on the Novel Four-Node Quadrilateral Element," European Journal of Computational Mechanics, vol. 29, pp. 139-172, 2020.

D. Chen, J. Yang, and S. Kitipornchai, "Elastic buckling and static bending of shear deformable functionally graded porous beam," Composite Structures, vol. 133, pp. 54-61, 2015.

A. Karamanli, "Static behaviour of two-directional functionally graded sandwich beams using various beam theories," New Trends in Mathematical Sciences, vol. 5, pp. 112-147, 2017.

B. Anirudh, M. Ganapathi, C. Anant, and O. Polit, "A comprehensive analysis of porous graphene-reinforced curved beams by finite element approach using higher-order structural theory: Bending, vibration and buckling," Composite Structures, vol. 222, p. 110899, 2019.

N. Wattanasakulpong, B. Gangadhara Prusty, and D. W. Kelly, "Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams," International Journal of Mechanical Sciences, vol. 53, pp. 734-743, 2011.

P. Sharma and R. Singh, "A numerical study on free vibration analysis of axial FGM beam," Materials Today: Proceedings, vol. 44, pp. 1664-1668, 2021.

H. L. Ton-That, "A new C0 third-order shear deformation theory for the nonlinear free vibration analysis of stiffened functionally graded plates," Facta Universitatis, Series: Mechanical Engineering, vol. 19, pp. 285-305, 2021.

F. Ebrahimi and E. Salari, "Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments," Composite Structures, vol. 128, pp. 363-380, 2015.

F. Z. Jouneghani, R. Dimitri, and F. Tornabene, "Structural response of porous FG nanobeams under hygro-thermo-mechanical loadings," Composites Part B: Engineering, vol. 152, pp. 71-78, 2018.

J. W. Lee and J. Y. Lee, "Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression," International Journal of Mechanical Sciences, vol. 122, pp. 1-17, 2017.

X. F. Li, "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams," Journal of Sound and Vibration, vol. 318, pp. 1210-1229, 2008.




How to Cite

L. Hoang That Ton, “Static bending analysis of two-directional functionally graded beam using simple Timoshenko beam elements”, International Journal of Engineering and Applied Physics, vol. 2, no. 1, pp. 394–401, Jan. 2022.