Refined analytical formulation for static bending and free vibrations of functionally graded beams
DOI:
https://doi.org/10.54355/tbus/6.1.2026.0096Keywords:
functionally graded beam, shear deformation theory, static bending analysis, free vibration analysis, transverse shear stress, refined beam theoryAbstract
This paper presents a refined analytical theory for the bending and free vibration analysis of functionally graded (FG) beams accounting for transverse shear deformation. The material properties are assumed to vary continuously through the thickness according to a power-law distribution. Based on the adopted kinematic assumptions, the governing differential equations and effective stiffness relations are derived for both static and dynamic problems. Analytical solutions for the static bending of an FG beam under a uniformly distributed load and for the natural frequencies of free vibrations are obtained. A parametric study is performed to evaluate the influence of the material gradient index ???? and the beam slenderness ratio ????/h. The results show that the dimensionless mid-span deflection increases from 6.2586 to 9.6986 when the gradient index changes from ???? = 1 to ???? = 5 for ????/ℎ = 5. The corresponding dimensionless natural frequency decreases from 3.9710 to 3.4025. Comparisons with the Timoshenko beam theory, Reddy’s higher-order theory, and other refined models demonstrate very good agreement, with differences typically not exceeding 3-5%. The proposed formulation provides a realistic distribution of transverse shear stresses and converges to classical beam solutions for slender beams. The developed model can be effectively applied in the analysis and design of engineering structures made of functionally graded materials.
Downloads
Metrics
References
V. Birman and L. W. Byrd, “Modeling and Analysis of Functionally Graded Materials and Structures,” Applied Mechanics Reviews, vol. 60, no. 5, pp. 195–216, Sep. 2007, doi: 10.1115/1.2777164. DOI: https://doi.org/10.1115/1.2777164
H.-T. Thai and T. P. Vo, “Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories,” International Journal of Mechanical Sciences, vol. 62, no. 1, pp. 57–66, Sep. 2012, doi: 10.1016/j.ijmecsci.2012.05.014. DOI: https://doi.org/10.1016/j.ijmecsci.2012.05.014
S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 41, no. 245, pp. 744–746, May 1921, doi: 10.1080/14786442108636264. DOI: https://doi.org/10.1080/14786442108636264
J. N. Reddy, “A Simple Higher-Order Theory for Laminated Composite Plates,” Journal of Applied Mechanics, vol. 51, no. 4, pp. 745–752, Dec. 1984, doi: 10.1115/1.3167719. DOI: https://doi.org/10.1115/1.3167719
M. Aydogdu and V. Taskin, “Free vibration analysis of functionally graded beams with simply supported edges,” Materials & Design, vol. 28, no. 5, pp. 1651–1656, Jan. 2007, doi: 10.1016/j.matdes.2006.02.007. DOI: https://doi.org/10.1016/j.matdes.2006.02.007
B. V. Sankar, “An elasticity solution for functionally graded beams,” Composites Science and Technology, vol. 61, no. 5, pp. 689–696, Apr. 2001, doi: 10.1016/S0266-3538(01)00007-0. DOI: https://doi.org/10.1016/S0266-3538(01)00007-0
T. P. Vo, H.-T. Thai, T.-K. Nguyen, and F. Inam, “Static and vibration analysis of functionally graded beams using refined shear deformation theory,” Meccanica, vol. 49, no. 1, pp. 155–168, Jan. 2014, doi: 10.1007/s11012-013-9780-1. DOI: https://doi.org/10.1007/s11012-013-9780-1
T. P. Vo, H.-T. Thai, T.-K. Nguyen, A. Maheri, and J. Lee, “Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory,” Engineering Structures, vol. 64, pp. 12–22, Apr. 2014, doi: 10.1016/j.engstruct.2014.01.029. DOI: https://doi.org/10.1016/j.engstruct.2014.01.029
V. Kahya and M. Turan, “Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory,” Composites Part B: Engineering, vol. 109, pp. 108–115, Jan. 2017, doi: 10.1016/j.compositesb.2016.10.039. DOI: https://doi.org/10.1016/j.compositesb.2016.10.039
H. Ziou, M. Himeur, H. Guenfoud, and M. Guenfoud, “An Efficient Finite Element Formulation Based on Deformation Approach for Bending of Functionally Graded Beams,” JSM, vol. 12, no. 2, 2020, doi: 10.22034/jsm.2019.1867884.1437.
M. Touratier, “An efficient standard plate theory,” International Journal of Engineering Science, vol. 29, no. 8, pp. 901–916, Jan. 1991, doi: 10.1016/0020-7225(91)90165-Y. DOI: https://doi.org/10.1016/0020-7225(91)90165-Y
M. Bourada, K. Abdelhakim, M. S. A. Houari, and A. Tounsi, “A new simple shear and normal deformations theory for functionally graded beams,” Steel and Composite Structures, vol. 18, pp. 409–423, Feb. 2015, doi: 10.12989/scs.2015.18.2.409. DOI: https://doi.org/10.12989/scs.2015.18.2.409
S. M. Ghumare and A. S. Sayyad, “A New Fifth-Order Shear and Normal Deformation Theory for Static Bending and Elastic Buckling of P-FGM Beams,” Lat. Am. j. solids struct., vol. 14, no. 11, pp. 1893–1911, 2017, doi: 10.1590/1679-78253972. DOI: https://doi.org/10.1590/1679-78253972
A. Sayyad and Y. Ghugal, “Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams,” MACS, vol. 5, no. 1, Apr. 2018, doi: 10.22075/macs.2018.12214.1117.
T.-K. Nguyen, T. Truong-Phong Nguyen, T. P. Vo, and H.-T. Thai, “Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory,” Composites Part B: Engineering, vol. 76, pp. 273–285, Jul. 2015, doi: 10.1016/j.compositesb.2015.02.032. DOI: https://doi.org/10.1016/j.compositesb.2015.02.032
D. Indronil and I. M. Nazmul, “A novel hyperbolic shear deformation beam theory for functionally graded nonlocal nanobeams,” Discov Appl Sci, vol. 7, no. 10, p. 1204, Oct. 2025, doi: 10.1007/s42452-025-07047-5. DOI: https://doi.org/10.1007/s42452-025-07047-5
Y. L. Pei and L. X. Li, “Comment on the Navier’s solution in ‘A sinusoidal beam theory for functionally graded sandwich curved beams’ (Composite Structures 226 (2019) 111246),” Composite Structures, vol. 243, p. 112248, Jul. 2020, doi: 10.1016/j.compstruct.2020.112248. DOI: https://doi.org/10.1016/j.compstruct.2020.112248
F. Rahmani, R. Kamgar, and R. Rahgozar, “Finite Element Analysis of Functionally Graded Beams using Different Beam Theories,” Civ Eng J, vol. 6, no. 11, pp. 2086–2102, Nov. 2020, doi: 10.28991/cej-2020-03091604. DOI: https://doi.org/10.28991/cej-2020-03091604
L. Hadji, N. Zouatnia, and F. Bernard, “An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models,” Structural Engineering & Mechanics, vol. 69, no. 2, pp. 231–241, Feb. 2019, doi: 10.12989/sem.2019.69.2.231.
A. S. Sayyad and Y. M. Ghugal, “Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams,” Asian J Civ Eng, vol. 19, no. 5, pp. 607–623, Jul. 2018, doi: 10.1007/s42107-018-0046-z. DOI: https://doi.org/10.1007/s42107-018-0046-z
K. A. Tursynov, “Arqalyqty esepteu teoriyalary,” in Textbook, Karaganda: Karaganda Buketov University, 2007.
B. Maxfield, “Essential PTC® Mathcad Prime® 3.0: a guide for new and current users,” Oxford: Academic Press, 2014, p. 563. DOI: https://doi.org/10.1016/B978-0-12-410410-5.00001-5
Published
How to Cite
License
Copyright (c) 2026 Sungat Akhazhanov, Aizhan Nurgoziyeva, Kiyas Kutimov
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Funding data
-
Ministry of Education and Science of the Republic of Kazakhstan
Grant numbers AP22684709
