AN ANALYTICAL METHOD FOR THE ANALYSIS OF FUNCTIONALLY GRADED BEAMS RESTING ON A WINKLER ELASTIC FOUNDATION
DOI:
https://doi.org/10.31650/2786-6696-2025-13-99-108Keywords:
functionally graded beam, elastic foundation, exact solution, analytical method.Abstract
This publication addresses the analysis of so-called functionally graded beams (FGBs), whose material properties vary along their length according to a specified gradient. The study is devoted to the development of an analytical method (AM) for bending analysis of such beams resting on a homogeneous, continuous Winkler elastic foundation. The material's modulus of elasticity and the external load are assumed to be arbitrary, continuously varying functions dependent on the coordinate of the beam’s neutral axis. The proposed method is based on an exact solution to the corresponding fourth-order ordinary differential equation of bending with variable coefficients. The unknown integration constants are expressed in terms of initial parameters, which are determined by applying the prescribed boundary conditions. The fundamental functions and the particular solution of the differential equation are represented as power series in a dimensionless parameter, with variable coefficients obtained through recursive integral relations. For practical convenience, the fundamental functions and the particular solution are transformed into power series format. This reduces the bending analysis of FGBs to a numerical implementation of explicit analytical expressions for stress-strain state parameters.
A numerical example demonstrates the practical application of the AM. A prismatic FGB with a parabolically varying modulus of elasticity is considered. The AM results are presented in both numerical and graphical formats for the case where the left end of the beam is simply supported and the right end is fixed. The obtained numerical values are treated as exact, since the analysis is based on an exact solution of the governing differential equation. The availability of such a method enables the assessment of the accuracy of solutions obtained using various approximate methods. For verification purposes, corresponding solutions obtained by the Finite Element Method (FEM) using the LIRA-SAPR software are also provided. The comparison confirms the validity of the proposed analytical method.
References
1. Calim, F.F. Transient analysis of axially functionally graded Timoshenko beams with variable cross-section. Composites Part B: Engineering. 2016. 98. 472–483. URL: https://doi.org/10.1016/j.compositesb.2016.05.040
2. Shafiei, N., Kazemi, M., & Ghadiri, M. Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science. 2016. 102. 12-26. URL: https://doi.org/10.1016/j.ijengsci.2016.02.007
3. Lee, J.W., & Lee, J.Y. Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression. International Journal of Mechanical Sciences. 2017. 122. 1-17. URL: https://doi.org/10.1016/j.ijmecsci.2017.01.011
4. Su, H., Banerjee, J.R., & Cheung, C.W. Dynamic stiffness formulation and free vibration analysis of functionally graded beams. Composite Structures. 2013. 106. 854-862. URL: https://doi.org/10.1016/j.compstruct.2013.06.029
5. Jing, L.L., Ming, P.J., Zhang, W.P., Fu, L.R., & Cao, Y.P. Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method. Composite structures. 2016. 138. 192-213. URL: https://doi.org/10.1016/j.compstruct.2015.11.027
6. Ait Atmane, H., Tounsi, A., Meftah, S.A., & Belhadj, H.A. Free vibration behavior of exponential functionally graded beams with varying cross-section. Journal of Vibration and Control. 2011. 17(2). 311-318. URL: https://doi.org/10.1177/1077546310370691
7. Cao, D., Wang, B., Hu, W., & Gao, Y. Free vibration of axially functionally graded beam. Mechanics of functionally graded materials and structures. IntechOpen. 2020. URL: https://doi.org/10.5772/intechopen.85835
8. Nguyen, D. K. Large displacement response of tapered cantilever beams made of axially functionally graded material. Composites Part B: Engineering. 2013. 55. 298–305. URL: https://doi.org/10.1016/j.compositesb.2013.06.024
9. Hieu, D.V., Chan, D.Q., & Sedighi, H.M. Nonlinear bending, buckling and vibration of functionally graded nonlocal strain gradient nanobeams resting on an elastic foundation. Journal of Mechanics of Materials and Structures. 2021. 16(3). 327-346. URL: https://doi.org/10.2140/jomms.2021.16.327
10. Liu, H., Huang, Y., & Zhao, Y. A unified numerical approach to the dynamics of beams with longitudinally varying cross-sections, materials, foundations, and loads using Chebyshev spectral approximation. Aerospace. 2023. 10(10). 842. URL: https://doi.org/10.3390/aerospace10100842
11. Ghazwani, M.H. New enriched beam element for static bending analysis of functionally graded porous beams resting on elastic foundations. Mechanics of Solids. 2023. 58(5). 1878-1893. URL: https://doi.org/10.3103/S0025654423600885
12. Kanığ, D., Bab, Y., Kutlu, A., & Omurtag, M. H. Mixed finite element formulation for functionally graded beams resting on elastic foundation using HSDT. In 4th international civil engineering & architecture conference (pp. 2177–2183). Golden Light Publishing. 2025. URL: https://doi.org/10.31462/icearc2025_ce_sme_531
13. Крутій Ю. С. Розробка методу розв’язання задач стійкості і коливань деформівних систем зі змінними неперервними параметрами: дис. д-ра техн. наук: 01.02.04. Луцьк, 2016. 273 с.
14. Krutii, Y., Suriyaninov, M., & Vandynskyi, V. Exact solution of the differential equation of transverse oscillations of the rod taking into account own weight. MATEC Web of Conferences. 2017. 116, 02022. URL: https://doi.org/10.1051/matecconf/201711602022
15. Krutii, Y.S. Construction of a solution of the problem of stability of a bar with arbitrary continuous parameters. Journal of Mathematical Sciences. 2018. 231(5). 665–677. URL: https://doi.org/10.1007/s10958-018-3843-8
16. Krutii, Y., Surianinov, M., Petrash, S., & Yezhov, M. Development of an analytical method for calculating beams on a variable elastic Winkler foundation. IOP Conference Series: Materials Science and Engineering. 2021. 1162(1). 012009. URL: https://doi.org/10.1088/1757-899x/1162/1/012009
Downloads
Published
Issue
Section
License
Copyright (c) 2025 MODERN CONSTRUCTION AND ARCHITECTURE
This work is licensed under a Creative Commons Attribution 4.0 International License.
