Free vibration and buckling analysis of functionally graded sandwich beams resting on a two-parameter elastic foundation using a quasi-3D theory

Year 2025, Volume: 43 Issue: 1, 47 - 61, 28.02.2025

Ibrahim Mohamed Sebahat Şimşek Volkan Kahya

Abstract

This study examines the free vibration and buckling behavior of functionally graded (FG) sandwich beams supported by a Winkler-Pasternak elastic foundation, utilizing a quasi-3D deformation theory. The material properties of the FG sandwich beams are modeled to vary continuously through the thickness according to a power-law distribution. Using Hamilton’s principle, the governing equations of motion are derived. Analytical solutions are obtained for simply supported FG sandwich beams with homogeneous cores by employing Navier’s method. The accuracy of the proposed model is demonstrated by comparing the current results with the higher-order deformation theories-based solutions available in literature. A comprehensive parametric study is also carried out to explore the effect of the skin-core-skin thickness ratio, the power-law index, beam span-to-depth ratio, normal strain, core material, and elastic foundation on fundamental natural frequencies and critical buckling loads.

Keywords

Buckling, Elastic foundation, FG Sandwich Beam, Free Vibration, Navier Method, Quasi-3D Theory

References

• REFERENCES
• [1] Garg A, Chalak HD. A review on analysis of laminated composite and sandwich structures under hygrothermal conditions. Thin-Walled Struct 2019;142:205–226. [CrossRef]
• [2] Saleh B, Jiang J, Ma A, Song D, Yang D, Xu Q. Review on the influence of different reinforcements on the microstructure and wear behavior of functionally graded aluminum matrix composites by centrifugal casting. Metals Mater Int 2020;26:933–960. [CrossRef]
• [3] Zhang N, Khan T, Guo H, Shi S, Zhong W, Zhang W. Functionally graded materials: an overview of stability, buckling, and free vibration analysis. Adv Mater Sci Eng 2019;2019:1–18. [CrossRef]
• [4] Sayyad AS, Ghugal YM. Modeling and analysis of functionally graded sandwich beams: a review. Mech Adv Mater Struct 2019;26:1776–1795. [CrossRef]
• [5] Garg A, Belarbi M-O, Chalak HD, Chakrabarti A. A review of the analysis of sandwich FGM structures. Compos Struct 2021;258:113427. [CrossRef]
• [6] Birman V, Kardomateas GA. Review of current trends in research and applications of sandwich structures. Compos B Eng 2018;142:221–240. [CrossRef]
• [7] Q. Yang, B. Zheng, and J. Zhu. Analytical solution of a bilayer functionally graded cantilever beam with concentrated loads. Arch Appl Mech 2013:455–466. [CrossRef]
• [8] Aydogdu M, Vedat Taskin. Free vibration analysis of functionally graded beams with simply supported edges. Mater Design 2007;28:1651–1656. [CrossRef]
• [9] Yang J, Chen Y. Free vibration and buckling analyses of functionally graded beams with edge cracks. Compos Struct 2008;83:48–60. [CrossRef]
• [10] Wang ZH, Wang XH, Xu GD, Cheng S, Zeng T. Free vibration of two-directional functionally graded beams. Compos Struct 2016;135:191–198. [CrossRef]
• [11] Şimşek M. Vibration analysis of a functionally graded beam under a moving mass by using different beam theories. Compos Struct 2010;92:904–917. [CrossRef]
• [12] Pradhan KK, Chakraverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos B Eng 2013;51:175–184. [CrossRef]
• [13] Giunta G, Belouettar S, Carrera E. Analysis of FGM beams by means of classical and advanced theories. Mech Adv Mater Struct 2010;17:622–635. [CrossRef]
• [14] Alshorbagy AE, Eltaher MA, Mahmoud FF. Free vibration characteristics of a functionally graded beam by finite element method. Appl Math Model 2011;35:412–425. [CrossRef]
• [15] Jin C, Wang X. Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method. Compos Struct 2015;125:41–50. [CrossRef]
• [16] Li SR, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Compos Struct 2013;95:5–9. [CrossRef]
• [17] Nguyen TK, Vo TP, Thai HT. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Compos B Eng 2013;55:147–157. [CrossRef]
• [18] Pradhan KK, Chakraverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method. Compos B Eng 2013;51:175–184. [CrossRef]
• [19] Chen D, Yang J, Kitipornchai S. Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct 2015;133:54–61. [CrossRef]
• [20] Tossapanon P, Wattanasakulpong N. Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation. Compos Struct 2016;142:215–225. [CrossRef]
• [21] Zhang Z, Li Y, Wu H, Zhang H, Wu H, Jiang S, et al. Mechanical analysis of functionally graded graphene oxide-reinforced composite beams based on the first-order shear deformation theory. Mechanics of Advanced Materials and Structures 2020;27:3–11. [CrossRef]
• [22] Nguyen T-K, Vo TP, Thai H-T. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Compos B Eng 2013;55:147–157. [CrossRef]
• [23] Kahya V, Turan M. Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Compos B Eng 2017;109:108–115. [CrossRef]
• [24] Turan M, Uzun Yaylacı E, Yaylacı M. Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods. Arch Appl Mech 2023;93:1351–1372. [CrossRef]
• [25] Turan M, Kahya V. Free vibration and buckling analysis of functionally graded sandwich beams by Navier’smethod. J Fac Eng Architect Gazi Univ 2021;36:743–758. [Turkish] [CrossRef]
• [26] Apetre NA, Sankar BV, Ambur DR. Analytical modeling of sandwich beams with functionally graded core. J Sandwich Struct Mater 2008;10:53–1074. [CrossRef]
• [27] Mousavi SB, Amir S, Jafari A, Arshid E. Analytical solution for analyzing initial curvature effect on vibrational behavior of PM beams integrated with FGP layers based on trigonometric theories. Adv Nano Res 2021;10:235–251.
• [28] Pradhan KK, Chakraverty S. Effects of different shear deformation theories on free vibration of functionally graded beams. Int J Mech Sci 2014;82:149–160. [CrossRef]
• [29] Tran TT, Nguyen NH, Do T Van, Minh P Van, Duc ND. Bending and thermal buckling of unsymmetric functionally graded sandwich beams in high-temperature environment based on a new third-order shear deformation theory. J Sandwich Struct Mater 2021;23:906–930. [CrossRef]
• [30] Ninh VTA, Anh LTN, Kien ND. Free vibration of a 2D-FGSW beam based on a shear deformation theory. Vietnam J Mech 2020;4:189–205. [CrossRef]
• [31] Trinh LC, Vo TP, Thai H-T, Nguyen T-K. An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads. Compos B Eng 2016;100:152–163. [CrossRef]
• [32] Hadji L, Daouadji TH, Tounsi A, Bedia EA. A n-order refined theory for bending and free vibration of functionally graded beams. Struct Eng Mech 2015;54:923–936. [CrossRef]
• [33] Nguyen T-K, Nguyen B-D. A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams. J Sandwich Struct Mater 2015;17:613–631. [CrossRef]
• [34] Hadji L, Khelifa Z, El Abbes AB. A new higher order shear deformation model for functionally graded beams. KSCE J Civ Eng 2016;20:1835–1841. [CrossRef]
• [35] Reddy JN. A simple higher order theory for laminated composite plates. ASME J Appl Mech 1984;51:745–752. [CrossRef]
• [36] Sayyad AS, Ghugal YM. Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams. Asian J Civ Eng 2018;19:607–623. [CrossRef]
• [37] Avcar M, Hadji L, Civalek Ö. Natural frequency analysis of sigmoid functionally graded sandwich beams in the framework of high order shear deformation theory. Compos Struct 2021;276:114564. [CrossRef]
• [38] Malhari Ramteke P, Mehar K, Sharma N, Panda S. Numerical prediction of deflection and stress responses of functionally graded structure for grading patterns (power-law, sigmoid and exponential) and variable porosity (even/uneven). Trans Mechanic Eng 2021;28:811–829.
• [39] Derikvand M, Farhatnia F, Hodges DH. Functionally graded thick sandwich beams with porous core: Buckling analysis via differential transform method. Mech Based Des Struct Mach 2023;51:3650–3677. [CrossRef]
• [40] Ramteke PM, Panda SK. Free vibrational behaviour of multi-directional porous functionally graded structures. Arab J Sci Eng 2021;46:7741–7756. [CrossRef]
• [41] Nguyen T-K, Truong-Phong Nguyen T, Vo TP, Thai H-T. Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Compos B Eng 2015;76:273–285. [CrossRef]
• [42] Vo TP, Thai H-T, Nguyen T-K, Maheri A, Lee J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct 2014;64:12–22. [CrossRef]
• [43] Ghumare SM, Sayyad AS. A new fifth-order shear and normal deformation theory for static bending and elastic buckling of P–FGM beams. Latin Am J Solids Struct 2017;14:1893–911. [CrossRef]
• [44] Bennai R, Atmane HA, Tounsi A. A new higher-order shear and normal deformation theory for functionally graded sandwich beams. Steel Compos Struct 2015;19:521–546. [CrossRef]
• [45] Sayyad AS, Shinde BM, Shinde BM, Sayyad AS. A quasi-3D polynomial shear and normal deformation theory for laminated composite, sandwich, and functionally graded beams. Mech Adv Compos Struct 2017;4:139–152.
• [46] Nguyen T-K, Vo TP, Nguyen B-D, Lee J. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Compos Struct 2016;156:238–252. [CrossRef]
• [47] Karamanli A, Aydogdu M. Free vibration and buckling analysis of laminated composites and sandwich microbeams using a transverse shear-normal deformable beam theory. J Vib Control 2020;26:214–228. [CrossRef]
• [48] Karamanlı A. Free vibration and buckling analysis of two directional functionally graded beams using a four-unknown shear and normal deformable beam theory. Anadolu Univ J Sci Technol A Appl Sci Eng 2018;19:375–406. [CrossRef]
• [49] Vo TP, Thai H-T, Nguyen T-K, Inam F, Lee J. A quasi- 3D theory for vibration and buckling of functionally graded sandwich beams. Compos Struct 2015;119:1–12. [CrossRef]
• [50] Vo TP, Thai H-T, Nguyen T-K, Inam F, Lee J. Static behaviour of functionally graded sandwich beams using a quasi–3D theory. Compos B Eng 2015;68:59–74. [CrossRef]
• [51] Osofero AI, Vo TP, Nguyen T-K, Lee J. Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi-3D theories. J Sandwich Struct Mater 2016;18:3–29. [CrossRef]
• [52] Winkler E. Theory of elasticity and strength. Dominicus Prague 1967.
• [53] Pasternak P. On a new method of analysis of an elastic foundation by means of two foundation constants. Gosudarstvennoe Izdatelstvo Literaturipo Stroitelstvu I Arkhitekture 1954.
• [54] Nguyen Thi H. On mechanical behavior of two-layer functionally graded sandwich curved beams resting on elastic foundations using an analytical solution and refined Timoshenko beam theory. Ain Shams Eng J 2022;13:101647. [CrossRef]
• [55] Zenkour A, Ebrahimi F, Barati MR. Buckling analysis of a size-dependent functionally graded nanobeam resting on Pasternak’s foundations. Int J Nano Dimens 2019;10:141–153.
• [56] Mohammed AT, Hareb MA, Eqal AK. Investigation on the analysis of bending and buckling for FGM Euler-Bernoulli beam resting on Winkler-Pasternak elastic foundation. J Phys Conf Ser 2021;1773:12027. [CrossRef]
• [57] Songsuwan W, Pimsarn M, Wattanasakulpong N. Dynamic responses of functionally graded sandwich beams resting on elastic foundation under harmonic moving loads. Int J Struct Stabil Dynamics 2018;18:1850112. [CrossRef]
• [58] Hung DX, Truong HQ. Free vibration analysis of sandwich beams with FG porous core and FGM faces resting on Winkler elastic foundation by various shear deformation theories. J Sci Technol Civ Eng 2018;12:23–33. [CrossRef]
• [59] Fahsi B, Bouiadjra RB, Mahmoudi A, Benyoucef S, Tounsi A. Assessing the effects of porosity on the bending, buckling, and vibrations of functionally graded beams resting on an elastic foundation by using a new refined quasi-3D theory. Mech Compos Mater 2019;55:219–230. [CrossRef]
• [60] Ait Atmane H, Tounsi A, Bernard F. Effect of thickness stretching and porosity on mechanical response of functionally graded beams resting on elastic foundations. Int J Mech Mater Design 2017;13:71–84. [CrossRef