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Free vibration analysis of Bi-Directional Functionally Graded Beams using a simple and efficient finite element model

  • Zakaria Belabed (Artificial Intelligence Laboratory for Mechanical and Civil Structures, and Soil, Institute of Technology, University Center of Naama) ;
  • Abdeldjebbar Tounsi (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes) ;
  • Abdelmoumen Anis Bousahla (Laboratoire de Modelisation et Simulation Multi-echelle, Faculty of Science & Technology, Mechanical Engineering Department, Universite de Sidi Bel Abbes) ;
  • Abdelouahed Tounsi (Department of Civil and Environmental Engineering, Lebanese American University) ;
  • Mohamed Bourada (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes) ;
  • Mohammed A. Al-Osta (Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals)
  • Received : 2023.03.20
  • Accepted : 2024.04.09
  • Published : 2024.05.10

Abstract

This research explores a new finite element model for the free vibration analysis of bi-directional functionally graded (BDFG) beams. The model is based on an efficient higher-order shear deformation beam theory that incorporates a trigonometric warping function for both transverse shear deformation and stress to guarantee traction-free boundary conditions without the necessity of shear correction factors. The proposed two-node beam element has three degrees of freedom per node, and the inter-element continuity is retained using both C1 and C0 continuities for kinematics variables. In addition, the mechanical properties of the (BDFG) beam vary gradually and smoothly in both the in-plane and out-of-plane beam's directions according to an exponential power-law distribution. The highly elevated performance of the developed model is shown by comparing it to conceptual frameworks and solution procedures. Detailed numerical investigations are also conducted to examine the impact of boundary conditions, the bi-directional gradient indices, and the slenderness ratio on the free vibration response of BDFG beams. The suggested finite element beam model is an excellent potential tool for the design and the mechanical behavior estimation of BDFG structures.

Keywords

References

  1. Ahmed, R.A., Moustafa, N.M., Faleh, N.M. and Fenjan, R.M. (2020), "Nonlocal nonlinear stability of higher-order porous beams via Chebyshev-Ritz method", Struct. Eng. Mech., 76(3), 413-420. https://doi.org/10.12989/sem.2020.76.3.413.
  2. Alimoradzadeh, M. and Akbas, S.D. (2022), "Nonlinear dynamic behavior of functionally graded beams resting on nonlinear viscoelastic foundation under moving mass in thermal environment", Struct. Eng. Mech., 81(6), 705-714. https://doi.org/10.12989/sem.2022.81.6.705.
  3. Alneamy A.M. and Ouakad H.M. (2024), "Modeling and structural analysis of MEMS shallow arch assuming multimodal initial curvature profiles", Math., 12(7), 970. https://doi.org/10.3390/math12070970.
  4. Alneamy, A.M. (2024), "Dynamic snap-through motion and chaotic attractor of electrostatic shallow arch micro-beams", Chaos Solit. Fractal., 182, 114777. https://doi.org/10.1016/j.chaos.2024.114777.
  5. Ansari, R., Oskouie, M.F. and Zargar, M. (2022), "Hygrothermally induced vibration analysis of bidirectional functionally graded porous beams", Transp. Porous. Med., 142, 41-62. https://doi.org/10.1007/s11242-021-01700-4.
  6. Balireddy, S.N. and Pitchaimani, J. (2022), "Stability and dynamic behaviour of bi-directional functionally graded beam subjected to variable axial load", Mater. Today Commun., 32, 104043. https://doi.org/10.1016/j.mtcomm.2022.104043.
  7. Belabed, Z., Selim, M.M., Slimani, O., Taibi, N., Tounsi, A. and Hussain, M. (2021), "An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells", Steel Compos. Struct., 40(2), 307-321. https://doi.org/10.12989/scs.2021.40.2.307.
  8. Berkia, A., Benguediab, S., Menasria, A., Bouhadra, A., Mamen, F.B.B., Tounsi, A., ... & Hussain, M. (2022), "Static buckling analysis of bi-directional functionally graded sandwich (BFGSW) beams with two different boundary conditions", Steel Compos. Struct., 44(4), 489-503. https://doi.org/10.12989/scs.2022.44.4.503.
  9. Bouzeriba, A. and Bouzrira, C. (2024), "The analysis of pressurized FGM cylinders with arbitrarily varying material properties using strain based approach", Mech. Bas. Des. Struct. Mach., 1-19. https://doi.org/10.1080/15397734.2024.2318737.
  10. Charef, T., Bachir Bouiadjra, R., Sekkal, M., Bachiri, A., Benyoucef, S., Saleh, M.M.S., ... & Hussain, M. (2023), "Assessing the impact of different foundations on the thermodynamic response of bidirectional FG porous beams", Arab. J. Geosci., 16, 48. https://doi.org/10.1007/s12517-022-11138-7.
  11. Chen, W.R. and Chang, H. (2020), "Vibration analysis of bi-directional functionally graded Timoshenko beams using Chebyshev collocation method", Int. J. Struct. Stab. Dyn., 21(01), 2150009. https://doi.org/10.1142/S0219455421500097.
  12. Chen, X., Huang, S., Zhu, B., Wu, R. and Ren, Z. (2022), "A domain decomposition method based vibration analysis of BDFGs imperfect beams with arbitrary boundary conditions", Compos. Struct., 284, 115115. https://doi.org/10.1016/j.compstruct.2021.115115.
  13. Chen, X., Lu, Y. and Li, Y. (2019), "Free vibration, buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium", Appl. Math. Model., 67, 430-448. https://doi.org/10.1016/j.apm.2018.11.004.
  14. Chen, X., Lu, Y., Wu, Z., Shao, Y., Xue, X. and Wu, Y. (2023), "Free vibration of in-plane bi-directional functionally graded materials rectangular plates with geometric imperfections and general elastic restraints", Aerosp. Sci. Technol., 132, 108045. https://doi.org/10.1016//10.1016/j.ast.2022.108045.
  15. Dangi, C., Saini, S., Lal, R. and Singh, I.V. (2020), "Size dependent FEM model for Bi-directional functionally graded nano-beams", Mater. Today: Proceed., 24, 1302-1311. https://doi.org/10.1016/j.matpr.2020.04.445.
  16. Deng, H. and Cheng, W. (2016), "Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams", Compos. Struct., 141, 253-263. https://doi.org/10.1016/j.compstruct.2016.01.051.
  17. Emadi, M., Nejad, M.Z., Ziaee, S. and Hadi, A. (2021), "Buckling analysis of arbitrary two-directional functionally graded nano-plate based on nonlocal elasticity theory using generalized differential quadrature method", Steel. Compos. Struct., 39(5), 565-581. https://doi.org/10.12989/scs.2021.39.5.565.
  18. Gao, Y., Xiao, W.S. and Zhu, H. (2019), "Nonlinear thermal buckling of bi-directional functionally graded nanobeams", Struct. Eng. Mech., 71(6), 669-682. https://doi.org/10.12989/sem.2019.71.6.669.
  19. Gautam, M., Sharma, P. and Chaturvedi, M. (2023), "Modeling of FGM beam under an extended exponential law", Int. J. Interact. Des. Manuf., 1-6. https://doi.org/10.1007/s12008-023-01239-2.
  20. Ghatage, P.S., Kar, V.R. and Sudhagar, P.E. (2020), "On the numerical modelling and analysis of multi-directional functionally graded composite structures: A review", Compos. Struct., 236, 111837. https://doi.org/10.1016/j.compstruct.2019.111837.
  21. Guo, Q., Yao, W., Li, W. and Gupta, N. (2021), "Constitutive models for the structural analysis of composite materials for the finite element analysis: a review of recent practices", Compos. Struct., 260, 113267. https://doi.org/10.1016/j.compstruct.2020.113267.
  22. Huang, Y. (2020b), "Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section", Appl. Math. Mech.-Engl. Ed., 41, 1497-1516. https://doi.org/10.1007/s10483-020-2670-6.
  23. Huang, Y. and Ouyang, Z.Y. (2020a), "Exact solution for bending analysis of two-directional functionally graded Timoshenko beams", Arch. Appl. Mech., 90, 1005-1023. https://doi.org/10.1007/s00419-019-01655-5.
  24. Huynh, T.A., Lieu, X.Q. and Lee, J. (2017), "NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem", Compos. Struct., 160, 117890. https://doi.org/10.1016/j.compstruct.2016.10.076.
  25. Kar, U.K. and Srinivas, J. (2023), "Dynamic analysis and identification of bi-directional functionally graded elastically supported cracked microbeam subjected to thermal shock loads", Eur. J. Mech.-A/Solid., 99, 104930. https://doi.org/10.1016/j.euromechsol.2023.104930.
  26. Karamanli, A. (2018), "Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory", Compos. Struct., 189, 127-136. https://doi.org/10.1016/j.compstruct.2018.01.060.
  27. Karamanli, A., Vo, T.P. and Civalek, O. (2023), "Higher order finite element models for transient analysis of strain gradient functionally graded microplates", Eur. J. Mech.-A/Solid., 99, 104933. https://doi.org/10.1016/j.euromechsol.2023.104933.
  28. Lal, R. and Dangi, C. (2019), "Thermomechanical vibration of bidirectional functionally graded non-uniform Timoshenko nanobeam using nonlocal elasticity theory", Compos. Part B: Eng., 172, 724-742. https://doi.org/10.1016/j.compositesb.2019.05.076.
  29. Le, C.I., Le, N.A.T. and Nguyen, D.K. (2020), "Free vibration and buckling of bidirectional functionally graded sandwich beams using an enriched third-order shear deformation beam element", Compos. Struct., 261, 113309. https://doi.org/10.1016/j.compstruct.2020.113309.
  30. Madenci, E. (2021), "Free vibration and static analyses of metal-ceramic FG beams via high-order variational MFEM", Steel Compos. Struct., 39(5), 493-509. https://doi.org/10.12989/scs.2021.39.5.493.
  31. Madenci, E. and Ozutok, A. (2020), "Variational approximate for high order bending analysis of laminated composite plates", Struct. Eng. Mech., 73(1), 97-108. http://doi.org/10.12989/sem.2020.73.1.097.
  32. Meksi, A., Benyoucef, S., Sekkal, M., Bouiadjra, R.B., Selim, M.M., Tounsi, A. and Hussain, M. (2021), "Influence of micromechanical models on the bending response of bidirectional FG beams under linear, uniform, exponential and sinusoidal distributed loading", Steel. Compos. Struct., 39(2), 215-228. https://doi.org/10.12989/scs.2021.39.2.215.
  33. Mesbah, A., Belabed, Z., Tounsi, A., Ghazwani, M.H., Alnujaie, A. and Aldosari, S.M. (2024), "Assessment of new Quasi-3D finite element model for free vibration and stability behaviors of thick functionally graded beams.", J. Vib. Eng. Technol., 12, 2231-2247. https://doi.org/10.1007/s42417-023-00976-8.
  34. Mohammadian, M. (2021), "Nonlinear free vibration of damped and undamped bi-directional functionally graded beams using a cubic-quintic nonlinear model", Compos. Struct., 255, 112866. https://doi.org/10.1016/j.compstruct.2020.112866.
  35. Mousavi, M.A., Sadeghi-Nik, A., Bahari, A., Jin, C., Ahmed, R., Ozbakkaloglu, T. and de Brito, J. (2021), "Strength optimization of cementitious composites reinforced by carbon nanotubes and titania nanoparticles", Constr. Build. Mater., 303(124), 510. https://doi.org/10.1016/j.conbuildmat.2021.124510.
  36. Nejad, M.Z., Hadi, A. and Farajpour, A. (2017), "Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials", Struct. Eng. Mech., 63(2), 161-169. https://doi.org/10.12989/sem.2017.63.2.161.
  37. Nejad, M.Z., Hadi, A., Omidvari, A. and Rastgoo, A. (2018), "Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's nonlocal elasticity theory", Struct. Eng. Mech., 67(4), 417-425. https://doi.org/10.12989/sem.2018.67.4.417.
  38. Nguyen, D.K., Vu, A.N.T., Pham, V.N. and Truong, T.T. (2022), "Vibration of a three-phase bidirectional functionally graded sandwich beam carrying a moving mass using an enriched beam element", Eng. Comput., 38, 4629-4650. https://doi.org/10.1007/s00366-021-01496-3.
  39. Ohab-Yazdi, S.M.K. and Kadkhodayan, M. (2021), "Free vibration of bi-directional functionally graded imperfect nanobeams under rotational velocity", Aerosp. Sci. Technol., 119, 107210. https://doi.org/10.1016/j.ast.2021.107210.
  40. Rajasekaran, S. and Khaniki, H.B. (2018), "Free vibration analysis of bi-directional functionally graded single/multi-cracked beams", Int. J. Mech. Sci., 144, 341-356. https://doi.org/10.1016/j.ijmecsci.2018.06.004.
  41. Rajasekaran, S. and Khaniki, H.B. (2019), "Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part II: Static, stability and free vibration studies", Thin Wall. Struct., 141, 646-674. https://doi.org/10.1016/j.tws.2019.02.005.
  42. Selmi, A. (2021), "Free vibration of bi-dimensional functionally graded simply supported beams", Adv. Concrete Constr., 12(3), 195-205. https://doi.org/10.12989/acc.2021.12.3.195.
  43. Sharma, P. and Khinchi, A. (2023), "Comparative analysis of the behavior of Bi-Directional Functionally Graded Beams: Numerical and parametric study", Int. J. Interact. Des. Manuf., 1-12. https://doi.org/10.1007/s12008-022-01191-7.
  44. Simsek, M. (2015), "Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions", Compos. Struct., 133, 968978. https://doi.org/10.1016/j.compstruct.2015.08.021.
  45. Tang, Y. and Yang, T. (2018), "Bi-directional functionally graded nanotubes: Fluid conveying dynamics", J. Appl. Mech., 10(04), 1850041. https://doi.org/10.1142/S1758825118500412.
  46. Tang, Y., Lv, X. and Yang, T. (2019), "Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration", Compos. Part B: Eng., 156, 319-331. https://doi.org/10.1016/j.compositesb.2018.08.140.
  47. Timesli, A. (2020), "Prediction of the critical buckling load of SWCNT reinforced concrete cylindrical shell embedded in an elastic foundation", Comput. Concrete, 26(1), 53-62. https://doi.org/10.12989/cac.2020.26.1.053.
  48. Turan, M. (2022) "Bending analysis of two-directional functionally graded beams using trigonometric series functions", Arch. Appl. Mech., 92, 1841-1858. https://doi.org/10.1007/s00419-022-02152-y.
  49. Turan, M. and Adiyaman, G. (2023), "Free vibration and buckling analysis of porous two-directional functionally graded beams using a higher-order finite element model", J. Vib. Eng. Technol., 1-20. https://doi.org/10.1007/s42417-023-00898-5.
  50. Viet, N.V., Zaki, W. and Wang, Q. (2020), "Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis", J. Appl. Math. Mech., 41, 1787-1804. https://doi.org/10.1007/s10483-020-2664-8.
  51. Vu, A.N.T., Le, N.A.T. and Nguyen, D.K. (2021), "Dynamic behaviour of bidirectional functionally graded sandwich beams under a moving mass with partial foundation supporting effect", Acta Mechanica, 232, 2853-2875. https://doi.org/10.1007/s00707-021-02948-z.
  52. Yaylaci, M., Adiyaman, G., Oner, E. and Birinci, A. (2021a), "Investigation of continuous and discontinuous contact cases in the contact mechanics of graded materials using analytical method and FEM", Comput. Concrete, 27(3), 199-210. https://doi.org/10.12989/cac.2021.27.3.199.
  53. Yaylaci, M., Yayli, M., Uzun Yaylaci, E., Olmez, H. and Birinci, A. (2021b), "Analyzing the contact problem of a functionally graded layer resting on an elastic half plane with theory of elasticity, finite element method and multilayer perceptron", Struct. Eng. Mech., 78(5), 585-597. https://doi.org/10.12989/sem.2021.78.5.585.
  54. Zhao, L., Chen, W.Q. and Lu, C.F. (2012), "Symplectic elasticity for bi-directional functionally graded materials", Mech. Mater., 54, 32-42. https://doi.org/10.1016/j.mechmat.2012.06.001.
  55. Zhao, L., Zhu, J. and Wen, X.D. (2016), "Exact analysis of bi-directional functionally graded beams with arbitrary boundary conditions via the symplectic approach", Struct. Eng. Mech., 59(1), 101-122. https://doi.org/10.12989/sem.2016.59.1.101.
  56. Zhao, S., Zhao, Z., Yang, Z., Ke, L.L., Kitipornchai, S. and Yang, J. (2020), "Functionally graded graphene reinforced composite structures: A review", Eng. Struct., 210, 110339. https://doi.org/10.1016/j.engstruct.2020.110339.
  57. Zhou, J., Moradi, Z., Safa, M. and Khadimallah, M.A. (2022), "Intelligent modeling to investigate the stability of a two-dimensional functionally graded porosity-dependent nanobeams", Comput. Concrete, 30(2), 85-97. https://doi.org/10.12989/cac.2022.30.2.085.