Geomechanics and Engineering

  • 제45권6호
  • /
  • Pages.723-751
  • /
  • 2026
  • /
  • 2005-307X(pISSN)
  • /
  • 2092-6219(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Effect of orthotropic foundation on the stability and bending analysis of FG-CNT porous beam with stretching effect

  • Kenza Djilali Djebbour (Laboratory of Structures, Geotechnics and Risks, Department of Civil Engineering, Hassiba Benbouali University of Chlef) ;
  • Mokhtar Nebab (Laboratory of Structures, Geotechnics and Risks, Department of Civil Engineering, Hassiba Benbouali University of Chlef) ;
  • Hassen Ait Atmane (Laboratory of Structures, Geotechnics and Risks, Department of Civil Engineering, Hassiba Benbouali University of Chlef) ;
  • Riadh Bennai (Laboratory of Structures, Geotechnics and Risks, Department of Civil Engineering, Hassiba Benbouali University of Chlef)
  • 투고 : 2024.05.04
  • 심사 : 2026.05.12
  • 발행 : 2026.06.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we investigate the displacement characteristics and buckling behavior of Functionally graded carbon nanotube (FG CNT) beams resting on an arbitrary orthotropic variable elastic foundation (AOVEF) using a micromechanical Eshelby-Mori-Tanaka approach with an extended rule of mixture. Four material distributions are introduced to describe the variation of carbon nanotube (CNT) volume fractions throughout the beam's thickness. Additionally, the influence of the porosity factor is considered, and a comparison between the Eshelby-Mori-Tanaka and extended rule of mixture is presented. To account for shear deformation and stretching effects, a Quasi-Three-Dimensional higher-order shear deformation beam theory (Quasi-3D HSDT) is employed. The novelty lies in the simultaneous consideration of: (i) an arbitrarily orthotropic variable elastic foundation (AOVEF) with four Winkler variations (parabolic, sinusoidal, exponential, and constant); (ii) a quasi-3D HSDT including stretching effects; (iii) a comparative micromechanical analysis (Eshelby-Mori-Tanaka vs. extended rule of mixture); and (iv) the combined influence of porosity, orthotropy angle, and CNT distributions. The governing equations for the FG-CNT beam with simply supported ends are derived using Hamilton's principle and solved via Navier's method. A comprehensive investigation into the critical buckling load is conducted, considering the influence of various factors including mode shapes, aspect ratio (L/h), as well as angle and porosity variations.

키워드

참고문헌

  1. Chaudhari, V.K., Lal, A. (2016). Nonlinear free vibration analysis of elastically supported nanotube reinforced composite beam in thermal environment. Procedia Engineering, 144, 928-935. https://doi.org/10.1016/j.proeng.2016.05.119.
  2. Iijima, S. (1991). Helical microtubules of graphitic carbon. Nature, 354(6348), 56-58. https://doi.org/10.1038/354056a0.
  3. Wattanasakulpong, N., Ungbhakorn, V. (2013). Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Computational Materials Science, 71, 201-208. https://doi.org/10.1016/j.commatsci.2013.01.028.
  4. Madenci, E. (2021). Free vibration analysis of carbon nanotube RC nanobeams with variational approaches. Advances in Nano Research, 11(2), 157-171. https://doi.org/10.12989/anr.2021.11.2.157.
  5. Shokravi, M. (2017). Buckling analysis of embedded laminated plates with agglomerated CNT-reinforced composite layers using FSDT and DQM. Geomechanics and Engineering, 12(2), 327-346. https://doi.org/10.12989/gae.2017.12.2.327.
  6. Xu, J., She, G.L. (2023). The effects of temperature and porosity on resonance behavior of graphene platelet reinforced metal foams doubly-curved shells with geometric imperfection. Geomechanics and Engineering, 35(1), 81-93. https://doi.org/10.12989/gae.2023.35.1.081.
  7. Zhang, Y.W., Ding, H.X., She, G.L., Tounsi, A. (2023). Wave propagation of CNTRC beams resting on elastic foundation based on various higher-order beam theories. Geomechanics and Engineering, 33(4), 381-391. https://doi.org/10.12989/gae.2023.33.4.381.
  8. Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R., Sadeghi, J. (2014). Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams. Composite Structures, 113, 316-327. https://doi.org/10.1016/j.compstruct.2014.03.015.
  9. Mohseni, A., Shakouri, M. (2019). Vibration and stability analysis of functionally graded CNT-reinforced composite beams with variable thickness on elastic foundation. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 233(12), 2478-2489. https://doi.org/10.1177/1464420719866222.
  10. Kiani, Y., Mirzaei, M. (2019). Nonlinear stability of sandwich beams with carbon nanotube reinforced faces on elastic foundation under thermal loading. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(5), 1701-1712. https://doi.org/10.1177/0954406218772613.
  11. Garg, A., Chalak, H., Zenkour, A., Belarbi, M.O., Sahoo, R. (2022). Bending and free vibration analysis of symmetric and unsymmetric functionally graded CNT reinforced sandwich beams containing softcore. Thin-Walled Structures, 170, 108626. https://doi.org/10.1016/j.tws.2021.108626.
  12. Mohammadimehr, M., Mohammadi-Dehabadi, A., Alavi, S.A., Alambeigi, K., Bamdad, M., Yazdani, R., et al. (2018). Bending, buckling, and free vibration analyses of carbon nanotube reinforced composite beams and experimental tensile test to obtain the mechanical properties of nanocomposite. Steel and Composite Structures, 29(3), 405-422. https://doi.org/10.12989/scs.2018.29.3.405.
  13. Tong, G., Liu, Y., Cheng, Q., Dai, J. (2020). Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect. European Journal of Mechanics - A/Solids, 80, 103929. https://doi.org/10.1016/j.euromechsol.2019.103929.
  14. Doğruoğlu, A.N., Omurtag, M.H. (2000). Stability analysis of composite-plate foundation interaction by mixed FEM. Journal of Engineering Mechanics, 126(9), 928-936. https://doi.org/10.1061/(ASCE)0733-9399(2000)126:9(928).
  15. Ermis, M., Kutlu, A., Eratlı, N., Omurtag, M.H. (2022). Free vibration of axially FG curved beam on orthotropic Pasternak foundation via mixed FEM. Journal of the Brazilian Society of Mechanical Sciences Engineering, 44(12), 597. https://doi.org/10.1007/s40430-022-03853-9.
  16. Çömez, İ., Omurtag, M.H. (2021). Contact problem between a rigid punch and a functionally graded orthotropic layer resting on a Pasternak foundation. Archive of Applied Mechanics, 91(9), 3937-3958. https://doi.org/10.1007/s00419-021-01988-0.
  17. Arefi, M., Mohammadi, M., Tabatabaeian, A., Rabczuk, T. (2020). Free vibration analysis of FG-CNTRC cylindrical pressure vessels resting on Pasternak foundation with various boundary conditions. Computers, Materials & Continua, 62(3), 1001-1023. https://doi.org/10.32604/cmc.2020.08052.
  18. Babaei, H, Kiani, Y., Eslami, M.R. (2021). Vibrational behavior of thermally pre-/post-buckled FG-CNTRC beams on a nonlinear elastic foundation: a two-step perturbation technique. Acta Mechanica, 232(10), 3897-3915. https://doi.org/10.1007/s00707-021-03027-z.
  19. Djebbour, K.D., Nebab, M., Avcar, M., Ait Atmane, H., Bernard, F., Bennai, R., et al. (2025). Stability and vibration analysis of shear deformable FG-CNTRC beams resting on elastic foundations. Advances in Nano Research, 205-221. https://doi.org/10.12989/anr.2025.18.3.205.
  20. Kehli, A., Nebab, M., Bennai, R., Ait Atmane, H., Dahmane, M. (2024). Dynamic characteristics analysis of functionally graded cracked beams resting on viscoelastic medium using a new quasi-3D HSDT. Mechanics of Advanced Materials and Structures, 1-14. https://doi.org10.1080/15376494.2024.2326983.
  21. Zarga, D., Nebab, M., Ait Atmane, H., Lazreg, H. (2025). Influence of nonlinear distribution CNT and variable viscoelastic foundation on wave propagation in composite beams. Advances in Nano Research, 19(4), 399-410. https://doi.org/10.12989/anr.2025.19.4.399.
  22. Berghouti, H., Adda Bedia, E., Benkhedda, A., Tounsi, A. (2019). Vibration analysis of nonlocal porous nanobeams made of functionally graded material. Advances in Nano Research, 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351.
  23. Duc, N.D., Cong, P.H., Tuan, N.D., Tran, P., Van Thanh, N. (2017). Thermal and mechanical stability of functionally graded carbon nanotubes (FG CNT)-reinforced composite truncated conical shells surrounded by the elastic foundations. Thin-Walled Structures, 115, 300-310. https://doi.org/10.1016/j.tws.2017.02.016.
  24. Khaniki, H.B., Ghayesh, M.H., Hussain, S., Amabili, M. (2022). Effects of geometric nonlinearities on the coupled dynamics of CNT strengthened composite beams with porosity, mass and geometric imperfections. Engineering with Computers, 38(4), 3463-3488. https://doi.org/10.1007/s00366-021-01474-9.
  25. Dat, N.D., Thanh, N.V., MinhAnh, V., Duc, N.D. (2022). Vibration and nonlinear dynamic analysis of sandwich FG-CNTRC plate with porous core layer. Mechanics of Advanced Materials and Structures, 29(10), 1431-1448. https://doi.org/10.1080/15376494.2020.1822476.
  26. Ramteke, P.M., Patel, B., Panda, S.K. (2020). Time-dependent deflection responses of porous FGM structure including pattern and porosity. International Journal of Applied Mechanics, 12(9), 2050102. https://doi.org/10.1142/S1758825120501021.
  27. Sayyad, A.S., Avhad, P.V., Hadji, L. (2022). On the static deformation and frequency analysis of functionally graded porous circular beams. Forces in Mechanics, 7, 100093. https://doi.org/10.1016/j.finmec.2022.100093.
  28. Safaei, B., Moradi-Dastjerdi, R., Behdinan, K., Chu, F. (2019). Critical buckling temperature and force in porous sandwich plates with CNT-reinforced nanocomposite layers. Aerospace Science Technology, 91, 175-185. https://doi.org/10.1016/j.ast.2019.05.020.
  29. Raut, A.S., Parker, C.B., Stoner, B.R., Glass, J.T. (2012). Effect of porosity variation on the electrochemical behavior of vertically aligned multi-walled carbon nanotubes. Electrochemistry Communications, 19, 138-141. https://doi.org/10.1016/j.elecom.2012.03.021.
  30. Turan, M., Uzun Yaylacı, E., Yaylacı, M. (2023). Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods. Archive of Applied Mechanics, 93(4), 1351-1372. https://doi.org/10.1007/s00419-022-02332-w.
  31. Wu, H., Yang, J., Kitipornchai, S. (2016). Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections. Composites Part B: Engineering, 90, 86-96. https://doi.org/10.1016/j.compositesb.2015.12.007.
  32. Shenas, A.G., Malekzadeh, P., Ziaee, S. (2017). Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment. Composite Structures, 162, 325-340. https://doi.org/10.1016/j.compstruct.2016.12.009.
  33. Shahraki, H., Riahi, H.T., Izadinia, M., Talaeitaba, S.B. (2021). Mindlin's strain gradient theory for vibration analysis of FG-CNT-reinforced composite nanoplates resting on Kerr foundation in thermal environment. Journal of Thermoplastic Composite Materials, 34(1), 68-101. https://doi.org/10.1177/0892705719843175.
  34. Polit, O., Anant, C., Anirudh, B., Ganapathi, M. (2019). Functionally graded graphene reinforced porous nanocomposite curved beams: Bending and elastic stability using a higher-order model with thickness stretch effect. Composites Part B: Engineering, 166, 310-327. https://doi.org/10.1016/j.compositesb.2018.11.074.
  35. Hamed, M., Abo-Bakr, R., Mohamed, S., Eltaher, M. (2020). Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core. Engineering with Computers, 36, 1929-1946. https://doi.org/10.1007/s00366-020-01023-w.
  36. Sayyad, A.S., Ghugal, Y.M. (2018). Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams. Mechanics of Advanced Composite Structures, 5(1), 13-24. https://doi.org/10.22075/macs.2018.12214.1117.
  37. Sayyad, A.S., Ghugal, Y.M. (2018). Effect of thickness stretching on the static deformations, natural frequencies, and critical buckling loads of laminated composite and sandwich beams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(6), 296. https://doi.org/10.1007/s40430-018-1222-5.
  38. Avhad, P.V., Sayyad, A.S. (2022). On the deformation of laminated composite and sandwich curved beams. Curved And Layered Structures, 9(1), 1-12. https://doi.org/10.1515/cls-2022-0001.
  39. Sayyad, A.S., Avhad, P.V. (2022). A new higher order shear and normal deformation theory for the free vibration analysis of sandwich curved beams. Composite Structures, 280, 114948. https://doi.org/10.1016/j.compstruct.2021.114948.
  40. Arshid, E., Maraghi, Z.K., Civalek, Ö. (2025). Variable-thickness higher-order sandwich beams with FG cellular core and CNT-RC patches: vibrational analysis in thermal environment. Archive of Applied Mechanics, 95(1), 2. https://doi.org/10.1007/s00419-024-02716-0.
  41. Draouche, K., Hadji, L., Madan, R., Zouatnia, N., Ait Atmane, H. (2026). Porosity effects on the thermal buckling of functionally graded plate: Modeling and analysis. Advances in Materials Research, 15(1), 83. https://doi.org/10.12989/amr.2026.15.1.083.
  42. Saad, M., Larbi, L.O., Hadji, L., Ait Atmane, H., Madan, R. (2025). The natural frequency analysis of in-plane bi-directional functionally graded plates under various boundary conditions. Advances in Concrete Construction, 20(3), 217.
  43. Ould Larbi, L., Saad, M., Zouatnia, N., Hadji, L., Sayyad, A.S. (2025). A simple refined plate theory for buckling problems of in-plane bi-directional functionally graded plates with porosity under various boundary conditions. Mechanics of Advanced Materials and Structures, 32(3), 403-412. https://doi.org/10.1080/15376494.2024.2346946.
  44. Amara, R., Riadh, B., Hassen, A.A., Mokhtar, N., Hadji, L. (2025). Hygrothermal effect of bio-inspired helicoid laminate plate for strengthening damaged RC beam. Mechanics of Advanced Materials and Structures, 32(14), 3319-3336. https://doi.org/10.1080/15376494.2024.2392623.
  45. Mehar, K., Panda, S.K., Dehengia, A., Kar, V.R. (2016). Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment. Journal of Sandwich Structures Materials, 18(2), 151-173. https://doi.org/10.1177/1099636215613324.
  46. Torabi, J., Ansari, R., Hassani, R. (2019). Numerical study on the thermal buckling analysis of CNT-reinforced composite plates with different shapes based on the higher-order shear deformation theory. European Journal of Mechanics - A/Solids, 73, 144-160. https://doi.org/10.1016/j.euromechsol.2018.07.009.
  47. Rouhi, M., Rais-Rohani, M. (2013). Modeling and probabilistic design optimization of a nanofiber-enhanced composite cylinder for buckling. Composite Structures, 95, 346-353. https://doi.org/10.1016/j.compstruct.2012.05.035.
  48. Yaylaci, E.U., Yaylaci, M., Ozdemir, M.E., Terzi, M., Ozturk, S. (2023). Analyzing the mechano-bactericidal effect of nano-patterned surfaces by finite element method and verification with artificial neural networks. Advances in Nano Research, 15(2), 165-174. https://doi.org/10.12989/anr.2023.15.2.165.
  49. Yaylaci, E.U., Ozdemir, M.E., Guvercin, Y., Ozturk, S., Yaylaci, M. (2023). Analysis of the mechano-bactericidal effects of nanopatterned surfaces on implant-derived bacteria using the FEM. Advances in Nano Research, 567-577. https://doi.org/10.12989/anr.2023.15.6.567.
  50. Yaylacı, M., Öner, E., Adıyaman, G., Öztürk, Ş., Uzun Yaylacı, E., Birinci, A. (2024). Analyzing of continuous and discontinuous contact problems of a functionally graded layer: theory of elasticity and finite element method. Mechanics Based Design of Structures and Machines, 52(8), 5720-5738. https://doi.org/10.1080/15397734.2023.2262562.
  51. Yaylacı, M., Yaylacı, E.U., Özdemir, M.E., Öztürk, Ş., Sesli, H. (2023). Vibration and buckling analyses of FGM beam with edge crack: Finite element and multilayer perceptron methods. Steel and Composite Structures, 46(4), 565-575. https://doi.org/10.12989/scs.2023.46.4.565.
  52. Ozdemir, M.E., Yaylaci, M. (2023). Research of the impact of material and flow properties on fluid-structure interaction in cage systems. Wind and Structures, 36(1), 31-40. https://doi.org/10.12989/was.2023.36.1.031.
  53. Mehar, K., Panda, S.K., Mahapatra, T.R. (2017). Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure. International Journal of Mechanical Sciences, 133, 319-329. https://doi.org/10.1016/j.ijmecsci.2017.08.057.
  54. Mori, T., Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21(5), 571-574. https://doi.org/10.1016/0001-6160(73)90064-3.
  55. Aragh, B.S., Barati, A.N., Hedayati, H. (2012). Eshelby-Mori-Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels. Composites Part B: Engineering, 43(4), 1943-1954. https://doi.org/10.1016/j.compositesb.2012.01.004.
  56. Asadi, H., Souri, M., Wang, Q. (2017). A numerical study on flow-induced instabilities of supersonic FG-CNT reinforced composite flat panels in thermal environments. Composite Structures, 171, 113-125. https://doi.org/10.1016/j.compstruct.2017.02.003.
  57. Wang, Z.X., Shen, H.S. (2011). Nonlinear vibration of nanotube-reinforced composite plates in thermal environments. Computational Materials Science, 50(8), 2319-2330. https://doi.org/10.1016/j.commatsci.2011.03.005.
  58. Shen, H.S., Zhang, C.L. (2010). Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Materials & Design, 31(7), 3403-3411. https://doi.org/10.1016/j.matdes.2010.01.048.
  59. Shi, D.L., Feng, X.Q., Huang, Y.Y., Hwang, K.C., Gao, H. (2004). The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. Journal of Engineering Materials and Technology, 126(3), 250-257. https://doi.org/10.1115/1.1751182.
  60. Yas, M., Heshmati, M. (2012). Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load. Applied Mathematical Modelling, 36(4), 1371-1394. https://doi.org/10.1016/j.apm.2011.08.037.
  61. Eshelby, J.D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences, 241(1226), 376-396. https://doi.org/10.1098/rspa.1957.0133.
  62. Moradi-Dastjerdi, R., Behdinan, K., Safaei, B., Qin, Z. (2020). Buckling behavior of porous CNT-reinforced plates integrated between active piezoelectric layers. Engineering Structures, 222, 111141. https://doi.org/10.1016/j.engstruct.2020.111141.
  63. Nguyen, L.B., Nguyen, N.V., Thai, C.H., Ferreira, A., Nguyen-Xuan, H. (2019). An isogeometric Bézier finite element analysis for piezoelectric FG porous plates reinforced by graphene platelets. Composite Structures, 214, 227-245. https://doi.org/10.1016/j.compstruct.2019.01.077.
  64. Frahlia, H., Bennai, R., Nebab, M., Ait Atmane, H., Tounsi, A. (2023). Assessing effects of parameters of viscoelastic foundation on the dynamic response of functionally graded plates using a novel HSDT theory. Mechanics of Advanced Materials and Structures, 30(13), 2765-2779. https://doi.org10.1080/15376494.2022.2062632.
  65. Nebab, M., Benguediab, S., Atmane, H.A., Bernard, F. (2020). A simple quasi-3D HDST for dynamic behavior of advanced composite plates with the effect of variables elastic foundations. Geomechanics and Engineering, 22(5), 415-431. https://doi.org/10.12989/gae.2020.22.5.415.
  66. Nebab, M., Dahmane, M., Belqassim, A., Atmane, H.A., Bernard, F., Benadouda, M., et al. (2023). Fundamental frequencies of cracked FGM beams with influence of porosity and Winkler/Pasternak/Kerr foundation support using a new quasi-3D HSDT. Mechanics of Advanced Materials and Structures, 1-13. https://doi.org/10.1080/15376494.2023.2294371.
  67. Aribas, U.N., Ermis, M., Eratli, N., Omurtag, M.H. (2019). The static and dynamic analyses of warping included composite exact conical helix by mixed FEM. Composites Part B: Engineering, 160, 285-297. https://doi.org/10.1016/j.compositesb.2018.10.018.
  68. Nebab, M., Atmane, H.A., Bennai, R., Tounsi, A., Bedia, E.A. (2019). Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT. Structural Engineering and Mechanics, 69(5), 511-525. https://doi.org10.12989/sem.2019.69.5.511.
  69. Djilali Djebbour, K., Mokhtar, N., Hassen, A.A., Alghanmi, R.A., Hadji, L., Riadh, B. (2025). An enhanced quasi-3D HSDT for free vibration analysis of porous FG-CNT beams on a new concept of orthotropic VE-foundations. Mechanics of Advanced Materials and Structures, 32(5), 893-909. https://doi.org10.1080/15376494.2024.2356728.
  70. Eghbali, M., Hosseini, S.A., Hamidi, B.A. (2022). Mantari's higher-order shear deformation theory of sandwich beam with cntrc face layers with porous core under thermal loading. International Journal of Structural Stability Dynamics, 22(16), 2250181. https://doi.org/10.1142/S0219455422501814.
  71. Yas, M., Samadi, N. (2012). Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation. International Journal of Pressure Vessels and Piping, 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012.