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A three-unknown refined shear beam model for the bending of randomly oriented FG-CNT/fiber-reinforced composite laminated beams rested on a new variable elastic foundation

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Abstract

Advanced composite materials are widely employed in several industrial structures such as turbine blades, cutting tools, and aircraft engines. Given the need for analytical and numerical analysis of these complex structures, a mathematical model of novel structures is presented in this paper. The main aim of the present work is to analyze the static bending response of laminated composite beams reinforced by both functionally graded (FG) fibers and randomly oriented single-walled carbon nanotubes rested on a new variable elastic foundation (EF). The fibers volume fraction is changed along the beam thickness from layer to layer in a linear manner, whereas the CNTs volume fraction is uniformly distributed. Three distribution patterns, namely FG-V, FG-O, and FG-X, are considered here to define the fiber-reinforced elements distribution in addition to the uniform distribution UD. A new shear deformation theory is proposed to depict the kinematic displacement field and the requirement of zero transverse shear stresses at the upper and the lower surfaces of the FG beam are satisfied. Further, the present theory obviates the use to shear correction factors as it satisfies the parabolic variation of through-thickness shear stress distribution. Three types of EFs (including linear, trigonometric and reverse trigonometric) are selected here for the analysis. Virtual work principle is exploited to derive the equilibrium equations and Fourier series is employed to get a numerical solution. A detailed parametric analysis was carried out to highlight the impact of various schemes of material distributions, volume fractions and the EF parameters on the deflection of FG CNTs/Fiber reinforced composite beam.

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References

  1. Daikh, A.A., Houari, M.S., Eltaher, M.A.: A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates. Compos. Struct. 262, 113347 (2021). https://doi.org/10.1016/j.compstruct.2020.113347

    Article  Google Scholar 

  2. Mohamed, N., Mohamed, S.A., Eltaher, M.A.: Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng. Comput. 37, 2823–2836 (2021). https://doi.org/10.1007/s00366-020-00976-2

    Article  Google Scholar 

  3. Zuo, Y.T., Liu, H.J.: Fractal approach to mechanical and electrical properties of graphene/sic composites. Facta Univ. Ser. Mech. Eng. 19, 271–284 (2021). https://doi.org/10.22190/FUME201212003Z

    Article  Google Scholar 

  4. He, C.H., Liu, S.H., Liu, C., Mohammad-Sedighi, H.: A novel bond stress-slip model for 3-D printed concretes. Discret. Contin. Dyn. Syst. S 14, 3459–3478 (2021). https://doi.org/10.3934/dcdss.2021161

    Article  Google Scholar 

  5. Ji, F.Y., He, C.H., Zhang, J.J., He, J.H.: A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar. Appl. Math. Model. 82, 437–448 (2020). https://doi.org/10.1016/j.apm.2020.01.027

    Article  MathSciNet  MATH  Google Scholar 

  6. Shen, H.S., Zhang, C.L.: Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Mater. Des. 31, 3403–3411 (2010). https://doi.org/10.1016/j.matdes.2010.01.048

    Article  Google Scholar 

  7. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.-O., Houari, M.-S.-A.: A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures. Arch. Comput. Methods Eng. 29(4), 2237–2270 (2022). https://doi.org/10.1007/s11831-021-09652-0

    Article  MathSciNet  Google Scholar 

  8. Garg, A., Chalak, H.D., Belarbi, M.O., Zenkour, A.M., Sahoo, R.: Estimation of carbon nanotubes and their applications as reinforcing composite materials—an engineering review. Compos. Struct. 272, 114234 (2021). https://doi.org/10.1016/j.compstruct.2021.114234

    Article  Google Scholar 

  9. Liew, K.M., Lei, Z.X., Zhang, L.W.: Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review. Compos. Struct. 120, 90–97 (2015). https://doi.org/10.1016/j.compstruct.2014.09.041

    Article  Google Scholar 

  10. Liu, D., Geng, T., Wang, H., Esmaeili, S.: Analytical solution for thermoelastic oscillations of nonlocal strain gradient nanobeams with dual-phase-lag heat conduction. Mech. Based Des. Struct. Mach. (2021). https://doi.org/10.1080/15397734.2021.1987261

    Article  Google Scholar 

  11. Yu, J.N., She, C., Xu, Y.P., Esmaeili, S.: On size-dependent generalized thermoelasticity of nanobeams. Waves Random Complex Media (2022). https://doi.org/10.1080/17455030.2021.2019351

    Article  Google Scholar 

  12. Kumar, P., Srinivas, J.: Free vibration, bending and buckling of a FG-CNT reinforced composite beam: Comparative analysis with hybrid laminated composite beam. Multidiscip. Model. Mater. Struct. 13(4), 590–611 (2017). https://doi.org/10.1108/MMMS-05-2017-0032

    Article  Google Scholar 

  13. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.: Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle. Steel Compos. Struct. 32(5), 595–610 (2019). https://doi.org/10.12989/scs.2019.32.5.595

    Article  Google Scholar 

  14. Yue, X., Yue, X., Borjalilou, V.: Generalized thermoelasticity model of nonlocal strain gradient Timoshenko nanobeams. Archiv. Civ. Mech. Eng 21, 124 (2021). https://doi.org/10.1007/s43452-021-00280-w

    Article  Google Scholar 

  15. Taati, E., Borjalilou, V., Fallah, F., Ahmadian, M.T.: On size-dependent nonlinear free vibration of carbon nanotube-reinforced beams based on the nonlocal elasticity theory: perturbation technique. Mech. Based Des. Struct. Mach. 50(6), 2124–2146 (2022). https://doi.org/10.1080/15397734.2020.1772087

    Article  Google Scholar 

  16. Kiani, Y., Mirzaei, M.: Nonlinear stability of sandwich beams with carbon nanotube reinforced faces on elastic foundation under thermal loading. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 233(5), 1701–1712 (2019). https://doi.org/10.1177/0954406218772613

    Article  Google Scholar 

  17. Belarbi, M.-O., Garg, A., Houari, M.-S.-A., Hirane, H., Tounsi, A., Chalak, H.D.: A three-unknown refined shear beam element model for buckling analysis of functionally graded curved sandwich beams. Eng. Comput. (2021). https://doi.org/10.1007/s00366-021-01452-1

    Article  Google Scholar 

  18. Belarbi, M.-O., Houari, M.S.A., Hirane, H., Daikh, A.A., Bordas, S.P.A.: On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory. Compos. Struct. 279, 114715 (2022). https://doi.org/10.1016/j.compstruct.2021.114715

    Article  Google Scholar 

  19. Belarbi, M.-O.A., Khechai, A., Bessaim, M.-S.-A., Houari, A., Garg, H., Chalak, H.: Finite element bending analysis of symmetric and non-symmetric functionally graded sandwich beams using a novel parabolic shear deformation theory. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 0(0), 14644207211005096 (2021). https://doi.org/10.1177/14644207211005096

    Article  Google Scholar 

  20. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Mohamed, S.A., Eltaher, M.A.: Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory. Defence Technol. (2021). https://doi.org/10.1016/j.dt.2021.09.011

    Article  Google Scholar 

  21. Van Vinh, P., Tounsi, A., Belarbi, M.-O.: On the nonlocal free vibration analysis of functionally graded porous doubly curved shallow nanoshells with variable nonlocal parameters. Eng. Comput. (2022). https://doi.org/10.1007/s00366-022-01687-6

    Article  Google Scholar 

  22. Vinh, P.V., Belarbi, M.-O., Tounsi, A.: Wave propagation analysis of functionally graded nanoplates using nonlocal higher-order shear deformation theory with spatial variation of the nonlocal parameters. Waves Random Complex Media (2022). https://doi.org/10.1080/17455030.2022.2036387

    Article  Google Scholar 

  23. Nikam, R.D., Sayyad, A.S.: A unified nonlocal formulation for bending, buckling and free vibration analysis of nanobeams. Mech. Adv. Mater. Struct. 27(10), 807–815 (2020). https://doi.org/10.1080/15376494.2018.1495794

    Article  Google Scholar 

  24. Belarbi, M.O., Daikh, A.A., Garg, A., Hirane, H., Houari, M.S.A., Civalek, Ö., Chalak, H.: D, Bending and free vibration analysis of porous functionally graded sandwich plate with various porosity distributions using an extended layerwise theory. Arch. Civ. Mech. Eng. 23(1), 15 (2022). https://doi.org/10.1007/s43452-022-00551-0

    Article  Google Scholar 

  25. Garg, A., Chalak, H.D., Belarbi, M.O., et al.: Finite element-based free vibration analysis of power-law, exponential and sigmoidal functionally graded sandwich beams. J. Inst. Eng. India Ser. C 102, 1167–1201 (2021). https://doi.org/10.1007/s40032-021-00740-5

    Article  Google Scholar 

  26. Garg, A., Belarbi, M.O., Chalak, H.D., Li, L., Sharma, A., Avcar, M., Sharma, N., Paruthi, S., Gulia, R.: Buckling and free vibration analysis of bio-inspired laminated sandwich plates with helicoidal/Bouligand face sheets containing softcore. Ocean Eng. 270, 113684 (2023). https://doi.org/10.1016/j.oceaneng.2023.113684

    Article  Google Scholar 

  27. Yas, M., Samadi, N.: Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation. Int. J. Press. Vessels Pip. 98, 119–128 (2012). https://doi.org/10.1016/j.ijpvp.2012.07.012

    Article  Google Scholar 

  28. Yang, J., Ke, L.-L., Feng, C.: Dynamic buckling of thermo-electro-mechanically loaded FG-CNTRC beams. Int. J. Struct. Stabil. 15(08), 1540017 (2015). https://doi.org/10.1142/S0219455415400179

    Article  MathSciNet  MATH  Google Scholar 

  29. Shi, Z., Yao, X., Pang, F., Wang, Q.: An exact solution for the free-vibration analysis of functionally graded carbon-nanotube-reinforced composite beams with arbitrary boundary conditions. Sci. Rep. 7(1), 1–18 (2017). https://doi.org/10.1038/s41598-017-12596-w

    Article  Google Scholar 

  30. Ebrahimi, F., Farazamandnia, N.: Thermo-mechanical analysis of carbon nanotube-reinforced composite sandwich beams. Coupled Syst. Mech. 6, 207–227 (2017). https://doi.org/10.12989/csm.2017.6.2.207

    Article  Google Scholar 

  31. Ebrahimi, F., Farazmandnia, N.: Vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in thermal environment. Adv. Aircr. Spacecr. Sci. 5(1), 107 (2018). https://doi.org/10.12989/aas.2018.5.1.107

    Article  Google Scholar 

  32. Belarbi, M.O., Salami, S.J., Garg, A., et al.: Mechanical behavior analysis of FG-CNT-reinforced polymer composite beams via a hyperbolic shear deformation theory. Contin. Mech. Thermodyn. 35, 497–520 (2023). https://doi.org/10.1007/s00161-023-01191-2

    Article  MathSciNet  Google Scholar 

  33. Ladmek, M., Belkacem, A., Daikh, A.A., Bessaim, A., Garg, A., Houari, M.S.A., Belarbi, M.O., Ouldyerou, A.: Free vibration of functionally graded carbon nanotubes reinforced composite nanobeams. Adv. Mater. Res. 12(2), 161–177 (2023). https://doi.org/10.12989/amr.2023.12.2.161

    Article  Google Scholar 

  34. Mohseni, A., Shakouri, M.: Vibration and stability analysis of functionally graded CNT-reinforced composite beams with variable thickness on elastic foundation. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 233(12), 2478–2489 (2019). https://doi.org/10.1177/1464420719866222

    Article  Google Scholar 

  35. Talebizadehsardari, P., Eyvazian, A., Asmael, M., Karami, B., Shahsavari, D., Mahani, R.B.: Static bending analysis of functionally graded polymer composite curved beams reinforced with carbon nanotubes. Thin Wall. Struct. 157, 107139 (2020). https://doi.org/10.1016/j.tws.2020.107139

    Article  Google Scholar 

  36. Tong, G., Liu, Y., Cheng, Q., Dai, J.: Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect. Eur. J. Mech. A Solids 80, 103929 (2020). https://doi.org/10.1016/j.euromechsol.2019.103929

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, Y.: Combined effect of carbon nanotubes distribution and orientation on functionally graded nanocomposite beams using finite element analysis. Mater. Res. Express (2020). https://doi.org/10.1088/2053-1591/abc773

    Article  Google Scholar 

  38. Volovoi, V.V., Hodges, D.H., Cesnik, C.E., Popescu, B.: Assessment of beam modeling methods for rotor blade applications. Math. Comput. Model. 33(10–11), 1099–1112 (2001). https://doi.org/10.1016/S0895-7177(00)00302-2

    Article  MATH  Google Scholar 

  39. Akbaş, ŞD., Fageehi, Y.A., Assie, A.E., Eltaher, M.A.: Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load. Eng. Comput. 38, 365–377 (2022). https://doi.org/10.1007/s00366-020-01070-3

    Article  Google Scholar 

  40. Patil, H.H., Pitchaimani, J., Eltaher, M.A.: Buckling and vibration of beams using Ritz method: effects of axial grading of GPL and axially varying load. Mech. Adv. Mater. Struct. (2023). https://doi.org/10.1080/15376494.2023.2185711

    Article  Google Scholar 

  41. Tagrara, S., Benachour, A., Bouiadjra, M.B., Tounsi, A.: On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams. Steel Compos. Struct. 19(5), 1259–1277 (2015). https://doi.org/10.12989/scs.2015.19.5.1259

    Article  Google Scholar 

  42. Shen, H.-S., He, X.-Q., Yang, D.-Q.: Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations. Int. J. Non-Linear Mech. 91, 69–75 (2017). https://doi.org/10.1016/j.ijnonlinmec.2017.02.010

    Article  Google Scholar 

  43. Mayandi, K., Jeyaraj, P.: Bending, buckling and free vibration characteristics of FG-CNT-reinforced polymer composite beam under non-uniform thermal load. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 229(1), 13–28 (2015). https://doi.org/10.1177/1464420713493720

    Article  Google Scholar 

  44. Salami, S.J.: Extended high order sandwich panel theory for bending analysis of sandwich beams with carbon nanotube reinforced face sheets. Physica E 76, 187–197 (2016). https://doi.org/10.1016/j.physe.2015.10.015

    Article  Google Scholar 

  45. Mohammadimehr, M., Alimirzaei, S.: Buckling and free vibration analysis of tapered FG-CNTRC micro Reddy beam under longitudinal magnetic field using FEM. Smart Struct. Syst. 19(3), 309–322 (2017). https://doi.org/10.12989/sss.2017.19.3.309

    Article  Google Scholar 

  46. Khelifa, Z., Hadji, L., Daouadji, T.H., Bourada, M.: Buckling response with stretching effect of carbon nanotube-reinforced composite beams resting on elastic foundation. Struct. Eng. Mech 67(2), 125–130 (2018). https://doi.org/10.12989/sem.2018.67.2.125

    Article  Google Scholar 

  47. Daikh, A.A., Drai, A., Houari, M.S.A., Eltaher, M.A.: Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes. Steel Compos. Struct. 36(6), 643–656 (2020). https://doi.org/10.12989/scs.2020.36.6.643

    Article  Google Scholar 

  48. Karami, B., Janghorban, M., Shahsavari, D., Dimitri, R., Tornabene, F.: Nonlocal buckling analysis of composite curved beams reinforced with functionally graded carbon nanotubes. Molecules 24(15), 2750 (2019). https://doi.org/10.3390/molecules24152750

    Article  Google Scholar 

  49. Nejati, M., Eslampanah, A., Najafizadeh, M.: Buckling and vibration analysis of functionally graded carbon nanotube-reinforced beam under axial load. Int. J. Appl. Mech. 08(01), 1650008 (2016). https://doi.org/10.1142/s1758825116500083

    Article  Google Scholar 

  50. Nazarenko, L., Chirkov, A.Y., Stolarski, H., Altenbach, H.: On modeling of carbon nanotubes reinforced materials and on influence of carbon nanotubes spatial distribution on mechanical behavior of structural elements. Int. J. Eng. Sci. 143, 1–13 (2019). https://doi.org/10.1016/j.ijengsci.2019.06.008

    Article  MathSciNet  MATH  Google Scholar 

  51. Fu, T., Chen, Z., Yu, H., Wang, Z., Liu, X.: Mechanical behavior of laminated functionally graded carbon nanotube reinforced composite plates resting on elastic foundations in thermal environments. J. Compos. Mater. 53(9), 1159–1179 (2019). https://doi.org/10.1177/0021998318796170

    Article  Google Scholar 

  52. Karamanli, A., Vo, T.P.: Finite element model for carbon nanotube-reinforced and graphene nanoplatelet-reinforced composite beams. Compos. Struct. 264, 113739 (2021). https://doi.org/10.1016/j.compstruct.2021.113739

    Article  Google Scholar 

  53. Garg, A., Chalak, H.D., Zenkour, A.M., Belarbi, M.O., Sahoo, R.: Bending and free vibration analysis of symmetric and unsymmetric functionally graded CNT reinforced sandwich beams containing softcore. Thin Wall. Struct. 170, 108626 (2022). https://doi.org/10.1016/j.tws.2021.108626

    Article  Google Scholar 

  54. Yas, M.H., Heshmati, M.: Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load. Appl. Math. Model. 36(4), 1371–1394 (2012). https://doi.org/10.1016/j.apm.2011.08.037

    Article  MathSciNet  MATH  Google Scholar 

  55. Melaibari, A., Daikh, A.A., Basha, M., Wagih, A., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A., Eltaher, M.A.: A dynamic analysis of randomly oriented functionally graded carbon nanotubes/fiber-reinforced composite laminated shells with different geometries. Math. Mech. Solids 10(3), 408 (2022). https://doi.org/10.3390/math10030408

    Article  Google Scholar 

  56. Melaibari, A., Daikh, A.A., Basha, M., Abdalla, A.W., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A., Eltaher, M.A.: Free vibration of FG-CNTRCs nano-plates/shells with temperature-dependent properties. Mathematics 10(4), 583 (2022). https://doi.org/10.3390/math10040583

    Article  Google Scholar 

  57. Kazakov, I.A., Krasnovskii, A.N., Kishuk, P.S.: The influence of randomly oriented CNTs on the elastic properties of unidirectionally aligned composites. Mech. Mater. 134, 54–60 (2019). https://doi.org/10.1016/j.mechmat.2019.04.002

    Article  Google Scholar 

  58. Bisheh, H., Wu, N.: Wave propagation in piezoelectric cylindrical composite shells reinforced with angled and randomly oriented carbon nanotubes. Compos. B Eng. 160, 10–30 (2019). https://doi.org/10.1016/j.compositesb.2018.10.001

    Article  Google Scholar 

  59. Yengejeh, S.I., Kazemi, S.A., Öchsner, A.: Carbon nanotubes as reinforcement in composites: A review of the analytical, numerical and experimental approaches. Comput. Mater. Sci. 136, 85–101 (2017). https://doi.org/10.1016/j.commatsci.2017.04.023

    Article  Google Scholar 

  60. Shahmohammadi, M.A., Azhari, M., Salehipour, H., Civalek, Ö.: A novel composite model for vibration of thin-walled layered composite panels incorporating the agglomeration of CNTs. Aerosp. Sci. Technol. 116, 106897 (2021). https://doi.org/10.1016/j.ast.2021.106897

    Article  Google Scholar 

  61. Belarbi, M.O., Salami, S., Garg, A., Hiran, H., Daikh, A., Houari, M.S.A.: Finite element bending and buckling analysis of functionally graded carbon nanotubes-reinforced composite beam under arbitrary boundary conditions. Steel Compos. Struct. 44(4), 451–471 (2022). https://doi.org/10.12989/scs.2022.44.4.451

    Article  Google Scholar 

  62. Bensaid, I., Daikh, A.A.: Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects. Proc. IMechE Part C J.. Mech. Eng. Sci. 234(18), 3667–3688 (2020). https://doi.org/10.1177/0954406220916481

    Article  Google Scholar 

  63. Alazwari, M.A., Daikh, A.A., Houari, M.S.A., Tounsi, A., Eltaher, M.A.: On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations. Steel Compos. Struct. 40(3), 389–404 (2021). https://doi.org/10.12989/scs.2021.40.3.389

    Article  Google Scholar 

  64. Wattanasakulpong, N., Ungbhakorn, V.: Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Comput. Mater. Sci. 71, 201–208 (2013). https://doi.org/10.1016/j.commatsci.2013.01.028

    Article  Google Scholar 

  65. Kumar, M., Sarangi, S.K.: Bending and vibration study of carbon nanotubes reinforced functionally graded smart composite beams. Eng. Res. Express 4, 025043 (2022). https://doi.org/10.1088/2631-8695/ac76a0

    Article  Google Scholar 

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Acknowledgements

This research was supported by The Algerian General Directorate of Scientific Research and Technological Development (DGRSDT) and University of Mohamed Khider of Biskra in Algeria.

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Authors and Affiliations

  1. Artificial Intelligence Laboratory for Mechanical and Civil Structures, and Soil, University Center of Naama, P.O. Box 66, Naama, 45000, Algeria

    Ahmed Amine Daikh

  2. Laboratoire d’Etude des Structures et de Mécanique des Matériaux, Département de Génie Civil, Faculté des Sciences et de la Technologie, Université Mustapha Stambouli, B.P. 305, R.P., 29000, Mascara, Algérie

    Ahmed Amine Daikh, Miloud Ladmek, Abdelkader Belkacem & Mohamed Sid Ahmed Houari

  3. Laboratoire de Recherche en Génie Civil, LRGC, Université de Biskra, B.P. 145, R.P., 07000, Biskra, Algeria

    Mohamed-Ouejdi Belarbi

  4. Department of Biomedical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran

    Sattar Jedari Salami

  5. Civil Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Hani Magdy Ahmed

  6. Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Mohamed A. Eltaher

  7. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Mohamed A. Eltaher

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Daikh, A.A., Belarbi, MO., Salami, S.J. et al. A three-unknown refined shear beam model for the bending of randomly oriented FG-CNT/fiber-reinforced composite laminated beams rested on a new variable elastic foundation. Acta Mech 234, 5171–5186 (2023). https://doi.org/10.1007/s00707-023-03657-5

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