Skip to main content
SpringerLink
Log in

On the vibration and buckling behaviors of porous FG beams resting on variable elastic foundation utilizing higher-order shear deformation theory

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Using a three-variable higher-order shear deformation theory (HSDT), this research proposes an analytical method for studying the free vibration and stability of perfect and imperfect functionally graded (FG) beams resting on variable elastic foundations (VEFs). Unlike the other HSDTs, in this study, the number of unknown functions involved is only three, while the other HSDTs include four unknown functions. Besides, this theory meets the boundary requirements of zero tension on the beam surfaces and allows for hyperbolic distributions of transverse shear stresses without the need for shear correction factors. The elastic medium is supposed to have two parameters (i.e., Winkler–Pasternak foundations), with the Winkler parameter in the longitudinal direction being variable variations (linear, parabolic, sinusoidal, cosine, exponential, and uniform) and the Pasternak parameter being fixed, at first.1 The effective material characteristics of the FG beam are assumed to follow a simple power-law distribution in the thickness direction. Furthermore, the influence of porosity is investigated by considering four distinct types of porosity distribution patterns. First, the equations of motion are derived using Hamilton’s principle, and then Navier’s method is used to solve the system of equations for the FG beam with simply supported ends analytically. The correctness of the current formulation is demonstrated by comparing them with the results of open literature. Finally, parametric studies are done to explore the impacts of various parameters on the free vibration and buckling behaviors of FG beams. The new theory is shown to be not only correct but also simple in predicting the free vibration and buckling responses of FG beams resting on VEFs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47, 663–684 (2000). https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3

    Article  MATH  Google Scholar 

  2. Malekzadeh, P.: Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations. Compos. Struct. 89, 367–373 (2009). https://doi.org/10.1016/J.COMPSTRUCT.2008.08.007

    Article  Google Scholar 

  3. Shen, H.S., Wang, Z.X.: Assessment of Voigt and Mori-Tanaka models for vibration analysis of functionally graded plates. Compos. Struct. 94, 2197–2208 (2012). https://doi.org/10.1016/J.COMPSTRUCT.2012.02.018

    Article  Google Scholar 

  4. Thai, H.T., Choi, D.H.: A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Compos. Part B Eng. 43, 2335–2347 (2012). https://doi.org/10.1016/J.COMPOSITESB.2011.11.062

    Article  Google Scholar 

  5. Bennai, R., Atmane, H.A., Tounsi, A.: A new higher-order shear and normal deformation theory for functionally graded sandwich beams. Steel Compos. Struct. 19, 521–546 (2015). https://doi.org/10.12989/scs.2015.19.3.521

    Article  Google Scholar 

  6. Mantari, J.L.: Free vibration of advanced composite plates resting on elastic foundations based on refined non-polynomial theory. Meccanica 50, 2369–2390 (2015). https://doi.org/10.1007/S11012-015-0160-X

    Article  MathSciNet  MATH  Google Scholar 

  7. Benferhat, R., Daouadji, T.H., Mansour, M.S., Hadji, L.: Effect of porosity on the bending and free vibration response of functionally graded plates resting on Winkler-Pasternak foundations. Earthq. Struct. 10, 1429–1449 (2016). https://doi.org/10.12989/EAS.2016.10.6.1429

    Article  Google Scholar 

  8. Avcar, M.: Free vibration of imperfect sigmoid and power law functionally graded beams. Steel Compos. Struct. 30, 603–615 (2019). https://doi.org/10.12989/SCS.2019.30.6.603

    Article  Google Scholar 

  9. Batou, B., Nebab, M., Bennai, R., Atmane, H.A., Tounsi, A., Bouremana, M.: Wave dispersion properties in imperfect sigmoid plates using various HSDTs. Steel Compos. Struct. 33, 699–716 (2019). https://doi.org/10.12989/scs.2019.33.5.699

    Article  Google Scholar 

  10. Frahlia, H., Bennai, R., Nebab, M., Atmane, H.A., Tounsi, A.: Assessing effects of parameters of viscoelastic foundation on the dynamic response of functionally graded plates using a novel HSDT theory. Mech. Adv. Mater. Struct. (2022). https://doi.org/10.1080/15376494.2022.2062632

    Article  Google Scholar 

  11. Dehghan, M., Baradaran, G.H.: Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method. Appl. Math. Comput. 218, 2772–2784 (2011). https://doi.org/10.1016/J.AMC.2011.08.020

    Article  MathSciNet  MATH  Google Scholar 

  12. Grover, N., Maiti, D.K., Singh, B.N.: A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates. Compos. Struct. 95, 667–675 (2013). https://doi.org/10.1016/J.COMPSTRUCT.2012.08.012

    Article  Google Scholar 

  13. Yaghoobi, H., Fereidoon, A.: Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: an assessment of a simple refined nth-order shear deformation theory. Compos. Part B Eng. 62, 54–64 (2014). https://doi.org/10.1016/J.COMPOSITESB.2014.02.014

    Article  Google Scholar 

  14. Barati, M.R., Sadr, M.H., Zenkour, A.M.: Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation. Int. J. Mech. Sci. 117, 309–320 (2016). https://doi.org/10.1016/J.IJMECSCI.2016.09.012

    Article  Google Scholar 

  15. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.: An analytical solution for bending, buckling and vibration responses of FGM sandwich plates. J. Sandw. Struct. Mater. 21, 727–757 (2019). https://doi.org/10.1177/1099636217698443

    Article  Google Scholar 

  16. Hajlaoui, A., Dammak, F.: A modified first shear deformation theory for three-dimensional thermal post-buckling analysis of FGM plates. Meccanica 57, 337–353 (2022). https://doi.org/10.1007/S11012-021-01427-Y/TABLES/5

    Article  MathSciNet  Google Scholar 

  17. Atmane, H.A., Tounsi, A., Mechab, I., Bedia, E.A.A.: Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory. Int. J. Mech. Mater. Des. 6, 113–121 (2010). https://doi.org/10.1007/S10999-010-9110-X

    Article  Google Scholar 

  18. Kumar, R., Patil, H., Thermoplastic, A.L.-J.: Hygrothermoelastic free vibration response of laminated composite plates resting on elastic foundations with random system properties: micromechanical. J. Thermoplast. Compos. Mater. 26, 573–604 (2013). https://doi.org/10.1177/0892705711425851

    Article  Google Scholar 

  19. Sobhy, M.: Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos. Struct. 99, 76–87 (2013). https://doi.org/10.1016/J.COMPSTRUCT.2012.11.018

    Article  Google Scholar 

  20. Akavci, S.S.: An efficient shear deformation theory for free vibration of functionally graded thick rectangular plates on elastic foundation. Compos. Struct. 108, 667–676 (2014). https://doi.org/10.1016/J.COMPSTRUCT.2013.10.019

    Article  Google Scholar 

  21. Han, S.C., Park, W.T., Jung, W.Y.: 3D graphical dynamic responses of FGM plates on Pasternak elastic foundation based on quasi-3D shear and normal deformation theory. Compos. Part B Eng. 95, 324–334 (2016). https://doi.org/10.1016/J.COMPOSITESB.2016.04.018

    Article  Google Scholar 

  22. Nebab, M., Atmane, H.A., Bennai, R., Tounsi, A., Bedia, E.A.A.: Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT. Struct. Eng. Mech. Int. J. 69, 511–525 (2019)

    Google Scholar 

  23. Ramteke, P.M., Panda, S.K., Sharma, N., Ramteke, P.M., Panda, S.K., Sharma, N.: Effect of grading pattern and porosity on the eigen characteristics of porous functionally graded structure. Steel Compos. Struct. 33, 865 (2019). https://doi.org/10.12989/SCS.2019.33.6.865

    Article  Google Scholar 

  24. AlSaid-Alwan, H.H.S., Avcar, M.: Analytical solution of free vibration of FG beam utilizing different types of beam theories: a comparative study. Comput. Concr. Int. J. 26, 285–292 (2020)

    Google Scholar 

  25. Ramteke, P.M., Mahapatra, B.P., Panda, S.K., Sharma, N.: Static deflection simulation study of 2D Functionally graded porous structure. Mater. Today Proc. 33, 5544–5547 (2020). https://doi.org/10.1016/J.MATPR.2020.03.537

    Article  Google Scholar 

  26. Ramteke, P.M., Patel, B., Panda, S.K.: Time-dependent deflection responses of porous FGM structure including pattern and porosity. Int. J. Appl Mech. (2020). https://doi.org/10.1142/S1758825120501021

    Article  Google Scholar 

  27. 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. 276, 114564 (2021). https://doi.org/10.1016/J.COMPSTRUCT.2021.114564

    Article  Google Scholar 

  28. Fan, F., Cai, X., Sahmani, S., Safaei, B.: Isogeometric thermal postbuckling analysis of porous FGM quasi-3D nanoplates having cutouts with different shapes based upon surface stress elasticity. Compos. Struct. 262, 113604 (2021). https://doi.org/10.1016/J.COMPSTRUCT.2021.113604

    Article  Google Scholar 

  29. Ramteke, P.M., Panda, S.K.: Free vibrational behaviour of multi-directional porous functionally graded structures. Arab. J. Sci. Eng. 46, 7741–7756 (2021). https://doi.org/10.1007/S13369-021-05461-6/TABLES/7

    Article  Google Scholar 

  30. Ramteke, P.M., Mehar, K., Sharma, N., Panda, S.K.: Numerical prediction of deflection and stress responses of functionally graded structure for grading patterns (power-law, sigmoid, and exponential) and variable porosity (even/uneven). Sci. Iran. 28, 811–829 (2021). https://doi.org/10.24200/SCI.2020.55581.4290

    Article  Google Scholar 

  31. Ramteke, P.M., Patel, B., Panda, S.K.: Nonlinear eigenfrequency prediction of functionally graded porous structure with different grading patterns. Waves Random Complex Media (2021). https://doi.org/10.1080/17455030.2021.2005850

    Article  Google Scholar 

  32. Rao, R., Sahmani, S., Safaei, B.: Isogeometric nonlinear bending analysis of porous FG composite microplates with a central cutout modeled by the couple stress continuum quasi-3D plate theory. Arch. Civ. Mech. Eng. 21, 1–17 (2021). https://doi.org/10.1007/S43452-021-00250-2/METRICS

    Article  Google Scholar 

  33. Song, R., Sahmani, S., Safaei, B.: Isogeometric nonlocal strain gradient quasi-three-dimensional plate model for thermal postbuckling of porous functionally graded microplates with central cutout with different shapes. Appl. Math. Mech. (Engl. Ed.) 42, 771–786 (2021). https://doi.org/10.1007/S10483-021-2725-7/METRICS

    Article  MathSciNet  MATH  Google Scholar 

  34. Hadji, L., Avcar, M., Zouatnia, N.: Natural frequency analysis of imperfect FG sandwich plates resting on Winkler–Pasternak foundation. Mater. Today Proc. (2022). https://doi.org/10.1016/J.MATPR.2021.12.485

    Article  Google Scholar 

  35. Choudhary, J., Patle, B.K., Ramteke, P.M., Hirwani, C.K., Panda, S.K., Katariya, P.V.: Static and dynamic deflection characteristics of cracked porous FG panels. Int. J. Appl. Mech. (2022). https://doi.org/10.1142/S1758825122500764

    Article  Google Scholar 

  36. Hissaria, P., Ramteke, P.M., Hirwani, C.K., Mahmoud, S.R., Kumar, E.K., Panda, S.K.: Numerical Investigation of eigenvalue characteristics (vibration and buckling) of damaged porous bidirectional FG panels. J. Vib. Eng. Technol. 1, 1–13 (2022). https://doi.org/10.1007/S42417-022-00677-8/TABLES/10

    Article  Google Scholar 

  37. Ramteke, P.M., Panda, S.K., Patel, B.: Nonlinear eigenfrequency characteristics of multi-directional functionally graded porous panels. Compos. Struct. 279, 114707 (2022). https://doi.org/10.1016/J.COMPSTRUCT.2021.114707

    Article  Google Scholar 

  38. Ramteke, P.M., Sharma, N., Choudhary, J., Hissaria, P., Panda, S.K.: Multidirectional grading influence on static/dynamic deflection and stress responses of porous FG panel structure: a micromechanical approach. Eng. Comput. 38, 3077–3097 (2022). https://doi.org/10.1007/S00366-021-01449-W/FIGURES/18

    Article  Google Scholar 

  39. Ramteke, P.M., Panda, S.K., Sharma, N.: Nonlinear vibration analysis of multidirectional porous functionally graded panel under thermal environment. AIAA J. 60, 4923–4933 (2022). https://doi.org/10.2514/1.J061635

    Article  Google Scholar 

  40. Ramteke, P.M., Kumar, V., Sharma, N., Panda, S.K.: Geometrical nonlinear numerical frequency prediction of porous functionally graded shell panel under thermal environment. Int. J. Non-Linear Mech. 143, 104041 (2022). https://doi.org/10.1016/J.IJNONLINMEC.2022.104041

    Article  Google Scholar 

  41. Sahoo, B., Sharma, N., Sahoo, B., Malhari Ramteke, P., Kumar Panda, S., Mahmoud, S.R.: Nonlinear vibration analysis of FGM sandwich structure under thermal loadings. Structures 44, 1392–1402 (2022). https://doi.org/10.1016/J.ISTRUC.2022.08.081

    Article  Google Scholar 

  42. Wang, J., Ma, B., Gao, J., Liu, H., Safaei, B., Sahmani, S.: Nonlinear stability characteristics of porous graded composite microplates including various microstructural-dependent strain gradient tensors. Int. J. Appl. Mech. (2022). https://doi.org/10.1142/S1758825121501295

    Article  Google Scholar 

  43. Safaei, B., Onyibo, E.C., Goren, M., Kotrasova, K., Yang, Z., Arman, S., Asmael, M.: Free vibration investigation on rve of proposed honeycomb sandwich beam and material selection optimization. Fact. Univ. Ser. Mech. Eng. 21, 031–050 (2023). https://doi.org/10.22190/FUME220806042S

    Article  Google Scholar 

  44. Feng, J., Safaei, B., Qin, Z., Chu, F.: Nature-inspired energy dissipation sandwich composites reinforced with high-friction graphene. Compos. Sci. Technol. 233, 109925 (2023). https://doi.org/10.1016/J.COMPSCITECH.2023.109925

    Article  Google Scholar 

  45. Ramteke, P.M., Panda, S.K.: Nonlinear static and dynamic (deflection/stress) responses of porous functionally graded shell panel and experimental validation. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. (2023). https://doi.org/10.1177/09544062231155099

    Article  Google Scholar 

  46. Ramteke, P.M., Panda, S.K.: Computational modelling and experimental challenges of linear and nonlinear analysis of porous graded structure: a comprehensive review. Arch. Comput. Methods Eng. 2023(1), 1–16 (2023). https://doi.org/10.1007/S11831-023-09908-X

    Article  Google Scholar 

  47. Ramteke, P.M., Panda, S.K.: Nonlinear thermomechanical static and dynamic responses of bidirectional porous functionally graded shell panels and experimental verifications. J. Press. Vessel Technol. (2023). https://doi.org/10.1115/1.4062154

    Article  Google Scholar 

  48. Malhari Ramteke, P., Kumar Panda, S.: Nonlinear static and dynamic response prediction of bidirectional doubly-curved porous FG panel and experimental validation. Compos. Part A Appl. Sci. Manuf. 166, 107388 (2023). https://doi.org/10.1016/J.COMPOSITESA.2022.107388

    Article  Google Scholar 

  49. Mantari, J.L., Granados, E.V.: An original FSDT to study advanced composites on elastic foundation. Thin-Walled Struct. 107, 80–89 (2016). https://doi.org/10.1016/J.TWS.2016.05.024

    Article  Google Scholar 

  50. Park, M., Choi, D.H.: A simplified first-order shear deformation theory for bending, buckling and free vibration analyses of isotropic plates on elastic foundations, vol. 22, pp. 1235–1249. Springer, Berlin (2018). https://doi.org/10.1007/s12205-017-1517-6

    Book  Google Scholar 

  51. Said, A., Ameur, M., Bousahla, A.A., Tounsi, A.: A new simple hyperbolic shear deformation theory for functionally graded plates resting on winkler-pasternak elastic foundations. Int. J. Comput. Methods (2014). https://doi.org/10.1142/S0219876213500989

    Article  MathSciNet  MATH  Google Scholar 

  52. Xiang, S., Kang, G.W., Liu, Y.: A nth-order shear deformation theory for natural frequency of the functionally graded plates on elastic foundations. Compos. Struct. 111, 224–231 (2014). https://doi.org/10.1016/J.COMPSTRUCT.2014.01.004

    Article  Google Scholar 

  53. Atmane, H.A., Tounsi, A., Bernard, F.: Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations. Int. J. Mech. Mater. Des. 13, 71–84 (2017). https://doi.org/10.1007/S10999-015-9318-X

    Article  Google Scholar 

  54. Benahmed, A., Houari, M.S.A., Benyoucef, S., Belakhdar, K., Tounsi, A.: A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation. Geomech. Eng. 12, 9–34 (2017). https://doi.org/10.12989/GAE.2017.12.1.009

    Article  Google Scholar 

  55. Shahsavari, D., Shahsavari, M., Li, L., Karami, B.: A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp. Sci. Technol. 72, 134–149 (2018). https://doi.org/10.1016/J.AST.2017.11.004

    Article  Google Scholar 

  56. Mellal, F., Bennai, R., Nebab, M., Atmane, H.A., Bourada, F., Hussain, M., Tounsi, A., Abbes, B., Sidi, B., Abbes, A.: Investigation on the effect of porosity on wave propagation in FGM plates resting on elastic foundations via a quasi-3D HSDT. Waves Random Complex Media (2021). https://doi.org/10.1080/17455030.2021.1983235

    Article  Google Scholar 

  57. Atmane, H.A., Tounsi, A., Bernard, F., Mahmoud, S.R., Tounsi, A., Bernard, F., Mahmoud, S.R.: A computational shear displacement model for vibrational analysis of functionally graded beams with porosities. Steel Compos. Struct. 19, 369 (2015). https://doi.org/10.12989/SCS.2015.19.2.369

    Article  Google Scholar 

  58. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Bég, O.A., Mahmoud, S.R.: Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel Compos. Struct. 18, 425–442 (2015). https://doi.org/10.12989/scs.2015.18.2.425

    Article  Google Scholar 

  59. Hadji, L., Daouadji, T.H., Tounsi, A., Bedia, E.A.: A higher order shear deformation theory for static and free vibration of FGM beam. Steel Compos. Struct. 16, 507–519 (2014). https://doi.org/10.12989/scs.2014.16.5.507

    Article  Google Scholar 

  60. Vo, T.P., 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. 119, 1–12 (2015). https://doi.org/10.1016/j.compstruct.2014.08.006

    Article  Google Scholar 

  61. Vo, T.P., Thai, H.-T., Nguyen, T.-K., Inam, F., Lee, J.: Static behaviour of functionally graded sandwich beams using a quasi-3D theory. Compos. Part B Eng. 68, 59–74 (2015). https://doi.org/10.1016/J.COMPOSITESB.2014.08.030

    Article  Google Scholar 

  62. Akbaş, ŞD.: Nonlinear static analysis of functionally graded porous beams under thermal effect. Coupled Syst. Mech. 6, 399–415 (2017). https://doi.org/10.12989/csm.2017.6.4.399

    Article  Google Scholar 

  63. Al-shujairi, M., Mollamahmutoğlu, Ç.: Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect. Compos. Part B Eng. 154, 292–312 (2018). https://doi.org/10.1016/J.COMPOSITESB.2018.08.103

    Article  Google Scholar 

  64. Sayyad, A.S., Ghugal, Y.M.: Effect of thickness stretching on the static deformations, natural frequencies, and critical buckling loads of laminated composite and sandwich beams. J. Braz. Soc. Mech. Sci. Eng. (2018). https://doi.org/10.1007/S40430-018-1222-5

    Article  Google Scholar 

  65. Zenkour, A.M., Radwan, A.F.: Compressive study of functionally graded plates resting on Winkler–Pasternak foundations under various boundary conditions using hyperbolic shear deformation theory. Arch. Civ. Mech. Eng. 18, 645–658 (2018). https://doi.org/10.1016/J.ACME.2017.10.003

    Article  Google Scholar 

  66. Bouiadjra, R.B., Bachiri, A., Benyoucef, S., Fahsi, B., Bernard, F.: An investigation of the thermodynamic effect on the response of FG beam on elastic foundation. Struct. Eng. Mech. 76, 115–127 (2020). https://doi.org/10.12989/SEM.2020.76.1.115

    Article  Google Scholar 

  67. Tsiatas, G.C.: Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta Mech. 209, 141–152 (2010). https://doi.org/10.1007/s00707-009-0174-3

    Article  MATH  Google Scholar 

  68. Foyouzat, M.A., Mofid, M., Akin, J.E.: On the dynamic response of beams on elastic foundations with variable modulus. Acta Mech. 227, 549–564 (2016). https://doi.org/10.1007/S00707-015-1485-1/METRICS

    Article  MathSciNet  MATH  Google Scholar 

  69. Froio, D., Rizzi, E.: Analytical solution for the elastic bending of beams lying on a variable Winkler support. Acta Mech. 227, 1157–1179 (2016). https://doi.org/10.1007/S00707-015-1508-Y/METRICS

    Article  MathSciNet  MATH  Google Scholar 

  70. Doeva, O., Masjedi, P.K., Weaver, P.M.: Static analysis of composite beams on variable stiffness elastic foundations by the Homotopy Analysis Method. Acta Mech. 232, 4169–4188 (2021). https://doi.org/10.1007/S00707-021-03043-Z/FIGURES/7

    Article  MathSciNet  MATH  Google Scholar 

  71. Daikh, A.A., Belarbi, M.O., Ahmed, D., Houari, M.S.A., Avcar, M., Tounsi, A., Eltaher, M.A.: Static analysis of functionally graded plate structures resting on variable elastic foundation under various boundary conditions. Acta Mech. 234, 775–806 (2023). https://doi.org/10.1007/S00707-022-03405-1/FIGURES/14

    Article  MathSciNet  Google Scholar 

  72. Nebab, M., Atmane, H.A., Bennai, R., Tahar, B.: Effect of nonlinear elastic foundations on dynamic behavior of FG plates using four-unknown plate theory. Earthq. Struct. 17, 447–462 (2019). https://doi.org/10.12989/EAS.2019.17.5.447

    Article  Google Scholar 

  73. Nebab, M., Atmane, H.A., Bennai, R., Tounsi, A.: Effect of variable elastic foundations on static behavior of functionally graded plates using sinusoidal shear deformation. Arab. J. Geosci. (2019). https://doi.org/10.1007/S12517-019-4871-5

    Article  Google Scholar 

  74. Merzoug, M., Bourada, M., Sekkal, M., Abir, A.C., Chahrazed, B., Benyoucef, S., Benachour, A.: 2D and quasi 3D computational models for thermoelastic bending of FG beams on variable elastic foundation: effect of the micromechanical models. Geomech. Eng. 22, 361–374 (2020). https://doi.org/10.12989/GAE.2020.22.4.361

    Article  Google Scholar 

  75. Bouiadjra, R.B., Mahmoudi, A., Sekkal, M., Benyoucef, S., Selim, M.M., Tounsi, A., Hussain, M.: A quasi 3D solution for thermodynamic response of FG sandwich plates lying on variable elastic foundation with arbitrary boundary conditions. Steel Compos. Struct. 41, 873–886 (2021). https://doi.org/10.12989/SCS.2021.41.6.873

    Article  Google Scholar 

  76. Benaberrahmane, I., Benyoucef, S., Sekkal, M., Mekerbi, M., Bouiadjra, R.B., Selim, M.M., Tounsi, A., Hussain, M.: Investigating of free vibration behavior of bidirectional FG beams resting on variable elastic foundation. Geomech. Eng. 25, 383–394 (2021). https://doi.org/10.12989/GAE.2021.25.5.383

    Article  Google Scholar 

  77. Atmane, R.A., Mahmoudi, N., Bennai, R., Atmane, H.A., Tounsi, A.: Investigation on the dynamic response of porous FGM beams resting on variable foundation using a new higher order shear deformation theory. Steel Compos. Struct. 39, 95–107 (2021). https://doi.org/10.12989/SCS.2021.39.1.095

    Article  Google Scholar 

  78. Giang, N.T., Hong, N.T.: Hygro-thermo-mechanical stability analysis of variable thickness functionally graded sandwich porous plates resting on variable elastic foundations using finite element method. J. Therm. Stress. 45, 641–668 (2022). https://doi.org/10.1080/01495739.2022.2089307

    Article  Google Scholar 

  79. Wattanasakulpong, N., Ungbhakorn, V.: Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp. Sci. Technol. 32, 111–120 (2014). https://doi.org/10.1016/J.AST.2013.12.002

    Article  Google Scholar 

  80. Rabia, B., Daouadji, T.H., Abderezak, R.: Effect of distribution shape of the porosity on the interfacial stresses of the FGM beam strengthened with FRP plate. Earthq. Struct. 16, 601–609 (2019). https://doi.org/10.12989/EAS.2019.16.5.601

    Article  Google Scholar 

  81. Avcar, M., Hadji, L., Akan, R.: The influence of Winkler–Pasternak elastic foundations on the natural frequencies of imperfect functionally graded sandwich beams. Geomech. Eng. 31, 99–112 (2022). https://doi.org/10.12989/GAE.2022.31.1.099

    Article  Google Scholar 

  82. Bennai, R., Atmane, R.A., Bernard, F., Nebab, M., Mahmoudi, N., Atmane, H.A., Aldosari, S.M., Tounsi, A.: Study on stability and free vibration behavior of porous FGM beams. Struct. Eng. Mech. 45, 67–82 (2022). https://doi.org/10.12989/SCS.2022.45.1.067

    Article  Google Scholar 

  83. Pradhan, S.C., Murmu, T.: Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method. J. Sound Vib. 321, 342–362 (2009). https://doi.org/10.1016/J.JSV.2008.09.018

    Article  Google Scholar 

  84. Sobhy, M.: Thermoelastic response of FGM plates with temperature-dependent properties resting on variable elastic foundations. Int. J. Appl. Mech. (2015). https://doi.org/10.1142/S1758825115500829

    Article  Google Scholar 

  85. Beldjelili, Y., Tounsi, A., Mahmoud, S.R.: Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory. Smart Struct. Syst. 18, 755–786 (2016). https://doi.org/10.12989/SSS.2016.18.4.755

    Article  Google Scholar 

  86. Attia, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R., Alwabli, A.S.: A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations. Struct. Eng. Mech. 65, 453–464 (2018). https://doi.org/10.12989/SEM.2018.65.4.453

    Article  Google Scholar 

  87. Ayache, B., Bennai, R., Fahsi, B., Fourn, H., Atmane, H.A., Tounsi, A.: Analysis of wave propagation and free vibration of functionally graded porous material beam with a novel four variable refined theory. Earthq. Struct. 15, 369–382 (2018). https://doi.org/10.12989/EAS.2018.15.4.369

    Article  Google Scholar 

  88. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29, 901–916 (1991). https://doi.org/10.1016/0020-7225(91)90165-Y

    Article  MATH  Google Scholar 

  89. Nedri, K., El Meiche, N., Tounsi, A.: Free vibration analysis of laminated composite plates resting on elastic foundations by using a refined hyperbolic shear deformation theory. Mech. Compos. Mater. 49, 629–640 (2014). https://doi.org/10.1007/S11029-013-9379-6

    Article  Google Scholar 

  90. Hadji, L., Avcar, M.: Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory. Adv. Nano Res. 10, 281–293 (2021). https://doi.org/10.12989/anr.2021.10.3.281

    Article  Google Scholar 

  91. Hadji, L., Avcar, M.: Free vibration analysis of FG porous sandwich plates under various boundary conditions. J. Appl. Comput. Mech. 7, 505–519 (2021). https://doi.org/10.22055/jacm.2020.35328.2628

    Article  Google Scholar 

  92. Sobhani, E., Avcar, M.: Natural frequency analysis of imperfect GNPRN conical shell, cylindrical shell, and annular plate structures resting on Winkler–Pasternak foundations under arbitrary boundary conditions. Eng. Anal. Bound. Elem. 144, 145–164 (2022). https://doi.org/10.1016/J.ENGANABOUND.2022.08.018

    Article  MathSciNet  Google Scholar 

  93. Nguyen, T.-K., Vo, T.P., Thai, H.-T.: Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Compos. Part B Eng. 55, 147–157 (2013). https://doi.org/10.1016/j.compositesb.2013.06.011

    Article  Google Scholar 

  94. Ibnorachid, Z., Boutahar, L., EL Bikri, K., Benamar, R.: Buckling temperature and natural frequencies of thick porous functionally graded beams resting on elastic foundation in a thermal environment. Adv. Acoust. Vib. 2019, 1–17 (2019)

    Google Scholar 

  95. Şimşek, M.: Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl. Eng. Des. 240, 697–705 (2010). https://doi.org/10.1016/J.NUCENGDES.2009.12.013

    Article  Google Scholar 

  96. Chen, W.Q., Lü, C.F., Bian, Z.G.: A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Appl. Math. Model. 28, 877–890 (2004). https://doi.org/10.1016/J.APM.2004.04.001

    Article  MATH  Google Scholar 

  97. Fahsi, B., Bouiadjra, R.B., 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. 55, 219–230 (2019). https://doi.org/10.1007/S11029-019-09805-0/FIGURES/5

    Article  Google Scholar 

  98. Chikh, A.: Investigations in static response and free vibration of a functionally graded beam resting on elastic foundations. Frat. Integr. Strutt. 14, 115–126 (2020). https://doi.org/10.3221/IGF-ESIS.51.09

    Article  Google Scholar 

  99. Vo, T.P., 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. 64, 12–22 (2014). https://doi.org/10.1016/J.ENGSTRUCT.2014.01.029

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Structures, Geotechnics and Risks, Department of Civil Engineering, Hassiba Benbouali University of Chlef, Ouled Fares, Algeria

    Fatma Mellal, Riadh Bennai, Mokhtar Nebab & Hassen Ait Atmane

  2. Department of Technology, Faculty of Science and Technology, Djilali Bounaama University, Khemis Miliana, Algeria

    Fatma Mellal

  3. Department of Civil Engineering, Faculty of Civil Engineering and Architecture, University Hassiba Benbouali of Chlef, Ouled Fares, Algeria

    Riadh Bennai & Hassen Ait Atmane

  4. Department of Civil Engineering, Faculty of Engineering, Suleyman Demirel University, Cunur, Isparta, Turkey

    Mehmet Avcar

  5. Department of Civil Engineering, Faculty of Sciences, University of M’Hamed Bougara Boumerdes, Boumerdes, Algeria

    Mokhtar Nebab

Authors
  1. Fatma Mellal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riadh Bennai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mehmet Avcar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mokhtar Nebab
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hassen Ait Atmane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehmet Avcar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mellal, F., Bennai, R., Avcar, M. et al. On the vibration and buckling behaviors of porous FG beams resting on variable elastic foundation utilizing higher-order shear deformation theory. Acta Mech 234, 3955–3977 (2023). https://doi.org/10.1007/s00707-023-03603-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03603-5

Search

Navigation