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