Geomechanics and Engineering

  • 제36권2호
  • /
  • Pages.183-204
  • /
  • 2024
  • /
  • 2005-307X(pISSN)
  • /
  • 2092-6219(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

On the elastic stability and free vibration responses of functionally graded porous beams resting on Winkler-Pasternak foundations via finite element computation

  • Zakaria Belabed (Artificial Intelligence Laboratory for Mechanical and Civil Structures, and Soil, Institute of Technology, University Center of Naama) ;
  • Abdelouahed Tounsi (Center for Engineering Application & Technology Solutions, Ho Chi Minh City Open University) ;
  • Mohammed A. Al-Osta (Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals) ;
  • Abdeldjebbar Tounsi (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Hoang-Le Minh (Center for Engineering Application & Technology Solutions, Ho Chi Minh City Open University)
  • 투고 : 2023.04.11
  • 심사 : 2024.01.03
  • 발행 : 2024.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

In current investigation, a novel beam finite element model is formulated to analyze the buckling and free vibration responses of functionally graded porous beams resting on Winkler-Pasternak elastic foundations. The novelty lies in the formulation of a simplified finite element model with only three degrees of freedom per node, integrating both C0 and C1 continuity requirements according to Lagrange and Hermite interpolations, respectively, in isoparametric coordinate while emphasizing the impact of z-coordinate-dependent porosity on vibration and buckling responses. The proposed model has been validated and demonstrating high accuracy when compared to previously published solutions. A detailed parametric examination is performed, highlighting the influence of porosity distribution, foundation parameters, slenderness ratio, and boundary conditions. Unlike existing numerical techniques, the proposed element achieves a high rate of convergence with reduced computational complexity. Additionally, the model's adaptability to various mechanical problems and structural geometries is showcased through the numerical evaluation of elastic foundations, with results in strong agreement with the theoretical formulation. In light of the findings, porosity significantly affects the mechanical integrity of FGP beams on elastic foundations, with the advanced beam element offering a stable, efficient model for future research and this in-depth investigation enriches porous structure simulations in a field with limited current research, necessitating additional exploration and investigation.

키워드

참고문헌

  1. Abdelbari, S., Attia, A., Bourada, F., Bousahla, A.A., Tounsi, A. and Ghazwani, M.H. (2023), "Investigation of dynamic characteristics of imperfect FG beams on the Winkler-Pasternak foundation under thermal loading", Phys. Mesomech., 26(5), 557-572. https://doi.org/10.1134/S1029959923050089.
  2. Abo-Bakr, R.M., Eltaher, M.A. and Attia, M.A. (2020), "Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects", Eng. Comput., 1-22. https://doi.org/10.1007/s00366-020-01146-0.
  3. Adiyaman, G., Oner, E., Yaylaci, M. and Birinci, A. (2023), "A study on the contact problem of a layer consisting of functionally graded material (FGM) in the presence of body force", J. Mech. Mat. Struct., 18(1), 125-141. https://doi.org/10.2140/jomms.2023.18.125.
  4. Akbas, S.D., Fageehi, Y.A., Assie, A.E. and Eltaher, M.A. (2020a), "Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load", Eng. Comput., 1-13. https://doi.org/10.1007/s00366-020-01070-3.
  5. Akbas, S.D., Bashiri, A.H., Assie, A.E. and Eltaher, M.A. (2020b), "Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support", J. Vib. Control, 1077546320947302. https://doi.org/10.1177/1077546320947302.
  6. Ahmed, R.A., Fenjan, R.M. and Faleh, N.M. (2019), "Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections", Geomech. Eng., 17(2), 175-180. https://doi.org/10.12989/gae.2019.17.2.175.
  7. Ait Atmane, H., Tounsi, A. and Bernard, F. (2015), "Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations", Int .J. Mech. Mater. Design, 13(1), 71-84. https://doi.org/10.1007/s10999-015-9318-x.
  8. Akbas, S.D. (2017), "Thermal effects on the vibration of functionally graded deep beams with porosity", Int. J. Appl. Mech., 9(5), 1750076. https://doi.org/10.1142/S1758825117500764.
  9. Al-Maliki, A.F., Faleh, N.M. and Alasadi, A.A. (2019), "Finite element formulation and vibration of nonlocal refined metal foam beams with symmetric and non-symmetric porosities", Struct. Monit. Maint., 6(2), 147-159. https://doi.org/10.12989/smm.2019.6.2.147.
  10. Al-shujairi, M. and Mollamahmutoglu, C. (2018), "Buckling and free vibration analysis of functionally graded sandwich microbeams 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. https://doi.org/10.1016/j.compositesb.2018.08.103.
  11. Al-Toki, M.H., Ali, H.A., Faleh, N.M. and Fenjan, R.M. (2022), "Numerical assessment of nonlocal dynamic stability of graded porous beams in thermal environment rested on elastic foundation", Geomech. Eng., 28(5), 455-461. https://doi.org/10.12989/gae.2022.28.5.455.
  12. Alnujaie, A., Akbas, S.D., Eltaher, M.A. and Assie, A. (2021), "Forced vibration of a functionally graded porous beam resting on viscoelastic foundation", Geomech. Eng., 24(1), 91-103. https://doi.org/10.12989/gae.2021.24.1.091.
  13. Ansari, R., Faraji Oskouie, M., Nesarhosseini, S. and Rouhi, H. (2022), "Nonlinear thermally induced vibration analysis of porous FGM Timoshenko beams embedded in an elastic medium". Transp. Porous Med., 142, 63-87. https://doi.org/10.1007/s00707-022-03455-5.
  14. Arefi, M. and Zur, K.K. (2020), "Free vibration analysis of functionally graded cylindrical nanoshells resting on Pasternak foundation based on two-dimensional analysis", Steel Compos. Struct., 34(4), 615-623. https://doi.org/10.12989/scs.2020.34.4.615.
  15. Avcar, M. and Mohammed, W.K.M. (2018), "Free vibration of functionally graded beams resting on Winkler-Pasternak foundation", Arab. J. Geosci., 11(10), 232. https://doi.org/10.1007/s12517-018-3579-2
  16. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603.
  17. Assie, A.E., Mohamed, S.A., Shanab, R.A., Abo-bakr, R.M. and Eltaher, M.A. (2023), "Static buckling of 2D FG porous plates resting on elastic foundation based on unified shear theories", J. Appl. Comput. Mech., 9(1), 239-258. https://doi.org/10.22055/jacm.2022.41265.3723.
  18. Babaei, H., Kiani, Y. and Zur, K.K. (2022), "New insights into nonlinear stability of imperfect nanocomposite beams resting on a nonlinear medium", Commun. Nonlinear. Sci. Numer. Simul., 118,106993. https://doi.org/10.1016/j.cnsns.2022.106993.
  19. Bashiri, A.H., Akbas, S.D., Abdelrahman, A.A., Assie, A., Eltaher, M.A. and Mohamed, E.F. (2021), "Vibration of multilayered functionally graded deep beams under thermal load", Geomech. Eng., 24(6), 545-557. https://doi.org/10.12989/gae.2021.24.6.545.
  20. Belabed, Z., Selim, M.M., Slimani, O., Taibi, N., Tounsi, A. and Hussain, M. (2021), "An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells", Steel Compos. Struct., 40(2), 307-321. https://doi.org/10.12989/scs.2021.40.2.307.
  21. Chen, W.Q., Lu, C.F. and Bian, Z.G. (2004), "A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation", Appl. Math. Model., 28(10), 877-890. https://doi.org/10.1016/j.apm.2004.04.001.
  22. Chen, D., Yang, J. and Kitipornchai, S. (2015), "Elastic buckling and static bending of shear deformable functionally graded porous beam", Compos. Struct., 133, 54-61. https://doi.org/10.1016/j.compstruct.2015.07.052.
  23. Chen, D., Yang, J. and Kitipornchai, S. (2016), "Free and forced vibrations of shear deformable functionally graded porous beams", Int. J. Mech. Sci., 108, 14-22. https://doi.org/10.1016/j.ijmecsci.2016.01.025.
  24. Dhatt, G., Lefrancois, E. and Touzot, G. (2012). Finite Element Method, ISTE Ltd and John Wiley & Sons Inc. https://doi.org/10.1002/9781118569764.
  25. Duc, N.D. and Thang, P.T. (2015), "Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations", Aerosp. Sci. Technol., 40, 115-127. https://doi.org/10.1016/j.ast.2014.11.005.
  26. Ebrahimi, F. and Jafari, A. (2016), "Thermo-mechanical vibration analysis of temperature-dependent porous FG beams based on Timoshenko beam theory", Struct. Eng. Mech., 59(2), 343-371. https://doi.org/10.12989/sem.2016.59.2.343.
  27. Ebrahimi, F. and Dabbagh, A. (2019) "Vibration analysis of graphene oxide powder-/carbon fiber-reinforced multi-scale porous nanocomposite beams: a finite-element study", Eur. Phys. J. Plus, 134, 225. https://doi.org/10.1140/epjp/i2019-12594-1
  28. Ebrahimi, F. and Seyfi, A. (2021) "A wave propagation study for porous metal foam beams resting on an elastic foundation", Waves Random Complex., https://doi.org/10.1080/17455030.2021.1905909.
  29. Esen, I., Abdelrhmaan, A.A. and Eltaher, M.A. (2021a), "Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields", Eng. Comput., 1-20. https://doi.org/10.1007/s00366-021-01389-5.
  30. Esen, I., O Zarpa, C. and Eltaher, M.A. (2021b), "Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment", Compos. Struct., 261, 113552. https://doi.org/10.1016/j.compstruct.2021.113552.
  31. Esen, I., Eltaher, M.A. and Abdelrahman, A.A. (2021c), "Vibration response of symmetric and sigmoid functionally graded beam rested on elastic foundation under moving point mass", Mech. Based Des. Struc., 1-25. https://doi.org/10.1080/15397734.2021.1904255.
  32. Fallah, A. and Aghdam, M.M., (2023), "Physics-informed neural network for bending and free vibration analysis of three-dimensional functionally graded porous beam resting on elastic foundation", Eng. Comput., 1-18. https://doi.org/10.1007/s00366-023-01799-7.
  33. Fan, F., Safaei, B. and Sahmani, S. (2021), "Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis", Thin Wall. Struct., 159, 107231. https://doi.org/10.1016/j.tws.2020.107231.
  34. Fazzolari, F.A. (2018), "Generalized exponential, polynomial and trigonometric theories for vibration and stability analysis of porous FG sandwich beams resting on elastic foundations", Compos. Part B: Eng., 136, 254-271. https://doi.org/10.1016/j.compositesb.2017.10.022.
  35. Fouaidi, M., Belfallah, K., Jamal, M., and Belouaggadia, N. (2023), "Transient analysis of functionally graded graphene oxide powders-reinforced porous composite beams resting on elastic foundations using the reproducing kernel particle meshless method", Eng. Anal. Bound. Elem., 146, 460-472. https://doi.org/10.1016/j.enganabound.2022.10.029.
  36. Gao, K., Li, R. and Yang, J. (2019), "Dynamic characteristics of functionally graded porous beams with interval material properties", Eng. Struct., 197, 109441. https://doi.org/10.1016/j.engstruct.2019.109441.
  37. Ghandourah, E. E., Ahmed, H. M., Eltaher, M. A., Attia, M. A. and Abdraboh, A. M. (2021), "Free vibration of porous FG nonlocal modified couple nanobeams via a modified porosity model", Adv. Nano Res., 11(4), 405-422. https://doi.org/10.12989/anr.2021.11.4.405.
  38. Ghorbanpour Arani, A., Khani, M. and Khoddami Maraghi, Z. (2018), "Dynamic analysis of a rectangular porous plate resting on an elastic foundation using high-order shear deformation theory", J. Vib. Control, 24, 3698-3713. https://doi.org/10.1177/1077546317709388.
  39. Huang, W. and Tahouneh, V. (2021), "Frequency study of porous FGPM beam on two-parameter elastic foundations via Timoshenko theory", Steel Compos. Struct., 40(1), 139-156. https://doi.org/10.12989/scs.2021.40.1.139.
  40. Jamshidi, M., Arghavani, J. and Maboudi, G. (2019), "Post-buckling optimization of two-dimensional functionally graded porous beams", Int. J. Mech. Mater., 15, 801-815. https://doi.org/10.1007/s10999-019-09443-3.
  41. Jamshidi, M. and Arghavani, J. (2018), "Optimal material tailoring of functionally graded porous beams for buckling and free vibration behaviors", Mech. Res. Commun., 88, 19-24. https://doi.org/10.1016/j.mechrescom.2018.01.006.
  42. Jalaei, M.H. and Civalek, O. (2019), "On dynamic instability of magnetically embedded viscoelastic porous FG nanobeams", Int. J. Eng. Sci., 143, 14-32. https://doi.org/10.1016/j.ijengsci.2019.06.013.
  43. Jena, S.K., Chakraverty, S. and Malikan, M. (2020), "Application of shifted Chebyshev polynomial-based Rayleigh-Ritz method and Navier's technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation", Eng. Comput., 1-21. https://doi.org/10.1007/s00366-020-01018-7.
  44. Jena, S. K., Chakraverty, S., Mahesh, V. and Harursampath, D., (2022), "Wavelet-based techniques for Hygro-Magneto-Thermo vibration of nonlocal strain gradient nanobeam resting on Winkler-Pasternak elastic foundation", Eng. Anal. Bound. Elem., 140, 494-506. https://doi.org/10.1016/j.enganabound.2022.04.037
  45. Hamed, M.A., Mohamed, S.A. and Eltaher, M.A. (2020), "Buckling analysis of sandwich beam rested on elastic foundation and subjected to varying axial in-plane loads", Steel Compos. Struct., 34(1), 75-89. https://doi.org/10.12989/scs.2020.34.1.075.
  46. Khoram, M.M., Hosseini, M., Hadi, A. and Shishehsaz, M. (2020), "Bending analysis of bidirectional FGM Timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler-Pasternak foundation", Int. J. Appl. Mech., 12(8), 2050093. https://doi.org/10.1142/S1758825120500933.
  47. Lee, W.H., Han, S.C. and Park, W.T. (2015), "A refined higher order shear and normal deformation theory for E-, P-, and SFGM plates on Pasternak elastic foundation", Compos. Struct., 122, 330-342. https://doi.org/10.1016/j.compstruct.2014.11.047.
  48. Liu, H., Liu, H. and Yang, J. (2018), "Vibration of FG magnetoelectro-viscoelastic porous nanobeams on visco-Pasternak foundation", Compos. Part B Eng., 155, 244-256. https://doi.org/10.1016/j.compositesb.2018.08.042
  49. Madenci, E. (2021), "Free vibration and static analyses of metal-ceramic FG beams via high-order variational MFEM", Steel Compos. Struct., 39(5), 493-509. https://doi.org/10.12989/scs.2021.39.5.493.
  50. Mehala, T., Belabed, Z., Tounsi, A. and Beg, O.A. (2018), "Investigation of influence of homogenization models on stability and dynamic of FGM plates on elastic foundations", Geomech. Eng., 16(3), 257-271. https://doi.org/10.12989/gae.2018.16.3.257.
  51. Melaibari, A., Abo-bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020), "Static stability of higher order functionally graded beam under variable axial load", Alex. Eng. J., 59(3), 1661-1675. https://doi.org/10.1016/j.aej.2020.04.012.
  52. Melaibari, A., Mohamed, S. A., Assie, A.E., Shanab, R.A. and Eltaher, M.A. (2022), "Free vibration characteristics of bidirectional graded porous plates with elastic foundations using 2D-DQM", Mathematics, 11(1), 46. https://doi.org/10.3390/math11010046.
  53. Mohamed, S.A., Mohamed, N. and Eltaher, M.A. (2022), "Snap-through instability of helicoidal composite imperfect beams surrounded by nonlinear elastic foundation", Ocean Eng., 263, 112171. https://doi.org/10.1016/j.oceaneng.2022.112171.
  54. Naidu, N.R. and Rao, G.V. (1995), "Stability behaviour of uniform columns on a class of two parameter elastic foundation", Comput. Struct., 57(3), 551-553. https://doi.org/10.1016/0045-7949(94)00636-H.
  55. Nguyen, N.D., Nguyen, T.N., Nguyen, T.K. and Vo, T.P. (2022), "A new two-variable shear deformation theory for bending, free vibration and buckling analysis of functionally graded porous beams", Compos. Struct., 282, 115095. https://doi.org/10.1016/j.compstruct.2021.115095.
  56. Nguyen, N.D., Nguyen, T.N., Nguyen, T.K. and Vo, T.P. (2023), "A Legendre-Ritz solution for bending, buckling and free vibration behaviours of porous beams resting on the elastic foundation". Struct., 50, 1934-1950. https://doi.org/10.1016/j.istruc.2023.03.018.
  57. Oner, E., Sengul Sabano, B., Uzun Yaylaci, E., Adiyaman, G., Yaylaci, M. and Birinci, A. (2022), "On the plane receding contact between two functionally graded layers using computational, finite element and artificial neural network methods", ZAMM - J. Appl. Math. Mech., 102(2), e202100287. https://doi.org/10.1002/zamm.202100287.
  58. Ozdemir, M.E. and Yaylaci, M, (2023), "Research of the impact of material and flow properties on fluid-structure interaction in cage systems", Wind Struct., 36(1), 31-40. https://doi.org/10.12989/was.2023.36.1.031.
  59. Pham, Q.H., Tran, V.K. and Nguyen, P.C. (2022), "Hygro-thermal vibration of bidirectional functionally graded porous curved beams on variable elastic foundation using generalized finite element method", Case Stud. Therm. Eng., 40, 102478. https://doi.org/10.1016/j.csite.2022.102478.
  60. Rachedi, M.A., Benyoucef, S., Bouhadra, A., Bachir Bouiadjra, R., Sekkal, M. and Benachour, A. (2020), "Impact of the homogenization models on the thermoelastic response of FG plates on variable elastic foundation",Geomech. Eng., 22(1), 65-80. https://doi.org/10.12989/gae.2020.22.1.065.
  61. Setoodeh, A. and Rezaei, M. (2017), "Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation", Struct. Eng. Mech., 61(2), 209-220. https://doi.org/10.12989/sem.2017.61.2.209.
  62. Shanab, R., Mohamed, S., Tharwan, M.Y., Assie, A.E. and Eltaher, M.A. (2022), "Buckling of 2D FG Porous unified shear plates resting on elastic foundation based on neutral axis", Steel Compos. Struct., 45(5), 729-747. https://doi.org/10.12989/scs.2022.45.5.729
  63. Turan, M., Uzun Yaylaci, E. and Yaylaci, M. (2023), "Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods", Arch. Appl. Mech., 93, 1351-1372. https://doi.org/10.1007/s00419-022-02332-w.
  64. Uzun, B., Civalek, O. and Yayli, M.O., (2020), "Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions", Mech. Based Des. Struc., Mach., 1-20. https://doi.org/10.1080/15397734.2020.1846560.
  65. Van Long, N., Nguyen, V.L., Tran, M.T. and Thai, D.K. (2022), "Exact solution for nonlinear static behaviors of functionally graded beams with porosities resting on elastic foundation using neutral surface concept", Proc. Inst. Mech. Eng. Part C, 236(1), 481-495. https://doi.org/10.1177/095440622110211.
  66. Wang, Y., Xie, K., Fu, T. and Zhang, W. (2021), "A third order shear deformable model and its applications for nonlinear dynamic response of graphene oxides reinforced curved beams resting on visco-elastic foundation and subjected to moving loads", Eng. Comput., 1-15. https://doi.org/10.1007/s00366-020-01238-x.
  67. Wu, D., Liu, A., Huang, Y., Huang, Y., Pi, Y. and Gao, W. (2018), "Dynamic analysis of functionally graded porous structures through finite element analysis", Eng. Struct., 165, 287-301. https://doi.org/10.1016/j.engstruct.2018.03.023.
  68. Xiao, H., Yan, K.M. and She, G. (2021), "Study on the characteristics of wave propagation in functionally graded porous square plates", Geomech. Eng., 26(6), 607-615. https://doi.org/10.12989/gae.2021.26.6.607.
  69. Xin, L. and Kiani, Y. (2023), "Vibration characteristics of arbitrary thick sandwich beam with metal foam core resting on elastic medium", Struct., 49, 1-11. https://doi.org/10.1016/j.istruc.2023.01.108.
  70. Yaylaci, M. (2016), "The investigation crack problem through numerical analysis", Struct. Eng. Mech., 57(6), 1143-1156. https://doi.org10.12989/sem.2016.57.6.1143.
  71. Yaylaci, M., Yayli, M., Yaylaci, E.U., Olmez, H. and Birinci, A. (2021), "Analyzing the contact problem of a functionally graded layer resting on an elastic half plane with theory of elasticity, finite element method and multilayer perceptron", Struct. Eng. Mech., 78(5), 585-597. https://doi.org/10.12989/sem.2021.78.5.585.
  72. Yaylaci, M., Sabano, B.S., Ozdemir, M.E. and Birinci, A. (2022a), "Solving the contact problem of functionally graded layers resting on a HP and pressed with a uniformly distributed load by analytical and numerical methods", Struct. Eng. Mech., 82(3), 401-416. https://doi.org/10.12989/sem.2022.82.3.401.
  73. Yaylaci, M., Abanoz, M., Yaylaci, E.U., Olmez, H., Sekban, D.M. and Birinci, A. (2022b), "Evaluation of the contact problem of functionally graded layer resting on rigid foundation pressed via rigid punch by analytical and numerical (FEM and MLP) methods", Arch. Appl. Mech., 92(6), 1953-1971. https://doi.org/10.1007/s00419-022-02159-5.
  74. Yaylaci, M. (2022c), "Simulate of edge and an internal crack problem and estimation of stress intensity factor through finite element method", Adv. Nano Res., 12(4), 405-414. https://doi.org/10.12989/anr.2022.12.4.405.
  75. Yaylaci, M., Yaylaci, E.U., Ozdemir, M.E., Ay, S. and Ozturk, S. (2022d), "Implementation of finite element and artificial neural network methods to analyze the contact problem of a functionally graded layer containing crack", Steel Compos. Struct., 45(4), 501-511. https://doi.org/10.12989/scs.2022.45.4.501.
  76. Yaylaci, E.U., Oner, E., Yaylaci, M., Ozdemir, M.E., Abushattal, A. and Birinci, A. (2022e), "Application of artificial neural networks in the analysis of the continuous contact problem", Struct. Eng. Mech., 84(1), 35-48. https://doi.org/10.12989/sem.2022.84.1.035.
  77. Yaylaci, M., Abanoz, M., Yaylaci, E.U., Olmez, H., Sekban, D.M. and Birinci, A. (2022f), "The contact problem of the functionally graded layer resting on rigid foundation pressed via rigid punch", Steel Compos. Struct., 43(5), 661-672. https://doi.org/10.12989/scs.2022.43.5.661.
  78. Yaylaci, M., Yaylaci, E.U., Ozdemir, M.E., Ozturk, S. and Sesli, H. (2023), "Vibration and buckling analyses of FGM beam with edge crack: Finite element and multilayer perceptron methods", Steel Compos. Struct., 46(4), 565-575. https://doi.org/10.12989/scs.2023.46.4.565.
  79. Ying, J., Lu, C.F. and Chen, W.Q. (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Compos. Struct., 84(3), 209-219. https://doi.org/10.1016/j.compstruct.2007.07.004.
  80. Zienkiewicz, O.Z., Taylor, R.L. and Zhu, J.Z. (2005), The Finite Element Method: Its Basis and Fundamentals, 6th Ed., Singapore.
  81. Zhang, B., He, Y., Liu, D., Shen, L. and Lei, J. (2015), "An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation", Appl. Math. Model., 39(13), 3814-3845. https://doi.org/10.1016/j.apm.2014.12.001.
  82. Zhang, L.H., Lai, S.K., Wang, C. and Yang, J. (2021), "DSC regularized Dirac-delta method for dynamic analysis of FG graphene platelet-reinforced porous beams on elastic foundation under a moving load", Compos. Struct., 255, 112865. https://doi.org/10.1016/j.compstruct.2020.112865.