DOI QR코드

DOI QR Code

Static buckling analysis of bi-directional functionally graded sandwich (BFGSW) beams with two different boundary conditions

  • Berkia, Abdelhak (Laboratory of Materials and Reactive Systems, Department of mechanical Engineering, University of Sidi Bel Abbes, Faculty of Technology) ;
  • Benguediab, Soumia (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Menasria, Abderrahmane (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Bouhadra, Abdelhakim (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Bourada, Fouad (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Mamen, Belgacem (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Benrahou, Kouider Halim (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Benguediab, Mohamed (Laboratory of Materials and Reactive Systems, Department of mechanical Engineering, University of Sidi Bel Abbes, Faculty of Technology) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad)
  • Received : 2022.02.06
  • Accepted : 2022.08.09
  • Published : 2022.08.25

Abstract

This paper presents the mechanical buckling of bi-directional functionally graded sandwich beams (BFGSW) with various boundary conditions employing a quasi-3D beam theory, including an integral term in the displacement field, which reduces the number of unknowns and governing equations. The beams are composed of three layers. The core is made from two constituents and varies across the thickness; however, the covering layers of the beams are made of bidirectional functionally graded material (BFGSW) and vary smoothly along the beam length and thickness directions. The power gradation model is considered to estimate the variation of material properties. The used formulation reflects the transverse shear effect and uses only three variables without including the correction factor used in the first shear deformation theory (FSDT) proposed by Timoshenko. The principle of virtual forces is used to obtain stability equations. Moreover, the impacts of the control of the power-law index, layer thickness ratio, length-to-depth ratio, and boundary conditions on buckling response are demonstrated. Our contribution in the present work is applying an analytical solution to investigate the stability behavior of bidirectional FG sandwich beams under various boundary conditions.

Keywords

References

  1. Abdellatif Selmi (2021), "Vibration behavior of bi-dimensional functionally graded beams", 77(5), 587-599. https://doi.org/10.12989/sem.2021.77.5.587.
  2. Abouelregal, A.E., Mohammed, W.W. and Sedighi, H.M. (2021), "Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dualphase lagse", Arch. Appl. Mech., 91(5), 2127-2142. https://doi.org/10.1007/s00419-020-01873-2.
  3. 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.
  4. Ahmed, R.A., Khalaf, B.S., Raheef, K.M., Fenjan, R.M. and Faleh, N.M. (2021), "Investigating dynamic response of nonlocal functionally graded porous piezoelectric plates in thermal environment", Steel Compos. Struct., 40(2), 243-254. https://doi.org/10.12989/scs.2021.40.2.243.
  5. Akbas, S.D. (2015), "Wave propagation of a functionally graded beam in thermal environments", Steel Compos. Struct., 19(6), 1421-1447. https://doi.org/10.12989/SCS.2015.19.6.1421.
  6. Akbas, S.D. (2021), "Dynamic analysis of axially functionally graded porous beams under a moving load", Steel Compos. Struct., 39(6), 811-821. https://doi.org/10.12989/scs.2021.39.6.811.
  7. Al-Basyouni, K.S., Ghandourah, E., Mostafa, H.M. and Algarni, A. (2020), "Effect of the rotation on the thermal stress wave propagation in non-homogeneous viscoelastic body", Geomech. Eng., 21(1), 1-9. https://doi.org/10.12989/GAE.2020.21.1.001.
  8. Alper Polat (2021), "Examination of contact problem between functionally graded punch and functionally graded layer resting on elastic plane", Struct. Eng. Mech., 78(2), 135-143. https://doi.org/10.12989/sem.2021.78.2.135.
  9. Arbind, A. and Reddy, J.N. (2013), "Nonlinear analysis of functionally graded microstructure-dependent beams", Compos. Struct., 98, 272-281. https://doi.org/10.1016/j.compstruct.2012.10.003.
  10. Asemi, K. and Shariyat, M. (2016), "Three-dimensional biaxial post-buckling analysis of heterogeneous auxetic rectangular plates on elastic foundations by new criteria", Comput. Meth. Appl. Mech. Eng., 302, 1-26. https://doi.org/10.1016/j.cma.2015.12.026.
  11. Attia, M.A. (2017), "On the mechanics of functionally graded nanobeams with the account of surface elasticity", Int. J. Eng. Sci., 115,73-101. https://doi.org/10.1016/j.ijengsci.2017.03.011.
  12. 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.
  13. Bakora, A. and Tounsi, A. (2015), "Thermo-mechanical postbuckling behavior of thick functionally graded plates resting on elastic foundations", Struct. Eng. Mech., 56(1), 85-106. https://doi.org/10.12989/sem.2015.56.1.085.
  14. Bhangale, R.K., and Ganesan, N. (2006), "Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core", J. Sound Vib., 295(1-2), 294-316. https://doi.org/10.1016/j.jsv.2006.01.026.
  15. Bharath, H.S., Waddar, S., Bekinal, S.I., Jeyaraj, P. and Doddamani, M. (2020), "Effect of axial compression on dynamic response of concurrently printed sandwich", Compos. Struct., 113223. https://doi.org/10.1016/j.compstruct.2020.113223.
  16. Bouhadra, A., Menasria, A. and Rachedi, M.A. (2021), "Boundary conditions effect for buckling analysis of porous functionally graded nanobeam", Adv. Nano Res., 10(4), 313-325. https://doi.org/10.12989/ANR.2021.10.4.313.
  17. Chakraborty, A., Gopalakrishnan, S. and Reddy, J.N. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539. https://doi.org/10.1016/s0020-7403(03)00058-4.
  18. Chehel Amirani, M., Khalili, S.M.R. and Nemati, N. (2009), "Free vibration analysis of sandwich beam with FG core using the element free Galerkin method", Compos. Struct., 90(3), 373-379. https://doi.org/10.1016/j.compstruct.2009.03.023.
  19. Chen, D., Yang, J. and Kitipornchai, S. (2017), "Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams", Compos. Sci. Technol., 142, 235-245. https://doi.org/10.1016/j.compscitech.2017.02.008.
  20. Daouadji, T.H. and Hadji, L. (2015), "Analytical solution of nonlinear cylindrical bending for functionally graded plates", Geomech. Eng., 9(5), 631-644. https://doi.org/10.12989/GAE.2015.9.5.631.
  21. Deng, H. and Cheng, W. (2016), "Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams", Compos. Struct., 141, 253-263. https://doi.org/10.1016/j.compstruct.2016.01.051.
  22. Farokhi, H., Ghayesh, M.H. and Gholipour, A. (2017), "Dynamics of functionally graded micro-cantilevers", Int. J. Eng. Sci., 115, 117-130. https://doi.org/10.1016/j.ijengsci.2017.01.004.
  23. Fekrar, A., El Meiche, N., Bessaim, A., Tounsi, A. and Adda Bedia, E.A. (2012), "Buckling analysis of functionally graded hybrid composite plates using a new four variable refined plate theory", Steel Compos. Struct., 13(1), 91-107. https://doi.org/10.12989/scs.2012.13.1.091.
  24. Fenjan, R.M., Faleh, N.M. and Ahmed, R.A. (2020), "Geometrical imperfection and thermal effects on nonlinear stability of microbeams made of graphene-reinforced nano-composites", Adv. Nano Res., 9(3), 147-156. https://doi.org/10.12989/ANR.2020.9.3.147.
  25. Fukui, Y. (1991), "Fundamental investigation of functionally gradient material manufacturing system using centrifugal force", JSME Int. J. Ser. 3, Vibration, Control Eng., Eng. Ind., 34(1), 144-148. https://doi.org/10.1299/jsmec1988.34.144.
  26. Gafour, Y., Hamidi, A., Benahmed, A., Zidour, M. and Bensattalah, T. (2020), "Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle", Adv. Nano Res., 8(1), 37-47. https://doi.org/10.12989/anr.2020.8.1.037.
  27. Garbin, F., Levano, A. and Arciniega, R. (2020), "Bending analysis of nonlocal functionally graded beams", IOP Conference Series: Materials Science and Engineering, 739, 012045. https://doi.org/10.1088/1757-899x/739/1/012045.
  28. Hadji, L. (2020), "Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model", Smart Struct. Sys., 26(2), 253-262. https://doi.org/10.12989/sss.2020.26.2.253.
  29. Huang, Y. and Li, X.F. (2010), "Buckling of functionally graded circular columns including shear deformation", Mater. Des., 31(7), 3159-3166. https://doi.org/10.1016/j.matdes.2010.02.032.
  30. Jia, X.L., Ke, L.L., Feng, C.B., Yang, J. and Kitipornchai, S. (2015), "Size effect on the free vibration of geometrically nonlinear functionally graded micro-beams under electrical actuation and temperature change", Compos. Struct., 133, 1137-1148. https://doi.org/10.1016/j.compstruct.2015.08.044.
  31. Karami, B. and Karami, S. (2019), "Buckling analysis of nanoplate-type temperature-dependent heterogeneous materials", Adv. Nano Res., 7(1), 51-61. https://doi.org/10.12989/ANR.2019.7.1.051.
  32. Ke, L.L., Yang, J., Kitipornchai, S. and Bradford, M.A. (2012), "Bending, buckling and vibration of size-dependent functionally graded annular microplates", Compos. Struct., 94(11), 3250-3257. https://doi.org/10.1016/j.compstruct.2012.04.037.
  33. Kiani, Y. (2019), "NURBS-based thermal buckling analysis of graphene platelet reinforced composite laminated skew plates", J. Thermal Stresses, 1-19. https://doi.org/10.1080/01495739.2019.1673687.
  34. Kiani, Y. and Eslami, M.R. (2010), "Thermal buckling analysis of functionally graded material beams", Int. J. Mech. Mater. Des., 6(3), 229-238. https://doi.org/10.1007/s10999-010-9132-4.
  35. Le, C. I., Le, N.A.T. and Nguyen, D.K. (2021), "Free vibration and buckling of bidirectional functionally graded sandwich beams using an enriched third-order shear deformation beam element", Compos. Struct., 261, 113309. https://doi.org/10.1016/j.compstruct.2020.113309.
  36. Lee, J.W. and Lee, J.Y. (2017), "Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression", Int. J. Mech. Sci., 122, 1-17. https://doi.org/10.1016/j.ijmecsci.2017.01.011.
  37. Li, L. and Hu, Y. (2017), "Post-buckling analysis of functionally graded nanobeams incorporating non-local stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., 120, 159-170. https://doi.org/10.1016/j.ijmecsci.2016.11.025.
  38. Lu, C.F., Chen, W.Q., Xu, R.Q. and Lim, C. W. (2008), "Semianalytical elasticity solutions for bi-directional functionally graded beams", Int. J. Solids Struct., 45(1), 258-275. https://doi.org/10.1016/j.ijsolstr.2007.07.018.
  39. Ma, X., Sahmani, S. and Safaei, B. (2021), "Quasi-3D large deflection nonlinear analysis of isogeometric FGM microplates with variable thickness via nonlocal stress-strain gradient elasticity", Eng. Comput., 38(4), 3691-3704. https://doi.org/10.1007/s00366-021-01390-y.
  40. Madenci, E. (2019), "A refined functional and mixed formulation to static analyses of fgm beams", Struct. Eng. Mech., 69(4), 427-437. https://doi.org/10.12989/sem.2019.69.4.427.
  41. Mehar, K. and Panda, S.K. (2019), "Multiscale modeling approach for thermal buckling analysis of nanocomposite curved structure", Adv. Nano Res., 7(3), 181-190. https://doi.org/10.12989/ANR.2019.7.3.181.
  42. Merzoug, M., Bourada, M., Sekkal, M., Ali Chaibdra, A., Belmokhtar, C., Benyoucef, S. and Benachour, A. (2020), "2D and quasi 3D computational models for thermoelastic bending of FG beams on variable elastic foundation: Effect of the micromechanical models", Geomech. Eng., 22(4), 361-374. https://doi.org/10.12989/gae.2020.22.4.361.
  43. Mozafari, H. and Ayob, A. (2012), "Effect of thickness variation on the mechanical buckling load in plates made of functionally graded materials", Proc. Tech., 1, 496-504. https://doi.org/10.1016/j.protcy.2012.02.108.
  44. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares, C.M.M. (2013), "Free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique", Compos. B, 44(1), 657-674. https://doi.org/10.1016/j.compositesb.2012.01.089.
  45. Nguyen, D.K., Vu, A.N.T., Le, N.A.T. (2020), "Dynamic behaviour of a bidirectional functionally graded sandwich beam under nouniform motion of a moving load", Shock Vib., https://doi.org/10.1155/2020/8854076.
  46. Nguyen, T.K., Vo, T.P., Nguyen, B.D. and Lee, J. (2016), "An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory", Compos. Struct., 156, 238-252. https://doi.org/10.1016/j.compstruct.2015.11.074.
  47. Oner (2022), "Study the double receding contact problem of two layers made of E-FG materials and supported by homogeneous half plane using different methods (CM, FEM and ANM)". https://doi.org/10.1002/zamm.202100287.
  48. Oner, E., Sengul, B., Ecren, S., Yaylaci, U., 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", J. Appl. Mathem. Mech.., 102(2), e202100287. https://doi.org/10.1002/zamm.202100287.
  49. Oner, E., Yaylaci, M. and Birinci, A. (2015), "Analytical solution of a contact problem and comparison with the results from FEM", Struct. Eng.Mech., 54(4), 607-622. https://doi.org/10.12989/sem.2015.54.4.000.
  50. Osofero, A.I., Vo, T.P., Nguyen, T.K. and Lee, J. (2015), "Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi-3D theories", J. Sandw. Struct. Mater., 18(1), 3-29. https://doi.org/10.1177/1099636215582217.
  51. Pradhan, S.C. and Murmu, T. (2009), "Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method", J. Sound Vib., 321(1-2), 342-362. https://doi.org/10.1016/j.jsv.2008.09.018.
  52. Priyanka, R., Twinkle, C.M. and Pitchaimani, J. (2021), "Stability and dynamic behavior of porous FGM beam: influence of graded porosity, graphene platelets, and axially varying loads", Eng. Comput., 1-20. https://doi.org/10.1007/s00366-021-01478-5.
  53. 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.
  54. Rajasekaran, S. and Khaniki, H.B. (2019), "Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass", Appl. Math. Model. 72, 129-154. https://doi.org/10.1016/j.apm.2019.03.021.
  55. Reddy, J.N. (1997), Mechanics of Laminated Composite Plate: Theory and analysis, CRC Press, New York, U.S.A.
  56. Reddy, J.N. (2011), "Microstructure-dependent couple stress theories of functionally graded beams", J. Mech. Phys. Solids, 59(11), 2382-2399. https://doi.org/10.1016/j.jmps.2011.06.008.
  57. Refrafi, S., Bousahla, A.A., Bouhadra, A., Menasria, A. Bourada, F., Tounsi, A. Adda Bedia, E.A., Mahmoud, S.R., Benrahou, K.H., and Tounsi, A. (2020), "Effects of hygro-thermomechanical conditions on the buckling of FG sandwich plates resting on elastic foundations", Comput. Concrete, 25(4), 311-325. https://doi.org/10.12989/cac.2020.25.4.311.
  58. Saha, R. and Maiti, P.R. (2012), "Buckling of simply supported FGM plates under uniaxial load", Int. J. Civil Struct. Eng., 2(4), 1036-1050.
  59. Sayyad, A.S. and Ghugal, Y.M. (2018), "Modeling and analysis of functionally graded sandwich beams: A review", Mech. Adv. Mater. Struct., 1-20. https://doi.org/10.1080/15376494.2018.1447178.
  60. Selmi, A. (2020), "Exact solution for nonlinear vibration of clamped-clamped functionally graded buckled beam", Smart Struct. Syst., 26(3), 361-371. https://doi.org/10.12989/SSS.2020.26.3.361.
  61. Shafiei, N. and Kazemi, M. (2017), "Nonlinear buckling of functionally graded nano-/micro-scaled porous beams", Compos. Struct., 178, 483-492. https://doi.org/10.1016/j.compstruct.2017.07.045.
  62. Shahsavari, D., Karami, B. and Janghorban, M. (2019), "Sizedependent vibration analysis of laminated composite plates", Adv. Nano Res., 7(5), 337-349. https://doi.org/10.12989/ANR.2019.7.5.337.
  63. Simsek, M. (2015), "Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions", Compos. Struct., 133, 968-978. https://doi.org/10.1016/j.compstruct.2015.08.021.
  64. Simsek, M. and Al-shujairi, M. (2017), "Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads", Compos. Part B: Eng., 108, 18-34. https://doi.org/10.1016/j.compositesb.2016.09.098.
  65. Simsek, M. and Yurtcu, H.H. (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Compos. Struct., 97, 378-386. https://doi.org/10.1016/j.compstruct.2012.10.038.
  66. Soncco, K., Jorge, X. and Arciniega, R. (2019), "Postbuckling Analysis of Functionally Graded Beams", IOP Conference Series: Materials Science and Engineering, 473, 012028. https://doi.org/10.1088/1757-899x/473/1/012028.
  67. Songsuwan, W., Pimsarn, M. and Wattanasakulpong, N. (2018), "Dynamic Responses of Functionally Graded Sandwich Beams Resting on Elastic Foundation Under Harmonic Moving Loads", Int. J. Struct. Stab. Dyn., 1850112. https://doi.org/10.1142/s0219455418501122.
  68. Su, Z., Jin, G., Wang, Y. and Ye, X. (2016), "A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations", Acta Mechanica, 227(5), 1493-1514. https://doi.org/10.1007/s00707-016-1575-8.
  69. Thai, H.T. and Vo, T.P. (2013), "A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates", Appl. Math. Model., 37(5), 3269-3281. https://doi.org/10.1016/j.apm.2012.08.008..
  70. Timesli, A. (2020), "Prediction of the critical buckling load of SWCNT reinforced concrete cylindrical shell embedded in an elastic foundation", Comput. Concrete., 26(1), 53-62. https://doi.org/10.12989/CAC.2020.26.1.053.
  71. Tossapanon, P. and Wattanasakulpong, N. (2016), "Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation", Compos. Struct., 142, 215-225. https://doi.org/10.1016/j.compstruct.2016.01.085.
  72. Trinh, L.C., Vo, T.P., Osofero, A.I. and Lee, J. (2016), "Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach", Compos. Struct., 156, 263-275. https://doi.org/10.1016/j.compstruct.2015.11.010.
  73. Uzun Yaylaci, E., Yaylaci, M., Olmez, H. and Birinci, A., (2020), "Artificial neural network calculations for a receding contact problem", Comput. Concrete, 25(6), https://doi.org/10.12989/cac.2020.25.6.551.
  74. Vedat Taskin and Pinar Aydan Demirhan (2021), "Static analysis of simply supported porous sandwich plates", 77(4), 549-557. https://doi.org/org/10.12989/sem.2021.77.4.549.
  75. Vinyas, M. (2020), "On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT", Compos. Struct., 240, 112044. https://doi.org/10.1016/j.compstruct.2020.112044.
  76. Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. and Lee, J. (2015), "A quasi-3D theory for vibration and buckling of functionally graded sandwich beams", Compos. Struct., 119, 1-12. https://doi.org/10.1016/j.compstruct.2014.08.006.
  77. Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. and Lee, J. (2015), "Static behaviour of functionally graded sandwich beams using a quasi-3D theory", Compos. Part B: Eng., 68, 59-74. https://doi.org/10.1016/j.compositesb.2014.08.030.
  78. Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A. and Lee, J. (2014), "Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory", Eng. Struct., 64, 12-22. https://doi.org/10.1016/j.engstruct.2014.01.029.
  79. Wang, Z., Wang, X., Xu, G., Cheng, S. and Zeng, T. (2016), "Free vibration of two-directional functionally graded beams", Compos. Struct., 135, 191-198. https://doi.org/10.1016/j.compstruct.2015.09.013.
  80. Wu, H., Kitipornchai, S. and Yang, J. (2017), "Imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite beams", Appl. Mathem. Modelling, 42, 735-752. https://doi.org/10.1016/j.apm.2016.10.045.
  81. Xiaohui Y., Saeid S., and Babak S. (2019), "Post buckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects", 37(2), 1549-1564. https://doi.org/10.1007/s00366-019-00901-2.
  82. Yang, J., Liew, K.M. and Kitipornchai, S. (2005), "Second-order statistics of the elastic buckling of functionally graded rectangular plates", Compos. Sci. Tech., 65(7-8), 1165-1175. https://doi.org/10.1016/j.compscitech.2004.11.012.
  83. Yang, J., Wu, H. and Kitipornchai, S. (2017), "Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams", Compos. Struct., 161, 111-118. https://doi.org/10.1016/j.compstruct.2016.11.048.
  84. Yang, T., Tang, Y., Li, Q., & Yang, X.D. (2018), "Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams", Compos. Struct., 204, 313-319. https://doi.org/10.1016/j.compstruct.2018.07.045.
  85. Yang, Y., Lam, C. C., Kou, K.P. and Iu, V. P. (2014), "Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method", Compos. Struct., 117, 32-39. https://doi.org/10.1016/j.compstruct.2014.06.016.
  86. Yarasca, J., Mantari, J.L. and Arciniega, R.A. (2016), "Hermite-Lagrangian finite element formulation to study functionally graded sandwich beams", Compos. Struct., 140, 567-581. https://doi.org/10.1016/j.compstruct.2016.01.015.
  87. Yaylaci M. (2022), "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.
  88. Yaylaci M., Abanoz M., Yaylaci E.U., Olmez H., Sekban D.M. and Birinci A., (2022), "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, 1953-1971. https://doi.org/10.1007/s00419-022-02159-5.
  89. Yaylaci M., Yayli M., Uzun Yaylaci E., 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.
  90. Yaylaci Murat and Birinci Ahmet (2013), "The receding contact problem of two elastic layers supported by two elastic quarter planes", Struct. Eng. Mech., 48(2), 241-255. http://doi.org/10.12989/sem.2013.48.2.241.
  91. Yaylaci, M. and Avcar, M. (2020), "Finite element modeling of contact between an elastic layer and two elastic quarter planes", Comput. Concrete, 26(2), 107-114. https://doi.org/10.12989/CAC.2020.26.2.107.
  92. Yaylaci, M., (2016), "The investigation crack problem through numerical analysis", Struct. Eng. Mech., 57(6), 1143-1156. https://doi.org/10.12989/sem.2016.57.6.1143.
  93. Yaylaci, M., Adiyaman, E., Oner, E. and Birinci, A., (2020). "Examination of analytical and finite element solutions regarding contact of a functionally graded layer", Struct. Eng. Mech., 76(3), 325-336. https://doi.org/10.12989/sem.2020.76.3.325.
  94. Yaylaci, M., Eyuboglu, A., Adiyaman, G., Uzun Yaylaci, E., Oner, E. and Birinci, A., (2021)"Assessment of different solution methods for receding contact problems in functionally graded layered mediums", Mech. Mater., https://doi.org/10.1016/j.mechmat.2020.103730.
  95. Yaylaci, M., Sabano, B.S., Ozdemir, M.E. and Birinci, A. (2022), "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.
  96. Yaylaci, M., Terzi, C. and Avcar, M. (2019), "Numerical analysis of the receding contact problem of two bonded layers resting on an elastic half plane", Struct. Eng. Mech., https://doi.org/10.12989/sem.2019.72.6.325.
  97. Zenkour, A. M.R and A.F. Radwan (2019), "Hygrothermomechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model", J. Comput. Meth. Eng. Sci. Mech., https://doi.org/10.1080/15502287.2019.1568618.
  98. Zenkour, A.M. (2006), "Generalized shear deformation theory for bending analysis of functionally graded plates", Appl. Math. Model., 30(1), 67-84. https://doi.org/10.1016/j.apm.2005.03.009.
  99. Zhao, X., Lee, Y.Y. and Liew, K.M. (2009), "Mechanical and thermal buckling analysis of functionally graded plates", Compos. Struct., 90(2), 161-171. https://doi.org/10.1016/j.compstruct.2009.03.005.