Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Exact third-order static and free vibration analyses of functionally graded porous curved beam

  • Beg, Mirza S. (Smart Structures Lab, Department of Mechanical Engineering, Z. H. College of Engineering and Technology, Aligarh Muslim University) ;
  • Khalid, Hasan M. (Smart Structures Lab, Department of Mechanical Engineering, Z. H. College of Engineering and Technology, Aligarh Muslim University) ;
  • Yasin, Mohd Y. (Smart Structures Lab, Department of Mechanical Engineering, Z. H. College of Engineering and Technology, Aligarh Muslim University) ;
  • Hadji, L. (Laboratory of Geomatics and Sustainable Development, Ibn Khaldoun University of Tiaret)
  • Received : 2019.12.28
  • Accepted : 2021.02.18
  • Published : 2021.04.10
⟨ Previous Next ⟩

Abstract

An exact solution based on refined third-order theory (TOT) has been presented for functionally graded porous curved beams having deep curvature. The displacement field of the refined TOT is derived by imposing the shear free conditions at the outer and inner surfaces of curved beams. The properties of the two phase composite are tailored according the power law rule and the effective properties are computed using Mori-Tanaka homogenization scheme. The equations of motion as well as consistent boundary conditions are derived using the Hamilton's principle. The curved beam stiffness coefficients (A, B, D) are obtained numerically using six-point Gauss integration scheme without compromising the accuracy due to deepness (1 + z/R) terms. The porosity has been modeled assuming symmetric (even) as well as asymmetric (uneven) distributions across the cross section of curved beam. The programming has been performed in MATLAB and is validated with the results available in the literature as well as 2D finite element model developed in ABAQUS. The effect of inclusion of 1 + z/R terms is studied for deflection, stresses and natural frequencies for FG curved beams of different radii of curvature. Results presented in this work will be useful for comparison of future studies.

Keywords

References

  1. Addou, F.Y., Meradjah, M., Bousahla, A.A., Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S. (2019), "Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT", Comput. Concrete, 24(4), 347-367. http://dx.doi.org/10.12989/cac.2019.24.4.347.
  2. Akbarzadeh, A., Abedini, A. and Chen, Z. (2015), "Effect of micromechanical models on structural responses of functionally graded plates", Compos. Struct., 119, 598-609. https://doi.org/10.1016/j.compstruct.2014.09.031.
  3. 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.
  4. Akbas, S.D. (2018), "Forced vibration analysis of functionally graded porous deep beams", Compos. Struct., 186, 293-302. https://doi.org/10.1016/j.compstruct.2017.12.013.
  5. Al Rjoub, Y.S. and Hamad, A.G. (2017), "Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method", KSCE J. Civil Eng., 21(3), 792-806. https://doi.org/10.1155/2017/8186976.
  6. Allahkarami, F., Nikkhah-Bahrami, M. and Saryazdi, M.G. (2018), "Nonlinear forced vibration of FG-CNTs-reinforced curved microbeam based on strain gradient theory considering out-ofplane motion", Steel Compos. Struct., 26(6), 673-691. https://doi.org/10.12989/scs.2018.26.6.673.
  7. Amir, M. and Talha, M. (2018), "Thermoelastic vibration of shear deformable functionally graded curved beams with microstructural defects", Int. J. Struct. Stab. Dynam., 18(11), 1850135. https://doi.org/10.1142/S0219455418501353.
  8. Amir, M. and Talha, M. (2019a), "Imperfection sensitivity in the vibration behavior of functionally graded arches by considering microstructural defects", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(8), 2763-2777. https://doi.org/10.1177/0954406218792584.
  9. Amir, M. and Talha, M. (2019b), "Nonlinear vibration characteristics of shear deformable functionally graded curved panels with porosity including temperature effects", Int. J. Pressure Vess. Piping, 172, 28-41. https://doi.org/10.1016/j.ijpvp.2019.03.008.
  10. Arefi, M. and Zenkour, A.M. (2018), "Thermal stress and deformation analysis of a size-dependent curved nanobeam based on sinusoidal shear deformation theory", Alexandria Eng. J., 57(3), 2177-2185. https://doi.org/10.1016/j.aej.2017.07.003.
  11. Atmane, H.A., Tounsi, A. and Bernard, F. (2017), "Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations", Int. J. Mech. Mater. Des., 13(1), 71-84. https://doi.org/10.1007/s10999-015-9318-x
  12. Bedia, W.A., Houari, M.S.A., Bessaim, A., Bousahla, A.A., Tounsi, A., Saeed, T. and Alhodaly, M.S. (2019), "A new hyperbolic two-unknown beam model for bending and buckling analysis of a nonlocal strain gradient nanobeams", J. Nano Res., 57, 175-191. https://doi.org/10.4028/www.scientific.net/JNanoR.57.175.
  13. Berghouti, H., Adda Bedia, E., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351
  14. Beg, M.S., Yasin, M.Y. and Khalid, H.M. (2018), "Analysis of laminated and FGM beams using various theories", IOP Conference Series: Materials Science and Engineering, Vol. 404. IOP Publishing, p. 012030. https://doi.org/10.1088/1757-899X/404/1/012030
  15. Bhattacharyya, M., Kapuria, S. and Kumar, A. (2007), "On the stress to strain transfer ratio and elastic deflection behavior for Al/SiC functionally graded material", Mech. Adv. Mater. Struct., 14(4), 295-302. https://doi.org/10.1080/15376490600817917.
  16. Birman, V. and Byrd, L.W. (2007), "Modeling and analysis of functionally graded materials and structures", Appl. Mech. Review., 60(5), 195-216. https://doi.org/10.1115/1.2777164.
  17. Boukhlif, Z., Bouremana, M., Bourada, F., Bousahla, A.A., Bourada, M., Tounsi, A. and Al-Osta, M.A. (2019), "A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation", Steel Compos.Struct., 31(5), 503-516. http://dx.doi.org/10.12989/scs.2019.31.5.503.
  18. Bourada, F., Bousahla, A.A., Bourada, M., Azzaz, A., Zinata, A. and Tounsi, A. (2019), "Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory", Wind Struct., 28(1), 19-30. http://dx.doi.org/10.12989/was.2019.28.1.019
  19. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., 7(3), 189-206. http://dx.doi.org/10.12989/anr.2019.7.3.191.
  20. Bouremana, M., Houari, M.S.A., Tounsi, A., Kaci, A., and Bedia, E.A.A. (2013), "A new first shear deformation beam theory based on neutral surface position for functionally graded beams", Steel Composi. Struct., 15(5), 467-479. https://doi.org/10.12989/scs.2013.15.5.467.
  21. Chaabane, L.A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F. Z., Tounsi, A., Derras, A., Bousahla, A.A. and Tounsi, A. (2019), "Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation", Struct. Eng. Mech., 71(2), 185-196. http://dx.doi.org/10.12989/sem.2019.71.2.185.
  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. 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.
  25. Cho, J. and Ha, D. (2001), "Averaging and finite element discretization approaches in the numerical analysis of functionally graded materials", Mater. Sci. Eng. A, 302(2), 187-196. https://doi.org/10.1016/S0921-5093(00)01835-9.
  26. Ebrahimi, F. and Daman, M. (2017), "Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment", Struct. Eng. Mech., 64(1), 121-133. https://doi.org/10.12989/sem.2017.64.1.121.
  27. Ebrahimi, F. and Jafari, A. (2016), "A higher-order thermomechanical vibration analysis of temperature-dependent FGM beams with porosities", J. Eng., Article ID 9561504. http://dx.doi.org/10.1155/2016/9561504.
  28. Ebrahimi, F. and Jafari, A. (2018), "A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities", Mech. Adv. Mater. Struct., 25(3), 212-224. https://doi.org/10.1080/15376494.2016.1255820
  29. Ebrahimi, F. and Mokhtari, M. (2015), "Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method", J. Braz. Soc. Mech. Sci. Eng., 37(4), 1435-1444. https://doi.org/10.1007/s40430-014-0255-7.
  30. Eltaher, M., Fouda, N., El-midany, T. and Sadoun, A. (2018), "Modified porosity model in analysis of functionally graded porous nanobeams", J. Braz. Soc. Mech. Sci. Eng., 40(3), 141. https://doi.org/10.1007/s40430-018-1065-0
  31. Fahsi, B., Bouiadjra, R.B., Mahmoudi, A., Benyoucef, S. and Tounsi, A. (2019), "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(2), 1-12. https://doi.org/10.1007/s11029-019-09805-0
  32. Fang, W., Yu, T., Van Lich, L. and Bui, T. Q. (2019), "Analysis of thick porous beams by a quasi-3D theory and isogeometric analysis", Compos. Struct., 221, 110890. https://doi.org/10.1016/j.compstruct.2019.04.062.
  33. 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.
  34. Fereidoon, A., Andalib, M. and Hemmatian, H. (2015), "Bending analysis of curved sandwich beams with functionally graded core", Mech. Adv. Mater. Struct., 22(7), 564-577. https://doi.org/10.1080/15376494.2013.828815.
  35. Filipich, C.P. and Piovan, M.T. (2010), "The dynamics of thick curved beams constructed with functionally graded materials", Mech. Res. Commun., 37(6), 565-570. https://doi.org/10.1016/j.mechrescom.2010.07.007.
  36. Gasik, M.M. (1998), "Micromechanical modelling of functionally graded materials", Comput. Mater. Sci., 13(1-3), 42-55. https://doi.org/10.1016/S0927-0256(98)00044-5.
  37. Hadji, L., Zouatnia, N. and Bernard, F. (2019), "An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models", Struct. Eng. Mech., 69(2), 231-241. https://doi.org/10.12989/sem.2019.69.2.231.
  38. Hajianmaleki, M. and Qatu, M. S. (2012), "Static and vibration analyses of thick, generally laminated deep curved beams with different boundary conditions", Compos. Part B: Eng., 43(4), 1767-1775. https://doi.org/10.1016/j.compositesb.2012.01.019.
  39. Haskul, M. (2020), "Elastic state of functionally graded curved beam on the plane stress state subject to thermal load", Mech. Based Des. Struct. Machines, 48(6), 739-754. https://doi.org/10.1080/15397734.2019.1660890
  40. Hosseini, S. and Rahmani, O. (2016), "Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model", Appl. Phys. A, 122(3), 169. https://doi.org/10.1007/s00339-016-9696-4.
  41. Huynh, T.A., Luu, A.T. and Lee, J. (2017), "Bending, buckling and free vibration analyses of functionally graded curved beams with variable curvatures using isogeometric approach", Meccanica, 52(11-12), 2527-2546. https://doi.org/10.1007/s11012-016-0603-z.
  42. Javani, M., Kiani, Y. and Eslami, M. (2019), "Free vibration of arbitrary thick FGM deep arches using unconstrained higherorder shear deformation theory", Thin-Wall. Struct., 136, 258-266. https://doi.org/10.1016/j.tws.2018.12.020.
  43. Jha, D., Kant, T. and Singh, R. (2013), "A critical review of recent research on functionally graded plates", Compos. Struct., 96, 833-849. https://doi.org/10.1016/j.compstruct.2012.09.001.
  44. Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A., Bedia, E. and Al-Osta, M.A. (2020), "A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: bending and free vibration analysis", Comput. Concrete, 25(1), 37-57. http://dx.doi.org/10.12989/cac.2020.25.1.037.
  45. Kapuria, S., Bhattacharyya, M. and Kumar, A. (2008a), "Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation", Compos. Struct., 82(3), 390-402. https://doi.org/10.1016/j.compstruct.2007.01.019
  46. Kapuria, S., Bhattacharyya, M. and Kumar, A. (2008b), "Theoretical modeling and experimental validation of thermal response of metal-ceramic functionally graded beams", J. Therm. Stresses, 31(8), 759-787. https://doi.org/10.1016/j.compstruct.2007.01.019.
  47. Kapuria, S., Patni, M. and Yasin, M.Y. (2015), "A quadrilateral shallow shell element based on the third-order theory for functionally graded plates and shells and the inaccuracy of rule of mixtures", Eur. J. Mech.-A/Solids, 49, 268-282. https://doi.org/10.1016/j.euromechsol.2014.06.010.
  48. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019), "Influence of homogenization schemes on vibration of functionally graded curved microbeams", Compos. Struct., 216, 67-79. https://doi.org/10.1016/j.compstruct.2019.02.089.
  49. Karami, B., Janghorban, M. and Tounsi, A. (2019a), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Eng. with Comput., 35(4), 1297-1316. http://dx.doi.org/10.1007/s00366-018-0664-9.
  50. Karami, B., Janghorban, M. and Tounsi, A. (2019d), "Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation", Struct. Eng. Mech., 70(1), 55-66. http://dx.doi.org/10.12989/sem.2019.70.1.055.
  51. Kendall, K., Howard, A.J. and Birchall, D.J. (1983), "The relation between porosity, microstructure and strength, and the approach to advanced cement-based materials", Philos. T. Roy. Soc. London. Series A, Math. Phys. Sci., 310(1511), 139-153. https://doi.org/10.1098/rsta.1983.0073.
  52. Khan, M.A., Yasin, M., Beg, M.S. and Khan, A. (2020), "Free and forced vibration analysis of functionally graded beams using finite element model based on refined third-order theory", Emerging Trends in Mechanical Engineering, 603-612. https://doi.org/10.1007/978-981-32-9931-3
  53. Khiloun, M., Bousahla, A.A., Kaci, A., Bessaim, A., Tounsi, A. and Mahmoud, S. (2020), "Analytical modeling of bending and vibration of thick advanced composite plates using a fourvariable quasi 3D HSDT", Eng. with Comput., 36(3), 807-821. https://doi.org/10.1007/s00366-019-00732-1.
  54. Kurtaran, H. (2015), "Large displacement static and transient analysis of functionally graded deep curved beams with generalized differential quadrature method", Compos. Struct., 131, 821-831. https://doi.org/10.1016/j.compstruct.2015.06.024.
  55. Levinson, M. (1981), "A new rectangular beam theory", J. Sound Vib., 74(1), 81-87. https://doi.org/10.1016/0022-460X(81)90493-4.
  56. Lim, C.W., Yang, Q. and Lu, C.F. (2009), "Two-dimensional elasticity solutions for temperature-dependent in-plane vibration of FGM circular arches", Compos. Struct., 90, 323-329. https://doi.org/10.1016/j.compstruct.2009.03.014.
  57. Malekzadeh, P. (2009), "Two-dimensional in-plane free vibrations of functionally graded circular arches with temperaturedependent properties", Compos. Struct., 91(1), 38-47. https://doi.org/10.1016/j.compstruct.2009.04.034.
  58. Malekzadeh, P., Atashi, M. and Karami, G. (2009), "In-plane free vibration of functionally graded circular arches with temperature-dependent properties under thermal environment", J. Sound Vib., 326, 837-851. https://doi.org/10.1016/j.jsv.2009.05.016.
  59. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle", Steel Compos. Struct., 32(5), 595-610. http://dx.doi.org/10.12989/scs.2019.32.5.595.
  60. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S. (2019), "An analytical solution for bending, buckling and vibration responses of FGM sandwich plates", J. Sandw. Struct. Mater., 21(2), 727-757. https://doi.org/10.1177/1099636217698443.
  61. Nguyen, D.K. and Tran, T.T. (2018), "Free vibration of tapered BFGM beams using an efficient shear deformable finite element model", Steel Compos. Struct., 29(3), 363-377. https://doi.org/10.12989/scs.2018.29.3.363.
  62. Piovan, M.T., Domini, S. and Ramirez, J. M. (2012), "In-plane and out-of-plane dynamics and buckling of functionally graded circular curved beams", Compos. Struct., 94(11), 3194-3206. https://doi.org/10.1016/j.compstruct.2012.04.032.
  63. Polit, O., Anant, C., Anirudh, B. and Ganapathi, M. (2019), "Functionally graded graphene reinforced porous nanocomposite curved beams: Bending and elastic stability using a higher-order model with thickness stretch effect", Compos. Part B: Eng., 166, 310-327. https://doi.org/10.1016/j.compositesb.2018.11.074.
  64. Pydah, A. and Batra, R. (2017), "Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams", Compos. Struct., 172, 45-60. https://doi.org/10.1016/j.compstruct.2017.03.072.
  65. Rahmani, O., Hosseini, S., Ghoytasi, I. and Golmohammadi, H. (2018), "Free vibration of deep curved FG nano-beam based on modified couple stress theory", Steel Compos. Struct., 26(5), 607-620. https://doi.org/10.12989/scs.2018.26.5.607.
  66. Reiter, T. and Dvorak, G. J. (1998), "Micromechanical models for graded composite materials: II. thermomechanical loading", J. Mech. Phys. Solid., 46(9), 1655-1673. https://doi.org/10.1016/S0022-5096(97)00039-2.
  67. Reiter, T., Dvorak, G.J. and Tvergaard, V. (1997), "Micromechanical models for graded composite materials", J. Mech. Phys. Solid., 45(8), 1281-1302. https://doi.org/10.1016/S0022-5096(97)00007-0.
  68. Sayyad, A.S. and Ghugal, Y.M. (2018), "Modeling and analysis of functionally graded sandwich beams: A review", Mech. Adv. Mater. Struct., 26(21), 1-20. https://doi.org/10.1080/15376494.2018.1447178.
  69. Sayyad, A.S. and Ghugal, Y.M. (2019), "A sinusoidal beam theory for functionally graded sandwich curved beams", Compos. Struct., 226, 111246. https://doi.org/10.1016/j.compstruct.2019.111246.
  70. Semmah, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2019), "Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., 7(2), 89-98. http://dx.doi.org/10.12989/anr.2019.7.2.089.
  71. Teraki, J., Hirano, T. and Wakashima, K. (1993), "Elastic-plastic analysis of thermal stresses in an FGM plate under cyclic thermal load", Ceramic Transactions, 34, 67-74.
  72. Tlidji, Y., Zidour, M., Draiche, K., Safa, A., Bourada, M., Tounsi, A., Bousahla, A.A. and Mahmoud, S. (2019), "Vibration analysis of different material distributions of functionally graded microbeam", Struct. Eng. Mech., 69(6), 637-649. http://dx.doi.org/10.12989/sem.2019.69.6.637.
  73. Tufekci, E., Eroglu, U. and Aya, S.A. (2016), "Exact solution for in-plane static problems of circular beams made of functionally graded materials", Mech. Based Des. Struct. Machines, 44(4), 476-494. https://doi.org/10.1080/15397734.2015.1121398.
  74. Vel, S.S. and Batra, R. (2004), "Three-dimensional exact solution for the vibration of functionally graded rectangular plates", J. Sound Vib., 272(3-5), 703-730. https://doi.org/10.1016/S0022-460X(03)00412-7.
  75. Wakashima, K. and Tsukamoto, H. (1991), "Mean-field micromechanics model and its application to the analysis of thermomechanical behaviour of composite materials", Mater. Sci. Eng: A, 146(1-2), 291-316. https://doi.org/10.1016/0921-5093(91)90284-T.
  76. Wattanasakulpong, N. and Chaikittiratana, A. (2015), "Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method", Meccanica, 50(5), 1331-1342. https://doi.org/10.1007/s11012-014-0094-8
  77. Wattanasakulpong, N., Chaikittiratana, A. and Pornpeerakeat, S. (2018), "Chebyshev collocation approach for vibration analysis of functionally graded porous beams based on third-order shear deformation theory", Acta Mechanica Sinica, 34(6), 1124-1135. https://doi.org/10.1007/s10409-018-0770-3.
  78. Wattanasakulpong, N., Prusty, B.G., Kelly, D.W. and Hoffman, M. (2012), "Free vibration analysis of layered functionally graded beams with experimental validation", Mater. Design, 36, 182-190. https://doi.org/10.1016/j.matdes.2011.10.049.
  79. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002.
  80. 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.
  81. Yasin, M.Y., Khalid, H.M. and Beg, M.S. (2020), "Exact solution considering layerwise mechanics for laminated composite and sandwich curved beams of deep curvatures", Compos. Struct., 244, 112258. https://doi.org/10.1016/j.compstruct.2020.112258.
  82. Yousefi, A. and Rastgoo, A. (2011), "Free vibration of functionally graded spatial curved beams", Compos. Struct., 93(11), 3048-3056. https://doi.org/10.1016/j.compstruct.2011.04.024.
  83. Zaoui, F.Z., Ouinas, D. and Tounsi, A. (2019), "New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations", Compos. Part B: Eng., 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051.
  84. Zhang, C. and Wang, Q. (2018), "Free vibration analysis of elastically restrained functionally graded curved beams based on the Mori-Tanaka scheme", Mech. Adv. Mater. Struct., 26(21), 1821-1831. https://doi.org/10.1080/15376494.2018.1452318.
  85. Zhao, J., Wang, Q., Deng, X., Choe, K., Xie, F. and Shuai, C. (2019), "A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams", Compos. Part B: Eng., 165, 155-166. https://doi.org/10.1016/j.compositesb.2018.11.080.
  86. Zouatnia, N. and Hadji, L. (2019), "Effect of the micromechanical models on the bending of FGM beam using a new hyperbolic shear deformation theory", Earthq. Struct., 16(2), 177-183. https://doi.org/10.12989/eas.2019.16.2.177.