Steel and Composite Structures

  • 제19권2호
  • /
  • Pages.369-384
  • /
  • 2015
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

A computational shear displacement model for vibrational analysis of functionally graded beams with porosities

  • Atmane, Hassen Ait (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) ;
  • Bernard, Fabrice (Laboratoire de Genie Civil et Genie Mecanique INSA de Rennes) ;
  • Mahmoud, S.R. (Department of Mathematics, Faculty of Science, King Abdulaziz University)
  • 투고 : 2014.11.18
  • 심사 : 2015.01.18
  • 발행 : 2015.08.25
⟨ 이전 논문 다음 논문 ⟩

초록

This work presents a free vibration analysis of functionally graded metal-ceramic (FG) beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a simple displacement field based on higher order shear deformation theory is implemented. The proposed theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. In addition, it has strong similarities with Euler-Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. By employing the Hamilton's principle, governing equations of motion for coupled axial-shear-flexural response are determined. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

키워드

참고문헌

  1. Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  2. Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech, Int. J., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  3. Amara, K., Tounsi, A., Mechab, I. and Adda Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Math. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  4. Attia, A, Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct, Int. J., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187
  5. Bachir Bouiadjra, R., Adda Bedia, E.A. and Tounsi, A. (2013), "Nonlinear thermal buckling behavior of functionally graded plates using an efficient sinusoidal shear deformation theory", Struct. Eng. Mech, Int. J., 48(4), 547-567. https://doi.org/10.12989/sem.2013.48.4.547
  6. Bedjilili, Y., Tounsi, A., Berrabah, H.M., Mechab, I., Adda Bedia, E.A. and Benaissa, S. (2009), "Natural frequencies of composite beams with a variable fiber volume fraction including rotary inertia and shear deformation", Appl. Math. Mech. - Engl. Ed., 30(6), 1-10. https://doi.org/10.1007/s10483-009-0101-z
  7. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos.: Part B, 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  8. Benatta, M.A., Mechab, I., Tounsi, A. and Adda Bedia, E.A. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci., 44(2), 765-773. https://doi.org/10.1016/j.commatsci.2008.05.020
  9. Benatta, M.A, Tounsi, A., Mechab, I. and Bachir Bouiadjra, M. (2009), "Mathematical solution for bending of short hybrid composite beams with variable fibers spacing", Appl. Math. Comput., 212(2), 337-348. https://doi.org/10.1016/j.amc.2009.02.030
  10. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  11. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N. and Boumia, L. (2008), "The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 41(22), 225404. https://doi.org/10.1088/0022-3727/41/22/225404
  12. Berrabah, H.M., Tounsi, A., Semmah, A. and Adda Bedia, E.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech, Int. J., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  13. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., Int. J., 14(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  14. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., Int. J., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  15. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Method., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  16. Bresse, J.A.C. (1859), Cours de Mecanique Applique, Mallet-Bachelier, Paris, France.
  17. Cowper, G.R. (1966), "The shear coefficients in Timoshenko beam theory", ASME, J. Appl. Mech., 33(2), 335-340. https://doi.org/10.1115/1.3625046
  18. Chakraverty, S. and Pradhan, K.K. (2014), "Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions", Aerosp. Sci. Technol., 36, 132-156. https://doi.org/10.1016/j.ast.2014.04.005
  19. 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
  20. Draiche, K., Tounsi, A. and Khalfi, Y. (2014), "A trigonometric four variable plate theory for free vibration of rectangular composite plates with patch mass", Steel Compos. Struct., Int. J., 17(1), 69-81. https://doi.org/10.12989/scs.2014.17.1.069
  21. Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), "A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates", Meccanica, 49(4), 795-810. https://doi.org/10.1007/s11012-013-9827-3
  22. Hadji, L., Daouadji, T.H., Tounsi, A. and Adda Bedia, E.A. (2014), "A higher order shear deformation theory for static and free vibration of FGM beam", Steel Compos. Struct., Int. J., 16(5), 507-519. https://doi.org/10.12989/scs.2014.16.5.507
  23. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  24. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., ASCE, 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  25. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E., 40(8), 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  26. Kadoli, R., Akhtar, K. and Ganesan, N. (2008), "Static analysis of functionally graded beams using higher order shear deformation theory", Appl. Math. Model., 32(12), 2509-2525. https://doi.org/10.1016/j.apm.2007.09.015
  27. Karama, M., Afaq, K.S. and Mistou, S. (2003), "Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity", Int. J. Solid. Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9
  28. Khalfi, Y., Houari, M.S.A. and Tounsi, A. (2014), "A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation", Int. J. Comput. Method., 11(5), 135007.
  29. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B, 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9
  30. Koochaki, G.R. (2011), "Free vibration analysis of functionally graded beams", World Acad. Sci., Eng. Technol., 74, 366-369.
  31. Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Beg, O. and Mahmoud, S.R. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., Int. J., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  32. Li, X.F. (2008), "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound Vib., 318(4-5), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056
  33. Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  34. Mahmoud, S.R., Abd-Alla, A.M., Tounsi, A. and Marin, M. (2014), "The problem of wave propagation in magneto-rotating orthotropic non-homogeneous medium", J. Vib. Control, 1077546314521443. https://doi.org/10.1177/1077546314521443
  35. Najafov, A.M., Sofivev, A.H., Hui, D., Karaca, Z., Kalpakci, V. and Ozcelik, M. (2014), "Stability of EG cylindrical shells with shear stresses on a Pasternak foundation", Steel Compos. Struct., Int. J., 17(4), 453-470. https://doi.org/10.12989/scs.2014.17.4.453
  36. Nedri, K., El Meiche, N. and Tounsi, A. (2014), "Free vibration analysis of laminated composite plates resting on elastic foundations by using a refined hyperbolic shear deformation theory", Mech. Compos. Mater., 49(6), 641-650. https://doi.org/10.1007/s11029-013-9380-0
  37. Ould Larbi, L., Kaci, A., Houari, M.S.A. and Tounsi, A. (2013), "An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams", Mech. Based Des. Struct. Mach., 41(4), 421-433. https://doi.org/10.1080/15397734.2013.763713
  38. Rayleigh, J.W.S. (1880), The Theory of Sound, Macmillan Publishers, London, UK.
  39. Reddy, J.N. (1984), "A simple higher-order theory for laminated composite plates", ASME, J. Appl. Mech. 51(4), 745-752. https://doi.org/10.1115/1.3167719
  40. Reddy, J.N. (1999), Theory and Anaslysis of Elastic Plates, Taylor & Francis Publication, PA, USA.
  41. Saidi, H., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2013), "Thermo-mechanical bending response with stretching effect of functionally graded sandwich plates using a novel shear deformation theory", Steel Compos. Struct., Int. J., 15(2), 221-245. https://doi.org/10.12989/scs.2013.15.2.221
  42. Sallai, B.O., Tounsi, A., Mechab, I., Bachir, B.M., Meradjah, M. and Adda, B.E.A. (2009), "A theoretical analysis of flexional bending of $Al/Al_2O_3$ S-FGM thick beams", Comput. Mater. Sci., 44(4), 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001
  43. Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696. https://doi.org/10.1016/S0266-3538(01)00007-0
  44. Semmah, A., Tounsi, A., Zidour, M., Heireche, H. and Naceri, M. (2014), "Effect of chirality on critical buckling temperature of a zigzag single-walled carbon nanotubes using nonlocal continuum theory", Fuller. Nanotub. Car. N., 23(6), 518-522. https://doi.org/10.1080/1536383X.2012.749457
  45. Shimpi, R.P. and Patel, H.G. (2006a), "A two variable refined plate theory for orthotropic plate analysis", Int. J. Solid. Struct., 43(22), 6783-6799. https://doi.org/10.1016/j.ijsolstr.2006.02.007
  46. Shimpi, R.P. and Patel, H.G. (2006b), "Free vibrations of plate using two variable refined plate theory", J. Sound Vib., 296(4-5), 979-999. https://doi.org/10.1016/j.jsv.2006.03.030
  47. Sina, S.A., Navazi, H.M. and Haddadpour H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Des., 30(3), 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
  48. Swaminathan, K. and Naveenkumar, D.T. (2014), "Higher order refined computational models for the stability analysis of FGM plates - Analytical solutions", Eur. J. Mech. A/Solids, 47, 349-361. https://doi.org/10.1016/j.euromechsol.2014.06.003
  49. Timoshenko, S.P. (1921), On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, Series 6, pp. 742-746.
  50. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013a), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  51. Tounsi, A., Semmah, A. and Bousahla, A.A. (2013b), "Thermal buckling behavior of nanobeam using an efficient higher-order nonlocal beam theory", J. Nanomech. Micromech., ASCE, 3(3), 37-42. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000057
  52. Tounsi, A., Benguediab, S., Adda Bedia, E.A., Semmah, A. and Zidour, M. (2013c), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  53. Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013d), "A new nonlocal beam theory with thickness stretching effect for nanobeams", Int. J. Nanosci., 12(4), 1350025. https://doi.org/10.1142/S0219581X13500257
  54. Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070
  55. Touratier, M. (1991), "An efficient standard plate theory", Int. J. Eng. Sci., 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y
  56. 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
  57. Wattanasakulpong, N., Prusty, B.G., Kelly, D.W. and Hoffman, M. (2012), "Free vibration analysis of layered functionally graded beams with experimental validation", Mater. Des., 36, 182-190. https://doi.org/10.1016/j.matdes.2011.10.049
  58. Yaghoobi, H. and Torabi, M. (2013a), "Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation", Appl. Math. Model., 37(18-19), 8324-8340. https://doi.org/10.1016/j.apm.2013.03.037
  59. Yaghoobi, H. and Fereidoon, A. (2014), "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, 62, 54-64 https://doi.org/10.1016/j.compositesb.2014.02.014
  60. Yaghoobi, H. and Torabi, M. (2013b), "An analytical approach to large amplitude vibration and post-buckling of functionally graded beams rest on non-linear elastic foundation", J. Theor. Appl. Mech., 51(1), 39-52.
  61. Yaghoobi, H., Valipour, M.S., Fereidoon, A. and Khoshnevisrad, P. (2014), "Analytical study on post-buckling and nonlinear free vibration analysis of FG beams resting on nonlinear elastic foundation under thermo-mechanical loading using VIM", Steel Compos. Struct., Int. J., 17(5), 753-776. https://doi.org/10.12989/scs.2014.17.5.753
  62. Zhong, Z. and Yu, T. (2007), "Analytical solution of a cantilever functionally graded beam", Compos. Sci. Technol., 67(3-4), 481-488. https://doi.org/10.1016/j.compscitech.2006.08.023
  63. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of $ZrO_2-NiCr$ functionally graded material by powder metallurgy", Mater. Chem. Phys., 68(1-3), 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2
  64. Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001

피인용 문헌

  1. Numerical analysis of acoustic radiation properties of laminated composite flat panel in thermal environment: A higher-order finite-boundary element approach 2018, https://doi.org/10.1177/0954406217735866
  2. Thermo-mechanical postbuckling of symmetric S-FGM plates resting on Pasternak elastic foundations using hyperbolic shear deformation theory vol.57, pp.4, 2016, https://doi.org/10.12989/sem.2016.57.4.617
  3. Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature vol.184, 2018, https://doi.org/10.1016/j.compstruct.2017.10.052
  4. A novel four variable refined plate theory for bending, buckling, and vibration of functionally graded plates vol.22, pp.3, 2016, https://doi.org/10.12989/scs.2016.22.3.473
  5. A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate vol.60, pp.4, 2016, https://doi.org/10.12989/sem.2016.60.4.547
  6. Aero-hygro-thermal stability analysis of higher-order refined supersonic FGM panels with even and uneven porosity distributions vol.73, 2017, https://doi.org/10.1016/j.jfluidstructs.2017.06.007
  7. Analysis of postbuckling of graded porous GPL-reinforced beams with geometrical imperfection 2019, https://doi.org/10.1080/15376494.2017.1400622
  8. Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory vol.21, pp.6, 2016, https://doi.org/10.12989/scs.2016.21.6.1287
  9. On thermal stability of plates with functionally graded coefficient of thermal expansion vol.60, pp.2, 2016, https://doi.org/10.12989/sem.2016.60.2.313
  10. An efficient shear deformation theory for wave propagation of functionally graded material plates vol.57, pp.5, 2016, https://doi.org/10.12989/sem.2016.57.5.837
  11. A novel four variable refined plate theory for laminated composite plates vol.22, pp.4, 2016, https://doi.org/10.12989/scs.2016.22.4.713
  12. Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory vol.22, pp.4, 2016, https://doi.org/10.12989/was.2016.22.4.429
  13. Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions vol.182, 2017, https://doi.org/10.1016/j.compstruct.2017.09.008
  14. A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates vol.22, pp.2, 2016, https://doi.org/10.12989/scs.2016.22.2.257
  15. A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates vol.184, 2018, https://doi.org/10.1016/j.compstruct.2017.10.047
  16. Thermal Effects on the Vibration of Functionally Graded Deep Beams with Porosity vol.09, pp.05, 2017, https://doi.org/10.1142/S1758825117500764
  17. Thermal post-buckling behavior of imperfect temperature-dependent sandwich FGM plates resting on Pasternak elastic foundation vol.22, pp.1, 2016, https://doi.org/10.12989/scs.2016.22.1.091
  18. On vibrations of porous nanotubes vol.125, 2018, https://doi.org/10.1016/j.ijengsci.2017.12.009
  19. Effect of porosity on the bending and free vibration response of functionally graded plates resting on Winkler-Pasternak foundations vol.10, pp.6, 2016, https://doi.org/10.12989/eas.2016.10.6.1429
  20. Dynamic modeling of porous heterogeneous micro/nanobeams vol.132, pp.12, 2017, https://doi.org/10.1140/epjp/i2017-11754-7
  21. Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory vol.18, pp.4, 2016, https://doi.org/10.12989/sss.2016.18.4.755
  22. A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation vol.72, 2018, https://doi.org/10.1016/j.ast.2017.11.004
  23. Thermal stability analysis of solar functionally graded plates on elastic foundation using an efficient hyperbolic shear deformation theory vol.10, pp.3, 2016, https://doi.org/10.12989/gae.2016.10.3.357
  24. A simple hyperbolic shear deformation theory for vibration analysis of thick functionally graded rectangular plates resting on elastic foundations vol.11, pp.2, 2016, https://doi.org/10.12989/gae.2016.11.2.289
  25. A refined theory with stretching effect for the flexure analysis of laminated composite plates vol.11, pp.5, 2016, https://doi.org/10.12989/gae.2016.11.5.671
  26. A general higher-order nonlocal couple stress based beam model for vibration analysis of porous nanocrystalline nanobeams vol.112, 2017, https://doi.org/10.1016/j.spmi.2017.09.010
  27. Analysis of postbuckling behavior of general higher-order functionally graded nanoplates with geometrical imperfection considering porosity distributions 2018, https://doi.org/10.1080/15376494.2018.1430280
  28. Forced vibration analysis of functionally graded porous deep beams vol.186, 2018, https://doi.org/10.1016/j.compstruct.2017.12.013
  29. Buckling of symmetrically laminated plates using nth-order shear deformation theory with curvature effects vol.21, pp.6, 2016, https://doi.org/10.12989/scs.2016.21.6.1347
  30. An efficient and simple shear deformation theory for free vibration of functionally graded rectangular plates on Winkler-Pasternak elastic foundations vol.22, pp.3, 2016, https://doi.org/10.12989/was.2016.22.3.329
  31. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory vol.58, pp.3, 2016, https://doi.org/10.12989/sem.2016.58.3.397
  32. Vibration analysis of functionally graded graphene platelet reinforced cylindrical shells with different porosity distributions pp.1537-6532, 2019, https://doi.org/10.1080/15376494.2018.1444235
  33. Modeling and analysis of functionally graded sandwich beams: A review pp.1537-6532, 2018, https://doi.org/10.1080/15376494.2018.1447178
  34. Fracture problems, vibration, buckling, and bending analyses of functionally graded materials: A state-of-the-art review including smart FGMS pp.1548-0046, 2018, https://doi.org/10.1080/02726351.2017.1410265
  35. Vibration and buckling analysis of a rotary functionally graded piezomagnetic nanoshell embedded in viscoelastic media vol.29, pp.11, 2018, https://doi.org/10.1177/1045389X18770856
  36. A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation vol.12, pp.1, 2015, https://doi.org/10.12989/gae.2017.12.1.009
  37. A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams vol.19, pp.2, 2017, https://doi.org/10.12989/sss.2017.19.2.115
  38. Non-linear study of mode II delamination fracture in functionally graded beams vol.23, pp.3, 2015, https://doi.org/10.12989/scs.2017.23.3.263
  39. A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation vol.23, pp.3, 2015, https://doi.org/10.12989/scs.2017.23.3.317
  40. Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT vol.19, pp.3, 2015, https://doi.org/10.12989/sss.2017.19.3.289
  41. A novel and simple HSDT for thermal buckling response of functionally graded sandwich plates vol.62, pp.4, 2017, https://doi.org/10.12989/sem.2017.62.4.401
  42. A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams vol.62, pp.6, 2017, https://doi.org/10.12989/sem.2017.62.6.695
  43. Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory vol.19, pp.6, 2015, https://doi.org/10.12989/sss.2017.19.6.601
  44. A new shear deformation plate theory with stretching effect for buckling analysis of functionally graded sandwich plates vol.24, pp.5, 2017, https://doi.org/10.12989/scs.2017.24.5.569
  45. Free vibrations of laminated composite plates using a novel four variable refined plate theory vol.24, pp.5, 2017, https://doi.org/10.12989/scs.2017.24.5.603
  46. An original single variable shear deformation theory for buckling analysis of thick isotropic plates vol.63, pp.4, 2017, https://doi.org/10.12989/sem.2017.63.4.439
  47. A simple analytical approach for thermal buckling of thick functionally graded sandwich plates vol.63, pp.5, 2015, https://doi.org/10.12989/sem.2017.63.5.585
  48. An efficient shear deformation theory for wave propagation in functionally graded material beams with porosities vol.13, pp.3, 2015, https://doi.org/10.12989/eas.2017.13.3.255
  49. A four variable refined nth-order shear deformation theory for mechanical and thermal buckling analysis of functionally graded plates vol.13, pp.3, 2015, https://doi.org/10.12989/gae.2017.13.3.385
  50. Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory vol.20, pp.3, 2015, https://doi.org/10.12989/sss.2017.20.3.369
  51. A new and simple HSDT for thermal stability analysis of FG sandwich plates vol.25, pp.2, 2015, https://doi.org/10.12989/scs.2017.25.2.157
  52. A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams vol.64, pp.2, 2015, https://doi.org/10.12989/sem.2017.64.2.145
  53. Free vibration of functionally graded plates resting on elastic foundations based on quasi-3D hybrid-type higher order shear deformation theory vol.20, pp.4, 2017, https://doi.org/10.12989/sss.2017.20.4.509
  54. An analytical solution for bending and vibration responses of functionally graded beams with porosities vol.25, pp.4, 2015, https://doi.org/10.12989/was.2017.25.4.329
  55. An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates vol.25, pp.3, 2015, https://doi.org/10.12989/scs.2017.25.3.257
  56. A novel and simple higher order shear deformation theory for stability and vibration of functionally graded sandwich plate vol.25, pp.4, 2017, https://doi.org/10.12989/scs.2017.25.4.389
  57. A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates vol.64, pp.4, 2015, https://doi.org/10.12989/sem.2017.64.4.391
  58. Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded porosities vol.5, pp.4, 2017, https://doi.org/10.12989/anr.2017.5.4.393
  59. A new quasi-3D HSDT for buckling and vibration of FG plate vol.64, pp.6, 2015, https://doi.org/10.12989/sem.2017.64.6.737
  60. An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.693
  61. A new simple three-unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.717
  62. An original HSDT for free vibration analysis of functionally graded plates vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.735
  63. Vibration analysis of thick orthotropic plates using quasi 3D sinusoidal shear deformation theory vol.16, pp.2, 2015, https://doi.org/10.12989/gae.2018.16.2.141
  64. Development of super convergent Euler finite elements for the analysis of sandwich beams with soft core vol.65, pp.6, 2015, https://doi.org/10.12989/sem.2018.65.6.657
  65. A novel four variable refined plate theory for wave propagation in functionally graded material plates vol.27, pp.1, 2018, https://doi.org/10.12989/scs.2018.27.1.109
  66. A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory vol.21, pp.4, 2015, https://doi.org/10.12989/sss.2018.21.4.397
  67. Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates vol.14, pp.6, 2015, https://doi.org/10.12989/gae.2018.14.6.519
  68. Bending of FGM rectangular plates resting on non-uniform elastic foundations in thermal environment using an accurate theory vol.27, pp.3, 2018, https://doi.org/10.12989/scs.2018.27.3.311
  69. Analytical investigation of bending response of FGM plate using a new quasi 3D shear deformation theory: Effect of the micromechanical models vol.66, pp.3, 2018, https://doi.org/10.12989/sem.2018.66.3.317
  70. Free vibration of FGM plates with porosity by a shear deformation theory with four variables vol.66, pp.3, 2015, https://doi.org/10.12989/sem.2018.66.3.353
  71. A novel four-unknown quasi-3D shear deformation theory for functionally graded plates vol.27, pp.5, 2015, https://doi.org/10.12989/scs.2018.27.5.599
  72. A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates vol.28, pp.1, 2015, https://doi.org/10.12989/scs.2018.28.1.099
  73. Geometrically nonlinear analysis of functionally graded porous beams vol.27, pp.1, 2015, https://doi.org/10.12989/was.2018.27.1.059
  74. Single variable shear deformation model for bending analysis of thick beams vol.67, pp.3, 2015, https://doi.org/10.12989/sem.2018.67.3.291
  75. Effect of homogenization models on stress analysis of functionally graded plates vol.67, pp.5, 2018, https://doi.org/10.12989/sem.2018.67.5.527
  76. A novel quasi-3D hyperbolic shear deformation theory for vibration analysis of simply supported functionally graded plates vol.22, pp.3, 2015, https://doi.org/10.12989/sss.2018.22.3.303
  77. Analysis of wave propagation and free vibration of functionally graded porous material beam with a novel four variable refined theory vol.15, pp.4, 2018, https://doi.org/10.12989/eas.2018.15.4.369
  78. A refined quasi-3D hybrid-type higher order shear deformation theory for bending and Free vibration analysis of advanced composites beams vol.27, pp.4, 2015, https://doi.org/10.12989/was.2018.27.4.269
  79. An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models vol.69, pp.2, 2015, https://doi.org/10.12989/sem.2019.69.2.231
  80. Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory vol.28, pp.1, 2015, https://doi.org/10.12989/was.2019.28.1.019
  81. Dynamic and wave propagation investigation of FGM plates with porosities using a four variable plate theory vol.28, pp.1, 2015, https://doi.org/10.12989/was.2019.28.1.049
  82. Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections vol.17, pp.2, 2019, https://doi.org/10.12989/gae.2019.17.2.175
  83. Strain gradient based dynamic response analysis of heterogeneous cylindrical microshells with porosities under a moving load vol.6, pp.3, 2015, https://doi.org/10.1088/2053-1591/aaf5a2
  84. Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT vol.69, pp.5, 2015, https://doi.org/10.12989/sem.2019.69.5.511
  85. Free vibration of imperfect sigmoid and power law functionally graded beams vol.30, pp.6, 2019, https://doi.org/10.12989/scs.2019.30.6.603
  86. Nonlocal strain gradient model for thermal stability of FG nanoplates integrated with piezoelectric layers vol.23, pp.3, 2015, https://doi.org/10.12989/sss.2019.23.3.215
  87. 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 vol.55, pp.2, 2015, https://doi.org/10.1007/s11029-019-09805-0
  88. Dynamic response of metal foam FG porous cylindrical micro-shells due to moving loads with strain gradient size-dependency vol.134, pp.5, 2015, https://doi.org/10.1140/epjp/i2019-12540-3
  89. Nonlocal strain gradient thermal vibration analysis of double-coupled metal foam plate system with uniform and non-uniform porosities vol.8, pp.3, 2015, https://doi.org/10.12989/csm.2019.8.3.247
  90. Finite element formulation and vibration of nonlocal refined metal foam beams with symmetric and non-symmetric porosities vol.6, pp.2, 2015, https://doi.org/10.12989/smm.2019.6.2.147
  91. Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions vol.33, pp.1, 2019, https://doi.org/10.12989/scs.2019.33.1.133
  92. Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT vol.24, pp.4, 2015, https://doi.org/10.12989/cac.2019.24.4.347
  93. Effect of nonlinear elastic foundations on dynamic behavior of FG plates using four-unknown plate theory vol.17, pp.5, 2015, https://doi.org/10.12989/eas.2019.17.5.447
  94. Effect of variable elastic foundations on static behavior of functionally graded plates using sinusoidal shear deformation vol.12, pp.24, 2019, https://doi.org/10.1007/s12517-019-4871-5
  95. Wave dispersion properties in imperfect sigmoid plates using various HSDTs vol.33, pp.5, 2015, https://doi.org/10.12989/scs.2019.33.5.699
  96. Nonlocal strain gradient effects on forced vibrations of porous FG cylindrical nanoshells vol.8, pp.2, 2015, https://doi.org/10.12989/anr.2020.8.2.149
  97. A numerical method for dynamic characteristics of nonlocal porous metal-ceramic plates under periodic dynamic loads vol.7, pp.1, 2020, https://doi.org/10.12989/smm.2020.7.1.027
  98. Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory vol.35, pp.1, 2020, https://doi.org/10.12989/scs.2020.35.1.147
  99. Transversely isotropic thin circular plate with multi-dual-phase lag heat transfer vol.35, pp.3, 2015, https://doi.org/10.12989/scs.2020.35.3.343
  100. Nonlocal nonlinear dynamic behavior of composite piezo-magnetic beams using a refined higher-order beam theory vol.35, pp.4, 2015, https://doi.org/10.12989/scs.2020.35.4.545
  101. Post-buckling of higher-order stiffened metal foam curved shells with porosity distributions and geometrical imperfection vol.35, pp.4, 2020, https://doi.org/10.12989/scs.2020.35.4.567
  102. Assessment of transient vibrations of graphene oxide reinforced plates under pulse loads using finite strip method vol.25, pp.6, 2015, https://doi.org/10.12989/cac.2020.25.6.575
  103. Dynamic response of size-dependent porous functionally graded beams under thermal and moving load using a numerical approach vol.7, pp.2, 2015, https://doi.org/10.12989/smm.2020.7.2.069
  104. Numerical investigation on scale-dependent vibrations of porous foam plates under dynamic loads vol.7, pp.2, 2020, https://doi.org/10.12989/smm.2020.7.2.085
  105. Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model vol.26, pp.2, 2020, https://doi.org/10.12989/sss.2020.26.2.253
  106. Porosity effects on post-buckling behavior of geometrically imperfect metal foam doubly-curved shells with stiffeners vol.75, pp.6, 2015, https://doi.org/10.12989/sem.2020.75.6.701
  107. Nonlocal nonlinear stability of higher-order porous beams via Chebyshev-Ritz method vol.76, pp.3, 2015, https://doi.org/10.12989/sem.2020.76.3.413
  108. A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory vol.136, pp.2, 2015, https://doi.org/10.1140/epjp/s13360-021-01238-w
  109. Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory vol.10, pp.3, 2015, https://doi.org/10.12989/anr.2021.10.3.281
  110. Investigation on the dynamic response of porous FGM beams resting on variable foundation using a new higher order shear deformation theory vol.39, pp.1, 2015, https://doi.org/10.12989/scs.2021.39.1.095
  111. Analyzing dynamic response of nonlocal strain gradient porous beams under moving load and thermal environment vol.26, pp.1, 2015, https://doi.org/10.12989/gae.2021.26.1.089
  112. Investigating dynamic response of nonlocal functionally graded porous piezoelectric plates in thermal environment vol.40, pp.2, 2021, https://doi.org/10.12989/scs.2021.40.2.243
  113. Numerical forced vibration analysis of compositionally gradient porous cylindrical microshells under moving load and thermal environment vol.40, pp.6, 2021, https://doi.org/10.12989/scs.2021.40.6.893