Steel and Composite Structures

  • 제30권6호
  • /
  • Pages.603-615
  • /
  • 2019
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Free vibration of imperfect sigmoid and power law functionally graded beams

  • Avcar, Mehmet (Department of Civil Engineering, Faculty of Engineering, Suleyman Demirel University)
  • 투고 : 2019.01.18
  • 심사 : 2019.03.14
  • 발행 : 2019.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

In the present work, free vibration of beams made of imperfect functionally graded materials (FGMs) including porosities is investigated. Because of faults during process of manufacture, micro voids or porosities may arise in the FGMs, and this situation causes imperfection in the structure. Therefore, material properties of the beams are assumed to vary continuously through the thickness direction according to the volume fraction of constituents described with the modified rule of mixture including porosity volume fraction which covers two types of porosity distribution over the cross section, i.e., even and uneven distributions. The governing equations of power law FGM (P-FGM) and sigmoid law FGM (S-FGM) beams are derived within the frame works of classical beam theory (CBT) and first order shear deformation beam theory (FSDBT). The resulting equations are solved using separation of variables technique and assuming FG beams are simply supported at both ends. To validate the results numerous comparisons are carried out with available results of open literature. The effects of types of volume fraction function, beam theory and porosity volume fraction, as well as the variations of volume fraction index, span to depth ratio and porosity volume fraction, on the first three non-dimensional frequencies are examined in detail.

키워드

참고문헌

  1. Abualnour, M., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates", Compos. Struct., 184, 688-697. https://doi.org/10.1016/j.compstruct.2017.10.047
  2. Abdelaziz, H.H., Meziane, M.A.A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2017), "An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions", Steel Compos. Struct., Int. J., 25(6), 693-704.
  3. Akbas, S.D. (2017), "Thermal effects on the vibration of functionally graded deep beams with porosity", Int. J. Appl. Mech., 09, 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-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
  6. 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. Civ. Eng., 21, 792-806. https://doi.org/10.1007/s12205-016-0149-6
  7. Aldousari, S.M. (2017), "Bending analysis of different material distributions of functionally graded beam", Appl. Phys. A, 123, 296. https://doi.org/10.1007/s00339-017-0854-0
  8. 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
  9. Atmane, H.A., Tounsi, A., Ziane, N. and Mechab, I. (2011), "Mathematical solution for free vibration of sigmoid functionally graded beams with varying cross-section", Steel Compos. Struct., Int. J., 11(6), 489-504. https://doi.org/10.12989/scs.2011.11.6.489
  10. Atmane, H.A., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., Int. J., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369
  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, 71-84. https://doi.org/10.1007/s10999-015-9318-x
  12. Avcar, M. and Alwan, H.H.A. (2017), "Free vibration of functionally graded Rayleigh beam", Int. J. Eng. Appl. Sci., 9, 127-137.
  13. Avcar, M. and Mohammed, W.K.M. (2018), "Free vibration of functionally graded beams resting on Winkler-Pasternak foundation", Arab. J. Geosci., 11, 232. https://doi.org/10.1007/s12517-018-3579-2
  14. Aydogdu, M. and Taskin, V. (2007), "Free vibration analysis of functionally graded beams with simply supported edges", Mater. Des., 28, 1651-1656. https://doi.org/10.1016/j.matdes.2006.02.007
  15. Bao, G. and Wang, L. (1995), "Multiple cracking in functionally graded ceramic/metal coatings", Int. J. Solids Struct., 32, 2853-2871. https://doi.org/10.1016/0020-7683(94)00267-Z
  16. Belabed, Z., Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate", Earthq. Struct., Int. J., 14(2), 103-115.
  17. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygrothermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct. Syst., Int. J., 18(4), 755-786. https://doi.org/10.12989/sss.2016.18.4.755
  18. Bellifa, H., Benrahou, K.H., Hadji, L., Houari, M.S.A. and Tounsi, A. (2016), "Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position", J. Braz. Soc. Mech. Sci. Eng., 38(1), 265-275. https://doi.org/10.1007/s40430-015-0354-0
  19. Ben-Oumrane, S., Abedlouahed, T., Ismail, M., Mohamed, B.B., Mustapha, M. and El Abbas, A.B. (2009), "A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams", Comput. Mater. Sci., 44, 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001
  20. 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
  21. Chakraverty, S. and Pradhan, K.K. (2016), Vibration of Functionally Graded Beams and Plates, Academic Press.
  22. Chen, W.R. and Chang, H. (2017), "Closed-form solutions for free vibration frequencies of functionally graded euler-bernoulli beams", Mech. Compos. Mater., 53, 79-98. https://doi.org/10.1007/s11029-017-9642-3
  23. Chen, W.R. and Chang, H. (2018), "Vibration analysis of functionally graded timoshenko beams", Int. J. Struct. Stab. Dyn., 18, 1850007. https://doi.org/10.1142/S0219455418500074
  24. Chi, S.H. and Chung, Y.L. (2002), "Cracking in sigmoid functionally graded coating", J. Mech., 18, 41-53.
  25. Chi, S. and Chung, Y.L. (2003), "Cracking in coating-substrate composites with multi-layered and FGM coatings", Eng. Fract. Mech., 70, 1227-1243. https://doi.org/10.1016/S0013-7944(02)00114-5
  26. Chi, S.H. and Chung, Y.L. (2006a), "Mechanical behavior of functionally graded material plates under transverse load-Part I: Analysis", Int. J. Solids Struct., 43, 3657-3674. https://doi.org/10.1016/j.ijsolstr.2005.04.011
  27. Chi, S.H. and Chung, Y.L. (2006b), "Mechanical behavior of functionally graded material plates under transverse load-Part II: Numerical results", Int. J. Solids Struct., 43, 3675-3691. https://doi.org/10.1016/j.ijsolstr.2005.04.010
  28. Chung, Y.L. and Chi, S.H. (2001), "The residual stress of functionally graded materials", J. Chinese Inst. Civ. Hydraul. Eng., 13, 1-9.
  29. Civalek, O. (2017), "Vibration of laminated composite panels and curved plates with different types of FGM composite constituent", Compos. B Eng., 122, 89-108. https://doi.org/10.1016/j.compositesb.2017.04.012
  30. Civalek, O. and Baltacioglu, A.K. (2019), "Free vibration analysis of laminated and FGM composite annular sector plates", Compos. B Eng., 157, 182-194. https://doi.org/10.1016/j.compositesb.2018.08.101
  31. Delale, F. and Erdogan, F. (1983), "The crack problem for a nonhomogeneous plane", J. Appl. Mech., 50, 609. https://doi.org/10.1115/1.3167098
  32. Ebrahimi, F. and Dashti, S. (2015), "Free vibration analysis of a rotating non-uniform functionally graded beam", Steel Compos. Struct., Int. J., 19(5), 1279-1298. https://doi.org/10.12989/scs.2015.19.5.1279
  33. Ebrahimi, F. and Habibi, S. (2016), "Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate", Steel Compos. Struct., Int. J., 20(1), 205-225. https://doi.org/10.12989/scs.2016.20.1.205
  34. Ebrahimi, F. and Hashemi, M. (2017), "Vibration analysis of nonuniform imperfect functionally graded beams with porosities in thermal environment", J. Mech., 33, 739-757. https://doi.org/10.1017/jmech.2017.81
  35. Fereidoon, A., Asghardokht S.M. and Mohyeddin, A. (2011), "Bending analysis of thin functionally graded plates using generalized differential quadrature method", Arch. Appl. Mech., 81, 1523-1539. https://doi.org/10.1007/s00419-010-0499-3
  36. Fouda, N., El-Midany, T. and Sadoun, A.M. (2017), "Bending, buckling and vibration of a functionally graded porous beam using finite elements", Shahid Chamran Univ. Ahvaz, 3, 274-282.
  37. Fourn, H., Atmane, H.A., Bourada, M., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel four variable refined plate theory for wave propagation in functionally graded material plates", Steel Compos. Struct., Int. J., 27(1), 109-122.
  38. Galeban, M.R., Mojahedin, A., Taghavi, Y. and Jabbari, M. (2016), "Free vibration of functionally graded thin beams made of saturated porous materials", Steel Compos. Struct., Int. J., 21, 999-1016. https://doi.org/10.12989/scs.2016.21.5.999
  39. Hamed, M.A., Eltaher, M.A., Sadoun, A.M. and Almitani, K.H. (2016), "Free vibration of symmetric and sigmoid functionally graded nanobeams", Appl. Phys. A, 122, 829. https://doi.org/10.1007/s00339-016-0324-0
  40. Han, S.C., Lee, W.H. and Park W.T. (2009), "Non-linear analysis of laminated composite and sigmoid functionally graded anisotropic structures using a higher-order shear deformable natural Lagrangian shell element", Compos. Struct., 89, 8-19. https://doi.org/10.1016/j.compstruct.2008.08.006
  41. Heshmati, M. and Daneshmand, F. (2018), "Vibration analysis of non-uniform porous beams with functionally graded porosity distribution", Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl., 146442071878090.
  42. Jing, L., Ming, P., Zhang, W., Fu, L. and Cao, Y. (2016), "Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method", Compos. Struct., 138, 192-213. https://doi.org/10.1016/j.compstruct.2015.11.027
  43. Kahya, V. and Turan, M. (2017), "Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory", Compos. Part B Eng., 109, 108-115. https://doi.org/10.1016/j.compositesb.2016.10.039
  44. Kieback, B., Neubrand, A. and Riedel, H. (2003), "Processing techniques for functionally graded materials", Mater. Sci. Eng. A, 362, 81-106. https://doi.org/10.1016/S0921-5093(03)00578-1
  45. Koizumi, M. (1993), "The concept of FGM", Ceram. Trans. Funct. Graded Mater., 34, 3-10.
  46. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B Eng., 28, 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9
  47. Lee, Y.D. and Erdogan, F. (1995), "Residual/thermal stresses in FGM and laminated thermal barrier coatings", Int. J. Fract., 69, 145-165. https://doi.org/10.1007/BF00035027
  48. 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, 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056
  49. Li, X.F., Wang, B.L. and Han, J.C. (2010), "A higher-order theory for static and dynamic analyses of functionally graded beams", Arch. Appl. Mech., 80, 1197-1212. https://doi.org/10.1007/s00419-010-0435-6
  50. Mahi, A., Adda Bedia, E.A., Tounsi, A. and Mechab, I. (2010), "An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions", Compos. Struct., 92, 1877-1887. https://doi.org/10.1016/j.compstruct.2010.01.010
  51. Meradjah, M., Bouakkaz, K., Zaoui, F.Z. and Tounsi, A. (2018), "A refined quasi-3D hybrid-type higher order shear deformation theory for bending and free vibration analysis of advanced composites beams", Wind Struct., Int. J., 27(4), 269-282.
  52. Mirjavadi, S.S., Afshari, B.M., Shafiei, N., Hamouda, A.M.S. and Kazemi, M. (2017), "Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct., Int. J., 25(4), 415-426.
  53. 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., Int. J., 29(3), 363-377.
  54. Nguyen, T.K., Vo, T.P. and Thai, H.T. (2013), "Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory", Compos. Part B Eng., 55, 147-157. https://doi.org/10.1016/j.compositesb.2013.06.011
  55. Nguyen, D.K., Nguyen, Q.H., Tran, T.T. and Bui, V.T. (2017), "Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load", Acta Mech., 228, 141-155. https://doi.org/10.1007/s00707-016-1705-3
  56. Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., Int. J., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239
  57. Pradhan, K.K. and Chakraverty, S. (2013), "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method", Compos. Part B Eng., 51, 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027
  58. Pradhan, K.K. and Chakraverty, S. (2014), "Effects of different shear deformation theories on free vibration of functionally graded beams", Int. J. Mech. Sci., 82, 149-160. https://doi.org/10.1016/j.ijmecsci.2014.03.014
  59. Rahmani, O., Hosseini, S.A.H., Ghoytasi, I. and Golmohammadi, H. (2018), "Free vibration of deep curved FG nano-beam based on modified couple stress theory", Steel Compos. Struct., Int. J., 26(5), 607-620.
  60. Simsek, M. (2010a), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nucl. Eng. Des., 240, 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013
  61. Simsek, M. (2010b), "Vibration analysis of a functionally graded beam under a moving mass by using different beam theories", Compos. Struct., 92, 904-917. https://doi.org/10.1016/j.compstruct.2009.09.030
  62. Sina, S.A., Navazi, H.M. and Haddadpour, H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Des., 30, 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
  63. Thai, H.T. and Vo, T.P. (2012), "Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories", Int. J. Mech. Sci., 62, 57-66. https://doi.org/10.1016/j.ijmecsci.2012.05.014
  64. Wang, Y.Q. and Zu, J.W. (2017), "Large-amplitude vibration of sigmoid functionally graded thin plates with porosities", Thin-Wall. Struct., 119, 911-924. https://doi.org/10.1016/j.tws.2017.08.012
  65. Wang, Y.Q. and Zu, J.W. (2018), "Vibration characteristics of moving sigmoid functionally graded plates containing porosities", Int. J. Mech. Mater. Des., 14, 473-489. https://doi.org/10.1007/s10999-017-9385-2
  66. Wattanasakulpong, N. and Chaikittiratana, A. (2015), "Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method", Meccanica, 50, 1331-1342. https://doi.org/10.1007/s11012-014-0094-8
  67. Wattanasakulpong, N. and Mao, Q. (2015), "Dynamic response of Timoshenko functionally graded beams with classical and nonclassical boundary conditions using Chebyshev collocation method", Compos. Struct., 119, 346-354. https://doi.org/10.1016/j.compstruct.2014.09.004
  68. Wattanasakulpong, N. and Ungbhakorn, V. (2012), "Free vibration analysis of functionally graded beams with general elastically end constraints by DTM", World J. Mech., 2, 297-310. https://doi.org/10.4236/wjm.2012.26036
  69. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32, 111-120. https://doi.org/10.1016/j.ast.2013.12.002
  70. Wattanasakulpong, N., Gangadhara Prusty, B., 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
  71. 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 Mech. Sin., 34, 1124-1135. https://doi.org/10.1007/s10409-018-0770-3
  72. Yahia, S.A., Atmane, H.A., Houari, M.S.A., 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
  73. Younsi, A., Tounsi, A., Zaoui, F.Z., Bousahla, A.A. and Mahmoud, S.R. (2018), "Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates", Geomech. Eng., Int. J., 14(6), 519-532.
  74. Zhou, Y. and Zhang, X. (2019), "Natural frequency analysis of functionally graded material beams with axially varying stochastic properties", Appl. Math. Model., 67, 85-100. https://doi.org/10.1016/j.apm.2018.10.011
  75. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of ZrO2-NiCr functionally graded material by powder metallurgy", Mater. Chem. Phys., 68, 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2

피인용 문헌

  1. A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation vol.31, pp.5, 2019, https://doi.org/10.12989/scs.2019.31.5.503
  2. Free vibration analysis of angle-ply laminated composite and soft core sandwich plates vol.33, pp.5, 2019, https://doi.org/10.12989/scs.2019.33.5.663
  3. Wave dispersion properties in imperfect sigmoid plates using various HSDTs vol.33, pp.5, 2019, https://doi.org/10.12989/scs.2019.33.5.699
  4. Vibration of angle-ply laminated composite circular and annular plates vol.34, pp.1, 2020, https://doi.org/10.12989/scs.2020.34.1.141
  5. Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model vol.34, pp.5, 2019, https://doi.org/10.12989/scs.2020.34.5.643
  6. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory vol.25, pp.3, 2020, https://doi.org/10.12989/cac.2020.25.3.225
  7. Effect of the rotation on the thermal stress wave propagation in non-homogeneous viscoelastic body vol.21, pp.1, 2019, https://doi.org/10.12989/gae.2020.21.1.001
  8. Eringen's nonlocal model sandwich with Kelvin's theory for vibration of DWCNT vol.25, pp.4, 2019, https://doi.org/10.12989/cac.2020.25.4.343
  9. Time-Dependent Deflection Responses of Porous FGM Structure Including Pattern and Porosity vol.12, pp.9, 2019, https://doi.org/10.1142/s1758825120501021
  10. Porosity-dependent mechanical behaviors of FG plate using refined trigonometric shear deformation theory vol.26, pp.5, 2020, https://doi.org/10.12989/cac.2020.26.5.439
  11. A mechanical model to investigate Aedesaegypti mosquito bite using new techniques and its applications vol.11, pp.6, 2019, https://doi.org/10.12989/mwt.2020.11.6.399
  12. Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory vol.37, pp.6, 2019, https://doi.org/10.12989/scs.2020.37.6.695
  13. Flow of casson nanofluid along permeable exponentially stretching cylinder: Variation of mass concentration profile vol.38, pp.1, 2019, https://doi.org/10.12989/scs.2021.38.1.033
  14. The nano scale buckling properties of isolated protein microtubules based on modified strain gradient theory and a new single variable trigonometric beam theory vol.10, pp.1, 2019, https://doi.org/10.12989/anr.2021.10.1.015
  15. Thermal stress effects on microtubules based on orthotropic model: Vibrational analysis vol.11, pp.3, 2019, https://doi.org/10.12989/acc.2021.11.3.255
  16. Effect of suction on flow of dusty fluid along exponentially stretching cylinder vol.10, pp.3, 2019, https://doi.org/10.12989/anr.2021.10.3.263
  17. Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory vol.10, pp.3, 2019, https://doi.org/10.12989/anr.2021.10.3.281
  18. Bending analysis of functionally graded plates using a new refined quasi-3D shear deformation theory and the concept of the neutral surface position vol.39, pp.1, 2021, https://doi.org/10.12989/scs.2021.39.1.051
  19. 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, 2019, https://doi.org/10.12989/scs.2021.39.1.095
  20. An analytical solution for equations and the dynamical behavior of the orthotropic elastic material vol.11, pp.4, 2021, https://doi.org/10.12989/acc.2021.11.4.315
  21. On the free vibration response of laminated composite plates via FEM vol.39, pp.2, 2019, https://doi.org/10.12989/scs.2021.39.2.149
  22. Influence of micromechanical models on the bending response of bidirectional FG beams under linear, uniform, exponential and sinusoidal distributed loading vol.39, pp.2, 2021, https://doi.org/10.12989/scs.2021.39.2.215
  23. Thermoelastic response of functionally graded sandwich plates using a simple integral HSDT vol.91, pp.7, 2019, https://doi.org/10.1007/s00419-021-01973-7
  24. An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells vol.40, pp.2, 2019, https://doi.org/10.12989/scs.2021.40.2.307
  25. A n-order refined theory for free vibration of sandwich beams with functionally graded porous layers vol.79, pp.3, 2019, https://doi.org/10.12989/sem.2021.79.3.279
  26. Mechanical and thermal buckling analysis of laminated composite plates vol.40, pp.5, 2019, https://doi.org/10.12989/scs.2021.40.5.697