Structural Engineering and Mechanics

  • 제80권3호
  • /
  • Pages.275-284
  • /
  • 2021
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear dynamic and stability of a beam resting on the nonlinear elastic foundation under thermal effect based on the finite strain theory

  • Alimoradzadeh, M. (Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University) ;
  • Akbas, S.D. (Department of Civil Engineering, Bursa Technical University) ;
  • Esfrajani, S.M. (Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University)
  • 투고 : 2021.03.03
  • 심사 : 2021.08.13
  • 발행 : 2021.11.10
⟨ 이전 논문 다음 논문 ⟩

초록

Nonlinear free vibration and thermal buckling of a beam resting on nonlinear are investigated in this study. In the nonlinear kinematic relations, the finite strain theory is used with Euler-Bernoulli and Hamilton's principle is employed to derive the nonlinear governing of motion. The Galerkin's method is applied to simplify the governing nonlinear partially differential equation to the nonlinear ordinary differential equation. In addition, He's variational method is employed to obtain an analytical expression for the nonlinear natural frequency and thermal buckling temperature. In this study, a comparison between the finite strain theory and the von Kármán Theory is presented and the results shows that the finite strain theory gives more accurate results than the von Kármán Theory for nonlinear natural frequency and thermal buckling temperature. In the numerical results, the effect of different parameters such as linear and nonlinear coefficients of the elastic foundation, boundary conditions and the amplitude of the oscillation on the thermal buckling temperature and the nonlinear natural frequency investigated.

키워드

참고문헌

  1. Abualnour, M., Chikh, A., Hebali, H., Kaci, A., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2019), "Thermomechanical analysis of antisymmetric laminated reinforced composite plates using a new four variable trigonometric refined plate theory", Comput. Concrete, 24(6), 489-498. https://doi.org/10.12989/cac.2019.24.6.489.
  2. Addou, F.Y., Meradjah, M., Bousahla, A.A., Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S.R. (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. https://doi.org/10.12989/cac.2019.24.4.347.
  3. Akbas, S.D. (2019b), "Nonlinear static analysis of laminated composite beams under hygro-thermal effect", Struct. Eng Mech., 72(4), 433-441. http://doi.org/10.12989/sem.2019.72.4.433.
  4. Akbas, S.D. (2014), "Free vibration of axially functionally graded beams in thermal environment", Int. J. Eng. Appl. Sci., 6(3), 37-51. https://doi.org/10.24107/ijeas.251224.
  5. Akbas, S.D. (2015a), "Post-buckling analysis of axially functionally graded three-dimensional beams", Int. J. Appl. Mech., 7(3), 1550047. https://doi.org/10.1142/S1758825115500477.
  6. Akbas, S.D. (2015b), "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.
  7. Akbas, S.D. (2015c), "Large deflection analysis of edge cracked simple supported beams", Struct. Eng Mech., 54(3), 433-451. http://doi.org/10.12989/sem.2015.54.3.433.
  8. Akbas, S.D. (2016), "Post-buckling analysis of edge cracked columns under axial compression loads", Int. J. Appl. Mech., 8(8), 1650086. https://doi.org/10.1142/S1758825116500861.
  9. Akbas, S.D. (2017), "Post-buckling responses of functionally graded beams with porosities", Steel Compos. Struct., 24(5), 579-589. https://doi.org/10.12989/scs.2017.24.5.579.
  10. Akbas, S.D. (2018a), "Thermal post-buckling analysis of a laminated composite beam", Struct. Eng Mech., 67(4), 337-346. http://doi.org/10.12989/sem.2018.67.4.337.
  11. Akbas, S.D. (2018b), "Nonlinear thermal displacements of laminated composite beams", Coupl. Syst. Mech., 7(6), 691-705. http://doi.org/10.12989/csm.2018.7.6.691.
  12. Akbas, S.D. (2018c), "Geometrically nonlinear analysis of functionally graded porous beams", Wind Struct., 27(1), 59-70. http://doi.org/10.12989/was.2018.27.1.059.
  13. Akbas, S.D. (2018d), "Large deflection analysis of a fiber reinforced composite beam", Steel Compos. Struct., 27(5), 567-576. https://doi.org/10.12989/scs.2018.27.5.567.
  14. Akbas, S.D. (2019a), "Hygrothermal post-buckling analysis of laminated composite beams", Int. J. Appl. Mech., 11(1), 1950009. https://doi.org/10.1142/S1758825119500091.
  15. Akbas, S.D. (2019c), "Hygro-thermal nonlinear analysis of a functionally graded beam", J. Appl. Comput. Mech., 5(2), 477-485. https://doi.org/10.22055/JACM.2018.26819.1360.
  16. Akbas, S.D. (2019d), "Post-buckling analysis of a fiber reinforced composite beam with crack", Eng. Fract. Mech., 212, 70-80. https://doi.org/10.1016/j.engfracmech.2019.03.007.
  17. Akbas, S.D. and Kocaturk, T. (2012), "Post-buckling analysis of Timoshenko beams with temperature-dependent physical properties under uniform thermal loading", Struct. Eng Mech., 44(1), 109-125. https://doi.org/10.12989/sem.2012.44.1.109.
  18. Akbas, S.D. and Kocaturk, T. (2013), "Post-buckling analysis of functionally graded three-dimensional beams under the influence of temperature", J. Therm. Stress., 36(12), 1233-1254. https://doi.org/10.1080/01495739.2013.788397.
  19. Al-Furjan, M.S.H., Habibi, M., Ni, J., won Jung, D. and Tounsi, A. (2020a), "Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems", Eng. Comput., 1-17. https://doi.org/10.1007/s00366-020-01200-x.
  20. Al-Furjan, M.S.H., Safarpour, H., Habibi, M., Safarpour, M. and Tounsi, A. (2020b), "A comprehensive computational approach for nonlinear thermal instability of the electrically FG-GPLRC disk based on GDQ method", Eng. Comput., 1-18. https://doi.org/10.1007/s00366-020-01088-7.
  21. Alimoradzadeh, M., Salehi, M. and Mohammadi Esfarjani, S. (2020), "Nonlinear vibration analysis of Axially Functionally Graded microbeams based on nonlinear elastic foundation using modified couple stress theory", Periodica Polytechnica, Mech. Eng., 64(2), 97-108. https://doi.org/10.3311/PPme.11684.
  22. Anandrao, K.S., Gupta, R.K., Ramchandran, P. and Rao, V. (2010), "Thermal post-buckling analysis of uniform slender functionally graded material beams", Struct. Eng Mech., 36(5), 545-560. https://doi.org/10.12989/sem.2010.36.5.545.
  23. Belbachir, N., Bourada, M., Draiche, K., Tounsi, A., Bourada, F., Bousahla, A.A. and Mahmoud, S.R. (2020), "Thermal flexural analysis of anti-symmetric cross-ply laminated plates using a four variable refined theory", Smart Struct. Syst., 25(4), 409-422. https://doi.org/10.12989/sss.2020.25.4.409.
  24. Bellal, M., Hebali, H., Heireche, H., Bousahla, A. A., Tounsi, A., Bourada, F. and Tounsi, A. (2020), "Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model", Steel Compos. Struct., 34(5), 643-655. https://doi.org/10.12989/scs.2020.34.5.643.
  25. Bendenia, N., Zidour, M., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H. and Tounsi, A. (2020), "Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation", Comput. Concrete, 26(3), 213-226. http://dx.doi.org/10.12989/cac.2020.26.3.213.
  26. Bourada, F., Bousahla, A.A., Tounsi, A., Bedia, E.A., Mahmoud, S.R., Benrahou, K.H. and Tounsi, A. (2020), "Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic-foundation", Comput. Concrete, 25(6), 485-495. https://doi.org/10.12989/cac.2020.25.6.485.
  27. Boussoula, A., Boucham, B., Bourada, M., Bourada, F., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2020), "A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates", Smart Struct. Syst., 25(2), 197-218. https://doi.org/10.12989/sss.2020.25.2.197.
  28. Chakraborty, S., Dey, T. and Kumar, R. (2019), "Stability and vibration analysis of CNT-Reinforced functionally graded laminated composite cylindrical shell panels using semi-analytical approach", Compos. Part B: Eng., 168, 1-14. https://doi.org/10.1016/j.compositesb.2018.12.051.
  29. Chikr, S.C., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Bedia, E.A. and Tounsi, A. (2020), "A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin's approach", Geomech. Eng., 21(5), 471-487. https://doi.org/10.12989/gae.2020.21.5.471.
  30. Civalek, O ., Uzun, B., Yayli, M. O . and Akgoz, B. (2020), "Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method", Eur. Phys. J. Plus, 135(4), 381. https://doi.org/10.1140/epjp/s13360-020-00385-w.
  31. Cui, D. and Hu, H. (2014), "Thermal buckling and natural vibration of the beam with an axial stick-slip-stop boundary", J. Sound Vib., 333(8), 2271-2282. https://doi.org/10.1016/j.jsv.2013.11.042.
  32. Ebrahimi, F., Shaghaghi, G.R. and Boreiry, M. (2016), "An investigation into the influence of thermal loading and surface effects on mechanical characteristics of nanotubes", Struct. Eng. Mech., 57(1), 179-200. http://doi.org/10.12989/sem.2016.57.1.179.
  33. Fernandes, R., Mousavi, S.M. and El-Borgi, S. (2016), "Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory", Acta Mechanica, 227(9), 2657-2670. https://doi.org/10.1007/s00707-016-1646-x.
  34. Fu, Y., Wang, J. and Mao, Y. (2012), "Nonlinear analysis of buckling. free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment", Appl. Math. Model., 36(9), 4324-4340. https://doi.org/10.1016/j.apm.2011.11.059.
  35. Ganapathi, M., Patel, B.P., Saravanan, J. and Touratier, M. (1998), "Application of spline element for large-amplitude free vibrations of laminated orthotropic straight/curved beams", Compos. Part B: Eng., 29(1), 1-8. https://doi.org/10.1016/S1359-8368(97)00025-5.
  36. Ghayesh, M.H., Kafiabad, H.A. and Reid, T. (2012), "Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam", Int. J. Solid. Struct., 49(1), 227-243. https://doi.org/10.1016/j.ijsolstr.2011.10.007.
  37. Guellil, M.., Saidi, H., Bourada, F., Bousahla, A.A., Tounsi, A., Al-Zahrani, M.M. and Mahmoud, S.R. (2021), "Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation", Steel Compos. Struct., 38(1), 1-15. http://doi.org/10.12989/scs.2021.38.1.001.
  38. Gunda, J.B. and Rao, G.V. (2013), "Post-buckling analysis of composite beams: A simple intuitive formulation", Sadhana, 38(3), 447-459. https://doi.org/10.1007/s12046-013-0144-2.
  39. He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos Soliton Fract., 34(5), 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026.
  40. Hellal, H., Bourada, M., Hebali, H., Bourada, F., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2019), "Dynamic and stability analysis of functionally graded material sandwich plates in hygro-thermal environment using a simple higher shear deformation theory", J. Sandw. Struct. Mater., 1099636219845841. https://doi.org/10.1177/1099636219845841.
  41. Jouneghani, F.Z., Dimitri, R. and Tornabene, F. (2018)., "Structural response of porous FG nanobeams under hygrothermo-mechanical loadings", Compos. Part B: Eng., 152, 71-78. https://doi.org/10.1016/j.compositesb.2018.06.023.
  42. Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A. 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. https://doi.org/10.12989/cac.2020.25.1.037.
  43. Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates", Struct. Eng Mech., 64(4), 391-402. http://doi.org/10.12989/sem.2017.64.4.391.
  44. Kocaturk, T. and Akbas, S.D. (2011), "Post-buckling analysis of Timoshenko beams with various boundary conditions under non-uniform thermal loading", Struct. Eng Mech., 40(3), 347-371. http://doi.org/10.12989/sem.2011.40.3.347.
  45. Kocaturk, T. and Akbas, S.D. (2012), "Post-buckling analysis of Timoshenko beams made of functionally graded material under thermal loading", Struct. Eng Mech., 41(6), 775-789. http://doi.org/10.12989/sem.2012.41.6.775.
  46. Kocaturk, T. and Akbas, S.D. (2013), "Thermal post-buckling analysis of functionally graded beams with temperature-dependent physical properties", Steel Compos. Struct., 15(5), 481-505. https://doi.org/10.12989/scs.2013.15.5.481.
  47. Kumar, R., Banerjee, B. and Ramachandra, L.S. (2016), "Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method", Mech. Res. Commun., 73, 76-90. https://doi.org/10.1016/j.mechrescom.2016.02.011.
  48. Kumar, R., Ramachandra, L.S. and Banerjee, B. (2017), "Semi-analytical approach for thermal buckling and postbuckling response of rectangular composite plates subjected to localized thermal heating", Acta Mechanica, 228(5), 1767-1791. https://doi.org/10.1007/s00707-016-1797-9.
  49. Matouk, H., Bousahla, A.A., Heireche, H., Bourada, F., Bedia, E.A., Tounsi, A. and Benrahou, K.H. (2020), "Investigation on hygro-thermal vibration of P-FG and symmetric S-FG nanobeam using integral Timoshenko beam theory", Adv. Nano Res., 8(4), 293-305. https://doi.org/10.12989/anr.2020.8.4.293.
  50. Mohammadi, H. and Mahzoon, M. (2013), "Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory", Compos. Struct., 106, 764-776. https://doi.org/1016/j.compstruct.2013.06.030. https://doi.org/10.16/j.compstruct.2013.06.030
  51. Mororo, L.A.T., Melo, A.M.C.D. and Parente Junior, E. (2015), "Geometrically nonlinear analysis of thin-walled laminated composite beams", Lat. Am. J. Solid. Struct., 12(11), 2094-2117. http://doi.org/10.1590/1679-78251782.
  52. Nayfeh, A.H. and Pai, P.F. (2004), Linear and Nonlinear Structural Mechanics, John Wiley and Sons.
  53. Pagani, A. and Carrera, E. (2017), "Large-deflection and postbuckling analyses of laminated composite beams by Carrera Unified Formulation", Compos. Struct., 170, 40-52. https://doi.org/10.1016/j.compstruct.2017.03.008.
  54. Patel, S.N. (2014), "Nonlinear bending analysis of laminated composite stiffened plates", Steel Compos. Struct., 17(6), 867-890. http://doi.org/10.12989/scs.2014.17.6.867.
  55. Rabhi, M., Benrahou, K.H., Kaci, A., Houari, M.S.A., Bourada, F., Bousahla, A.A. and Tounsi, A. (2020), "A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions", Geomech. Eng., 22(2), 119-132. https://doi.org/10.12989/gae.2020.22.2.119.
  56. Refrafi, S., Bousahla, A.A., Bouhadra, A., Menasria, A., Bourada, F. and Tounsi, A. (2020), "Effects of hygro-thermo-mechanical 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.
  57. Rouabhia, A., Chikh, A., Bousahla, A.A., Bourada, F., Heireche, H., Tounsi, A. and Al-Zahrani, M.M. (2020), "Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory", Steel Compos. Struct., 37(6), 695-709. https://doi.org/10.12989/scs.2020.37.6.695.
  58. Shariati, A., Ghabussi, A., Habibi, M., Safarpour, H., Safarpour, M., Tounsi, A. and Safa, M. (2020), "Extremely large oscillation and nonlinear frequency of a multi-scale hybrid disk resting on nonlinear elastic foundation", Thin Wall. Struct., 154, 106840. https://doi.org/10.1016/j.tws.2020.106840.
  59. Shi-rong L, Chang-jun C, You-he Z (2003), "Thermal postbuckling of an elastic beams subjected to a transversely nonuniform temperature rising", Appl Math Mech-Engl., 24(5):514-520. https://doi.org/10.1007/BF02435863.
  60. Simsek, M. (2014), "Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He's variational method", Compos. Struct., 112, 264-272. https://doi.org/10.1016/j.compstruct.2014.02.010.
  61. Singh, V., Kumar, R. and Patel, S.N. (2021a), "Non-linear vibration and instability of multi-phase composite plate subjected to non-uniform in-plane parametric excitation: Semi-analytical investigation", Thin Wall. Struct., 162, 107556. https://doi.org/10.1016/j.tws.2021.107556.
  62. Singh, V., Kumar, R., Patel, S.N., Dey, T. and Kumar Panda, S. (2021b), "Instability and vibration analyses of functionally graded carbon nanotube-reinforced laminated composite plate subjected to localized in-plane periodic loading", J. Aerosp. Eng., 34(6), 04021072. https://doi.org/10.1061/(ASCE)AS.1943-5525.0001302.
  63. Solano, R. and Vaz, M. (2018), "Thermal post-buckling of slender elastic rods with different boundary conditions", Revista de Engenharia Termica, 5(2), 50-57. https://doi.org/10.5380/reterm.v5i2.61842.
  64. Stoykov, S. and Margenov, S. (2014), "Nonlinear vibrations of 3D laminated composite beams", Math. Prob. Eng., 2014, Article ID 892782. https://doi.org/10.1155/2014/892782.
  65. Tounsi, A., Al-Dulaijan, S.U., Al-Osta, M.A., Chikh, A., Al-Zahrani, M.M., Sharif, A. and Tounsi, A. (2020), "A four variable trigonometric integral plate theory for hygro-thermomechanical bending analysis of AFG ceramic-metal plates resting on a two-parameter elastic foundation", Steel Compos. Struct., 34(4), 511-524. https://doi.org/10.12989/scs.2020.34.4.511.
  66. Yadav, A., Amabili, M., Panda, S.K., Dey, T. and Kumar, R. (2021), "Nonlinear damped vibrations of three-phase CNT-FRC circular cylindrical shell", Compos. Struct., 255, 112939. https://doi.org/10.1016/j.compstruct.2020.112939.