Smart Structures and Systems

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Exact solution for nonlinear vibration of clamped-clamped functionally graded buckled beam

  • Selmi, Abdellatif (Department of Civil Engineering, College of Engineering in Al-Kharj, Prince Sattam Bin Abdulaziz University)
  • Received : 2020.01.03
  • Accepted : 2020.03.27
  • Published : 2020.09.25
⟨ Previous Next ⟩

Abstract

Exact solution for nonlinear behavior of clamped-clamped functionally graded (FG) buckled beams is presented. The effective material properties are considered to vary along the thickness direction according to exponential-law form. The in-plane inertia and damping are neglected, and hence the governing equations are reduced to a single nonlinear fourth-order partial-integral-differential equation. The von Kármán geometric nonlinearity has been considered in the formulation. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. Based on the mode of the corresponding linear problem, which readily satisfy the boundary conditions, the frequencies for the nonlinear problem are obtained using the Jacobi elliptic functions. The effects of various parameters such as the Young's modulus ratio, the beam slenderness ratio, the vibration amplitude and the magnitude of axial load on the nonlinear behavior are examined.

Keywords

Acknowledgement

This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project No: 11267/01/2019.

References

  1. Abdelghany, S.M., Ewis, K.M., Mahmoud, A.A. and Nassar, M.M. (2015), "Dynamic response of non-uniform beam subjected to moving load and resting on nonlinear viscoelastic foundation", Aust. J. Basic Appl. Sci., 4(3), 192-199. https://doi.org/10.1016/j.bjbas.2015.05.007
  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, Int. J., 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347
  3. Ali, G.A, Mahmoud, P. and Mohammad, A. (2018), "Non-linear free and forced vibration analysis of sandwich nano-beam with FG-CNTRC face-sheets based on nonlocal strain gradient theory", Smart Struct. Syst., Int. J., 22(11), 105-120. https://doi.org/10.12989/sss.2018.22.1.105
  4. Ansari, M., Esmailzadeh, E. and Younesian, D. (2011), "Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads", J. Sound Vib., 330(7), 1455-1471. https://doi.org/10.1016/j.jsv.2010.10.005
  5. Ansari, R., Pourashraf, T. and Gholami, R. (2015), "An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surfaceelasticity theory", Thin Wall. Struct., 93, 169-176. https://doi.org/10.1016/j.tws.2015.03.013
  6. 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
  7. Balubaid, M., Tounsi, A., Dakhel, B. and Mahmoud, S.R. (2019), "Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory", Comput. Concrete, Int. J., 24(6), 579-586. https://doi.org/10.12989/cac.2019.24.6.579
  8. Barari, A., Omidvar, M., Abdoul, R.G. and Ganji, D.D. (2008), "Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations", Acta. Appl. Math., 104, 161-171. https://doi.org/10.1007/s10440-008-9248-9
  9. 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. https://doi.org/10.12989/eas.2018.14.2.103
  10. Belbachir, N., Draich, K., Bousahla, A., Bourada, M., Tounsi, A. and Mohammadimehr, M. (2019), "Bending analysis of anti-symmetric cross-ply laminated plates under nonlinear thermal and mechanical loadings", Steel Compos. Struct., Int. J., 33(1), 913-924. https://doi.org/10.12989/scs.2019.33.1.081
  11. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygro-thermo-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
  12. Berghouti, H., Bedia, E.A.A., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., Int. J., 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351
  13. Bodaghi, M., Damanpack, A.R., Aghdam, M.M. and Shakeri, M. (2012), "Non-linear active control of FG beams in thermal environments subjected to blast loads with integrated FGP sensor/actuator layers", Compos. Struct., 94(12), 3612-3623. https://doi.org/10.1016/j.compstruct.2012.06.001
  14. Bodaghi, M., Damanpack, A.R., Aghdam, M.M. and Shakeri, M. (2014), "Geometrically non-linear transient thermo-elastic response of FG beams integrated with a pair of FG piezoelectric sensors", Compos. Struct., 107, 48-59. https://doi.org/10.1016/j.compstruct.2013.07.045
  15. Bouafia, K., Kaci, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., Int. J., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115
  16. 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., Int. J., 28(1), 19-30. https://doi.org/10.12989/was.2019.28.1.019
  17. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S.R. (2019), "Dynamic Analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., Int. J., 7(3), 189-206. https://doi.org/10.12989/anr.2019.7.3.189
  18. Byrd, M.D. and Friedman, P.F. (1971), Handbook of elliptic integrals for engineers and physicists, Springer, Berlin.
  19. Damanpack, A.R., Bodaghi, M., Ghassemi, H. and Sayehbani, M., (2013a), "$^2$Boundary element method applied to the Bending Analysis of Thin Functionally Graded Plates", Lat. Am. J. Solids Stru., 10(3), 549-570. http://dx.doi.org/10.1590/S1679-78252013000300006
  20. Damanpack, A.R., Bodaghi, M., Aghdam, M.M. and Shakeri, M. (2013b), "Active control of geometrically non-linear transient response of sandwich beams with a flexible core using piezoelectric patches", Compos. Struct., 100, 517-531. https://doi.org/10.1016/j.compstruct.2012.12.029
  21. Daouadji, T.H. and Tounsi, A. (2013), "A new higher order shear deformation model for static behavior of functionally graded plates", Adv. Appl. Math Mech., 5(3), 351-364. https://doi.org/10.1017/S2070073300002721
  22. Ding, J., Chu, L., Xin, L. and Dui, G. (2018), "Nonlinear Vibration Analysis of Functionally Graded Beams Considering the Influences of the Rotary Inertia of the Cross Section and Neutral Surface Position", Mech. Based Des. Struct., 46(2), 225-237. https://doi.org/10.1080/15397734. 2017.1329020
  23. Duc, N.D., Thang, P.T., Dao, N.T. and Tac, H.V. (2015), "Nonlinear buckling of higher deformable S-FGM thick circular cylindrical shells with metal-ceramic-metal layers surrounded on elastic foundations in thermal environment", Compos. Struct., 121, 134-141. https://doi.org/10.1016/j.compstruct.2014.11.009
  24. Fereidoon, A., Ghadimi, M., Barari, A., Kaliji, H.D. and Domairry, G. (2011), "Nonlinear vibration of oscillation systems using frequency-amplitude formulation", Shock Vib., 19(3), 323-332. https://doi.org/10.3233/SAV-2011-0633
  25. Fouladi, F., Hosseinzadeh, E., Barari, A. and Domairry, G. (2010), "Highly Nonlinear Temperature-Dependent Fin Analysis by Variational Iteration Method", Heat Transf. Res., 41(2), 155-165. https://doi.org/10.1615/HeatTransRes. v41.i2.40
  26. Fourn, H., Ait Atmane, H., 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. https://doi.org/10.12989/scs.2018.27.1.109
  27. Ganapathi, M. and Prakash, T. (2006), "Thermal buckling of simply supported functionally graded skew plates", Compos. Struct., 74(2), 247-250. https://doi.org/10.1016/j.compstruct. 2005. 04.004
  28. Ganji, S.S., Barari, B. and Ganji, D.D. (2011), "Approximate analyses of two mass-spring systems and buckling of a column", Comput. Math. Appl., 61(4), 1088-1095. https://doi.org/10.1016/j.camwa.2010.12.059
  29. Ganji, S.S., Barari, A., Ibsen, L.B. and Domairry, G. (2012), "Differential transform method for mathematical modeling of jamming transition problem in traffic congestion flow", Cent. Eur. J. Oper. Res., 20, 87-100. https://doi.org/10.1007/s10100 -010-0154-7
  30. He, J.H. (2006), "Some asymptotic methods for strongly nonlinear equations", Int. J. Mod. Phys. B, 20(10), 1141-1199. https://doi.org/10.1142/S0217979206033796
  31. He, J.H. and Wu, X.H. (2007), "Variational iteration method: new development and applications", Comput. Math. Appl., 54, 881-894. https://doi.org/10.1016/j.camwa.2006.12.083
  32. Hosseini, H.S., Fadaee, M. and Atashipour, S.R. (2011), "A new exact analytical approach for free vibration of reissner-mindlin functionally graded rectangular plates", Int. J. Mech. Sci., 53(1), 11-22. https://doi:10.1016/j.ijmecsci.2010.10.002
  33. Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., Int. J., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257
  34. Huang, J.L., Su, R.K.L., Lee, Y.Y. and Chen, S.H. (2011), "Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities", J. Sound Vib., 330, 5151-5164. https://doi.org/10.1016/j.jsv. 2011.05.023
  35. Hussain, M. and Naeem, M. (2019a), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes", Appl. Math. Model., 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039
  36. Hussain, M. and Naeem, M.N. (2019b), "Vibration characteristics of zigzag and chiral FGM rotating carbon nanotubes sandwich with ring supports", J. Mech. Eng. Sci., Part C., 233(16), 5763-5780. https://doi.org/10.1177/095440 6218802320
  37. Hussain, M., Naeem, M.N., and Isvandzibaei, M. (2018a), "Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., 232(24), 4564-4577. https://doi.org/10.1177/0954406217753459
  38. Hussain, M., Naeem, M.N., Shahzad, A., He, M. and Habib, S. (2018b), "Vibrations of rotating cylindrical shells with FGM using wave propagation approach", IMechE Part C: J. Mech. Eng. Sci., 232(23), 4342-4356. https://doi.org/10.1177/09544 06218802320
  39. Ibsen, L.B., Barari, A. and Kimiaeifar, A.S. (2010), "Analysis of highly nonlinear oscillation systems using He's max-min method and comparison with homotopy analysis and energy balance methods", Sadhana, 35(4), 433-448. https://doi.org/ 10.1007/s12046-010-0024-y
  40. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., Int. J., 28(1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099
  41. Kamarian, S., Bodaghi, M., Pourasghar, A. and Talebi, S. (2016), "Vibrational behavior of non-uniform piezoelectric sandwich beams made of CNT-reinforced polymer nanocomposite by considering the agglomeration effect of CNTs", Polym. Compos., 38, E553-E562. https://doi.org/10.1002/pc.23933
  42. Ke, L.L., Yang, J. and Kitipornchai, S. (2010), "An analytical study on the nonlinear vibration of functionally graded beams", Meccanica, 45(6), 743-752. https://doi.org/10.1007/s11012-009-9276-1
  43. Khoa, N.D., Thiem, H.T. and Duc, N.D. (2019), "Nonlinear buckling and postbuckling of imperfect piezoelectric S-FGM circular cylindrical shells withmetal-ceramic-metal layers in thermal environment using Reddy's third-order shear deformation shell theory", Mech. Adv. Mater. Struct., 26(3), 248-259. https://doi.org/10.1080/15376494.2017.1341583
  44. Lee, Y.Y., Li, Q.S., Leung, A.Y.T. and Su, R.K.L. (2012), "The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity", Nonlinear Dyn., 69, 99-116. https://doi.org/10.1007/ s11071-011-0249-2
  45. Leung, A.Y.T., Guo, Z.J. and Fung, T.C. (2010), "The multi parameter homotopy harmonic balance method for steady state problems", Int. J. Comput. Math., 87(5), 1158-1177. https://doi.org/10.1080/00207160903229899
  46. Mahmoudi, A., Benyoucef, S., Tounsi, A., Benachour, A., Bedia, E.A. and Mahmoud, S.R. (2019), "A refined quasi-3D shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations", J. Sandw. Struct. Mater., 21(6), 1906-1926. https://doi.org/10.1177%2F1099636217727577 https://doi.org/10.1177/1099636217727577
  47. Moeenfard, H., Mojahedi, M. and Ahmadian, M.T. (2011), "A homotopy perturbation analysis of nonlinear free vibration of Timoshenko microbeams", J. Mech. Sci. Technol., 25, 557-565. https://doi.org/10.1007/s12206-011-0130-8
  48. Pirbodaghi, T., Ahmadian, M.T. and Fesanghary, M. (2009), "On the homotopy analysis method for non-linear vibration of beams", Mech. Res. Commun., 36, 143-148. https://doi.org/10.1016/j.mechrescom.2008.08.001
  49. Praveen, G.N. and Reddy, J.N. (1998), "Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates", Int. J. Solids Struct., 35(3), 4457-4476. https://doi.org/10.1016/S0020-7683(97)00253-9
  50. Rafei, M., Ganji, D., Daniali, H. and Pashaei, H. (2007), "Thevariational iteration method for nonlinear oscillators with discontinuities", J. Sound Vib., 305, 614-620. https://doi.org/10.1016/j.jsv.2007.04.020
  51. Selmi, A. (2019), "Buckling capacity of functionally graded beams: A comparative study of different shear deformation beam theories", Int. J. Computat. Mater. Sci. Eng., 8(2), 1950003. https://doi.org/10.1142/S2047684119500039
  52. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nucl. Eng. Des., 240(4), 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013
  53. Simsek, M. (2015), "Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He's variational method", Compos. Struct., 131, 207-214. https://doi.org/10.1016/j.compstruct.2015.05.004
  54. Sofiyev, A.H., Hui, D. and Valiyev, A.A. (2016), "Effects of shear stresses and rotary inertia on the stability and vibration of sandwich cylindrical shells with FGM core surrounded by elastic medium", Mech. Based Des. Struct., 44(4), 384-404. https://doi.org/10.1080/15397734.2015.1083870
  55. Sundararajan, N., Prakash, T. and Ganapathi, M. (2005), "Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments", Finite Elem. Anal. Des., 42(2), 152-168. https://doi.org/10.1016/j.finel.2005.06.001
  56. Tounsi, A., Houari, M.S.A. and Bessaim, A. (2016), "A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate", Struct. Eng. Mech., Int. J., 60(4), 547-565. https://doi.org/10.12989/sem.2016. 60.4.547
  57. 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., 44(4), 476-494. https://doi.org/10.1080/15397734. 2015.1121398
  58. Udupa, G., Rao, S.S. and Gangadharan, K.V. (2014), "Functionally graded composite materials: an overview", Procedia Mater. Sci., 5, 1291-1299. https://doi.org/10.1016/j.mspro.2014.07.442
  59. Yaghoobi, H. and Yaghoobi, P. (2013), "Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach", Meccanica, 48, 2019-2035. https://doi.org/10.10 07/s11012-013-9720-0 https://doi.org/10.1007/s11012-013-9720-0
  60. Yang, J. and Chen, Y. (2008), "Free vibration and buckling analysesof functionally graded beams with edge cracks", Compos. Struct., 83(1), 48-60. https://doi.org/10.1016/j.compstruct.2007.03.006
  61. Yazdi, A.A. (2013), "Homotopy perturbation method for nonlinear vibration analysis of functionally graded plate", J. Vib. Acoust., 135(2), 021012 (6 pages). https://doi.org/10.1115/1.4023252
  62. Younesian, D., Sadri, M. and Esmailzadeh, E. (2014), "Primary and secondary resonance analyses of clamped-clamped microbeams", Nonlinear Dyn., 76, 1867-1884. https://doi.org/10.1007/s11071-014-1254-z
  63. 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, 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051.https://doi.org/10.1007/s11071-010-9790-7

Cited by

  1. Buckling analysis of functionally graded plates using HSDT in conjunction with the stress function method vol.27, pp.1, 2020, https://doi.org/10.12989/cac.2021.27.1.073
  2. 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
  3. On the free vibration response of laminated composite plates via FEM vol.39, pp.2, 2020, https://doi.org/10.12989/scs.2021.39.2.149
  4. 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
  5. Thermoelastic response of functionally graded sandwich plates using a simple integral HSDT vol.91, pp.7, 2020, https://doi.org/10.1007/s00419-021-01973-7
  6. An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells vol.40, pp.2, 2020, https://doi.org/10.12989/scs.2021.40.2.307
  7. An innovative system for novel vibration of rotating FG shell with combination of fraction laws vol.12, pp.2, 2020, https://doi.org/10.12989/acc.2021.12.2.157