Steel and Composite Structures

  • 제31권3호
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  • Pages.261-276
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  • 2019
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  • 1229-9367(pISSN)
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  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

An extension of a high order approach for free vibration analysis of the nano-scale sandwich beam with steel skins for two types of soft and stiff cores

  • Marandi, S. Masoud (Faculty of Engineering, Shahrekord University) ;
  • Dehkordi, Mohsen Botshekanan (Faculty of Engineering, Shahrekord University) ;
  • Nourbakhsh, S. Hassan (Faculty of Engineering, Shahrekord University)
  • 투고 : 2018.07.29
  • 심사 : 2019.03.20
  • 발행 : 2019.05.10
⟨ 이전 논문 다음 논문 ⟩

초록

The study investigates the free vibration of a nano-scale sandwich beam by an extended high order approach, which has not been reported in the existing literature. First-order shear deformation theory for steel skins and so-called high-order sandwich panel theory for the core are applied. Next, the modified couple stress theory is used for both skins and cores. The Hamilton principle is utilized for deriving equations and corresponding boundary conditions. First, in the study the three-mode shapes natural frequencies for various material parameters are investigated. Also, obtained results are evaluated for two types of stiff and soft cores and isotropic, homogenous steel skins. In the research since the governing equations and also the boundary conditions are nonhomogeneous, therefore some closed-form solutions are not applicable. So, to obtain natural frequencies, the boundary conditions are converted to initial conditions called the shooting method as the numerical one. This method is one of the most robust approaches to solve complex equations and boundary conditions. Moreover, three types of simply supported on both sides of the beam (S-S), simply on one side and clamp supported on the other one (S-C) and clamped supported on both sides (C-C) are scrutinized. The parametric study is followed to evaluate the effect of nano-size scale, geometrical configurations for skins, core and material property change for cores as well. Results show that natural frequencies increase by an increase in skins thickness and core Young modulus and a decrease in beam length, core thickness as well. Furthermore, differences between obtained frequencies for soft and stiff cores increase in higher mode shapes; while, the more differences are evaluated for the stiff one.

키워드

참고문헌

  1. Akgoz, B. and Civalek, O . (2013), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004
  2. Allahkarami, F., Nikkhah-Bahrami, M. and Saryazdi, M.G. (2017), "Damping and vibration analysis of viscoelastic curved microbeam reinforced with FG-CNTs resting on viscoelastic medium using strain gradient theory and DQM", Steel Compos. Struct., Int. J., 25(2), 141-155.
  3. Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R. and Sadeghi, F. (2014), "Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams", Compos. Struct., 113, 316-327. https://doi.org/10.1016/j.compstruct.2014.03.015
  4. Asghari, M., Kahrobaiyan, M. and Ahmadian, M. (2010), "A nonlinear Timoshenko beam formulation based on the modified couple stress theory", Int. J. Eng. Sci., 48(12), 1749-1761. https://doi.org/10.1016/j.ijengsci.2010.09.025
  5. Aziz, A.K. (1975), Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations.
  6. Baltacioglu, A., Civalek, O., Akgoz, B. and Demir, F. (2011), "Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution", Int. J. Press. Vessels Pip., 88(8-9), 290-300. https://doi.org/10.1016/j.ijpvp.2011.06.004
  7. Barretta, R., Feo, L., Luciano, R., de Sciarra, F.M. and Penna, R. (2016), "Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation", Compos. Part B: Eng., 100, 208-219. https://doi.org/10.1016/j.compositesb.2016.05.052
  8. Bennai, R., Atmane, H.A. and Tounsi, A. (2015), "A new higherorder shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., Int. J., 19(3), 521-546. https://doi.org/10.12989/scs.2015.19.3.521
  9. Bock, H.G. (1983), Recent Advances in Parameteridentification Techniques for O.D.E, Birkhauser Boston, Boston, MA, USA.
  10. Bock, H.G. and Plitt, K.-J. (1984), "A multiple shooting algorithm for direct solution of optimal control problems", IFAC Proceedings Volumes, 17(2), 1603-1608. https://doi.org/10.1016/S1474-6670(17)61205-9
  11. Chandrashekhar, M. and Ganguli, R. (2010), "Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties", Int. J. Mech. Sci., 52(7), 874-891. https://doi.org/10.1016/j.ijmecsci.2010.03.002
  12. Chavan, S.G. and Lal, A. (2017), "Bending behavior of SWCNT reinforced composite plates", Steel Compos. Struct., Int. J., 24(5), 537-548.
  13. Dai, H. and Wang, X. (2006), "Non-Linear dynamic response of a single wall carbon nanotube subjected to radial impulse", Arch. Appl. Mech., 76(3-4), 145-158. https://doi.org/10.1007/s00419-006-0011-2
  14. Damanpack, A. and Khalili, S. (2012), "High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method", Compos. Struct., 94(5), 1503-1514. https://doi.org/10.1016/j.compstruct.2011.08.023
  15. Deuflhard, P. and Bader, G. (1983), Multiple Shooting Techniques Revisited, Birkhauser Boston, Boston, MA, USA.
  16. Duan, G., Wang, X. and Jin, C. (2014), "Free vibration analysis of circular thin plates with stepped thickness by the DSC element method", Thin-Wall. Struct., 85, 25-33. https://doi.org/10.1016/j.tws.2014.07.010
  17. Frostig, Y. and Baruch, M. (1994), "Free Vibrations Of Sandwich Beams With A Transversely Flexible Core: A High Order Approach", J. Sound Vib., 176(2), 195-208. https://doi.org/10.1006/jsvi.1994.1368
  18. Frostig, Y. and Thomsen, O.T. (2009), "On the free vibration of sandwich panels with a transversely flexible and temperaturedependent core material-Part I: Mathematical formulation", Compos. Sci. Technol., 69(6), 856-862. https://doi.org/10.1016/j.compscitech.2008.03.003
  19. Frostig, Y., Baruch, M., Vilnay, O. and Sheinman, I. (1992), "High-order theory for sandwich-beam behavior with transversely flexible core", J. Eng. Mech., 118(5), 1026-1043. https://doi.org/10.1061/(ASCE)0733-9399(1992)118:5(1026)
  20. Ganapathi, M., Merzouki, T. and Polit, O. (2018), "Vibration study of curved nanobeams based on nonlocal higher-order shear deformation theory using finite element approach", Compos. Struct., 184(Supplement C), 821-838. https://doi.org/10.1016/j.compstruct.2017.10.066
  21. Gurses, M., Civalek, O ., Korkmaz, A.K. and Ersoy, H. (2009), "Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first-order shear deformation theory", Int. J. Numer. Methods Eng., 79(3), 290-313. https://doi.org/10.1002/nme.2553
  22. Hadji, L., Atmane, H.A., Tounsi, A., Mechab, I. and Bedia, E.A. (2011), "Free vibration of functionally graded sandwich plates using four-variable refined plate theory", Appl. Math. Mech., 32(7), 925-942. https://doi.org/10.1007/s10483-011-1470-9
  23. He, X., Rafiee, M. and Mareishi, S. (2015), "Nonlinear dynamics of piezoelectric nanocomposite energy harvesters under parametric resonance", Nonlinear Dyn., 79(3), 1863-1880. https://doi.org/10.1007/s11071-014-1780-8
  24. Heshmati, M., Yas, M.H. and Daneshmand, F. (2015), "A comprehensive study on the vibrational behavior of CNTreinforced composite beams", Compos. Struct., 125(Supplement C), 434-448. https://doi.org/10.1016/j.compstruct.2015.02.033
  25. Jensen, A.E. and Irgens, F. (1999), "Thickness vibrations of sandwich plates and beams and delamination detection", J. Intel. Mater Syst. Struct., 10(1), 46-55. https://doi.org/10.1177/1045389X9901000106
  26. Kahrobaiyan, M., Asghari, M., Rahaeifard, M. and Ahmadian, M. (2011), "A nonlinear strain gradient beam formulation", Int. J.of Eng. Sci., 49(11), 1256-1267. https://doi.org/10.1016/j.ijengsci.2011.01.006
  27. Kar, V.R. and Panda, S.K. (2015), "Large deformation bending analysis of functionally graded spherical shell using FEM", Struct. Eng. Mech., Int. J., 53(4), 661-679. https://doi.org/10.12989/sem.2015.53.4.661
  28. Kavalur, P., Jeyaraj, P. and Babu, G.R. (2014), "Static behaviour of visco-elastic sandwich plate with nano-composite facings under mechanical load", Procedia Mater. Sci., 5, 1376-1384. https://doi.org/10.1016/j.mspro.2014.07.455
  29. Khalili, S., Dehkordi, M.B., Carrera, E. and Shariyat, M. (2013), "Non-linear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element theory", Compos. Struct., 96, 243-255. https://doi.org/10.1016/j.compstruct.2012.08.020
  30. Li, S.-R. and Zhou, Y.-H. (2001), "Shooting method for non-linear vibration and thermal buckling of heated orthotropic circular plates", J. Sound Vib., 248(2), 379-386. https://doi.org/10.1006/jsvi.2001.3665
  31. Lin, C.-C. and Tseng, C.-S. (1998), "Free vibration of polar orthotropic laminated circular and annular plates", J. Sound Vib., 209(5), 797-810. https://doi.org/10.1006/jsvi.1997.1293
  32. Lin, F. and Xiang, Y. (2014), "Numerical analysis on nonlinear free vibration of carbon nanotube reinforced composite beams", Int. J. Struct. Stabil. Dyn., 14(01), 1350056. https://doi.org/10.1142/S0219455413500569
  33. Loos, M.R. and Manas-Zloczower, I. (2012), "Reinforcement efficiency of carbon nanotubes-myth and reality", Macromol. Theory Simulat., 21(2), 130-137. https://doi.org/10.1002/mats.201100099
  34. Lubin, G. (2013), Handbook of Composites, Springer Science & Business Media.
  35. Marshall, I., Rhodes, J. and Banks, W. (1977), "Experimental snap-buckling behaviour of thin GRP curved panels under lateral loading", Composites, 8(2), 81-86. https://doi.org/10.1016/0010-4361(77)90063-5
  36. Marzulli, P. and Gheri, G. (1989), "Estimation of the global discretization error in shooting methods for linear boundary value problems", J. Computat. Appl. Math., 28, 309-314. https://doi.org/10.1016/0377-0427(89)90342-7
  37. McFarland, A.W. and Colton, J.S. (2005), "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", J. Micromech. Microeng., 15(5), 1060. https://doi.org/10.1088/0960-1317/15/5/024
  38. Mehar, K., Panda, S.K., Dehengia, A. and Kar, V.R. (2016), "Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment", J. Sandw. Struct. Mater., 18(2), 151-173. https://doi.org/10.1177/1099636215613324
  39. Naghipour, M. and Mehrzadi, M. (2007), "Evaluation of dynamic properties of extra light weight concrete sandwich beams reinforced with CFRP", Steel Compos. Struct., Int. J., 7(6), 457-468. https://doi.org/10.12989/scs.2007.7.6.457
  40. Nguyen, T.-K., Nguyen, T.T.-P., Vo, T.P. and Thai, H.-T. (2015), "Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory", Compos. Part B: Eng., 76, 273-285. https://doi.org/10.1016/j.compositesb.2015.02.032
  41. Rahmani, O., Khalili, S., Malekzadeh, K. and Hadavinia, H. (2009), "Free vibration analysis of sandwich structures with a flexible functionally graded syntactic core", Compos. Struct., 91(2), 229-235. https://doi.org/10.1016/j.compstruct.2009.05.007
  42. Saidi, A., Baferani, A.H. and Jomehzadeh, E. (2011), "Benchmark solution for free vibration of functionally graded moderately thick annular sector plates", Acta Mechanica, 219(3-4), 309-335. https://doi.org/10.1007/s00707-011-0459-1
  43. Sankar, A., Natarajan, S., Haboussi, M., Ramajeyathilagam, K. and Ganapathi, M. (2014), "Panel flutter characteristics of sandwich plates with CNT reinforced facesheets using an accurate higher-order theory", J. Fluids Struct., 50, 376-391. https://doi.org/10.1016/j.jfluidstructs.2014.06.028
  44. Shen, H.-S. and Zhu, Z. (2012), "Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations", Eur. J. Mech.-A/Solids, 35, 10-21. https://doi.org/10.1016/j.euromechsol.2012.01.005
  45. Sokolinsky, V.S., Nutt, S.R. and Frostig, Y. (2002), "Boundary condition effects in free vibrations of higher-order soft sandwich beams", AIAA J., 40(6), 1220-1227. https://doi.org/10.2514/2.1774
  46. Szekrenyes, A. (2014), "Stress and fracture analysis in delaminated orthotropic composite plates using third-order shear deformation theory", Appl. Math. Model., 38(15-16), 3897-3916. https://doi.org/10.1016/j.apm.2013.11.064
  47. Tagrara, S., Benachour, A., Bouiadjra, M.B. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., Int. J., 19(5), 1259-1277. https://doi.org/10.12989/scs.2015.19.5.1259
  48. Tahouneh, V. (2017), "Vibration and mode shape analysis of sandwich panel with MWCNTs FG-reinforcement core", Steel Compos. Struct., Int. J., 25(3), 347-360.
  49. Tong, L. (1994), "Free vibration of laminated conical shells including transverse shear deformation", Int. J. Solids Struct., 31(4), 443-456. https://doi.org/10.1016/0020-7683(94)90085-X
  50. Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016), "Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells", Compos. Part B: Eng., 89, 187-218. https://doi.org/10.1016/j.compositesb.2015.11.016
  51. Vinson, J.R. (1999), The Behavior of Sandwich Structures of Isotropic and Composite Materials, Technomic Publishing Co. Inc., Lancaster, UK.
  52. Viswanathan, K., Kim, K.S. and Lee, J.H. (2009), "Asymmetric free vibrations of laminated annular cross-ply circular plates including the effects of shear deformation and rotary inertia: spline method", Forschung im Ingenieurwesen, 73(4), 205-217. https://doi.org/10.1007/s10010-009-0106-3
  53. Wan, H., Delale, F. and Shen, L. (2005), "Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites", Mech. Res. Commun., 32(5), 481-489. https://doi.org/10.1016/j.mechrescom.2004.10.011
  54. Wang, Z.-X. and Shen, H.-S. (2012), "Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets", Compos. Part B: Eng., 43(2), 411-421. https://doi.org/10.1016/j.compositesb.2011.04.040
  55. Wang, Y.-G., Lin, W.-H. and Liu, N. (2013), "Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory", Int. J. Mech. Sci., 71, 51-57. https://doi.org/10.1016/j.ijmecsci.2013.03.008
  56. Wang, Q., Shi, D., Liang, Q. and Ahad, F. (2016), "An improved Fourier series solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions", J. Compos. Mater., 50(30), 4199-4233. https://doi.org/10.1177/0021998316635240
  57. Wu, H., Kitipornchai, S. and Yang, J. (2015), "Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets", Int. J. Struct. Stabil. Dyn., 15(7), 1540011. https://doi.org/10.1142/S0219455415400118
  58. Xie, W. and Pang, H. (2016), "The shooting method and integral boundary value problems of third-order differential equation", Adv. Differ. Eq., 2016(1), 138. https://doi.org/10.1186/s13662-016-0824-4
  59. Xie, X., Jin, G., Ye, T. and Liu, Z. (2014), "Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method", Appl. Acoustics, 85, 130-142. https://doi.org/10.1016/j.apacoust.2014.04.006
  60. Yang, Y., Lam, C.C., Kou, K.P. and Iu, V.P. (2014), "Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method", Compos. Struct., 117(Supplement C), 32-39. https://doi.org/10.1016/j.compstruct.2014.06.016
  61. Zhang, L., Lei, Z. and Liew, K. (2015), "Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLS-Ritz method", Appl. Math. Computat., 256, 488-504. https://doi.org/10.1016/j.amc.2015.01.066