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A new semi-analytical approach for bending, buckling and free vibration analyses of power law functionally graded beams

  • Du, Mengjie (State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology) ;
  • Liu, Jun (State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology) ;
  • Ye, Wenbin (State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology) ;
  • Yang, Fan (State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology) ;
  • Lin, Gao (State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)
  • Received : 2021.01.13
  • Accepted : 2021.10.22
  • Published : 2022.01.25

Abstract

The bending, buckling and free vibration responses of functionally graded material (FGM) beams are investigated semi-analytically by the scaled boundary finite element method (SBFEM) in this paper. In the concepts of the SBFEM, the dimension of computational domain can be reduced by one, therefore only the axial dimension of the beam is discretized using the higher order spectral element, which reduces the amount of calculation and greatly improves the calculation efficiency. The governing equation of FGM beams is derived in detail by the means of the principle of virtual work. Compared with the higher-order beam theory, fewer parameters and simpler control equations are used. And the governing equation is transformed into a first-order ordinary differential equation by introducing intermediate variables. Analytical solutions of the governing equation can be obtained by pade series expansion in the direction of thickness. Numerical example are compared with the numerical solutions provided by the previous researchers to verify the accuracy and applicability of the proposed method. The results show that the proposed formulations can quickly converge to the reference solutions by increasing the order of higher order spectral elements, and high accuracy can be achieved by using a small number of the elements. In addition, the influence of the structural sizes, material properties and boundary conditions on the mechanical behaviors of FG beams subjected to different load types is discussed.

Keywords

Acknowledgement

This research was supported by Grant 51779033 from the National Natural Science Foundation of China for which the authors are grateful.

References

  1. Akavci, S.S. and Tanrikulu, A.H. (2015), "Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories", Compos. Part B-Eng., 83, 203-215. https://doi.org/10.1016/j.compositesb.2015.08.043.
  2. Aldousari, S.M. (2017), "Bending analysis of different material distributions of functionally graded beam", Mater. Sci. Proc., 123, 123-296. https://doi.org/10.1007/s00339-017-0854-0.
  3. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006.
  4. Anandrao, K.S., Gupta, R.K., Ramachandran, P. and Rao, G.V. (2012), "Free vibration analysis of functionally graded beams", Defence Sci. J., 62(3), 139-146. http://doi.org/10.14429/dsj.62.1326.
  5. Aubad, M.J., Khafaji, S.O.W., Hussein, M.T. and Al-Shujairi, M.A. (2019), "Modal analysis and transient response of axially functionally graded (AFG) beam using finite element method", Mater. Res. Exp., 6(10), 1065G4. https://doi.org/10.1088/2053-1591/ab4234.
  6. Aydogdu, M. and Taskin, V. (2007), "Free vibration analysis of functionally graded beams with simply supported edges", Mater. Des., 28(5), 1651-1656. https://doi.org/10.1016/j.matdes.2006.02.007.
  7. Bazyar, M.H. and Talebi, A. (2015), "Scaled boundary finite-element method for solving non-homogeneous anisotropic heat conduction problems", Appl. Math. Model., 39(23), 7583-7599. https://doi.org/10.1016/j.apm.2015.03.024.
  8. Bekhadda, A., Cheikh, A. and Bensaid, I. (2019), "A novel first order refined shear-deformation beam theory for vibration and buckling analysis of continuously graded beams", Adv. Aircraft Spacecraft Sci., 6(3), 189-206. https://doi.org/10.12989/aas.2019.6.3.189.
  9. Celebi, K. and Tutuncu, N. (2014), "Free vibration analysis of functionally graded beams using an exact plane elasticity approach", Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., 228(14), 2488-2494. https://doi.org/10.1177/0954406213519974.
  10. Celebi, K., Yarimpabuc, D. and Tutuncu, N. (2017), "Free vibration analysis of functionally graded beams using complementary functions method", Arch. Appl. Mech., 88(5), 729-739. https://doi.org/10.1007/s00419-017-1338-6.
  11. Celebi, K., Yarimpabuc, D. and Tutuncu, N. (2018), "Free vibration analysis of functionally graded beams using complementary functions method", Arch. Appl. Mech., 88(5), 729-739. https://doi.org/10.1007/s00419-017-1338-6.
  12. Chaabane, L.A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F.Z., Tounsi, A., Derras, A., Bousahla, A.A. and Tounsi, A. (2019), "Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation", Struct. Eng. Mech., 71(2), 185-196. https://doi.org/10.12989/sem.2019.71.2.185.
  13. Chakraborty, A., Gopalakrishnan, S. and Reddy, J.N. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539. https://doi.org/10.1016/S0020-7403(03)00058-4.
  14. Claim, F.F. (2016), "Free and forced vibration analysis of axially functionally graded Timoshenko beams on two-parameter viscoelastic foundation", Compos. Part B-Eng., 103(103), 98-112. https://doi.org/10.1016/j.compositesb.2016.08.008.
  15. Deeks, A.J. and Wolf, J.P. (2002), "A virtual work derivation of the scaled boundary finite-element method for elastostatics", Comput. Mech., 28(6), 489-504. https://doi.org/10.1007/s00466-002-0314-2.
  16. Fouda, N., Elmidany, T.T., Sadoun, A.M. and Polit, O. (2017), "Bending, buckling and vibration of a functionally graded porous beam using finite elements", Appl. Comput. Mech., 3(4), 274-282. https://doi.org/10.22055/JACM.2017.21924.1121.
  17. Ghayesh, M.H. and Farokhi, H. (2018), "Bending and vibration analyses of coupled axially functionally graded tapered beams", Nonlin. Dyn., 91(1), 17-28. https://doi.org/10.1007/s11071-017-3783-8.
  18. Gravenkamp, H. and Natarajan, S. (2018), "Scaled boundary polygons for linear elastodynamics", Comput. Meth. Appl. Mech. Eng., 333, 238-256. https://doi.org/10.1016/j.cma.2018.01.031.
  19. He, Y.Q., Yang, H.T. and Deeks, A.J. (2013), "An element-free galerkin scaled boundary method for steady-state heat transfer problems", Numer. Heat Transf. Part B-Fundament., 64(3), 199-217. https://doi.org/10.1080/10407790.2013.791777.
  20. Hebbar, N., Hebbar, I., Ouinas, D. and Bourada, M. (2020), "Numerical modeling of bending, buckling, and vibration of functionally graded beams by using a higher-order shear deformation theory", Frattura ed Integrita Strutturale, 14(52), 230-246. https://doi.org/10.3221/IGF-ESIS.52.18.
  21. 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.
  22. Kapuria, S., Bhattacharyya, M. and Kumar, A.N. (2008), "Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation", Compos. Struct., 82(3), 390-402. https://doi.org/10.1016/j.compstruct.2007.01.019.
  23. Khan, M.A., Yasin, M.Y., Beg, M.S. and Khan, A.H. (2020), "Free and forced vibration analysis of functionally graded beams using finite element model based on refined third-order theory", Emerg. Trend. Mech. Eng., 603-612. https://doi.org/10.1007/978-981-32-9931-3_58.
  24. Li, C. and Tong, L.Y. (2015), "2D fracture analysis of magnetoelectroelastic composites by the SBFEM", Compos. Struct., 132, 984-994. https://doi.org/10.1016/j.compstruct.2015.07.015.
  25. Li, C., Ooi, E.T., Song, C.M. and Natarajan, S. (2015), "SBFEM for fracture analysis of piezoelectric composites under thermal load", Int. J. Solid. Struct., 52, 114-129. https://doi.org/10.1016/j.ijsolstr.2014.09.020.
  26. Li, L. and Hu, Y.J. (2016), "Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 107, 77-97. https://doi.org/10.1016/j.ijengsci.2016.07.011.
  27. Li, L., Li, X.B. and Hu, Y.J. (2016), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 102, 77-92. https://doi.org/10.1016/j.ijengsci.2016.02.010.
  28. Li, L., Li, X.B. and Hu, Y.J. (2018), "Nonlinear bending of a two-dimensionally functionally graded beam", Compos. Struct., 184, 1049-1061. https://doi.org/10.1016/j.compstruct.2017.10.087.
  29. Li, P., Liu, J., Lin, G., Zhang, P.C. and Yang, G.T. (2017), "A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces", Int. J. Heat Mass Transf., 113, 764-779. https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.065.
  30. Li, S.M. (2007), "Coupled finite element-scaled boundary finite element method for transient analysis of dam-reservoir interaction", International Conference on Computational Science and Its Applications, Berlin, Heidelberg. June.
  31. Li, S.R. and Batra, R.C. (2013), "Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams", Compos. Struct., 95, 5-9. https://doi.org/10.1016/j.compstruct.2012.07.027.
  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), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056.
  33. Lin, G., Liu, J., Li, J.B. and Hu, Z.Q. (2015), "A scaled boundary finite element approach for sloshing analysis of liquid storage tanks", Eng. Anal. Bound. Elem., 56, 70-80. https://doi.org/10.1016/j.enganabound.2015.02.006.
  34. Liu, J. and Lin, G. (2020), "A scaled boundary finite element method applied to electrostatic problems", Eng. Anal. Bound. Elem., 36(12), 1721-1732. https://doi.org/10.1016/j.enganabound.2012.06.010.
  35. Man, H., Song, C.M., Gao, W. and Tin-Loi, F. (2020), "A unified 3D-based technique for plate bending analysis using scaled boundary finite element method", Int. J. Numer. Meth. Eng., 91(5), 491-515. https://doi.org/10.1002/nme.4280.
  36. Nejad, M.Z. and Hadi, A. (2016), "Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 106, 1-9. https://doi.org/10.1016/j.ijengsci.2016.05.005.
  37. Nguyen, D.K. and Gan, B.S. (2014), "Large deflections of tapered functionally graded beams subjected to end forces", Appl. Math. Model., 38(11), 3054-3066. https://doi.org/10.1016/j.apm.2013.11.032.
  38. Nguyen, H.N., Hong, T.T., Vinh, P.V. and Van, T.D. (2019), "An efficient beam element based on Quasi-3D theory for static bending analysis of functionally graded beams", Mater., 12(13), 2198. https://doi.org/10.3390/ma12132198.
  39. Piovan, M.T. and Sampaio, R. (2009), "A study on the dynamics of rotating beams with functionally graded properties", J. Sound Vib., 327(1), 134-143. https://doi.org/10.1016/j.jsv.2009.06.015.
  40. 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(51), 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027.
  41. Reddy, J.N. (2011), "Microstructure-dependent couple stress theories of functionally graded beams", J. Mech. Phys. Solid., 59(11), 2382-2399. https://doi.org/10.1016/j.jmps.2011.06.008.
  42. 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.
  43. Sayyad, A.S. and Ghugal, Y.M. (2017), "A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates", Int. J. Appl. Mech., 9(01), 1750007. https://doi.org/10.1142/S1758825117500077.
  44. Sayyad, A.S. and Ghugal, Y.M. (2018), "Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams", Mech. Adv. Compos. Struct., 5, 13-24. https://doi.org/10.22075/MACS.2018.12214.1117.
  45. Schauer, M., Roman, J.E., Quintanaorti, E.S. and Langer, S. (2002), "Parallel computation of 3-d soil-structure interaction in time domain with a coupled FEM/SBFEM approach", J. Scientif. Comput., 52(2), 446-467. https://doi.org/10.1007/s10915-011-9551-x.
  46. Shahba, A., Attarnejad, R. and Hajilar, S. (2011), "Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams", Shock Vib., 18(5), 683-696. https://doi.org/10.3233/SAV-2010-0589.
  47. 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.
  48. Simsek, M. and Reddy, J.N. (2013), "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci., 64(64), 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002.
  49. 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.
  50. Soldatos, K.P. (1992), "A transverse shear deformation theory for homogeneous monoclinic plates", Acta Mechanica, 94, 195-200. https://doi.org/10.1007/BF01176650.
  51. Song, C.M. and Wolf, J.P. (1997), "The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics", Comput. Meth. Appl. Mech. Eng., 147(3-4), 329-355. https://doi.org/10.1016/S0045-7825(97)00021-2.
  52. Song, C.M. and Wolf, J.P. (2000), "The scaled boundary finiteelement method-A primer: Solution procedures", Comput. Struct., 78(1), 211-225. https://doi.org/10.1016/S0045-7949(00)00100-0.
  53. Song, H. and Tao, L.B. (2009), "Hydroelastic response of a circular plate in waves using scaled boundary FEM", Omae, 1, 247-254. https://doi.org/10.1115/OMAE2009-79271.
  54. Sun, Z.C., Ooi, E.T. and Song, C.M. (2020), "Finite fracture mechanics analysis using the scaled boundary finite element method", Eng. Fract. Mech., 134, 330-353. https://doi.org/10.1016/j.engfracmech.2014.12.002.
  55. Thai, H. 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(1), 57-66. https://doi.org/10.1016/j.ijmecsci.2012.05.014.
  56. Trinh, L.C., Vo, T.P., Thai, H.T. and Nguyen, T.K. (2016), "An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads", Compos. Part B-Eng., 100, 152-163. https://doi.org/10.1016/j.compositesb.2016.06.067.
  57. Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. and Lee, J. (2015), "A quasi-3D theory for vibration and buckling of functionally graded sandwich beams", Compos. Struct., 119, 1-12. https://doi.org/10.1016/j.compstruct.2014.08.006.
  58. Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. and Lee, J. (2015), "Static behaviour of functionally graded sandwich beams using a quasi-3D theory", Compos. Part B-Eng., 58, 59-74. https://doi.org/10.1016/j.compositesb.2014.08.030.
  59. Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A. and Lee, J. (2014), "Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory", Eng. Struct., 64(64), 12-22. https://doi.org/10.1016/j.engstruct.2014.01.029.
  60. Wang, W.Y., Guo, Z.J., Peng, Y. and Zhang, Q. (2016), "A numerical study of the effects of the T-shaped baffles on liquid sloshing in horizontal elliptical tanks", Ocean Eng., 111, 543-568. https://doi.org/10.1016/j.oceaneng.2015.11.020.
  61. Wang, W.Y., Peng, Y., Zhang, Q., Ren, L. and Jiang, Y. (2017), "Sloshing of liquid in partially liquid filled toroidal tank with various baffles under lateral excitation", Ocean Eng., 146, 434-456. https://doi.org/10.1016/j.oceaneng.2017.09.032.
  62. Wang, W.Y., Peng, Y., Zhou, Y. and Zhang, Q. (2016), "Liquid sloshing in partly-filled laterally-excited cylindrical tanks equipped with multi baffles", Appl. Ocean Res., 59, 543-563. https://doi.org/10.1016/j.apor.2016.07.009.
  63. Wang, W.Y., Tang, G.L., Song, X. and Zhou, Y. (2017), "Transient sloshing in partially filled laterally excited horizontal elliptical vessels with T-shaped baffles", J. Press. Ves. Technol., 139(2), 021302. https://doi.org/10.1115/1.4034148.
  64. Wang, W.Y., Zhang, Q., Ma, Q. and Ren, L. (2018), "Sloshing effects under longitudinal excitation in horizontal elliptical cylindrical containers with complex baffles", J. Waterw. Port Coast. Ocean Eng., 144(2), 04017044. https://doi.org/10.1061/(ASCE)WW.1943-5460.0000433.
  65. Xie, K., Wang, Y., Fan, X. and Fu, T. (2020), "Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories", Appl. Math. Model., 77, 1860-1880. https://doi.org/10.1016/j.apm.2019.09.024.
  66. Xue, L.J., Bian, X.Y., Feng, J.J. and Liu, J.N. (2020), "Elastoplastic analysis of a functionally graded material beam subjected to uniformly distributed load", J. Mech., 36(1), 73-85. https://doi.org/10.1017/jmech.2019.40.
  67. Yang, Z.J. and Deeks, A.J. (2007), "A frequency-domain approach for modelling transient elastodynamics using scaled boundary finite element method", Comput. Mech., 40, 725-738. https://doi.org/10.1007/s00466-006-0135-9.
  68. Ye, W.B., Liu, J., Lin, G., Xu, B. and Yu, L. (2018), "Application of scaled boundary finite element analysis for sloshing characteristics in an annular cylindrical container with porous structures", Eng. Anal. Bound. Elem., 97, 94-113. https://doi.org/10.1016/j.enganabound.2018.09.013.
  69. Yildirim, S. (2020), "Free vibration analysis of sandwich beams with functionally-graded-cores by complementary functions method", AIAA J., 58(12), 5431-5439. https://doi.org/10.2514/1.J059587.
  70. Ying, J., Lu, C.F. and Chen, W.Q. (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Compos. Struct., 84(3), 209-219. https://doi.org/10.1016/j.compstruct.2007.07.004.
  71. Zghal, S., Ataoui, D. and Dammak, F. (2020), "Static bending analysis of beams made of functionally graded porous materials", Mech. Bas. Des. Struct. Mach., 1-18. https://doi.org/10.1080/15397734.2020.1748053.
  72. Zhang, Z.H., Yang, Z.J. and Li, J.H. (2016), "An adaptive polygonal scaled boundary finite element method for elastodynamics", Int. J. Comput. Meth., 13(02), 164001. https://doi.org/10.1142/S0219876216400156.
  73. Zhong, Z. and Yu, T. (2006), "Two-dimensional analysis of functionally graded beams", AIAA J., 44(12), 3160-3160. https://doi.org/10.2514/1.26674.
  74. Zhu, H. and Sankar, B.V. (2007), "Analysis of sandwich TPS panel with functionally graded foam core by Galerkin method", Compos. Struct., 77(3), 280-287. https://doi.org/10.1016/j.compstruct.2005.07.005.