Computers and Concrete

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Combined effects of material properties and boundary conditions on the large deflection bending analysis of circular plates on a nonlinear elastic foundation

  • Altekin, Murat (Department of Civil Engineering, Yildiz Technical University)
  • Received : 2020.01.11
  • Accepted : 2020.05.18
  • Published : 2020.06.25
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Abstract

Geometrically nonlinear axisymmetric bending analysis of shear deformable circular plates on a nonlinear three-parameter elastic foundation was made. Plates ranging from "thin" to "moderately thick" were investigated for three types of material: isotropic, transversely isotropic, and orthotropic. The differential equations were discretized by means of the finite difference method (FDM) and the differential quadrature method (DQM). The Newton-Raphson method was applied to find the solution. A parametric investigation using seven unknowns per node was presented. The novelty of the paper is that detailed numerical simulations were made to highlight the combined effects of the material properties and the boundary conditions on (i) the deflection, (ii) the stress resultants, and (iii) the external load. The formulation was verified through comparison studies. It was observed that the results are highly influenced from the boundary conditions, and from the material properties.

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References

  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. Akgoz, B. and Civalek, O. (2011), "Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations", Steel Compos. Struct., 11(5), 403-421. https://doi.org/10.12989/scs.2011.11.5.403.
  4. Alibeigloo, A. and Simintan, V. (2011), "Elasticity solution of functionally graded circular and annular plates integrated with sensor and actuator layers using differential quadrature", Compos. Struct., 93(10), 2473-2486. https://doi.org/10.1016/j.compstruct.2011.04.003.
  5. Altekin, M. (2018), "Bending of super-elliptical Mindlin plates by finite element method", Teknik Dergi/Tech. J. Turkish Chamb. of Civil Eng., 29(4), 8469-8496. https://doi.org/10.18400/tekderg.332384.
  6. Altekin, M. (2019), "Axisymmetric nonlinear bending of shear deformable orthotropic circular plates on elastic foundation", PCM-CMM 2019, Krakow, Poland.
  7. Alwar, R.S. and Reddy, S. (1979), "Large deflection static and dynamic analysis of isotropic and orthotropic annular plates", Int. J. Nonlin. Mech., 14(5-6), 347-359. https://doi.org/10.1016/0020-7462(79)90008-8.
  8. Anh, V.T.T. and Duc, N.D. (2016), "Nonlinear response of a shear deformable S-FGM shallow spherical shell with ceramic-metal-ceramic layers resting on an elastic foundation in a thermal environment", Mech. Adv. Mater. Struct., 23(8), 926-934. https://doi.org/10.1080/15376494.2015.1059527.
  9. Bert, C.W. and Malik, M. (1996), "Differential quadrature in computational mechanics: A review", Appl. Mech. Rev., 49(1), 1-28. https://doi.org/10.1115/1.3101882.
  10. Bhardwaj, N., Gupta, A.P. and Choong, K.K. (2007), "Effect of elastic foundation on the vibration of orthotropic elliptic plates with varying thickness", Meccanica, 41(4), 341-358. https://doi.org/10.1007/s11012-007-9059-5.
  11. Bourad, F., Amara, K. and Tounsi, A. (2016), "Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory", Steel Compos. Struct., 21(6), 1287-1306. https://doi.org/10.12989/scs.2016.21.6.1287.
  12. Chien, R.D. and Chen, C.S. (2005), "Nonlinear vibration of laminated plates on a nonlinear elastic foundation", Compos. Struct., 70(1), 90-99. https://doi.org/10.1016/j.compstruct.2004.08.015.
  13. Civalek, O. and Catal, H.H. (2003), "Linear static and vibration analysis of circular and annular plates by the harmonic differential quadrature (HDQ) method", Eskisehir Osmangazi Universitesi Muhendislik ve Mimarlik Fakultesi Dergisi, 16(1), 43-71.
  14. Civalek, O. and Ersoy, H. (2009), "Free vibration and bending analysis of circular Mindlin plates using singular convolution method", Commun. Numer. Meth. Eng., 25(8), 907-922. https://doi.org/10.1002/cnm.1138.
  15. Civalek, O. (2013), "Nonlinear dynamic response of laminated plates on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches", Compos.: Part B, 50, 171-179. https://doi.org/10.1016/j.compositesb.2013.01.027.
  16. Draiche, K., Bousahla, A.A., Tounsi, A., Alwabli, A.S., Tounsi, A. and Mahmoud, S.R. (2019), "Static analysis of laminated reinforced composite plates using a simple first-order shear deformation theory", Comput. Concrete, 24(4), 369-378. https://doi.org/10.12989/cac.2019.24.4.369.
  17. Dumir, P.C. (1985), "Non-linear axisymmetric response of orthotropic thin spherical caps on elastic foundations", Int. J. Mech. Sci., 27(11-12), 751-760. https://doi.org/10.1016/0020-7403(85)90007-4.
  18. Dumir, P.C. and Shingal, L. (1986), "Nonlinear analysis of thick circular plates", J. Eng. Mech., 112(3), 260-272. https://doi.org/10.1061/(ASCE)0733-9399(1986)112:3(260).
  19. Eftekhari, S.A. (2016), "A modified differential quadrature procedure for numerical solution of moving load problem", Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., 230(5), 715-731. https://doi.org/10.1177/0954406215584630.
  20. Guminiak, M. and Knitter-Piatkowska, A. (2018), "Selected problems of damage detection in internally supported plates using one-dimensional discrete wavelet transform", J. Theor. Appl. Mech., 56(3), 631-644. https://doi.org/10.15632/jtam-pl.56.3.631.
  21. Haciyev, V.C., Sofiyev, A.H. and Kuruoglu, N. (2019), "On the free vibration of orthotropic and inhomogeneous with spatial coordinates plates resting on the inhomogeneous viscoelastic foundation", Mech. Adv. Mater. Struct., 26(10), 886-897. https://doi.org/10.1080/15376494.2018.1430271.
  22. Han, J.B. and Liew, K.M. (1997), "An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates", Comput. Meth. Appl. Mech. Eng., 141(3-4), 265-280, https://doi.org/10.1016/S0045-7825(96)01115-2.
  23. Han, J.B. and Liew, K.M. (1999), "Axisymmetric free vibration of thick annular plates", Int. J. Mech. Sci., 41(9), 1089-1109. https://doi.org/10.1016/S0020-7403(98)00057-5.
  24. Hejripour, F. and Saidi, A. (2011), "Nonlinear free vibration analysis of annular sector plates using differential quadrature method", Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., https://doi.org/10.1177/0954406211414517.
  25. Hsu, M.H. (2007), "Vibration analysis of annular plates", Tamkang J. Sci. Eng., 10(3), 193-199. https://doi.org/10.6180/jase.2007.10.3.02.
  26. Jayachandran, S.A., Seetharaman, S. and Abraham, S. (2008), "Simple formulation for the flexure of plates on nonlinear foundation", J. Eng. Mech., 134(1), 110-115. https://doi.org/10.1061/ASCE0733-93992008134:1110.
  27. Jemielita, G. and Kozyra, Z. (2018), "Modelling of FGM plates", J. Theor. Appl. Mech., 56(3), 829-839. https://doi.org/10.15632/jtam-pl.56.3.829.
  28. Jedrysiak, J. and Kazmierczak-Sobinska, M. (2015), "On free vibrations of thin functionally graded plate bands resting on an elastic foundation", J. Theor. Appl. Mech., 53(3), 629-642. https://doi.org/10.15632/jtam-pl.53.3.629.
  29. Kargaudas, V., Adamukaitis, N., Zmuida, M., Pakalnis, A. and Zadlauskas, S. (2019), "Elastic foundation displacement approximations", Baltic J. Road Bridge Eng., 14(2), 125-135. https://doi.org/10.7250/bjrbe.2019-14.436.
  30. Katsikadelis, J.T. (2014), The Boundary Element Method for Plate Analysis, Academic Press.
  31. Keleshteri, M.M., Asadi, H. and Aghdam, M.M. (2019), "Nonlinear bending analysis of FG-CNTRC annular plates with variable thickness on elastic foundation", Thin Wall. Struct., 135, 453-462. https://doi.org/10.1016/j.tws.2018.11.020.
  32. Kutlu, A. and Omurtag, M.H. (2012), "Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method", Int. J. Mech. Sci., 65(1), 64-74. https://doi.org/10.1016/j.ijmecsci.2012.09.004.
  33. Lal, R. and Ahlawat, N. (2015), "Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method", Eur. J. Mech.-A/Solid., 52, 85- 94. http://dx.doi.org/10.1016/j.euromechsol.2015.02.004.
  34. Lamacchia, E., Pirrera, A., Chenchiah, I.V. and Weaver, P.M. (2014), "Non-axisymmetric bending of thin annular plates due to circumferentially distributed moments", Int. J. Solid. Struct., 51, 622-632. http://dx.doi.org/10.1016/j.ijsolstr.2013.10.028.
  35. Levinson, M. (1983), "Generalized Vlasov-Jones foundation model: A foundation of grade 4", Int. J. Mech. Sci., 25(2), 149-154. https://doi.org/10.1016/0020-7403(83)90007-3.
  36. Li, X.Y., Ding, H.J. and Chen, W.Q. (2008), "Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load $qr^{k}$", Int. J. Solid. Struct., 45(1), 191-210. https://doi.org/10.1016/j.ijsolstr.2007.07.023.
  37. Mathews, J.H. (1992), Numerical Methods for Mathematics, Science and Engineering, Prentice-Hall International Inc., USA.
  38. Muradova, A.D. and Stavroulakis, G.E. (2012), "Buckling and postbuckling analysis of rectangular plates resting on elastic foundations with the use of the spectral method", Comput. Meth. Appl. Mech. Eng., 205-208, 213-220. https://doi.org/10.1016/j.cma.2011.02.013.
  39. Najafi, F., Shojaeefard, M.H. and Googarchin, H.S. (2016), "Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field", Compos. Part B, 107, 123-140. https://doi.org/10.1016/j.compositesb.2016.09.070.
  40. Nassar, M. and Labib A.M. (1988), "Vibrations of circular plates with linearly varying thickness resting on a nonlinear elastic foundation", Proc. Ind. Nat. Sci. Acad. Part A. Phys. Sci., 54(1), 88-94.
  41. Rad, A.B. (2012), "Static analysis of two directional functionally graded circular plate under combined axisymmetric boundary conditions", Int. J. Eng. Appl. Sci., 4(3), 36-48.
  42. Rades, M. (1971), "Dynamic response of a rigid circular plate on a Kerr-type foundation model", Int. J. Eng. Sci., 9(11), 1061-1073. https://doi.org/10.1016/0020-7225(71)90002-4.
  43. Rajasekaran, S. and Varghese, S.P. (2005), "Damage detection in beams and plates using wavelet transforms", Comput. Concrete, 2(6), 481-498. http://dx.doi.org/10.12989/cac.2005.2.6.481.
  44. Sepahi, O., Forouzan, M.R. and Malekzadeh, P. (2010), "Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM", Compos. Struct., 92(10), 2369-2378. https://doi.org/10.1016/j.compstruct.2010.03.011.
  45. Shu, C. and Du, H. (1997), "A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates", Int. J. Solid. Struct., 34(7), 837-846. https://doi.org/10.1016/S0020-7683(96)00056-X.
  46. Shu, C. (2000), Differential Quadrature and its Application in Engineering, Springer-Verlag, London.
  47. Singh, B.N., Lal, A. and Kumar, R. (2008), "Nonlinear bending response of laminated composite plates on nonlinear elastic foundation with uncertain system properties", Eng. Struct., 30(4), 1101-1112. https://doi.org/10.1016/j.engstruct.2007.07.007.
  48. Sofiyev, A.H., Karaca, Z. and Zerin, Z. (2017), "Non-linear vibration of composite orthotropic cylindrical shells on the non-linear elastic foundations within the shear deformation theory", Compos. Struct., 159, 53-62. http://dx.doi.org/10.1016/j.compstruct.2016.09.048.
  49. Sofiyev, A.H. and Kuruoglu, N. (2017), "Combined effects of transverse shear stresses and nonlinear elastic foundations on the nonlinear dynamic response of heterogeneous orthotropic cylindrical shells", Compos. Struct., 166, 153-162. http://dx.doi.org/10.1016/j.compstruct.2017.01.058.
  50. Striz, A.G., Jang, S.K. and Bert, C.W. (1988), "Nonlinear bending analysis of thin circular plates by differential quadrature", Thin Wall. Struct., 6(1), 51-62. https://doi.org/10.1016/0263-8231(88)90025-0.
  51. Szilard, R. (2004), Theories and Applications of Plate Analysis, John Wiley & Sons Inc., New Jersey.
  52. Tajeddini, V., Ohadi, A. and Sadighi, M. (2011), "Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation", Int. J. Mech. Sci., 53(4), 300-308. https://doi.org/10.1016/j.ijmecsci.2011.01.011.
  53. Temel, B. and Sahan, M.F. (2013), "Transient analysis of orthotropic, viscoelastic thick plates in the Laplace domain", Eur. J. Mech. A-Solid., 37, 96-105. https://doi.org/10.1016/j.euromechsol.2012.05.008.
  54. Temel, B. and Noori, A.R. (2020), "A unified solution for the vibration analysis of two-directional functionally graded axisymmetric Mindlin-Reissner plates with variable thickness", Int. J. Mech. Sci., 174, 105471. https://doi.org/10.1016/j.ijmecsci.2020.105471.
  55. Thai, H.T. and Kim, S.E. (2012), "Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory", Appl. Math. Model., 36(8), 3870-3882. https://doi.org/10.1016/j.apm.2011.11.003.
  56. Vivio, F. and Vullo, V. (2010), "Closed form solutions of axisymmetric bending of circular plates having non-linear variable thickness", Int. J. Mech. Sci., 52(9), 1234-1252. https://doi.org/10.1016/j.ijmecsci.2010.05.011
  57. Wang, X. (2015), Differential Quadrature and Differential Quadrature Based Element Methods Theory and Applications, Elsevier, USA.
  58. Wang, Y., Ding, H. and Xu, R. (2016), "Three-dimensional analytical solutions for the axisymmetric bending of functionally graded annular plates", Appl. Math. Model., 40(9-10), 5393-5420. http://dx.doi.org/10.1016/j.apm.2015.11.051.
  59. Wu, X. and Ren, Y. (2007), "Differential quadrature method based on the highest derivative and its applications", J. Comput. Appl. Math., 205(1), 239-250. https://doi.org/10.1016/j.cam.2006.04.055.
  60. Yas, M.H. and Tahouneh, V. (2012), "3-D free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM)", Acta Mechanica, 223(1), 43-62. https://doi.org/10.1007/s00707-011-0543-6.
  61. Yildirim, S. and Tutuncu, N. (2018), "Axisymmetric plane vibration analysis of polar anisotropic disks", Compos. Struct., 194, 509-515. https://doi.org/10.1016/j.compstruct.2018.04.040.
  62. Zenkour, A.M. (2011), "Bending of orthotropic plates resting on Pasternak's foundations using mixed shear deformation theory", Acta Mechanica Sinica, 27(6), 956-962. https://doi.org/10.1007/s10409-011-0515-z.
  63. Zhao, S. and Wei, G.W. (2009), "Matched interface and boundary (MIB) for the implementation of boundary conditions in high-order central finite differences", Int. J. Numer. Meth. Eng., 77(12), 1690-1730. https://doi.org/10.1002/nme.2473.
  64. Zheng, J. and Zhou, X. (2007), "A new numerical method for axisymmetrical bending of circular plates wih large deflection", Key Eng. Mater., 353-358(4), 2699-2702. https://doi.org/10.4028/www.scientific.net/KEM.353-358.2699.
  65. Zong, Z. and Zhang, Y. (2009), Advanced Differential Quadrature Methods, Chapman & Hall/CRC, Boca Raton.