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Shear locking-free earthquake analysis of thick and thin plates using Mindlin's theory

  • Ozdemir, Y.I. (Civil Engineering, Department of Civil Engineering, Karadeniz Technical University) ;
  • Ayvaz, Y. (Civil Engineering, Department of Civil Engineering, Karadeniz Technical University)
  • Received : 2009.05.13
  • Accepted : 2009.09.07
  • Published : 2009.10.20

Abstract

The purpose of this paper is to study shear locking-free parametric earthquake analysis of thick and thin plates using Mindlin's theory, to determine the effects of the thickness/span ratio, the aspect ratio and the boundary conditions on the linear responses of thick and thin plates subjected to earthquake excitations. In the analysis, finite element method is used for spatial integration and the Newmark-${\beta}$ method is used for the time integration. Finite element formulation of the equations of the thick plate theory is derived by using higher order displacement shape functions. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. In the analysis, 17-noded finite element is used. Graphs are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that 17-noded finite element can be effectively used in the earthquake analysis of thick and thin plates. It is also concluded that, in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio.

Keywords

Acknowledgement

Supported by : Karadeniz Technical University

References

  1. Ayvaz, Y. (1992), Parametric Analysis of Reinforced Concrete Slabs Subjected to Earthquake Excitation, Ph. D. Thesis, Graduate School of Texas Tech University, Lubbock, Texas
  2. Ayvaz, Y. and Durmu , A. (1995), 'Earhquake analysis of simply supported reinforced concrete slabs', J. Sound Vib., 187(3), 531-539 https://doi.org/10.1006/jsvi.1995.0539
  3. Ayvaz, Y., Dalo lu, A. and Do angün, A. (1998), 'Application of a modified Vlasov model to earthquake analysis of the plates resting on elastic foundations', J. Sound Vib., 212(3), 499-509 https://doi.org/10.1006/jsvi.1997.1394
  4. Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall, Upper Saddle River, New Jersey
  5. Belounar, L. and Guenfoud, M. (2005), 'A new rectangular finite element based on the strain approach for plate bending', Thin Wall. Struct., 43(1), 47-63 https://doi.org/10.1016/j.tws.2004.08.003
  6. Bergan, P.G. and Wang, X. (1984), 'Quadrilateral plate bending elements with shear deformations', Comput. Struct., 19(1-2), 25-34 https://doi.org/10.1016/0045-7949(84)90199-8
  7. Brezzi, F. and Marini, L.D. (2003), 'A nonconforming element for the Reissner-Mindlin plate', Comput. Struct., 81, 515-522 https://doi.org/10.1016/S0045-7949(02)00418-2
  8. Cai, L., Rong, T. and Chen, D. (2002), 'Generalized mixed variational methods for reissner plate and its application', Comput. Mech., 30, 29-37 https://doi.org/10.1007/s00466-002-0364-5
  9. Caldersmith, G.W. (1984), 'Vibrations of orthotropic rectangular plates', ACUSTICA, 56, 144-152
  10. Cen, S., Long, Y., Yao, Z. and Chiew, S. (2006), 'Application of the quadrilateral area coordinate method', Int. J. Numer. Eng., 66, 1-45 https://doi.org/10.1002/nme.1533
  11. Cook, R.D., Malkus, D.S. and Michael, E.P. (1989), Concepts and Applications of Finite Element Analysis. John Wiley & Sons, Inc., Canada
  12. Grice, R.M. and Pinnington, R.J. (2002), 'Analysis of the flexural vibration of a thin-plate box using a combination of finite element analysis and analytical impedances', J. Sound Vib., 249(3), 499-527 https://doi.org/10.1006/jsvi.2001.3847
  13. Hinton, E. and Huang, H.C. (1986), 'A family of quadrilateral mindlin plate element with substitute shear strain fields', Comput. Struct., 23(3), 409-431 https://doi.org/10.1016/0045-7949(86)90232-4
  14. Hughes, T.J.R., Taylor, R.L. and Kalcjai, W. (1977), 'Simple and efficient element for plate bending', Int. J. Numer. Meth. Eng., 11, 1529-1543 https://doi.org/10.1002/nme.1620111005
  15. Leissa, A.W. (1969), Vibration of Plates, NASA, sp. 160
  16. Leissa, A.W. (1973), 'The free vibration of rectangular plates', J. Sound Vib., 31(3), 257-294 https://doi.org/10.1016/S0022-460X(73)80371-2
  17. Liew, K.M. and Teo, T.M. (1999), 'Three-dimentional vibration analysis of rectangular plates based on differential quadrature method', J. Sound Vib., 22(4), 577-599
  18. Lok, T.S. and Cheng, Q.H. (2001), 'Free and forced vibration of simply supported, orthotropic sandwich panel', Comput. Struct., 79(3), 301-312 https://doi.org/10.1016/S0045-7949(00)00136-X
  19. Mindlin, R.D. (1951), 'Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates', J. Appl. M., 18, 31-38
  20. Morais, M.V.G., Pedroso, L.J. and Da Silva, S.F. (2005), 'Vibrations of thick plates using lagrangean quadrilateral finite element with 16 nodes', Passager de Paris, 1, 238-250
  21. Ozkul, T.A. and Ture, U. (2004), 'The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem', Thin Wall. Struct., 42, 1405-1430 https://doi.org/10.1016/j.tws.2004.05.003
  22. Ozdemir Y.I., Bekiro lu, S. and Ayvaz, Y. (2007), 'Shear locking-free analysis of thick plates using Mindlin's theory', Struct. Eng. Mech., 27(3), 311-331 https://doi.org/10.12989/sem.2007.27.3.311
  23. Ozdemir, Y.I. (2007), 'Parametric Analysis of Thick Plates Subjected to Earthquake Excitations by Using Mindlin’s Theory', Ph. D. Thesis, Karadeniz Technical University, Trabzon
  24. Providakis, C.P. and Beskos, D.E. (1989), 'Free and forced vibrations of plates by boundary elements', Comput. Method. Appl. M., 74, 231-250 https://doi.org/10.1016/0045-7825(89)90050-9
  25. Providakis, C.P. and Beskos, D.E. (1989), 'Free and forced vibrations of plates by boundary and interior elements', Int. J. Numer. Meth. Eng., 28, 1977-1994 https://doi.org/10.1002/nme.1620280902
  26. Qian, L.F., Batra, R.C. and Chen, L.M. (2003), 'Free and forced vibration of thick rectangular plates using higher-orde shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method', Comp. Model Eng., 4(5), 519-534
  27. Qiu, J. and Feng, Z.C. (2000), 'Parameter dependence of the impact dynamics of thin plates', Comput. Struct., 75(5), 491-506 https://doi.org/10.1016/S0045-7949(99)00106-6
  28. Raju, K.K. and Hinton, E. (1980), 'Natural frequencies and modes of rhombic Mindlin plates', Earhq. Eng. Struct. D., 8, 55-62 https://doi.org/10.1002/eqe.4290080106
  29. Reissner, E. (1945), 'The effect of transverse shear deformation on the bending of elastic plates', J. Appl. Mech. ASME, 12, A69-A77
  30. Reissner, E. (1947), 'On bending of elastic plates', Q. Appl. Math., 5, 55-68 https://doi.org/10.1090/qam/20440
  31. Reissner, E. (1950), 'On a variational theorem in elasticity', J. Math. Phys., 29, 90-95 https://doi.org/10.1002/sapm195029190
  32. Sakata, T. and Hosokawa, K. (1988), 'Vibrations of clamped orthotropic rectangular plates', J. Sound Vib., 125(3), 429-439 https://doi.org/10.1016/0022-460X(88)90252-0
  33. Shen, H.S., Yang, J. and Zhang, L. (2001), 'Free and forced vibration of Reissner-Mindlin plates with free edges resting on elastic foundation', J. Sound Vib., 244(2), 299-320 https://doi.org/10.1006/jsvi.2000.3501
  34. Si, W.J., Lam, K.Y. and Gang, S.W. (2005), 'Vibration analysis of rectangular plates with one or more guided edges via bicubic B-spline method', Shock Vib., 12(5)
  35. Soh, A.K., Cen, S., Long, Y. and Long, Z. (2001), 'A new twelve DOF quadrilateral element for analysis of thick and thin plates', Eur. J. Mech., A-Solids, 20, 299-326 https://doi.org/10.1016/S0997-7538(00)01129-3
  36. Timoshenko, S. and Woinowsky-Krieger, S. (1959), Theory of Plates and Shells. Second edition, McGraw-Hill., New York
  37. Tedesco, J.W., McDougal, W.G. and Ross, C.A. (1999), Structural Dynamics, Addison Wesley Longman Inc., California
  38. Ugural, A.C. (1981), Stresses in Plates and Shells, McGraw-Hill., New York
  39. Wanji, C. and Cheung, Y.K. (2000), 'Refined quadrilateral element based on Mindlin/Reissner plate theory', Int. J. Numer. Meth. Eng., 47, 605-627 https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<605::AID-NME785>3.0.CO;2-E
  40. Warburton, G.B. (1954), 'The vibration of rectangular plates', Proc. of the Institude of Mechanical Engineers, 371-384 https://doi.org/10.1243/PIME_PROC_1954_168_040_02
  41. Weaver, W. and Johnston, P.R. (1984), Finite Elements for Structural Analysis, Prentice Hall, Inc., Englewood Cliffs, New Jersey
  42. Woo, K.S., Hong, C.H., Basu, P.K. and Seo, C.G. (2003), 'Free vibration of skew Mindlin plates by p-version of F.E.M.', J. Sound Vib., 268, 637-656 https://doi.org/10.1016/S0022-460X(02)01536-5
  43. Zhu, J. and Gu, P. (1991), 'Dynamic response of orthotropic plate using BEM with approximate fundamental solution', J. Sound Vib., 151(2), 203-211 https://doi.org/10.1016/0022-460X(91)90852-B
  44. Zienkiewich, O.C., Taylor, R.L. and Too, J.M. (1971), 'Reduced integration technique in general analysis of plates and shells', Int. J. Numer. Meth. Eng., 3, 275-290 https://doi.org/10.1002/nme.1620030211

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