International Journal of Naval Architecture and Ocean Engineering

The Society of Naval Architects of Korea (대한조선학회)
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A new finite element formulation for vibration analysis of thick plates

  • Senjanovic, Ivo (University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture) ;
  • Vladimir, Nikola (University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture) ;
  • Cho, Dae Seung (Pusan National University, Dept. of Naval Architecture and Ocean Engineering)
  • Published : 2015.03.31
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Abstract

A new procedure for determining properties of thick plate finite elements, based on the modified Mindlin theory for moderately thick plate, is presented. Bending deflection is used as a potential function for the definition of total (bending and shear) deflection and angles of cross-section rotations. As a result of the introduced interdependence among displacements, the shear locking problem, present and solved in known finite element formulations, is avoided. Natural vibration analysis of rectangular plate, utilizing the proposed four-node quadrilateral finite element, shows higher accuracy than the sophisticated finite elements incorporated in some commercial software. In addition, the relation between thick and thin finite element properties is established, and compared with those in relevant literature.

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References

  1. Auricchio, F. and Taylor, R.L., 1995. A triangular thick plate finite element with an exact thin limit. Finite Elements in Analysis and Design, 19, pp.57-68. https://doi.org/10.1016/0168-874X(94)00057-M
  2. Bathe, K.J., 1996. Finite element procedures. Englewood Cliffs: Prentice-Hall/MIT.
  3. Bletzinger, K., Bischoff, M. and Ramm E., 2000. A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers and Structures, 75(3), pp.321-334. https://doi.org/10.1016/S0045-7949(99)00140-6
  4. Cheung, Y.K. and Zhou, D., 2000. Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers and Structures, 78(6), pp.757-768. https://doi.org/10.1016/S0045-7949(00)00058-4
  5. Cho, D.S., Vladimir, N. and Choi, T.M., 2013. Approximate natural vibration analysis of rectangular plates with openings using assumed mode method. International Journal of Naval Architecture and Ocean Engineering, 5(3), pp.478-491. https://doi.org/10.3744/JNAOE.2013.5.3.478
  6. Cho, D.S., Vladimir, N. and Choi, T.M., 2014. Natural vibration analysis of stiffened panels with arbitrary edge constraints using the assumed mode method. Proceedings of the IMechE, Part M: Journal of Engineering for the Maritime Environment. DOI:10.1177/1475090214521179 (published online).
  7. Dassault Systemes, 2008. Abaqus analysis user's manual. Version 6.8. Providence, RI, USA: Dassault Systemes.
  8. Dawe, D.J. and Roufaeil, O.L., 1980. Rayleigh-Ritz vibration analysis of Mindlin plates. Journal of Sound and Vibration, 69(3), pp.345-359. https://doi.org/10.1016/0022-460X(80)90477-0
  9. Eftekhari, S.A. and Jafari A.A., 2013. A simple and accurate mixed FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions. Meccanica, 48, pp.1139-1160. https://doi.org/10.1007/s11012-012-9657-8
  10. Endo, M. and Kimura, N., 2007. An alternative formulation of the boundary value problem for the Timoshenko beam and Mindlin plate. Journal of Sound and Vibration, 301(1-2), pp.355-373. https://doi.org/10.1016/j.jsv.2006.10.005
  11. Falsone, G. and Settineri, D., 2011. An Euler-Bernoulli-like finite element method for Timoshenko beams. Mechanics Research Communications, 38(1), pp.12-16. https://doi.org/10.1016/j.mechrescom.2010.10.009
  12. Falsone, G. and Settineri, D., 2012. A Kirchoff-like solution for the Mindlin plate model: A new finite element approach. Mechanics Research Communications, 40, pp.1-10. https://doi.org/10.1016/j.mechrescom.2011.11.008
  13. Hashemi, S.H. and Arsanjani, M., 2005. Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plate. International Journal of Solids and Structures, 42(3-4), pp.819-853. https://doi.org/10.1016/j.ijsolstr.2004.06.063
  14. Holand, I. and Bell, K., 1970. Finite element method in stress analysis. Trondheim: Tapir Forlag.
  15. Hughes, T.J.R., Taylor, R.L. and Kanoknukulchai, W., 1977. Simple and efficient element for plate bending. International Journal for Numerical Methods in Engineering, 11(10), pp.1529-1543. https://doi.org/10.1002/nme.1620111005
  16. Hughes, T.J.R. and Tezduyar, T., 1981. Finite elements based upon Mindlin plate theory with particular reference to the four-node isoparametric element. Journal of Applied Mechanics, 48, pp.587-596. https://doi.org/10.1115/1.3157679
  17. Kim, K., Kim, B.H., Choi, T.M. and Cho, D.S., 2012. Free vibration analysis of rectangular plate with arbitrary edge constraints using characteristic orthogonal polynomials in assumed mode method. International Journal of Naval Architecture and Ocean Engineering, 4(3), pp.267-280. https://doi.org/10.3744/JNAOE.2012.4.3.267
  18. Lee, S.W. and Wong, X., 1982. Mixed formulation finite elements for Mindlin theory plate bending. International Journal for Numerical Methods in Engineering, 18, pp.1297-1311. https://doi.org/10.1002/nme.1620180903
  19. Liew, K.M., Xiang, Y. and Kitipornchai, S., 1993. Transverse vibration of thick plates - I. Comprehensive sets of boundary conditions. Computers and Structures, 49, pp.1-29. https://doi.org/10.1016/0045-7949(93)90122-T
  20. Liew, K.M., Xiang, Y. and Kitipornchai, S., 1995. Research on thick plate vibration: a literature survey. Journal of Sound and Vibration, 180, pp.163-176. https://doi.org/10.1006/jsvi.1995.0072
  21. Lim, C.W., Li, Z.R., Xiang, Y., Wei, G.W. and Wang, C.M. 2005. On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates. Advances in Vibration Engineering, 4(3), pp.221-248.
  22. Liu, G.R., Nguyen-Thoi, T. and Lam, Y.K., 2009. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration, 320(4-5), pp.1100-1130. https://doi.org/10.1016/j.jsv.2008.08.027
  23. Lovadina, C., 1998. Analysis of a mixed finite element method for the Reissner-Mindlin plate problems. Computer Methods in Applied Mechanics and Engineering, 163, pp.71-85. https://doi.org/10.1016/S0045-7825(98)00003-6
  24. Mindlin, R.D., 1951. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics, 18(1), pp.31-38.
  25. MSC, 2010. MD Nastran 2010 Dynamic analysis user's guide. Newport Beach, California, USA: MSC Software.
  26. Nguyen-Xuan, H., Liu, G.R., Thai-Hoang, C. and Nguyen-Thoi, T., 2010. An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 199(9-12), pp.471-489. https://doi.org/10.1016/j.cma.2009.09.001
  27. Reissner, E., 1945. The effect of transverse shear deformation on the bending of elastic plate. Transactions of ASME Journal of Applied Mechanics, 12, pp.A69-A77.
  28. Senjanovic, I. and Fan, Y., 1989. Investigation of effective bending and shear stiffness of thin-walled girders related to ship hull vibration analysis. Journal of Ship Research, 33(4), pp.298-309.
  29. Senjanovic, I. and Grubisic, R., 1991. Coupled horizontal and torsional vibrations of a ship hull with large hatch openings. Computers and Structures, 41(2), pp.213-226. https://doi.org/10.1016/0045-7949(91)90425-L
  30. Senjanovic, I., Tomasevic, S. and Vladimir, N., 2009. An advanced theory of thin-walled girders with application to ship vibrations. Marine Structures, 22(3), pp.387-437. https://doi.org/10.1016/j.marstruc.2009.03.004
  31. Senjanovic, I., Vladimir, N. and Tomic, M., 2013a. An advanced theory of moderately thick plate vibrations. Journal of Sound and Vibration, 332(7), pp.1868-1880. https://doi.org/10.1016/j.jsv.2012.11.022
  32. Senjanovic, I., Tomic, M., Vladimir, N. and Cho, D.S., 2013b. Analytical solution for free vibrations of a moderately thick rectangular plate. Mathematical Problems in Engineering, 2013, pp.13.
  33. Senjanovic, I. and Vladimir, N., 2013. Physical insight into timoshenko beam theory and its modification with extension. Structural Engineering and Mechanics, 48(4), pp.519-545. https://doi.org/10.12989/sem.2013.48.4.519
  34. Shimpi, R.P. and Patel, H.G., 2006. Free vibrations of plate using two variable refined plate theory, Journal of Sound and Vibration, 296, pp.979-999. https://doi.org/10.1016/j.jsv.2006.03.030
  35. Szilard, R., 2004. Theories and applications of plate analysis. Hoboken, New Jersey, USA: John Wiley & Sons.
  36. Wang, C.M., 1994. Natural frequencies formula for simply supported Mindlin plates. ASME Journal of Vibration and Acoustics, 116, pp.536-540. https://doi.org/10.1115/1.2930460
  37. Xing, Y. and Liu, B., 2009. Characteristic equations and closed-form solution for free vibrations of rectangular Mindlin plates. Acta Mechanica Solida Sinica, 22(2), pp.125-136. https://doi.org/10.1016/S0894-9166(09)60097-5
  38. Zienkiewicz, O.C., Taylor, R.L. and To, J.M., 1971. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3, pp.275-290. https://doi.org/10.1002/nme.1620030211
  39. Zienkiewicz, O.C., 1971. The finite element method in engineering science. London: McGraw-Hill.
  40. Zienkiewicz, O.C. and Taylor, R.L., 2000. The finite element method. 5th ed. Oxford: Butterworth-Heinemann.

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