ࡱ> 685M @QbjbjVV Yr<r<    DPttt|pDPycBBBB!!!bbbbbbb$YfRhb] ( !( ( b BB3c8#8#8#( x B Bb8#( b8#8#.Uh 4ZB ŕt!ppX@,aIc0ycXi"i4ZPP i 4Z!8#clY!!!bbPP$ t"dPPtLINKED INTERPOLATION IN HIGHER-ORDER TRIANGULAR MINDLIN PLATE FINITE ELEMENTS Dragan Ribari, Gordan Jeleni University of Rijeka, Faculty of Civil Engineering, V.C. Emina 5, 51000 Rijeka, Republic of Croatia,  HYPERLINK "mailto:dragan.ribaric@gradri.hr" dragan.ribaric@gradri.hr,  HYPERLINK "mailto:gordan@gradri.hr" gordan@gradri.hr Introduction In this work higher-order linked interpolation applied to triangular plate finite elements is analyzed in detail. Benefits of the linked interpolation are well known in the Timoshenko (thick) beam finite elements, where not only is the shear locking with such an interpolation completely avoided, but also the exact solution is achieved provided the applied loading is static and expressed in terms of the polynomial functions [1]. In the case of Mindlin (thick) plate finite elements, the exact solution for a genuine planar behaviour cannot be obtained in terms of a finite number of parameters. Additionally, with linked interpolation shear locking may be considerably reduced, but it cannot be completely avoided [2, 3, 4]. In this paper the basis for development of higher-order triangular Mindlin plate elements is found in the analogy between the Timoshenko beam theory and the Mindlin plate theory in a manner closely related to our earlier studies of arbitrary quadrilateral plate elements [5]. The results obtained on standard test examples are compared and numerically assessed against the reference results from literature and from FEAP program by R.L.Taylor [6], using various mesh densities and various order of interpolation. Linked interpolation formulation The well known differential equations of the Timoshenko beam:  EMBED Equation.3 ,  EMBED Equation.3  relating the transversal load q with the second derivative of the bending moment  EMBED Equation.3 and the first derivative of the shear force  EMBED Equation.3 , can be exactly integrated for the transverse displacement w and the sections rotation  EMBED Equation.3 , without involving bending and shear stiffnesses EI and GAs. The expressions that are exact for the beam differential equations are now used to interpolate the displacement w and the two sections rotations x and y of the Mindlin plate trianglular element. In the linked interpolation formulation we propose here, the expression for shear along an element side is taken to be a polynomial two orders lower than the displacement interpolation polynomial. Every direction parallel to the element side retains this property. In contrast to the procedure proposed by Auricchio and Taylor [3], we remain within the realm of pure displacement-based finite-element design procedures. The elements obtained in this way are compared with the Liu-Riggs family of elements [7], also developed on the pure displacement-based concept and we find out that the two families are very similar (equal for the lowest-order element). However, the generation of the higher-order elements is easier using the linked interpolation formulation than the Liu-Riggs concept. A number of standard tests are performed (the patch test for constant bending and constant shear, the clamped and simply supported square plate, the skew Morley plate and circular plate) for the moderate and extremely small plate thicknesses and compared with some of the known elements based on the mixed formulation. Results and discussion The elements based on the linked interpolation formulation are capable of reproducing the exact analytical result for the case of cylindrical bending of certain order (quadratic for the three-node elements, cubic for the six-node elements and so on). This means that the six-node and ten-node triangular elements are able to reproduce the exact solution in the constant shear patch test [4] regardless of the plate thickness (Table 1). The lowest-order (three-node) element of the family suffers from the shear-locking effect for very coarse meshes and thin plates, but its performance improves with mesh refinement. Performance of the elements derived is generally good for a number of standard benchmark tests as the finite-element mesh is refined, one notable exception being the skewed plate, for which the presented elements do not perform so well as the element of Auricchio and Taylor [3]. Otherwise, the six-node and the ten-node element, in general, are clearly successful when compared to the lower-order elements for the problems with the same total number of the degrees of freedom. Concluding remarks The triangular elements based on the linked interpolation derived here perform comparably well for a number of standard benchmark tests and are competitive to the elements based on the mixed formulation in the Mindlin-type plate analysis. Acknowledgements The results shown here have been obtained within the scientific project No 114-0000000-3025: Improved accuracy in non-linear beam elements with finite 3D rotations financially supported by the Ministry of Science, Education and Sports of the Republic of Croatia. Table 1: The standard patch test [4] results for the proposed elements (displacement at control point) ElementsPatch test for constant curvaturePatch test for constant shear Thick plateThin plateresultThick plateThin plateresult3-node0.54140000.5414000pass-0.24558710.000318188fail6-node0.54140000.5414000pass-0.24509330.000215467pass10-node0.54140000.5414000pass-0.24509330.000215467passAnalytical solution0.54140.5414 -0.2450933330.000215467  References [1] G. Jeleni, E. Papa, 2011. Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order, Archive of Applied Mechanics 18: 171-183. [2] A. Tessler, T.J.R. Hughes, 1985. A three-node Mindlin plate element with improved transverse shear, Computer Methods in Applied Mechanics and Engineering 50: 71-101. [3] F. Auricchio, R.L. Taylor, 1995. A triangular thick plate finite element with an exact thin limit, Finite Elements in Analysis and Design 19: 57-68. [4] W.J. Chen, Y.K. Cheung, 2001. Refined 9-dof triangular Mindlin plate elements, International Journal for Numerical Methods in Engineering 51: 1259-1281. [5] *02<>HJ^`n|     ֻ{neWNWCWh1oh!0JaJh1oh!aJjh!UaJmHsHh1oh9aJh!h!aJmH sH h!h!CJaJmH sH  h?_`ϻzzzzzsl hjjlhw` hjjlh! hjjlhGqaJmH sH hjjlhKaJmH sH hjjlh4^aJmH sH hjjlhjjlaJmH sH hjjlh aJmH sH hjjlhXaJmH sH  hjjlh5A hjjlh9h1oh!aJ h9aJhmh!0JaJjh!UaJ h!aJ(`̴̇tfXK=fXhjjlh;6OJQJaJhjjlh;OJQJaJhjjlh! 6OJQJaJhjjlh g6OJQJaJ%j^hjjlh! EHOJQJUaJ3jFP hjjlh! CJOJQJUVaJnHtH%jhjjlh! EHOJQJUaJ/jͲ#P hjjlh! OJQJUVaJnHtHhjjlh! OJQJaJ!jhjjlh! OJQJUaJ)hjjlh! 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Ribari, G. Jeleni, Higher-order linked interpolation in quadrilateral thick plate finite elements, in print in Finite Elements in Analysis and Design, 2011. [6] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, Volume 1: The Basis, Elsevier Butterworth-Heinemann, Oxford, 2005. [7] Y.J. Liu, H.R Riggs, 2005. The MIN-N family of pure-displacement, triangular, Mindlin plate elements, Structural Engineering and Mechanics 19: 297-320.     BODOLOOOOFPNPP|Q~QQQQQQQQQmY@84848484hsjhsU0hhq/6B*CJOJQJaJmH phsH 'hh;6CJOJQJaJmH sH 'hVh;6CJOJQJaJmH sH 'hVhK6CJOJQJaJmH sH *hF6B*CJOJQJaJmH phsH *h i=6B*CJOJQJaJmH phsH 0hFhF6B*CJOJQJaJmH phsH 0hFhK6B*CJOJQJaJmH phsH hjjlhKaJmH sH QQQQQQQQQQQQQQQQ) & Fed^e`gdQQQQ0hhq/6B*CJOJQJaJmH phsH hThzi:&P 1h/R :p0K. A!n"n#$n% A 0&P 1h/R :p. A!"# $n% P = 0&P 1h/R :p2s. A!"# $n% 7A 0&P 1h/R :pzG. 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