Recent Improvements in Hu-Washizu Shell Elements with Drilling Rotation

  • Krzysztof WiśniewskiEmail author
  • Ewa Turska
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)


Enhanced four-node shell elements based on the Hu-Washizu (HW) functional are developed for the Green strain and constitutive laws modified by the zero-normal-stress condition. They have six dofs/node, i.e. the drilling rotation is included. The key features of the approach are as follows: 1. For the membrane part of HW shell elements, we use the 7-parameters stress, 9-parameter strain and 2-parameter EADG enhancement, which are selected as optimal for the 2D elements with drilling rotations. This ensures excellent accuracy and robustness to shape distortions for coarse meshes and very good performance for finite deformations. 2. The assumed representations of stress and strain are defined in skew coordinates, which not only improves accuracy, but also implies, as checked for 2D elements, that the homogenous equilibrium equations and the compatibility condition are satisfied point-wise for an arbitrary shape of an element. The skew coordinates are also easy to implement. 3. The bending/twisting and transverse shear parts are treated either by the HW functional or by the potential energy functional, in order to select the formulation which performs best for a reduced number of parameters. The performance of the shell HW elements is illustrated on several numerical examples and compared to the EADG elements in such aspects as: accuracy, radius of convergence, required number of iterations of the Newton method, and time of computation.


Drilling rotation Hu-Washizu functional Finite element method Four-node shell element 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This research was partially supported by the Polish Committee for Scientific Research (KBN) under grant No. N501-290234.


  1. 1.
    Badur, J., Pietraszkiewicz, W.: On geometrically non-linear theory of elastic shells derived from pseudo-Cosserat continuum with constrained micro-rotations. In: Pietraszkiewicz, W. (ed.) Finite Rotations in Structural Mechanics , pp. 19–32. Springer, Berlin (1986)Google Scholar
  2. 2.
    Bathe, K-J., Dvorkin, E.N.: A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. Int. J. Num. Meth. Engng., 21, 367–383 (1985)CrossRefGoogle Scholar
  3. 3.
    Cook, R.D: A plane hybrid element with rotational d.o.f. and adjustable stiffness. Int. J. Num. Meth. Engng., 24, 1499–1508 (1987)CrossRefGoogle Scholar
  4. 4.
    Chróścielewski, J., Makowski, J., Stumpf, H.: Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. Int. J. Num. Meth. Engng., 35, 63–94 (1992)CrossRefGoogle Scholar
  5. 5.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and dynamics of multi-segmented shells. Nonlinear theory and finite element method. IFTR PAS Publisher, Warsaw(in Polish) (2004)Google Scholar
  6. 6.
    Flanagan, D.P., Belytschko, T.: A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int. J. Num. Meth. Engng., 17, 679–706 (1981)CrossRefGoogle Scholar
  7. 7.
    Gruttmann, F., Wagner, W.: Structural analysis of composite laminates using a mixed hybrid shell element, Comput. Mech., 37, 479–497 (2006)Google Scholar
  8. 8.
    Kosloff, D., Frazier, G.A.: Treatment of hourglass pattern in low order finite element codes. Int. J. Numer. Analyt. Meths. Geomech. 2, 57–72 (1978)CrossRefGoogle Scholar
  9. 9.
    MacNeal, R.H., Harder, R.L.: A refined four-noded membrane element with rotational degrees of freedom. Computers & Structures, 28(1), 75–84 (1998)CrossRefGoogle Scholar
  10. 10.
    Panasz, P., Wisniewski, K.: Nine-node shell elements with 6 dofs/node based on two-level approximations. Finite Elements in Analysis and Design, 44, 784–796 (2008)CrossRefGoogle Scholar
  11. 11.
    Pian, T.H.H., Sumihara, K.: Rational approach for assumed stress finite elements. Int. J. Num. Meth. Engng., 20, 1685–1695 (1984)CrossRefGoogle Scholar
  12. 12.
    Piltner, R., Taylor, R.L.: A quadrilateral mixed finite element with two enhanced strain modes. Int. J. Num. Meth. Engng., 38, 1783–1808 (1995)CrossRefGoogle Scholar
  13. 13.
    Simo, J.C., Armero, F.: Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Num. Meth. Engng., 33, 1413–1449 (1992)CrossRefGoogle Scholar
  14. 14.
    Taylor, R.L., Beresford, P.J., Wilson, E.L.: A non-conforming element for stress analysis. Int. J. Num. Meth. Engng., 10, 1211–1220 (1976)CrossRefGoogle Scholar
  15. 15.
    Wagner, W., Gruttmann, F.: A robust nonlinear mixed hybrid quadrilateral shell element. Int. J. Num. Meth. Engng., 64, 635–666 (2005)CrossRefGoogle Scholar
  16. 16.
    Wilson, E.L., Taylor, R.L., Doherty, W.P., Ghaboussi, J.: Incompatible displacement models. In: Fenves, S.J., Perrone, N., Robinson, A.R., Schnobrich , W.C. (eds.) Numerical and Computer Methods in Finite Element Analysis. pp. 43–57. Academic Press, New York (1973)Google Scholar
  17. 17.
    Wisniewski, K. (2010) Recent Improvements in formulation of mixed and mixed/enhanced shell elements. SHELL STRUCTURES: THEORY AND APPLICATIONS. Proceedings of 9th Conference SSTA’2009, Jurata, October 14-16, 2009. Pietraszkiewicz W, Kreja I, (Eds.) General Lecture, pp.35-44, ISBN 978-0-415-54883-0, Taylor & FrancisGoogle Scholar
  18. 18.
    Wisniewski, K.: Finite Rotation Shells. CIMNE-Springer, (2010)Google Scholar
  19. 19.
    Wisniewski, K., Turska, E.: Kinematics of finite rotation shells with in-plane twist parameter. Comput. Methods Appl. Mech. Engng., 190(8-10), 1117–1135 (2000)CrossRefGoogle Scholar
  20. 20.
    Wisniewski, K., Turska, E.: Enhanced Allman quadrilateral for finite drilling rotations. Comput. Methods Appl. Mech. Engng., 195(44-47), 6086–6109 (2006)CrossRefGoogle Scholar
  21. 21.
    Wisniewski, K., Turska, E.: Improved four-node Hellinger-Reissner elements based on skew coordinates. Int. J. Num. Meth. Engng., 76, 798–836 (2008)CrossRefGoogle Scholar
  22. 22.
    Wisniewski, K., Turska, E.: Improved four-node Hu-Washizu elements based on skew coordinates. Computers & Structures, 87, 407–424 (2009)CrossRefGoogle Scholar
  23. 23.
    Wisniewski, K., Wagner, W., Turska, E., Gruttmann, F.: Four-node Hu-Washizu elements based on skew coordinates and contravariant assumed strain. Computers & Structures, 88, 1278–1284 (2010)CrossRefGoogle Scholar
  24. 24.
    Wisniewski, K., Turska, E.: Four-node Hu-Washizu shell elements with drilling rotation. In preparation, (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Polish Academy of SciencesInstitute of Fundamental Technological ResearchWarsawPoland
  2. 2.Polish Japanese Institute of Information TechnologyWarsawPoland

Personalised recommendations