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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 22
Representing Traction Free Boundaries using Drilling Degrees of Freedom A.A. Groenwold+, Q.Z. Xiao* and N.J. Theron+
+Department of Mechanical and Aeronautical Engineering, University of Pretoria, South Africa
A.A. Groenwold, Q.Z. Xiao, N.J. Theron, "Representing Traction Free Boundaries using Drilling Degrees of Freedom", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 22, 2002. doi:10.4203/ccp.75.22
Keywords: traction free, finite element, assumed stress, drilling degrees of freedom.Summary
This paper investigates new methodologies for the accurate representation
of traction free sides using membrane finite elements with drilling degrees
of freedom and an assumed stress interpolation.
Firstly, a method based on
direct enforcement of the traction free condition through
manipulation of the assumed stress field of an element is presented
(e.g. see Xiao
The methodologies are applied to the families of assumed stress membrane finite
elements with drilling degrees of freedom recently proposed by Geyer and Groenwold
[3], for which the potential energy is given by
This formulation results in three independent interpolation fields arising from the translations, rotations and the stress assumption. Rather conventional, the stress field is constructed as , resulting in the and families presented by Geyer and Groenwold. However, in both families for irregular geometries, is not optimal.
This makes these
elements susceptible to loss of accuracy as a result of element distortions.
This lack of robustness may mostly be overcome using the techniques already mentioned
in the preamble, and which are
briefly summarized in the following sections.
where represents the 2-D differential operator.
Using matrix notation, the potential energy of the elements under consideration
becomes
with , and
Four elements are formulated, namely HB12, , and . All the elements pass the patch test, are rank sufficient and invariant. When applied to the pure bending problem depicted in Figure 22.1, we note that the and elements developed are highly accurate, as is graphically depicted in Figure 22.1.
References
- 1
- Q.Z. Xiao, B.L. Karihaloo, and F.W. Williams. Application of penalty-equilibrium hybrid stress element method to crack problems. Engng. Fract. Mech., 63:1-22, 1999. doi:10.1016/S0013-7944(99)00015-6
- 2
- C.-C. Wu and Y.K. Cheung. On optimization approaches of hybrid stress elements. J. Finite Elem. Anal. Des., 21:111-128, 1995. doi:10.1016/0168-874X(95)00023-0
- 3
- S. Geyer and A.A. Groenwold. Two hybrid stress membrane finite element families with drilling rotations. Int. J. Num. Meth. Eng., 53:583-601, 2002. doi:10.1002/nme.287
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