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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 52
Difficulties with Non-Homogeneous Failure Criteria Like Tsai-Wu for Composite Laminates A.A. Groenwold+ and R.T. Haftka*
+Department of Mechanical and Aeronautical Engineering, University of Pretoria, South Africa
Full Bibliographic Reference for this paper
A.A. Groenwold, R.T. Haftka, "Difficulties with Non-Homogeneous Failure Criteria Like Tsai-Wu for Composite Laminates", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 52, 2004. doi:10.4203/ccp.79.52
Keywords: optimization, composite laminate, failure criterion, homogeneity, Tsai-Wu, Tsai-Hill.
Summary
In designing composite laminates,
minimization of a suitable failure criterion is sometimes
selected as the objective function.
However, for inhomogeneous criteria, e.g. the Tsai-Wu criterion,
this objective function may be inappropriate when the ratio
of the applied load to the critical load is not close to, or
for some materials even larger than, unity.
We suggest that the use of a safety factor for the objective function
is more appropriate, and demonstrate numerically that the use of the
failure index may reduce the safety factor to first-ply-failure.
(The dependence of the optimal results on the applied load level
rather complicates the design process. Ideally,
one would like to have the same result, irrespective of the
applied load level.)
While the selection of a suitable criterion is never straightforward, it turns out that industry in general prefers the interactive Tsai-Hill and Tsai-Wu theories. In particular, the Tsai-Wu criterion (e.g. see Tsai [1], and Wu [2]) seems to be very popular, and is frequently used, since a number of practical experiments have shown good agreement with this theory. The Tsai-Hill criterion is a statement of maximum allowable work or deviatoric strain energy. As such the criterion also has a physical basis. The Tsai-Wu criterion on the other hand is an interactive tensor polynomial expression, and empirical. A physical basis seems to be lacking. In addition, the Tsai-Hill criterion is observed to be homogeneous, while the Tsai-Wu criterion, (as a number of similar criteria), is inhomogeneous.
For reasons of brevity, we cannot give the criteria here.
Nevertheless, it is easy to see that if the failure index is
homogeneous (Tsai-Hill),
the load factor will multiply all the terms uniformly, so that the
maximization of the safety factor will be equivalent to the minimization
of the failure index. With an inhomogeneous criterion (Tsai-Wu), at low
load factors, the linear terms will be more important than at high load
factors, and so the optimum stress distribution in the plies, and hence
the optimum ply angles will depend on the load factor Structural Optimization: In illustrating the above, we study the implications of the optimal design of composite structures with failure considerations as the objective. In particular, we considered two different criteria, namely
a)
minimization of the maximum value of the failure index
b)
maximization of the minimum safety factor
Examples: Firstly, we study the failure of a
simple symmetric and balanced uni-axially loaded tensile specimen.
We show that, when minimization of the Tsai-Hill failure criterion
is selected as
the objective, the vector of optimal ply angles
Secondly, we present results for the effect of a variation in material properties on the deviation of the solution.
Thirdly, we demonstrate the detrimental effect of homogeneity on
safety factor,
(in particular for applied load fractions notably less than
unity),
through
anti-optimization of a 5 layer laminate
subjected
to in-plane average stresses
References
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