Keywords: optimization, composite laminate, failure criterion, homogeneity, Tsai-Wu, Tsai-Hill.

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 , and

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 coincides with the solution obtained using the criterion of maximum stiffness: The optimal angle is not influenced by the applied load fraction, nor the material used. However, when minimization of the Tsai-Wu criterion is selected as the objective, the results reveal a marked sensitivity to both the applied load fraction, and the material used. For the Tsai-Hill criterion, minimization of the maximum failure index is fully equivalent to maximization of the minimum safety factor . For Tsai-Wu, this is not necessarily the case. When maximizing the safety factor for Graphite/Epoxy using Tsai-Wu, the optimum angle is 5.6 degrees. If the failure index is minimized at an applied load fraction of 0.6 of the failure load, the optimum angle increases to 10.4 degrees.

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 , and . The problem is formulated as follows: Let equal the safety factor obtained through maximization of the minimum safety factor , and the safety factor obtained through minimization of the maximum value of the failure index . We then maximize the relative safety factor discrepancy , by varying the applied loads , and . This constitutes a two-level optimization problem, with three design variables (the generalized loads) in the outer level, and the two ply angles in the inner level. For some combinations of loading, even at an applied load fraction of 0.75 of the failure load, minimization of the failure index leads to a loss of more than 40% of the safety factor.

1

S.W. Tsai. "Strength theories of filament structures." In R.T. Schwartz and H.S. Schwartz, editors, Fundamental aspects of fiber reinforced plastic composites, pages 3-11. Wiley Interscience, New York, 1968.

2

E.M. Wu. "Strength and fracture of composites", In L.J. Broutman and R.H. Krock, editors, Composite materials, volume 5, pages 191-247. Academic Press, New York, 1974.

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