Keywords: random gust, power spectral density, probability of exceedance, Monte Carlo simulation, stacking sequence, genetic algorithm.

The use of composite materials in aerospace vehicles can result in a significant increase in payload, weight reduction and fuel efficiency. For example, advanced composite materials are widely used for modern wing designs to reduce the structural weight. Thus research efforts have been devoted to the optimal design of wing structures in connection with various objectives and constraints [1,2].

In this paper, the stacking sequence of composite wing subject to random gust loading is optimized to have maximum strength. By the atmospheric turbulence during flight, the wing may experience the excessive bending moment which cause structural failure. Thus, the failure index of Tsai-Hill criterion at the wing root is selected as the fitness function of the current optimization problem. The induced bending moment has probabilistic characteristics because the gust is random and irregular. So the probabilistic approach is needed for calculation of the fitness function, and Monte Carlo simulation is used in this research.

To evaluate the fitness function via Monte Carlo simulation, first, the probabilistic model for random variables should be defined. The probabilistic distribution of induced moment can be obtained by applying the concept of frequency of exceedance to the results of power spectral density analysis. The material properties of composite wing also have uncertainty and are assumed to show normal distribution. With the random variables for the bending moment and material properties, the maximum failure index at the wing root can be found by Monte Carlo simulation.

As next stage, the maximum failure index is minimized by using the ply angles as design variables. In practical applications, ply angles are limited to a fixed set of angles such as , , . Thus, combinatorial optimization methodology is needed for stacking sequence optimization to handle discrete ply angles. It is known that genetic algorithm (GA) is one of the suitable choices of the methodology for discretized optimization problems. GA is based on the mechanics of natural selection, crossover and mutation, and searches the optimal solution through random probability methods without auxiliary information such as derivatives or intelligently chosen starting points. GA has been successfully used in composite structural design. Riche and Haftka [3] proposed genetic algorithm to optimize the stacking sequence of composite laminate for buckling load maximization. For the same problem, Liu et al. [4] has provided permutation genetic algorithm. A recessive gene repair strategy was introduced by Todoroki and Haftka [5] for satisfaction of given constraints.

In the present study, GA with a repair strategy is adopted for the optimization of layup design. The balanced symmetric layup constraint and limitation of four contiguous layers are implemented by the repair strategy.

For numerical examples, the optimization process was applied to the simple cantilevered wing model for several gust intensities. GA with the population size of 10, probability of crossover lager than 0.8 and probability of mutation in the range of 0.0 0.1, showed high reliability for convergence and the same global optima were obtained irrespective of the gust intensities.

The anti-optimal solutions were also evaluated by maximizing the failure index for the purpose of comparison. The maximum failure index of optimal solutions was much smaller than that of anti-optimum. To demonstrate the superiority of the optimal solutions, the failure probability was evaluated for the initial design and optimization results. Monte Carlo simulation was used to evaluate the failure probability which was defined as the ratio of the number of cycles when failure criterion violated to the total number of simulation cycles. The failure probability of the wing decreased to almost zero after optimization process.

The presented methodology can be efficient way to improve the safety of aircraft structures subject to random gust. For practical use, more complex loading conditions and realistic flight profile should be considered in optimization process and this is currently under study.

1

J.H. Starnes, R.T. Haftka, "Preliminary Design of Composite Wings for Buckling, Strength, and Displacement Constraints", Journal of Aircraft, 16(8), 564-570, 1979. doi:10.2514/3.58565

2

I.W. Liu, C.C. Lin, "Optimum Design of Composite Wing Structures by a Refined Optimality Criterion", Composite Structures, 17, 51-65, 1991. doi:10.1016/0263-8223(91)90060-C

3

R. Riche, R.T. Haftka, "Optimization of Laminate Stacking Sequence for Buckling Load Maximization by Genetic Algorithm", AIAA Journal, 31(5), 951-956, 1993. doi:10.2514/3.11710

4

B. Liu, R.T. Haftka, M.A. Akgun, A. Todoroki, "Permutation Genetic Algorithm for Stacking Sequence Design of Composite Laminates", Computer Methods in Applied Mechanics and Engineering, 186, 357-372, 2000. doi:10.1016/S0045-7825(99)90391-2

5

A. Todoroki, R.T. Haftka, "Stacking Sequence Optimization by a Genetic Algorithm with a New Recessive Gene Like Repair Strategy", Composite Part B, 29(3), 277-285, 1998. doi:10.1016/S1359-8368(97)00030-9

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