Stacking sequence optimization to maximize the buckling load of blade-stiffened panels with strength constraints using the iterative fractal branch and bound method
Introduction
Laminated carbon fiber reinforced polymer (CFRP) composites have widespread applications in aerospace structures, and optimizations of the corresponding stacking sequences are indispensable. Miki [1] and Fukunaga and Vanderplaats [2] both proposed a graphical optimization method by means of the lamination parameters. For practical laminated CFRP structures, however, the available fiber angles are limited to a small set of fiber angles, due to the lack of experimental data. Moreover, several constraints in the fiber angles exist due to certain empirical rules, such as the four-contiguous ply rule to prevent large matrix cracking. These facts make the optimizations of the stacking sequences a combinatorial optimization problem with combinatorial constraints.
Genetic algorithms (GAs) are adopted for optimization of the stacking sequences [3], [4], [5], [6], [7], [8], [9]. Since GAs are one of the stochastic search approaches, they require several parameter tuning processes to avoid any reduction in the computational performance. Moreover, it is not easy to implement constraints into the algorithms for these GAs. In the evaluation of chromosomes, these GAs involve high computational costs. Narita has proposed a layerwise optimization method for the stacking sequence design [10].
Here, we have proposed a fractal branch and bound method (FBB) for optimizing the stacking sequence [11], [12]. This method employs a quadratic polynomial for the response surface using the lamination parameters, such as buckling load, to approximate the objective functions. This method involves only low computational costs, and the optimal result can be obtained by means of the deterministic process in milliseconds. The FBB method is based on the discovery that plots of the feasible laminates create fractal patterns in the lamination parameter space. Since this method is one such branch and bound approach, tuning of the parameters is not required. This method has been successfully applied to the problem of establishing the maximization of the buckling load of a laminate [11], [12], and for determining the maximization of the flutter limit [13], [14] with constraints. The FBB method has also been applied to unsymmetrical laminates, such as composite cylinders [15].
For the practical stacking sequence optimizations, it is necessary to optimize more than two laminates, because a practical aerospace structural component usually comprises one panel and a number of stiffeners made from composite laminates. Since the stacking sequences of the skin panel and stiffeners affect the buckling load of the stiffened panel, the optimization of both laminates must be performed simultaneously. The FBB method has therefore been extended to the optimization of multiple laminates [16].
In the present study, a new method to implement a constraint of strength for the FBB method is proposed for the optimization of more than two laminates, such as the panel and stiffeners, simultaneously. The constraint of the buckling load has previously been implemented in the FBB method [13] to address the single panel flutter speed limit maximization problem. This method, however, requires a modification for the stiffened, panel because the multiple stacking sequence optimizations are indispensable for practical stiffened composite structures. In the present method, a quadratic polynomial objective function is adopted along with the lamination parameter variables of the two laminates: the stiffeners and the panel. The strength constraint is implemented by means of a response surface. The new method is applied to the buckling load maximization of a blade-stiffened composite panel, in which the strength constraint is demonstrated as a feasibility study.
Section snippets
Optimization problem
A panel with four blade-type stiffeners is adopted as a target structure for optimization, as shown in Fig. 1. The configuration of the blade-stiffened panel is decided based on the optimized result performed by Nagendra et al. [17]. The panel width is 4b = 0.813 m and the height is a = 0.762 m. The panel comprises 40 plies and the panel is made from a symmetric laminate.
The detailed configuration of the blade-type stiffener is shown in Fig. 2. The length of the blade is h = 0.0781 m, and the width of
FBB for a single laminate
By considering the case of the thickness of each ply, t, the integral formulae are rewritten as summations using the half number of plies (N) and the fiber angle of the kth ply (θk) from the outer plyEqs. (5), (6) are rewritten by replacement of the coefficients as follows:
Results and discussion
First of all, to make response surfaces of the buckling load ratio λb and the strength factor λs, the design of experiments requires 90 FEM analyses to be performed. To reduce the bias at the origin, the isotropic laminate is added three times: , , , , , , . Totally, 93 results are used to obtain the unknown coefficients of the response surfaces. The obtained response surfaces are as follows:
For the buckling load ratio λb (=yb):
Concluding remarks
The present study proposes a new optimization method of multiple stacking sequences. A stiffened composite panel is one of the examples that require optimizations of multiple stacking sequences. In the present study, an alternative method is adopted to optimize multiple stacking sequences, in which the strength constraint is applied. For the constraint, the present method uses a response surface of a quadratic polynomial, and the method is easily applied to the fractal branch and bound method.
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