Keywords: genetic algorithms, finite element method, optimisation, minimum weight.

It is well known that designs for structures to be fabricated from fibre-reinforced laminated composites can be tailored for additional advantage. Thus, for example, by using the ply angles as design variables and determining the optimal values to maximise or minimise criterion like strength or mass the most benefit can be obtained from these materials. Due to manufacturing constraints, the sets of values from which the ply angles and layer thicknesses can be selected are generally discrete, and in such cases, the optimisation problem becomes one of finding the best permutation of these.

An important failure mode for laminated structures is bending under transverse loading. By selecting the stacking sequence and layer thicknesses optimally, the mass of a structure can be minimised for a given design strength constraint. Symmetrically laminated angle ply configurations are often used as they avoid bending-stretching effects by virtue of mid-plane symmetry. One phenomenon associated with symmetric angle-ply configurations is the occurrence of bending-twisting coupling, which may cause significantly different results as compared to cases in which this coupling is exactly zero [1]. The effect of bending-twisting coupling becomes even more pronounced for few layers. Due to this coupling, closed-form solutions cannot be obtained even for simple laminated plates, and thus many studies involving design optimisation of composite structures have neglected the effect. This study adopts a numerical approach to include the effect of bending-twisting coupling.

Genetic algorithms, which can be used to find the global solution of discrete optimisation problems, simulate the mechanics of natural genetics for artificial systems based on operations, which are the counterparts of the natural type [2]. They use techniques derived from nature, and rely on Darwin's principle of survival of the fittest. When a population of biological species evolves over generations, characteristics that are useful for survival tend to be passed on to future generations, because individuals carrying them get more chances to breed. Individual characteristics in biological populations are stored in chromosomal strings. The mechanics of natural genetics are based on operations that result in structural yet randomised exchange of genetic information (ie. useful traits) between the chromosomal strings of reproducing parents, and consist of crossover and occasional mutation of the chromosomal strings.

A technique for using GAs together with finite element analysis to minimise the mass of fibre reinforced symmetrically laminated structures with several discrete design variables is described. The design constraint implemented is based on the Tsai-Wu failure criterion, although any suitable failure criterion can be implemented. Rectangular plates are used to demonstrate the method, have eight layers, and are symmetric about the midplane. Thus, the four fibre orientations and laminae thicknesses are to be determined optimally. To determine the best configuration, optimal ply angles for each layer are selected from amongst a predefined set of fibre orientations, commonly used in industry. This approach leads to cost-effective designs by virtue of allowing the use of standard composite plies. The most common orientations are 0^o, 30^o, 45^o, 60^o, and 90^o, which are the ones used in the present study. Also, the laminae thicknesses must be multiples of a standard ply, thickness; eg. 0.001m.

Results are presented for different load distributions, and various combinations of clamped, simply supported and free boundary conditions are considered. The effect of aspect ratio is also investigated. Initially, the mass of the structure was taken as the GA fitness parameter, but the number of initial genes required for convergence near the global minimum was large, and thus other measures were investigated. Some of these are presented, all demonstrating enhanced efficiency, but introduce problems to the study. Previous work on discrete optimisation of composite laminates includes Reference [3].

1

R.M. Jones, "Mechanics of Composite Materials", Chap. 4, 166, McGraw Hill, 1975.

2

D.E. Goldberg, "Genetic algorithms in search optimisation and machine learning", Addison-Wesley Publishing Company, Inc., New York, 1989.

3

N. Kogiso, L.T. Watson, Z. Gurdal and R.T. Haftka, "Genetic algorithms with local improvement for composite laminate design", Structural Optimization, 7, pp. 207-218, 1994. doi:10.1007/BF01743714

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