Keywords: manufacturing tolerance, laminated structures, optimal design, genetic algorithm, buckling load.

Accurate optimal design solutions for most engineering structures present considerable difficulties due to the complexity and multi-modality of the functional design space. The situation is made even more complex when potential manufacturing tolerances must be accounted for in the optimizing process. When engineering structures are manufactured, the design parameters may deviate from their intended design values. These deviations are usually referred to as manufacturing tolerances. Though the deviations can be relatively small, their impact on the overall performance of the structure can be significant. Thus, determining the optimal design with the manufacturing tolerances accounted for can help to predict accurately such optimal characteristics like maximum applied load or minimum thickness, and by doing so save materials and, perhaps, save the structure from failure.

The present study provides an in-depth analysis of the multi-dimensional problem and then a technique for determining the optimal design of laminated structures, with manufacturing tolerances in the design variables accounted for, is proposed and demonstrated. The technique is simple, easy to implement and, at the same time, very efficient. In addition, the technique is non-probabilistic. A genetic algorithm with fitness sharing, including a micro-genetic algorithm, has been found to be very suitable to use, and implemented in the technique. Fibre-reinforced laminated structures are probably the most tolerance-prone of engineering structures because of the fabrication technologies used. These structures are manufactured as the material is fabricated from constituents, viz. reinforcing fibres and a plastic matrix, which must be cured as part of the processing. This can often lead to deviations from the intended design, particularly with regard the fibre orientations. In order to illustrate the methodology described we consider the design optimisation of symmetrically laminated orthotropic thin plates subject to buckling loads.

The objective of the design problem is to maximize the buckling loads N_x and N_y for a given plate thickness h by optimally determining the fibre orientations with the manufacturing tolerances in the orientation accounted for.

Although the shape of the buckling function might not look too complex, it cannot be easily optimized by using traditional gradient based methods. Therefore, the genetic algorithm is used for this and the other problems considered here since it proved to be effective in exploring complicated fitness landscapes and converging populations of candidate solutions to a single global optimum. The difference between the nominal and actual values is small, it is important to take the tolerances into account.

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