Keywords: manufacturing tolerance, optimal design, anisotropic pressure vessel, genetic algorithm.

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. Though the deviations can be relatively small, their impact on the overall performance of the structure can be significant.

A few researchers have described methods for dealing with manufacturing tolerances. For example, Bauer and Latalski [1] consider the issue of manufacturing tolerances in dimensions with regard design optimization, when the objective is minimum weight. In two papers by Walker and Hamilton [2,3], a technique for optimally designing laminated plates with manufacturing tolerances present in the design variable (which is the fibre orientation) is described.

The present study provides an in-depth analysis of the multi-dimensional problem and then a technique for determining the optimal design of engineering structures, with manufacturing tolerances in the design variables accounted for, is proposed and demonstrated. The examples used to demonstrate the technique involve the design optimisation of anisotropic laminated pressure vessels based on an exact three-dimensional elasticity solution [4]. The technique is simple, easy to implement and, at the same time, very efficient. It is assumed that the probability of any tolerance value occurring within the tolerance band, compared with any other, can be equal, and thus it is a worst-case scenario approach. 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. Numerical examples clearly demonstrate the impact of manufacturing tolerances on the overall performance of a structure and emphasize the importance of accounting for such tolerances in the design optimisation phase. The results show that when the example tolerances are accounted for, the maximum design pressure can be reduced by more than 60% if the nominal fibre orientations are implemented and the example tolerances are incurred during fabrication.

For the purpose of visual illustration one- and two-layered cylinders are used here. The material properties are those for a typical T300/5208 graphite/epoxy material and the ratios of the external radius to internal in the cylinder used in the example are and , while the length is arbitrary. The Tsai-Wu failure criterion is used to calculate the maximum burst pressure with respect to the fibre orientations in the layers and taking into account the manufacturing tolerances.

The exact values of the maximum critical pressure are MPa at and MPa at , respectively. Applying the manufacturing tolerances and we discover that the actual value of the burst pressure drop from the nominal one by 59% to MPa in the case of thin cylinder and by 42% to MPa in the case of the thick one. Furthermore, if we were to specify the nominal value of the orientation and yet during fabrication tolerances were incurred, the burst pressure could be as low as MPa (at ), which is 67% lower than the expected (nominal) value. And in the case of the thick cylinder we have MPa which is 51% lower than the nominal value. The similiar picture is also observed in the case of the two-layered pressure vessels.

It is assumed in this research that there can be an upper and lower tolerance in each case, and thus for a problem with dimensions, it is demonstrated that the solution lies within the common domain of the possible hyper-surfaces.

These examples clearly demonstrate how important it is to take the manufacturing tolerances into account in the design optimisation stage. It also illustrates that it is much safer to use a few layers instead of one. However, calculations show that after about 10 layers there is not much improvement in the performance of the pressure vessel.

1

J. Bauer, J. Latalski, "Manufacturing tolerances of truss members' lengths in minimum weight design", Computer Assisted Mechanics and Engineering Sciences, 7(4), 461-469, 2000.

2

M. Walker, R. Hamilton, "A methodology for optimally designing fibre-reinforced laminated structures with design variable tolerances for maximum buckling strength", Thin Walled Structures, 43, 161-174, 2005. doi:10.1016/j.tws.2004.07.001

3

M Walker, R. Hamilton, "A technique for optimally designing fibre-reinforced laminated plates with manufacturing uncertainties for maximum buckling strength", Engineering Optimisation, 37(2), 135-144, 2005. doi:10.1080/03052150412331298371

4

P.Y. Tabakov, E.B. Summers, "Lay-up optimization of multilayered anisotropic cylinders based on 3D elasticity solution", Computers and Structures, 84, 374-384, 2006. doi:10.1016/j.compstruc.2005.09.023

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