Keywords: shape, size, optimisation, concrete, shells, design.

The structural behaviour of shells, compared to other types of structures, is characterised by a higher mechanical efficiency. Concrete shells depend on their configuration, not on their mass, for stability. If appropriate designs are carried out, shells can support high loads and cover important spaces using little material and-or thickness. Moreover, shells present an attractive lightness and elegance from an aesthetic point of view, leading some authors to referring them as the "structural elegance" [1].

Since the main structural behaviour of shells is developed by their form, it would be interesting to find small modifications in the geometry that, without altering the initial aesthetic configuration too much, would improve that mechanical behaviour even further, whilst complying with the design conditions at the same time. This may be achieved by shape optimisation, which implies a better quality in design processes due to the attempt to find structures with quasi-perfect behaviour.

The development of optimisation techniques has been strongly boosted by the tremendous increase in computational and graphical capacities. These techniques represent an effective means to obtain alternative geometric forms of shells and improve their mechanical behaviour, complying with the design conditions (stress constraints, construction conditions, etc.) in an optimum way (minimum weight, maximum stiffness, minimum stress level, etc.) [1,2].

In this paper, optimisation techniques were applied to find optimum geometric designs that were close to a preconceived design, i.e. the final geometry should have an aesthetic shape similar to the initial one created by the designer. To achieve this task, different objective functions can be used, such as the strain energy, the weight of the structure, or the tensile stress in both faces of the shell. The parameters that govern the geometry of the structure and the thickness of the shell were used as variables. The constraints referred to the minimum thickness of the shell, the tensile stresses in concrete, and several parameters of geometric control. The optimisation of a shell based on Candela's blueprints is given as an example [3]. Several optimisation processes under predominant gravitational loads have been carried out. From the results obtained, it can be concluded that with slight changes in the shape, considerable improvements are obtained in the mechanical behaviour, and if necessary, the membrane state of compressive stresses may be achieved. The behaviour of the shell against instability improves significantly, since the buckling load is approximately double the initial value.

1

E. Ramm, R. Kemmler, S. Schwarz, "Formfinding and optimization of shell structures", in "Proceedings of the International Colloquium on Computation of Shell and Spatial Structures", Chania-Crete, Greece, 2000.

2

K.-U. Bletzinger, E. Ramm, "Form finding of shells by structural optimization", Engineering with Computers, 9, 27-35, 1993. doi:10.1007/BF01198251

3

A. Tomás, "Optimum design of shape and reinforcement for concrete shells", PhD. Thesis, Technical University of Cartagena, Cartagena, Spain, 2007.

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