Form finding of revolution shells by analysing mechanical behaviour by means of the finite element method

A methodology suitable for finding optimal forms of shells of revolution by considering shell structural mechanics features is presented. In this case shells of revolution that compose the walls of reinforced concrete pressure vessels used to contain fluids are considered. In general, typical design objectives are to determine the shell stress, maximum load bearing capacity, minimum material needed, etc. The work here is limited to a study of the forms in which bending stresses are minimized. The proposed methodology, which enables approximately bending free forms to be obtained, is based on the mathematical optimization concept, which combines the use of geometrical modelling tools, structural analysis, and mathematical optimization. The optimization schemes are applied using a function called ‘volumetric flexure’ as an objective, and bounds the stresses to a certain limit. This objective function has been introduced with the intent of minimizing bending throughout the structure. It is to be noted that minimizing bending means minimizing the strain energy. In order to show the versatility of the model thus developed two different applications are presented. Both refer to reinforced concrete water storage facilities of important aesthetic value, but have a relatively complex geometry. By applying this technique the intent is to minimize flexure and/or tension acting on the shell inner surface, with the aim of reducing crack generation that would provoke containment loss of the vessel walls in order to obtain a reasonable useful life of the vessel. It needs to be mentioned that crack generation resulting from rheological phenomena that occur in concrete are unavoidable, but when the shell has a mechanical behaviour where compression prevails, these cracks tend to close and their propagation is more limited.