Graphic methodologies, based in equilibrium hypotheses, for the structural analysis of masonry domes. Application to a historical building

Resumen

This dissertation presents a new graphic methodology for the structural analysis of domes and other surfaces of revolution, based in the combined use of funicular and projective geometry. The new methodology is presented through its application to a hemispherical brick dome of small thickness. The analysed model has been referenced to the inner brick dome in Basilica of San Juan de Dios in Granada (Spain). This basilica is regarded as a benchmark in the Spanish Baroque. For the first time, it is set out a detailed constructive analysis and a geometric modelling of the dome over the transept, which was built according to the “encamonada dome” typology, divulged by Fray Lorenzo de San Nicolás. The structural analysis of the inner brick dome is carried out by applying different equilibrium methodologies of increasing refinement. Firstly, an approximation is made by applying the Guastavino´s formulas. This is an experimental methodology that can be applied very quickly. Secondly, the application of the slicing technique in the frame of limit analysis. This is a very well known, and contrasted methodology, for the stability analysis of domes, which does not take into account the influence of the parallel internal forces. Finally, this dissertation presents a new graphic methodology, which allows to graphically determine internal forces in the dome, imposing the equilibrium in both vertical and horizontal planes. The dome is simplified into a network of longitudinal and latitudinal lines. The equilibrium in the vertical plane is assured by fitting the weight force polygon to the polar rays, so the dome geometry is a closed antifunicular polygon of the system of forces. The equilibrium in the horizontal plane is guaranteed by the formation of closed force polygons in the dual figure. The application of the new graphic methodology considers three structural situations: complete hemisphere, hemisphere with oculus, and hemisphere with lantern. Different inclinations of the reactions at the support are considered as well as various hypotheses for the structural behaviour of the backfilling. Multiple solutions with acceptable tensional values for the brick masonry have been obtained, which makes consideration of the backfilling on the extrados unnecessary. The results have been contrasted with the ones from the application of membrane analysis, and it indicates a strong coincidence if the number of sectors in the discretisation of the graphic methodology are high enough. Additionally, the new methodology allows its application to different structural situations, it is easy to understand, easy to program, and can be applied in domes of arbitrary geometry.