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The Prinzipalbogen Method
Vault design in the Germanic world has been widely researched by Werner Müller (1990). One of the methods listed by Müller, and undoubtedly the most peculiar, is the method of the main arch or Prinzipalbogen, so called in German. Here the whole vault is designed with the same arch and the height of the keystones is determined by a route the architect chooses following the tracing of the ribs in the horizontal projection of the vault, from the starting point to its highest point. This method has the peculiarity that the height of the keystones and therefore the shape of the vault can change according to the route (Figs. 8, 9 and 10). The sum of the lengths of every section of the plan route is used as a radius of a big arch, which is precisely the main arch. This big round arch determines the heights of the keystones and the shape of the different arches. This method establishes a direct link between plan and volumetry and allows us to standardize the curvatures of the arches defining a clear and simple protocol of carving and execution, a fact which makes its organization significantly easier.
The Prinzipalbogen method is a very rigorous and at the same time very flexible system which allowed medieval architects to adapt the vault to their own esthetic taste, always under a rigorous constructive rationality. Consequently, this system could result in very different grid crossing vaults, characterized by ribbed springing-stones starting from different heights and keystones, or rather arch intersections that intersect at different heights, giving rise to the characteristic superposed ribs and other striking inconsistencies of the German Gothic style (Fig. 11). In the process of reconsideration of Ribes's vault, the virtual modeling has allowed us to appreciate the different configurations that the vault might have adopted according to the route chosen to determine the main arch (Figs. 8, 9 and 10).
Therefore, it is interesting to point out that the Prinzipalbogen system is not a rigid method at all. Depending on the route chosen in the vault plan drawing, the final length, that is the radius of the arch, will vary significantly. A variation of this length means an important change in the curvature of the arch and, consequently, in the shape of the vault; also, a change of the main arch will make important changes both in the springing-stones, in which the arches will start at different heights, and in the keystones, where the intersections of the arches can change their height considerably. Figures 8, 9 and 10 show the changes in the volumetry of Ribes's vault when choosing different routes in the ground plan; observe the important variations of the springing-stones and the intersections of the arches.
In order to build this vault, we chose a route and, consequently, a main arch with which to build a harmonious and predominantly rounded volumetry, as Joseph Rives liked. The keystones, that is, the point in which the arches intersect with each other, meet in the Spanish way, that is, in one point; the springing-stones of this vault, although the diagonal arch starts at a different height, is also similar to the esthetics of the Spanish vault, that is with a clear impost plan.
So, The main arch is determined by the sum of the ground plan distances in between the keystones of the route which starts from point 0, passes through keystones 1 and 2, up to keystone 3 (Figs, 12, 13 and 14). In our case, with the aim of creating a rounded intrados, we have assumed that keystone 4 is at the same height as keystone 3. And so, the distances previously mentioned will be placed in line and, with the sum of all of them as a radius, we will draw a rounded arch, which is precisely the main arch; all the vault arches are equal and have this same curvature. From here we go on to make certain adjustments, for example, in order to determine rib 5-0 and to maintain the tangency with the springing-stones, we have lowered the impost plan of the whole vault, resulting in a camber of the starting point of rib 0-1. We have also decided to tilt ribs 1-5 and 2-6 in order to avoid an intersection of superposed arches. According to Ribes's design, all the vault keystones have been placed facing the vault centre; that is, they are radial keystones. The result achieved is shown in figure 15.[....]
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