Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 The sail-vault: a survey of constructive techniques to stabilize a sophisticated structure M. CobrerosO, E. Vazquez^ ^Profof School of Architecture of Seville. Avda Reina Mercedes 2. Cp 41002. Sevilla, Spain E-mail: [email protected] w Assistant Prof of School of Architecture of Seville. Avda Reina Mercedes 2. Cp 41002. Sevilla, Spain E-mail: [email protected] Abstract The Andalusian Renaissance architecture has been well studied but there is still a lack of information on the complicated structures that architects used in that period. The sail-vault was used in profusion, but their particular mode of operation has not been studied in detail. Current mathematical analyses systematically ignore these structures. This paper introduces an attempt to determine more information on these structures. 1 Introduction The geometry of the sail-vault is very well known and they can be visualised as a sphere cap resting on four pendatives, although this image does not work in the case of vaults with low rise/span ratio or vaults over a rectangular plan. The first image is that of a Byzantine dome while the others refer to a wide range of possibilities for use in architectures on a smaller scale. In fact, the plan form and the relationship between the rise and span are, together with the border conditions, the three aspects that will determine the way of working. While the semicircular sail-vault over a square plan is known, that is its Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 600 Structural Studies, Repairs and Maintenance of Historical Buildings way of working and its constructives techniques, the other cases presents several fields of research. In the Renaissance architecture of Andalusia the use of these vaults was conditioned to the writings and buildings of the architects A. Vandelvira who worked in this region during the XVI century. The work of Vandelvira overcomes the Gothic and can be registered in the Renaissance current. The treaty of architecture of Vandelvira was written at the end of the sixteen century and is a guide for its partners and pupils. This treaty is basically a study on the geometry and on the way of build very complex surfaces. The sail-vaults are studied by Vandelvira from several points of view: Figure 1: Church at Cazalla, Sevilla. Figure 2: Church at Cazalla, Sevilla. 1540. 1540. - The grilled and domed pattern for building. - The vaults over rectangular and square plan. - The reduced vaults. - The ribs vaults. In this last case the ribs constitutes an ingenious system of arches based on crossings carved in stone which support „ ^, , , • ^,1. i_ x T the rest of the structure. V/flgure 1 and Figure 3: Vandelvira Church at Jaen. * Vandelvira's analyses were put in practical by himself and for other architects of the Andalusian Renaissance, such as Hernan Ruiz and Diego de Riafio. v/figure 3 and 4. These examples are fundamentally sail-vaults for churches and cathedrals and they are distributed by the whole territory of the south of Spain. Starting from these examples the possibilities of the sail-vaults are enlarged until the field of the civil architecture. A fundamental example of this use is the second floor of the sevillian building of La Lonja projected by the architect Juan of Herrera at the end of the XVI century. It is a variation on the pattern of Italian palace with vaulted ground floor and wooden covered first floor. In this case the sail-dome is used as series of vaults for nine meters span over a Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 Structural Studies, Repairs and Maintenance of Historical Buildings 601 ground floor covered with stilted vaults The novelty of the solution consists on getting two superimposed and vaulted floors supported by walls with the usuals thickness, in the environment of fifth of the span. This solution is used in civil architecture starting from this building. General Analysis For this analysis we will compare two ways possible of work: The model of vault with meridians and parallel, and the pattern of arches grill. These two models can be combined with the border £ conditions and the rise/span ratio. Figure 4 Vandelvira model for a sail- If we consider first the sail-vault vault. supported in four columns with free Figure 5: Church at Cazalla, Sevilla. Figure 6: Church at Azpeitia, Guipuzcoa. 1540. Figure 8: J. Bautista de Toledo, Saul- Figure 7: Elpar. Membrane stresses. vault, The Escorial, Madrid. Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 602 Structural Studies, Repairs and Maintenance of Historical Buildings Figure 9 borders it is immediate to deduce that the pattern of arches grill is not possible 0 given the lack of the thrust factor. The domed pattern has in this case strong r% tension stresses in the pendentives according to the address of the parallel. The only possible model is that secondary arches are supported in two diagonal arches, in a similar way to the work of a groin vault. If we consider the borders coerced horizontally, that is to say admitting the thrust factor, both models are possible. It is also this way if the borders have horizontal and vertical coercion. This types of border conditions represent the vaults series and the supporting by transverse arch respectively, v/figures 5 Figure 10: Vandelvira model for grilled ribbed saul-vault. and 6. Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 Structural Studies, Repairs and Maintenance of Historical Buildings 603 Figure 11 Now we will must introduce en this discussion the ratio rise/span and the plan proportion. But before we will differentiate something more both models. For what we can introduce an approach from Salvador! to the sail-vaults based on the elliptic paraboloide. v/figure 7. Salvador! accepts the solutions of the elpar for the sail-vaults when they are reduced, that is to say when the rise/span ratio is smaller than 0.2 in the borders and 0.4 in total. The consequence is the arches grill pattern can be analyzed as traslation membrane in these cases. In any case the grill pattern need all the surface in compression while the domed pattern admits the tension stresses Figure 12: Vandelvira model for domed according to the parallel in the sail-vault. Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 604 Structural Studies, Repairs and Maintenance of Historical Buildings Figure 13 Figure 14 pendentives area in this case the final model is a compressive cap dome supported by arches in the pendentives area. All of that depend on the border conditions and the rise/span ratio. 'r Figure 15: Vandelvira model for diagonal ribbed sail-vault. Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 Structural Studies, Repairs and Maintenance of Historical Buildings 605 The Vandelvira models We will analyze the five basic models that Vandelvira settles down, starting from space model of bars, settling down for each one the variations of stresses, the tension areas and the thrust variation in the borders. The first case is a domed pattern with 0.7 rise/span ratio with semicircular arches in the borders, v/figure 8. Figure 16 The analytical model studies the two border condition and for both of then settle down tension areas in the pendentives according to the parallel direction. The variation of efforts among the meridian arches is low. If we eliminate the tension bar the variation of distribution of the border thrust is strong, higher in the corner than in the center. It is a pattern not very sensitive to this border condition, v/ figure 9. The same model designed with grill of arches have no tension areas and works better with the supporting of transverse arches. In this case the thrust is higher in the center than in the corner of the borders, v/figures 10 and 11. The third case is a domed pattern with 0.4 rise/span ratio with 0.2 rise/span ratio arches in the borders, v/figure 12 and 13. Transactions on the Built Environment vol 39 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 606 Structural Studies, Repairs and Maintenance of Historical Buildings This case does not present tension areas. The thrust distribution changes with the border conditions, being more uniform with the transverse arch condition. The same model with the grill pattern also improvement their behavior regarding the pattern more banked. The border condition change radically the thrust distribution as we can see in the figureH. With the vaults series condition the thrust increase from the corner to the center an in the transverse arch condition the thrust distribution is, basically, the inverse. The last Vandelvira examples is a diagonal grill pattern of arches with the same rise/ span ratio than the two previous. Their main characteristic is the smallest quantity in thrustes on the borders in its center in exchange for the increase in the corners. This model needs more of the transverse arch that the previous ones. It is the model whose constructives techniques represents better the elpar way of working, applied to this strcutures. v/figures 15 and 16. Conclusions Althought the sail-vaults geometry is very problematic because its lack of uniformity in stresses, the dificulty of tension in the pendentives areas and the differents thrusts conditions in the borders, it has been an usual solution for covering differents types of plan from Vandelvira and now on.