3.1 The tapered vault
Unfortunately there is no graphical evidence about the construction shape of the pre-1832 vault. However, in di Boveglio & d'Isola Maggiore (1834, pl. 7; Figure 2) the mark of vault with the triumphal arch has a constant thickness. Moreover, Perilli (1842, 6) explicitly states that the previous vault had a constant thickness.
Poletti's drawing dated July 19, 1836 shows the cross section of his new vault (Figure 3). Although this is of some interest, it has dragged little attention so far. Poletti designed a tapered vault (i.e. a vault with non constant thickness). The actual vault geometry is slightly different from the one originally drawn (Lunghi & Lunghi 189, 194-195). The vault has a crown thickness of 32 cm, and (through three 16 cm steps) a haunch thickness of about 80 cm. Therefore, the span (at springers) to crown thickness ratio is about 57.7, the span (at haunches) to crown thickness ratio is about 45.7, while the crown to haunch thicknesses ratio equals to 2.5.
It will be shown that such a profile has several advantages and it is therefore legitimate that Poletti purposely pursued it in order to reduce the vault's thrust, considered among the reasons of the 1832 collapses.
There is no much attention in the historical literature to tapered arches or vaults. Probably the first reference is due to architect Andrea Palladio (1508-1580) in a 1567 appraisal on the new Cathedral of Brescia in Northern Italy (Puppi 1988, 123-125). Writing about the dome, Palladio states that the thickness should grow from the lantern to the drum in order to reduce the load at mid-span. Such thickness reduction is obtained by using two spheres, of diverse radii, with their centres at different heights. According to Huerta (2004, 200-201), Palladio used this method to design all the domes in his treatise (Palladio 1570), and the same was done previously by architect Sebastiano Serlio (14751554). Palladio's appraisal was reported by Zamboni (1778) and later by Rondelet (1832-1835, Book 9, Sec. 6, Ch. 4, Notes and Pl. 195), so it is possible that Poletti was aware of it.
Architect Carlo Fontana (1638-1714), in is famous treatise on the Vatican Temple (1694, 361-367) suggested a partially different rule for pointed domes, using four different centres, at the same height, in order to get a thickness decreasing from the lantern to the drum. This rule was based on the observation of several existing domes and on the exam of previous geometric rules (Huerta 2004, 272).
The first reference to tapered arches and barrel vaults is probably due to the French military engineer Amedee-Frangois Frezier (1682-1773). In his treatise (1737-1739, 2: 87) he describes a method to draw arches of non constant thickness based on the first scientific theories on arches due to La Hire, Parent and Couplet (for such theories refer e.g. to: Benvenuto 1991, 321-326, 331-336, 338-344). Frezier's method is rather similar to that by Serlio and Palladio, since the intrados and extrados curves are obtained considering two centres of different height. Once the crown thickness has been established, their positions are fixed either by fixing the springer thickness or by lowering the centre of the intrados curve by a fraction of its radius. In Frezier's examples the springer/crown thickness ratio equals 3 and extrados/intrados radius ratio equals 7/6. The French engineer gave also rules for vaults bearing heavy loads (e.g. bridges), vaults bearing small loads (e.g. vaults loaded by some wooden truss), and vaults subjected to self weight. In the latter case he recommends a thickness at the crown equal to 1/24 of the span, a thickness which should be doubled in the 30° section close to the haunches (2: 96-97).
In 1748 in an unpublished manuscript (Huerta 2004, 358-360) the French engineer Jean Rodolphe Per-ronet (1708-1794) suggested to design bridges with a thickness doubling from the crown to the haunches. In the case of semi-circular arches the crown thickness should be equal to 5/144 of the span plus 1 foot. A rather similar rule was suggested by the French Engineer Bernard Forest de Belidor (1697-1761) in 1750-1753 (2: 445).
Frezier's and Perronet's approach was followed by the French architect Jean Baptiste Rondelet (17341829) in his famous treatise (French edition 18121817, Italian translation 1832-1835). Therefore, he recommends to design arches of non constant thickness by lowering the centre of the extrados curve (Pl. 27 and Book 9, Sec. 6, Ch. 1, Art. 1), and to differentiate the crown thickness as a function of the span and of the type of live load (heavy, medium and zero; Book 3, Sec. 3, Ch. 1). However, also based on some experiments he performed, the thicknesses suggested are sometimes much smaller. For the case at hand, and considering an average stone, he suggests a thickness of 25 cm at the crown which should be doubled at the haunches, by means of a linear increase. Moreover, Rondelet clearly states that a tapered vault is much more convenient than a constant-thickness one. As a matter of fact, he writes that a semi-circular arch of constant thickness subjected to its self-weight needs a thickness equal to 1/17 of the span, while a tapered one needs a crown thickness which is one fifth of the previous (ibidem). Furthermore, he declares that a tapered vault will exert a much lower thrust of the abutments than a constant thickness one of equal span (Book 9, Sec. 6, Ch. 1, Art. 2; and ibidem Ch. 3). In this last chapter Rondelet gives tables for the crown thickness of semicircular vaults with horizontal extrados, with solid infill up to half height and thickness constant or tapered. The crown thickness of the constant one is one third bigger than the crown thickness of the variable vault.
Rondelet's approach had a great fortune, as evidenced by other widespread manuals of 19th century (Breymann 1885, Vol. 1, Ch. 8, Sec. 9 and Pl. 100).
The vaults drawn by Rondelet or Breymann are meant to have a smooth thickness increase. Such a result can be obtained if the vault is meant to be made of cut stones. However, if bricks are used (as in Santa Maria degli Angeli) it is easier to has discrete steps in the cross section. As far as it is known to the authors, such case is rarely considered in the literature. Körner (1895, 286) recommends to avoid such a design, since the thrust line will be very close to the extrados of central (thinnest) segment, while cracks and bulging will appear on the two sides of the vault close to the first thickness variation. As Rondelet and Breymann he endorses an even growth of the cross section.
In order to better understand such favour for tapered vaults as well as Poletti's design, his vault is analysed here by means of the safe theorem of limit analysis (Heyman 1966). Two lines of thrust are drawn, minimising either maximum top and bottom stresses or the thrust exerted on the haunches (Mery 1840). In the second case maximum compression stress was assumed 2.2 MPa (OPCM 2005). Three different geometries are considered: (1) Poletti's, (2) constant thickness t = 0.54 m giving a vault of same weight, (3) constant thickness 1/3 bigger than Poletti's crown thickness (t = 0.43 m), thus following Rondelet's abovementioned tables. Two load cases are taken into account: (1) self-weight only, (2) self-weight plus a 18.5 kN load applied at mid span over a 0.5 m length (representing the load of the roof resting on the vault, whose load-per-unit-area has been estimated equal to 2.0kN/m2).
The analyses are performed with a computer code, drawing the aforementioned thrust lines and the associated stress fields (Dr. Cesare Tocci, personal communication). Sample outputs are presented in Figure 4 and Figure 5.
In Table 1 is presented the comparison between Poletti's vault and the two vaults of constant thickness. In terms of optimal stress Poletti's solution grants a 3 geometrical safety coefficient (Heyman 1982, 24) with a 62.0 kN thrust. The 0.54 m constant thickness vault gives the same geometrical coefficient with a much higher, 74.6 kN, thrust.
The 0.43 m constant thickness exerts a little smaller horizontal force, 62.0 kN, but with a much lower, 2.5 geometrical coefficient.
In terms of minimum thrust, Poletti's vault gives again the best value, 53.6 kN, with the 0.43 m vault granting a similar 54.6 kN. However, such figure is also due to the lower self-weight. Therefore, the ratio between horizontal / vertical components is 0.64 in the
Figure 4. Poletti's vault: optimal thrust line under self weight.
Figure 4. Poletti's vault: optimal thrust line under self weight.
Table 1. Analysis of Poletti's tapered vault compared to two constant thickness vaults. Self weight only considered.
Table 2. Analysis of Poletti 's tapered vault compared to two constant thickness vaults. Self weight + roof.
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