Mechanics of masonry vaults: The equilibrium approach

Santiago Huerta

2001

Abstract
AI

The paper re-evaluates the mechanics of masonry vaults through the lens of both traditional geometrical approaches and modern theories, particularly focusing on the equilibrium approach. Historical insights from prominent figures, such as Father Vicente Tosca and Professor Heyman, are highlighted to assert that the fundamental principles of masonry design rely heavily on geometry and the states of equilibrium in compressed masonry. The paper emphasizes the importance of these historical methods in the effective design and analysis of masonry structures.

Figures (22)

What we have is a pile of stones, bricks or rub brium under the force of gravity. The mortar, when exists, is in such a way that they are in equili very weak in tension, so the interaction compressive forces. Besides, it is a angle of friction for stone is 30°-35° he danger of a failure by crushing i modern architect or engineer, consid fact that force of friction between the different elements is sufficient . Finally, the compressive stresses are usually very low, so that ble, received wit between the di s negligi plus drum weight 400.000 kN) is on h mortar or with dry joints, disposed fferent elements must be through he building maintains its form during the years: the y high to avoid sliding (the typical ble. Maybe this nave in Beauvais the mean stress was only of 1.3 N/mm, etc. We have. then. a comnosite heterogeneous material. with a creat streneth in compression. verv ast remark will appear unlikely to a ering the enormous size, which some masonry structures have. But, in fact, in this huge buildings the stresses are an order of magnitude below the crushing values. The mean stress at the base of the columns which support the dome of St. Peters in Rome (dome y 1.7 N/mm. Benouville found that in the pillars of the central

Figure 2 : Etruscan voussoir arch (Durm, 1885)

Figure 4 : Possible line of thrust of a semicircular arch (Heyman, 1995) Figure 3 : "Hanging" masonry arch (Robison, 1851)

Figure 5 : Displacement of the line of thrust due to some yielding of the buttresses. Devices to "fix" the line of thrust during construction (Winkler, 1880) But it was Winkler (1879) who made he first discussion in depth of the elastic approach to masonry arch analysis. After a revision of all the contemporary theories, he concluded that elastic analysis was the best option. However, he tha can affect the position of the line of t centering during construction, the yield of of temperature. All these perturbations wi | aware this will affect notably the position of the line of thrust, which could be very different from the calculated (elastically); he, then, suggested some means of controlling the position of the we line of thrust by inserting internal hinges d added a discussion on the "St6rungen" (perturbations) hrust. Their main origins were: the deformation of the he buttresses under the thrust and the effect of changes | produce some cracking of the arch and Winkler was uring construction, Figure 5.

Figure 6 : Test to destruction of a concrete vault (Ost. Ing.- und Arch.-Verein 1895)

ee ee A Se, ee Oe a RE a See eR Rte eat ae ee Re Tea ARM ae! See ea eee ee a ete ee a Let us consider a masonry arch over a centering, Fig. 7. After the decentering the arch begin to thrust against the abutments. Real abutments are not rigid and they will yield a certain amount. The span, then, increases and the arch must accommodate itself to this increment of the span. In what way could an arch (made of a the rigid-unilateral material described above) do this? The arch cracks. A crack opens at the keystone (downwards) and two other cracks open at the abutments (upwards).

Figure 8 : Different patterns of cracking due to movements of the abutments (observed in a cardboard model Huerta 1990, Huerta 1996) The arch becomes tri-articulated and a unique line of thrust is possible. But it may be, that the movement is not symmetrical: perhaps the right abutment besides yielding horizontally, yields vertically. To every possible movement corresponds a certain cracking, and cracks open and close to permit the arch to respond to this aggression of the environment. This may be observed using models. Even simple, "plane", cardboard models give very good results, Fig. 8.

Figure 9 : Semicircular arch under its own weight. a) Minimum thrust; b) maximum thrust (Heyman 1995)

Figure 10 : Barrel vault with gross deformations. The deformed state cannot be explained "elastically" (Huerta/Lopez 1997) ON EO In particular, deformations are not "elastic" in any sense; they are the result of the division of the structure in ascertain number of parts which, connected through the hinges, allow certain movements. In Fig.10, the original semicircular barrel vault is severely distorted due to an increase of 250 mm of the original span of 6.5 m. The movements were stopped with the addition of massive buttresses, and it is clear the danger of collapse by "snap-through". Cracks are not dangerous, but great unrestricted displacements of the abutments can, of course, lead to the catastrophic collapse of the structure.

Figure 11 : Collapse of a semicircular masonry arch under a point load (Heyman 1995 mechanism which will collapse. Therefore an increase of the load which will lead to the formation of four hinges will lead to collapse without crushing of the material. This can occur in a stable arch with addition of load, which deforms sufficiently the line of thrust. Again the hanging chain analogy makes the process clear, Figure 11.

Figure 12 : Semicircular arch: a) stable; b) of limit minimum thickness

Figure : 13 Typical cracking of a dome (Heyman 1988) Figure 14 : Partial cracking of a dome (Heyman 1988) Figure 15 : Stability of a masonry dome built following Fontana’ s geometrical rules

Figure 17 : Hanging model of a typical cross-vault (Beranek 1988) The first graphical analysis of cross vaults was made, apparently, by Wittmann (1879). Graphical statics allowed the engineers to make complex equilibrium analysis. Hanging models can also be used, either to understand physically the principle or as a tool for the architectural design, and Fig. 17 shows the essentials of a gothic vault equilibrium.

Figure 18 : Gaudi=s funicular design for the church Figure Colonia Giiell (Rubio 1913) colu Figure 19 : Gaudi=s graphical design for the columns of the Park Giiell (Rubid 1913)

Figure 20 : Different patterns of slicing in the analysis of gothic vaults (Ungewitter and Mohrmann 1890)

Figure 21 : Equilibrium analysis of the late-gothic vaults of the convent of Santo Domingo in Medina de Rioseco, Valladolid annulled). The curve of loading is almost horizontal with a peak very near the buttress. It is evident hat we can work with a uniform distribution and neglect this peak. Besides, the weight of the vault will be an order of magnitude greater that the weight of the transverse rib and we can check the stability of a "weightless" arch supporting a uniform load (obtained dividing the total weight of the bay by the span of the transverse arch). Of course, the line of thrust is parabolic and the thrust can be immediately calculated. The final check of the stability of the buttress can now be made. In Figure 21, can also be appreciated the relation between the material of the vault and that of the buttress system. The vault represents less than 10% of the masonry structure (this figure is typically between 5-10%, depending on the type of structure). The buttresses are important not only because hey assure the global safety, most of the material consumption (and also the money) goes to them. Old master builders were right, again, considering buttress design the more important part of structural design (Huerta 1990).

Figure 22 : Cracks in a gothic quadripartite vault

Figure 25 : Vista of the cathedral of Palma de Mallorca Figure 24 : Weights over the central keystone ‘Palma de Mallorca (Rubio, 1912) and transverse arches (Rubio, 1912

As in Palma de Mallorca the gothic master is controlling the equilibrium playing with the loads as in a balance. In can be mentioned that usually neo-gothic churches are in a worse state as gothic churches. Bollig (1975) attributed this to the attitude of neo-gothic architects who copied gothic structures from the inside being ignorant of the necessity of superincumbent weights. Figure 26 : Equilibrium analysis of the church of Sankt Martin in Landshut (Zorn 1933)