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The first tile-vault of the SUDU has a 5.8 meter span and consists of a floor system for a second story occupancy. Thus far, we have looked at the first stage of construction, which employs plaster mortar to build the first layer of the vault out into space without formwork. Such a vault cannot, however, remain only one tile thick. Let us look more closely at the problem of static equilibrium in arches to understand some of the factors involved in the structural system of the SUDU vault.
The shape of a hanging chain is the most efficient geometry to resist loads, since it acts in pure axial tension, with no bending moments. As first identified by Hooke’s 2nd Law, if the geometry of this chain is frozen and inverted, then it describes the form of an arch in pure axial compression. Below, this principle is shown as it was first utilized by Poleni to analyze the stability of the dome of St. Peter’s Cathedral (1748).
This “catenary” or “funicular” geometry indicates a theoretical “line of thrust” which must exist in a masonry structure; this describes the compressive forces in the arch as they travel through the masonry system. Within any arch, a catenary line with a range of minimum and maximum thrusts may be found. The shallowest catenary indicates an arch of maximum thrust (which pushes more substantially outward on its supports), whereas the steepest catenary indicates an arch of minimal thrust (pushing outwards least on its supports). Whatever the geometry of the line of thrust, outward thrust of a masonry arch is inevitable.
If an arch takes precisely the geometry of the hanging chain – as in the tile-vault of the SUDU – then it may be very thin, since the catenary thrust line need only stay within the cross section of the material.
A shallow catenary may be used to describe a vault for a floor system, such as the one which was designed for the SUDU. With a span of 5.8 meters, a rise of 0.5 meters, and a thickness of 10 cm, the SUDU vault is a shallow, funicular barrel vault with a catenary curvature in only one direction – a simple catenary arch which is extruded out into space.
When an arch is subjected to a point load, it catenary thrust-line becomes deformed, just like a chain upon which a single weight is hung. As soon as this line of thrust touches the outside of the masonry (either intrados or extrados), cracks may be formed. When the line of thrust exits the masonry arch, failure mechanisms are formed which will cause it to collapse. Asymmetrical loading of a masonry arch is the most common failure mechanism, since the line of thrust very rapidly exits the cross section of the masonry.
The thrust-line of the shallow tile-vault, when evenly loaded by the fill for a floor system, will remain within the very thin geometry of the vault. A thin-shell barrel vault with a single degree of curvature, however, like the example of the arch above, is particularly structurally weak when asymmetrically loaded. Since it is catenary in one direction, there is a very limited load path for the compressive forces in the masonry. Asymmetrical loading – as in the case when a group of people all stand together on one side of the vault – will cause a ‘kink’ in the line of thrust, which can cause it to exit the vault surface. For this reason, diaphragms (or vertical walls) spaced approximately 0.9 meters apart are built above the masonry surface. These stiffening ribs create alternate load paths for the masonry vault when it is asymmetrically loaded, and combine with the semi-rigid fill of the floor system, to allow the line of thrust to travel through the floor system.
Lastly, as noted previously, very shallow arches and vaults have an increased horizontal thrust, meaning that the shallow SUDU vault ‘pushes’ horizontally outward on its supports. This thrust could of course be reduced by building a much deeper vault and floor system; such an approach, however, would require a lot of material (both more surface area of masonry and more fill for the floor system) and would require more labor and time to build. For this reason, the shallow vault is much more practical – yet its horizontal thrust must be contained. This outward thrust may be countered by ‘tieing’ the arch with a steel tension tie.
Thus, we have the structural logic for the SUDU vault: a very simple vault which must be tied in and supported with diaphragms to create a floor system for a second story occupancy. Again, it is important to stress here that this example is for the most simplified vault of a single degree of curvature. A double-curved vault has a greatly improved stability by virtue of the multiple load paths possible to be taken throughout the surface. In the case of the floor system for the SUDU, however, the design has included a funicular, vaulted stair which cuts up through the floor system. In this case, it is important to insure that the thrust of the masonry travels only towards the supports, and that there is no horizontal thrust directed into the stair well. If a double-curved vault were employed for the floor system, the problem of thrust would have to be addressed by increased reinforcing at the edge of the vault.
Below, we can clearly see the floor system of the SUDU, within which a semi-rigid fill and stabilizing ribs serve to establish the floor surface for upper story and provide alternate load paths for asymmetrical loading. Tension ties are employed to counter the horizontal thrust, and a reinforced ring-beam is also employed to resist deflection of the beam along the base structure of rammed earth.
In summary, this structural system is a simple case, which carefully considers the behavior of masonry vaults. Also of great interest with respect to principles of applied structures, is that the structural system and the constructional system of the SUDU must go hand in hand for the SUDU to be stable – not only in its final, completed state – but during all states of construction. Next, we shall demonstrate how these principles may be applied directly in the construction of the vault.
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