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I always write a post to thank my crews for their work – but in this case, I would like to thank them individually. They are my most honored invisible laborers and heros of the SUDU project.
Daily laborer crew manager
My core team pictured above, however, were not the only workers on the SUDU. There was a crew of 30 daily laborers that each at times participated in the the full scale construction – laying foundations, rammed earth walls, the ringbeam, excavating, sifting soil, pressing tiles, and so on – and who each contributed enormously. I am afraid that I do not remember all of their names – and it is only recorded, to my knowledge, in Amharic, on the lists of Abinet and in his informal networks of Ethiopian laborers looking for work.
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Working from either side of the structure, the vault is closing in…
All of the major new tasks have been accomplished and now repeated – the laborers may now work more rapidly with better confidence, and they are at this point much better prepared to correct their mistakes without excessive supervision.
The scaffolding is drawn closer, until workers can cross from one to the other without climbing down. The terminal walls of each 1/2 vault will then be built, so that the funicular stair can cut up through the vault to the second floor.
And now the first steps can be made to load the vault – however, until the floor fill is added, only crown loading and not asymmetrical loading of the vault may be permitted.
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It has already been briefly described how the principles from ‘Applied Structures I’ are carried out in the fabrication of the guide-work for the vault; however, this will be addressed again here so that the construction process, A-Z, can be contextualized in terms of structural behavior.
It is obvious now that Hooke’s 2nd Law enables the rapid construction of our efficient masonry geometry – “As Hangs the Flexible Line, so but Inverted will stand the Rigid Arch”. The hanging chain is set up with the corresponding span and rise of the full-scale vault, and is traced onto paper to create a template. This template, when flipped, is the most efficient geometry for a vault with this span and rise. Perhaps this seems a crude method, yet the accuracy and ease of the hanging chain lends itself to building in contexts with limited resources.
The guide-work is fabricated directly on top of the template and is installed – at opposite ends of the vault – to describe the curvature of the masonry surface. The guides must be installed so that they may be ‘dropped’ down and away from the vault after the masonry is completed; otherwise the guides may be accidentally pushed ‘up’ into the masonry, putting a very thin, compression-only structure into tension and risking collapse. Particularly as the first courses are being laid, motion of the guide-work can compromise the weaker bond of the first cantilevering masonry units, so it is critical that it is installed very securely. Since milled lumber is extremely expensive and limited in Ethiopia, Eucalypis branches are the primary multi-purpose construction material, serving as scaffolding and guide-stabilizer in our case.
Mason’s line (string) is tied to connect the guides, ‘lofting’ a surface with only string so the mason’s know where to lay the tiles in space. The first vaulted layer is set in plaster without formwork – cantilevering out into space while following the mason’s lines. Here, the masons must understand the accuracy required for this singly-curved geometry. Each line is spaced at roughly a half-meter, and the mason’s must develop the ability to see a continuous catenary surface, rather than building straight lines of masonry between each string. If only one tile is set too low over the mason’s line, pushing the entire length downwards, then each subsequent row will have an inaccurate geometry. Alternately, if straight courses are built from line to line, kinks will occur in the surface. Since our vault is so thin, and since it is only curved in one direction, there is no redundancy of the load paths – and such mistakes can have catastrophic effects, creating points at which stresses are collected and hinges can be formed which cause the structure to collapse.
The masons must understand how a single curved vault is weak (particularly when it is still only 1 layer with a 2 centimeter thickness), so they can understand how to load the surface without compromising the vault. A bucket, cut in half (which typically serves as the mixing vessel for the gypsum mortar), is appropriated as a teaching tool. It is a single-curved geometry which can be bent to be roughly the geometry of the SUDU. A since point load at the crown of the bucket vault shows how strong it is; hands wrapped over the surface show how well it performs with an evenly distributed load; but even one finger pushing at a quarter point demonstrates its deformation from an asymmetrical point load. Thus, workers are instructed to be careful with the first 2 cm layer, to wear hardhats when working below the worker scaffolding, and to be responsible for who is working beneath them.
The following layers of masonry are set ‘oblique’ to each other (generally rotated by 45 degrees) so that the joints between each layer are ‘broken’ (meaning, so that a joint does not overlap between multiple layers). This creates a very strong ‘sandwich’ of masonry, which gives us a reasonable assumption that ‘no sliding’ will occur in the material. Once the thickness of the vault – and thus its self-weight – is increased, it is more resilient against point loads.
Once a meter of the first and second course of masonry is laid, we may begin to build the stabilizing diaphragm walls that were described in the last post. The diaphragm walls are space 0.9 meters apart, creating regular stiffeners against asymmetrical loading. One the floor system fill has been added the diaphragms and the fill will work together to allow the load paths to travel through the floor system in the case of asymmetrical loading.
The sequencing of this whole series is important so that this single curved vault is not loaded – particularly asymmetrically – until these alternate load paths are provided for by the diaphragm and floor system fill. In our case, since the materials acquisition and testing of the semi-rigid fill was delayed, we had to continue construction while limiting access and loading to the unfinished floor system. Ideally, each section between diaphragms would have been filled, allowing on team to complete their work from the floor surface of the completed vault.
The tension ties must be specified with sufficient strength steel, at the proper gauge, and spaced accurately to tie the outward thrust of the single curved masonry shell. Special attention must be paid to the detailing of these members so that welded bonds with short overlaps – such as below – do not fail in shear at the connection.
This vault was designed so that all the load paths would be directed to the ring-beam, in the direction of the curvature of the vault. This will allow a funicular stair to cut up through the vault. Nevertheless, this detailing at the edge is important to insure that there is no critical lateral loading exerted upon the terminating wall of the vault.
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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.