Building the SUDU

Applied Structures I – SUDU Design
November 12, 2010, 3:21 pm
Filed under: Uncategorized

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.

(The following drawings, after Jacques Heyman, demonstrate some aspects of modern structural theory of masonry arches.)

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.


10 Comments so far
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Fabulous…thank you for sharing this level of detail…it’s very educational and appreciated.

Comment by David Komet

If i have two intersecting catenary vaults, what about the forces in the groin. Is the curve made by the intersection catenary?

Comment by Erik rowberg

This is not really the right forum for questions on geometry; but yes, the intersection at the intrados would be catenary, of a wider span with the same rise. Groins however are normally thickened somewhat (as in the case of a typical ribbed groin vault), because the forces will accumulate in groins or in any folded section of masonry.

Comment by limacon24

thank you for answering. As i moved from single brick vaults to cohesive tile your work was helpful. thanks for putting it out there. I still have a million questions, like waterproofing of the roof vault.

Comment by Erik

Hi, What a Fantastic Project! I’m looking into a project in Kigali adopting a similar system so i have a few questions, What is the material either side of the vault, are they concrete ring beams?? Do you simply put timber floorboards ontop of the vault? How did you waterproof the roof and Also how do these vaults perform under earthquake movement? You’re answers would be most helpful. Many Thanks! Max

Comment by Max

Answer in brief: 1. Yes, concrete ring beams, sized to counter thrust/loads from calculation of the principle vaulted structure. 2. No timber floorboards; it is a full masonry floor system. 3. I believe that the roof was not properly waterproofed, owing to how extensively it was leaking when I last saw the building. Further, the building in general is very poorly designed for an earth building in a monsoon climate like Ethiopia’s, and therefore the wetting related pathologies are expected to be quite extensive (e.g. oversights in detailing of waterproofing, coping/overhangs, drainage system, sills, damp-proof course, basement/substructure, plasters, etc.). 4. This structure was not designed with sufficient consideration for seismic activity in the rift valley; therefore, the model should not be considered as a reproduceable model without extensive further research in structural design and detailing for seismic conditions. 5. Steel reinforcement was retrofitted, as the Hydraform interlocking block was not properly specified for the building (the Hydraform system does not interlock in more than one direction, therefore it is not appropriate for walls of more than one wythe/course in thickness).

Comment by limacon24

Thanks ever so much for your reply lara, very helpful indeed. I’ve emailed your gmail as i may try to visit eiabc in a few weeks via kigali.

Comment by Max

Also is there any steel reinforcement or cement binder used in the loam walls?

Comment by Max

Hey there,i’m 5th year architectural student from mekelle, architecture school and i just wanted to say your project is very inspirational, helpful,detailed and over all updated .i would just like to say keep it up it’s really helpful and that your efforts are really appropriated here,not only by me but generally your project is really famous among me and my friends…..and the inverted chain idea is real one for the books….hats off to you.

Comment by Bisrat Yifalign

Thanks very much, Bisrat. Please be critical of this project and work with local engineers to develop solutions which also address the seismic requirements in Mekele and northern Ethiopia. Best of luck to you! I hope very much to return in future.

Comment by limacon24

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