What is double curvature?
Double curvature (also, but rarely known as, anticlastic curvature) is used in nearly all fabric structures to give stability to the membrane. However, whilst it has its uses, it is not actually an essential requirement for smaller tensile structures.
This article explains more about the principles behind tensioned membranes and the tensioning methods most frequently used.
Imagine four people holding out a large thin square rubber or cotton sheet, one at each corner, pulling fairly hard. The sheet is flat and tight around the edges, and yet the sheet has very little resistance to movement in the central area. For example, even a light wind will cause it to bulge up or down, and a ball thrown onto the surface will deflect it significantly.
Now imagine that two diagonally opposing people peg their corners to the ground, and the other two maintain their position.
The middle of the sheet is now halfway between the low points and high points. From this middle point, fabric is curving both downwards toward the corners on the ground, and upwards toward the people holding out the other two corners. This is a much more stable shape which will inherently resist movement from download or uplift. The fabric is in double curvature, the form in this instance is a “hypar”.
However what is also important here is the nature of the material itself. You could perform this trick with rubber because it’s very stretchy, so when you distorted it from being a flat square, it can accommodate the distortion quite comfortably. If you tried it with say a very large piece of paper, it wouldn’t really take up the changed shape at all well.
Although the rubber could easily adopt a double curvature, in reality it wouldn’t work for a tensile structure as the following will hopefully make clear.
Why do we need to tension it in the first place? Well if the fabric was literally cut to fit exactly its final position, it would move around very easily in even light winds, and after a buffeting from very high winds or a heavy snowfall, the fabric would have stretched enough to not even return to its original installed size.
Why shouldn’t the fabric move around in the wind? The fabric used typically weighs around 1kg/m2. Even a small structure will have enough fabric that if it were flapping and thumping in strong winds, there would be quite a significant amount of wear to all the connections, as well as fabric strain at corners and other highly stressed locations. Not to mention the (not unreasonable) concern that the public would have for its and their own safety!
What we are looking for is a material that, once tensioned, has just enough stretch and elasticity to be able to absorb higher stresses with mimimal stretch, and which will return to this “prestress” condition time and time again after high winds or heavy snow load. This is really the crux of tensile fabric structures. We have to pull this “non elastic” stretch out of the fabric before we put the canopy in. We do it by reducing the fabric size when it’s cut out, and we call it compensating the patterns.
The other factor not to forget is that although the fabrics we use have a certain amount of stretchiness, it’s generally nowhere near stretchy enough to create a large doubly curved surface in a single piece. We have to cut the fabric into narrow strips, and join them up to create the appropriate 3d form to fit to the structure. As mentioned before, you couldn’t create a hypar form with a single sheet of paper, but you could recreate the shape with lots of thin strips.
In fact the fabrics we use are a lot more forgiving than paper, and can accommodate a lot of “shear”, partly because the basecloth is simply a woven cloth, like a sheet, (albeit an extremely strong one) and partly because the coating is flexible and fully interlocked with the basecloth.
Hopefully by now you’re clear about what we’re trying to achieve with a tensioned membrane, and how we have to make it from strips of fabric, slightly reduced in size, so that when the canopy is installed, it takes a smooth rounded form, and can resist all but the highest winds with virtually no movement.
There a couple of other things I want to discuss, because they relate to this field although not specifically to the double curvature title. Firstly I want to talk about fabric stresses in canopies, and secondly, the techniques employed to achieve those stresses.
Consider the original flat square, held out by four people. If they gradually increase the pulling force, what is likely to happen? Well assuming they have the strength, the corners will tear off where they are holding it, and yet the fabric in the centre is nowhere near breaking strain.
Effectively, the “overall” stresses in the fabric are concentrated more and more towards the corners. If a heavy object was placed in the centre of the sheet, the stresses in the fabric in that location would of course increase dramatically.
This shows us that the stresses in the fabric vary across the surface. If we know that we need to stretch a particular fabric to a particular stress in order to 
ensure all the “non elastic” stretch is taken out, how would that be achieved on a large canopy?
The sheet analogy is again useful. A square sheet, pulled hard at the corners, will not be especially tight in the centre. The stresses are concentrated around the edges, which would tend to develop ripples parallel with the edge. There is another problem with the sheet – at the corners, the tension force is diagonal to the main warp and fill directions. That means the fabric will stretch enormously in the bias direction, and transmit virtually no load to the central area of the sheet.
Now imagine taking a pair of scissors and cutting a curved edge (a hollow, or scollop) into each side, from corner to corner. Throw away the offcuts and turn the cut edge back on itself and stitch it to form a perimeter pocket on all four 
edges. Thread a non stretchy rope through these pockets and fix them to the fabric at each corner. Now what happens when you pull the corners?
As you would imagine, the rope tries to pull straight, but it can’t because the fabric won’t let it. The corners of the sheet don’t stretch and stretch – in fact the harder you pull, the tighter the ropes get, and this in turn provides the tension to enable the central area of the fabric to be pulled tight. In real structures, we use cables, not rope, for their durability, very low stretch properties and high load capacity.
These “boundary” cables transfer the tension forces in the fabric into the supporting steelwork at each end via the corner plate assembly, which why the corner details are so important to design correctly.
So how do we actually put all that tension into the system (in order to take out the non-elastic stretch)? You can’t pull corner plates into position by hand – even on a 3m square flat canopy, the corners could need up to 500kg of pull to fit the canopy to a suitable tension.
It’s quite simple really, somewhere along the design process, you have to decide on a method by which you plan to tension the canopy. In general, if you had a 3m square canopy, it would be easy to connect up the first three corners. You could pull by hand the last corner to within say 100mm of its fixing, and then you’d have to use a mechanical method to pull the last bit. This could be as simple as a lever, or a ratchet strap, but these are only useful for relatively light tension loads.
For bigger structures, the loads are correspondingly higher, so you need more powerful equipment. Chain lever hoists and tirfor winches are both extremely efficient when high pulling capacities are 
required.
But not all canopies can be tensioned just by pulling on a corner. For example, a typical cone structure is one of the easiest canopy types to tension. Imagine your 3m square canopy, but cone shaped. If you assume all four corners are fixed at 2.0m height, it’s very easy to push the middle up with a long post, which extends to the ground. If you get the pole length right, it would be just vertical when the fabric is properly tensioned.
As you push the middle up, the fibres running from the centre to the perimeter try to straighten and as they do so, the fibres running around the canopy come tight, so the result of pushing up the middle is to apply tension to the whole surface of the canopy.
This technique- to apply a concentrated and controlled direct push on a specific part of the canopy or structure, can be applied in many different ways using different bits of equipment. Although a great way to tension conic forms, this wouldn’t be a suitable approach for a barrel vault form.
Barrel vaults usually rely on a sliding plate at each corner which can be pulled out along the arches to apply tension to the whole membrane. The details on small structures are fairly straightforward, but on larger canopies, there is a lot of careful engineering and detailing required to ensure the end plates can be pulled into position while under the high loads generated by the boundary cable.
Many structures often contain more than one way to apply the tension, and in some cases, the easiest way is to make a simple but effective device to enable a quick and safe installation. The important thing is that the tensioning method is given due consideration early in the design phase and to incorporate the method into the final structure details.
If you’ve absorbed all the information above, well done and congratulations, you now have a great understanding about tensile fabric!!
But wait, there’s still more...! A further insight can be gained from taking a more careful look at how the fabric behaves when under active loading. It’s a very interesting aspect of fabric structures, because it has a bearing on the whole design of the canopy structure.
If you want to read about it, look at some typical loading figures, or see what’s involved in a structural analysis, please have a look at our OTHER STUFF place.
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