[Ferro List] Catenary Dome

[Keith B]

This follows on from thread "[Ferro List] subterranean - catenary 
strongest shape & tensile    vs compressive forces", but I've changed 
the subject as "Catenary Dome" seems likely to be a more useful search 
term for the archives in the future.

One can't make a "Catenary Dome" in the same sense that one can a 
circular/spherical/sub-spherical one.  Both are examples of double 
curvature, so they both are related to arch structures used for bridges 
and barrel vaults, but fundamentally different because of the double 
curvature.  What's not obvious though, is that the two kinds of dome are 
much less similar than are their circular and catenary arch equivalents.

A dome based on spherical curvature has the same curvature in all 
directions.  A "Catenary Dome" cannot be similarly symmetrical.  It has 
spherical curvature horizontally, catenary curvature vertically, and 
intermediate curvature at all other orientations.  A moment's reflection 
is enough to see the theoretical superiority of the  catenary dome.  
Rings are the best shape for resisting stress in any dome, and both 
kinds have them for horizontal stresses.  Vertical stresses reflect 
gravity though, for which the catenary curve is optimal.  It's also 
quickly apparent that, for a given height and diameter, lateral stress 
at the base should be less for the catenary dome as it reaches its base 
at a steeper angle.  Less obvious are some practical concerns.

For a wide but shallow dome, there's not much difference, but for taller 
ones, the catenary dome may be much more attractive.  Often with a 
sphere based dome, to achieve adequate height requires the dome to be 
mounted on a ring wall.   Lloyd's example of a 20' high dome on a 20' 
base can't be done with a spherical dome, try and you end up with a 20' 
diameter ball or a 10' high ring wall topped by a 10' radius hemisphere 
(with lateral thrust at the junction).  With a catenary dome, it can be 
done in a single piece.  Further, such can be made arbitrarily high and 
still be done in one piece, e.g. a 30' high dome on a 20' footprint. 

Further again, the catenary lends itself well to developed surfaces.  An 
oval footprint can be topped with catenary dome and arch segments or a 
rounded corner rectangular footprint with similar and a central flat 
roof, for example. 

kb

Lloyd Turner wrote:
> Gents -
>
> Earlier this winter my son and I had some fun with this catenary  
> phenomenon.  I made a catenary shaped airform made of light weight  
> Tyvek fabric (5 mils thick) and shipped it to him in Anchorage.  He  
> and a friend anchored it down to a 20 foot diameter circular base,  
> inflated it with a small centrifugal fan, then sprayed it with several  
> passes of water mist.  After just misting on the first coat the  
> structure was self supporting and the fan was turned off.  At this  
> point the dome was no more than a  frozen wet rag in a 20 foot high  
> catenary dome shape.  It was amazingly strong.  By using a garden hose  
> on this already rigid structure the ice thickness was built up to  
> about an inch.  It lasted for weeks until the first rainstorm and 43  
> degree (F) temperature.
>
> Lloyd -
>