[Ferro List] subterranean - catenary strongest shape & tensile vs compressive forces

[Keith B]

Chuck,

The Catenary Arch/Curve is interesting, but of limited use and subject 
to some misconceptions.  It's what you get if a long member with tensile 
but no bending strength is restrained at its ends and allowed to sag 
under gravity to its naturally stable position.  String or cable can 
closely approximate it, but conceptually, it's best to think of a length 
of chain, since it's clearer that neither compressive nor bending 
strength can be involved.

As tension is increased, the curve flattens.  Decrease in tension 
produces a deeper sag, but it's important to note that the force vector 
at the ends can NEVER become parallel with that of gravity.  There is 
always some inwards component.  It follows that, for the 
reverse/inverted case of the Catenary Arch, the compressive force vector 
at the foot ALWAYS has some outward component.  Whether that can be 
dealt with by an angled block and a simple foundation or needs a tension 
ring or beam depends on the design.  For comparable height and width, 
the difference in that respect between Catenary and circular or 
spherical is minor to negligible.

More important is the possible misconception that the Catenary Arch is 
per se intrinsically self stable.  That's only true in one rarely met 
case.  By definition, that's where the member is freestanding with no 
other forces acting but gravity on the element alone and resisted solely 
by the element's compressive strength.  (Like the chain, it's fine for 
the arch to be made of discontinuous parts, so the element as a whole 
has zero tensile strength but has the compressive strength of the parts.) 

It's fine if you want to live under that, but if you want to also 
support lateral thrust from being partially or fully buried, or if you 
wish to vertically load it with weights not in proportion to the 
projected mass distribution in the arch itself, or, for that matter, to 
support floors, resist wind or seismic forces, then the Catenary is 
probably not the optimal curve and what you use MUST meet the general 
stability requirements for arches.  The last requires contribution from 
section thickness and may rule out a thin FC skin, for that reason.

kb