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

[Keith B]

Hi Chuck

The St Louis arch looks like a catenary but the only pure inverted 
catenaries I know of are kilns.  The insulation brick for those is so 
weak it can't resist anything but compression, and it can't be bonded or 
thermal cycling would wreck it.  Inverted near catenary combinations 
work well where the arch is cheap, mostly mass concrete (usable because 
it's notionally only in compression and needs no lateral support above 
the base), and the loading from roadbed and connecting structure is 
relatively low and transferred down by typically thin pillars or other 
means which transfer only vertical loads. 

"..outward forces at the base of an arch can be controlled by the angle 
of the arch at the base..."   One has to be careful with semantics 
here.  The angle results from the curve chosen.  The angle determines 
the ratio of all the forces at the base (but not their magnitude), so 
the lateral force(s) cannot be independently controlled by the angle.  
To independently control any of the three principal logical forces, one 
must vary the curve, so it's better to memorize something like, "Forces 
at the base of an arch are set by the curve selected."

Lovely and delicate engineering like Lance's dam (or Lloyd's ice dome - 
another gem from a prince of elegant design) is only possible because 
the stresses are so evenly distributed.  Thin sections fail too easily 
in compression by buckling, and often are embarrassingly long in flexure 
and/or short in shear strength.  It's not the gross shape of  an 
inverted boat that's problematic, it's the thinness of section for such 
a shell.  Boats, if you don't run into something or something doesn't 
run into you, in principle need very little structure.  Water gives 
distributed loads.  It's only when you add stress makers like sails, 
masts, rigging and keels that things begin to become complicated, but 
the real killer is addition of internal masses that give pseudo-static 
loads plus dynamic ones from inertia.

I certainly hear you when you want to minimize use of material to 
minimize carbon footprint and also seek long term energy efficiency.   
Thin shell or minimal material approaches aren't ruled out, but you have 
to think much harder before you use them underground because of 
potential lateral and local forces.  Gravel and well designed drainage 
alongside you becomes man rated structure to preclude unexpected force 
excursions.   An eps buffer might be considered to help even stress and 
guard against a sharp rock concentrating a destabilizing point load.  
Internal structure could be designed as bulkheading - or held separate 
from the shell but designed to provide support and safe egress in the 
event of incipient shell failure. 

A logically thicker shell may be vastly more forgiving but not use 
drastically more material.  A barrel vault of thin gauge flat sheet 
steel would be obviously and fatally unstable.  Change to corrugated 
iron and it's suitable for bomb proof military bunkers - at the cost of 
only about 70% increase in steel usage.  Go corrugated and halve the 
gauge - strength and stability are good with less material...  A ribbed 
barrel vault would suit LFC, for instance.  LFC is also well suited to 
arch curve optimization.  Reconsider aspects of Hait's PAHS approach.  
Umbrella and dry gravel have great value for their engineering virtues, 
independent of any use for thermal storage.

kb