H.Baker: There is a principle of "inversion" for structures in gravity: under the transformation y <-> -y, compression becomes tension & vice versa. I think that Hooke remarked on this principle.
So, the catenary (y=cosh(x)) curve assumed by a hanging chain becomes the St. Louis arch or the Gaudi cathedral (Gaudi actually designed his cathedral upside down with weights & strings tied together; these models are on view in the basement of his cathedral in Barcelona).
This principle must have been known to the Greeks & Romans; they need only look at the necklaces on their spouses, or the chains holding their slaves together.
--well... no. Perhaps the inverted catenary is indeed an optimum arch, but definitely NOT for a constant-cross-section thin beam... instead the beam would have to become thicker near its base, with a certain roughly exponential dependence of thickness on height. This should be obvious by considering infinitely large arch, obviously there is a finite upper limit to size at given strength, which the catenary fails to tell you about hence it is wrong. Put another way, obviously the catenary does not have constant tension, it grows along the chain; hence cannot be optimum in the sense of having constant stress. Wikipedia says: "a catenary is the ideal shape for a freestanding arch of constant thickness" which my analysis would claim was a flat-out lie. http://en.wikipedia.org/wiki/Catenary#The_inverted_catenary_arch To try to get constant stress, cross-sectional-area instead would have to vary proportional to total mass of portion of arch lying above that point, times the |y'| / sqrt(1+y'^2) geometry factor. The St Louis arch indeed does have such thickening, but the trouble is that when you put in such thickening, that destroys the validity of the "inversion principle" since you'd then instead want a hanging chain with lighter links near the middle... Wikipedia says the St Louis arch actually is "close to a more general curve called a flattened catenary." So... what is going on here? How can we reconcile conflicting claims? I think answer is this. The cycloid is the optimal freestanding arch of constant thickness IF we are only interested in uniformizing *compressive* (along beam) stress. But it suffers bending-type stresses which are not uniform. The inverted catenary is optimum IF we are only interested in keeping the bending stress zero everywhere but do not care about having highly nonuniform along-arch compressive (or along-chain tensional) stress. I think either goal could be correct. I.e. it could easily be that bending is not a concern, for example because arch is thick enough, in which case only compression is a concern. If you want to design a thin-beam arch which has uniform bending-type stress everywhere AND also uniform compressive-along-beam stress, that'd require a coupled system of 2 differential equations to tell you both the beam thickness (which varies along the arch) and also the curve shape. Oho! Wikipedia http://en.wikipedia.org/wiki/Catenary#Catenary_of_equal_strength seems to realize that (rather deeply buried) claiming that for hanging chain of small variable thickness where strength is proportional locally to mass per unit length of chain, the uniform-tensional-stress chain shape, with zero bending-type-stress, has shape y = C * ln(sec(x/C)) where C=constant. See http://books.google.com/books?id=3N5JAAAAMAAJ&pg=PA329#v=onepage&q&f=false where this is derived and attributed to Davies Gilbert, Philosophical Transactions part iii (1826) page 202. And I compute that this would have CrossSectionalArea = A*sec(x/C)*|tan(x/C)|^(-B) where A,B constants and ArcLength(From 0 to x) = C*ln(sec(x/C) + tan(x/C)). Note |x|<pi*C/2 otherwise we get singularities, i.e. the span cannot be too long. Note that these are quite unusual curves... You may also enjoy the exercise of proving that there is no optimum dome shape for a constant-thickness thin-shell dome (if "optimum" means "constant stress"). I.e, thickness variation with height is necessary for an optimal dome. Hence describing it requires a coupled system of two differential equations one for thickness and one for location... and there are really two kinds of stress, compressive along a meridian, and tensional along a line of latitude, of a dome, and by "uniform" we need to mean some real-valued function of these two kinds of stress numbers, is constant everywhere on the dome surface. Wikipedia http://en.wikipedia.org/wiki/Catenary#Suspension_bridge_curve also claims the best shape for a suspension bridge is a parabolic cable for constant thickness cable in heavy-deck light-cable limit. They cite Routh, Edward John (1891). "Chapter X: On Strings". A Treatise on Analytical Statics. University Press. But this parabola claim is also false, as you again can see by considering the infinite-x limit which clearly cannot exist, the cable would break beyond a certain critical bridge length. Huh? So again it appears that the parabola claim is true if you do not care about nonuniform tensional stress in the cable (or, if inverted, compressional stress along arch is nonuniform); but as I derived in earlier post, if you do want that uniform, then optimal shape for thin-beam arch of constant thickness is circular arc, BUT this will have bending stresses in the arch which will not be uniform! (The fact is, the Greeks and Romans did not build catenary arches, so you seem wrong about them appreciating that.) The Eiffel tower is an example of an exponential pillar, roughly speaking -- that is the optimum shape for a thin vertical beam in a constant gravity field (constant stress) under vertical gravity forces only. However, in reality, tall towers' main problem usually is not collapsing vertically like the world trade center, it instead is wind loads and buckling instabilities. Hence the "optimality" of the exponential pillar is largely irrelevant. Optimal pillar shapes with respect to buckling (i.e. maximizing a buckling eigenvalue) have in fact been found in closed form and the cycloid shape arises in that problem too. I consider solving that to have been quite an impressive feat...