From: Henry Baker <hbaker1@pipeline.com> I've always wondered about the shape of the ideal self-supported masonry dome -- e.g., the Roman Pantheon, Hagia Sophia in Istanbul, St. Peter's Rome, etc. "Self-supported" here means compression forces only, as masonry has very little strength in tension or shear.
Here's the solution, which involves the erf & erg functions, instead of (exp(x)+exp(-x))/2=cosh(x) (2D catenary).
z_r = 1/2 r_0 [erg(r/r_0) - sqrt(pi)/2 erf(r/r_0)]
(equation (9) in [Heyman1998] below)
Note the obvious analogy to cosh/sinh functions.
The Roman Pantheon is hemispherical, which is too narrow near the top and too wide near the bottom. All domes since the Pantheon incorporate one or more circular iron/steel chains horizontally around the lower portions of the dome to counteract the spreading forces;
--well, now you've blown it. Your stated goal was to get pure compression. But as you just admitted, there is horizontal tension. But in your defense, I claim this is impossible to avoid, at least for a convex dome that eventually has vertical tangent; you unavoidably get compression along vertical "meridians" but tension along "lines of latitude." Jacques Heyman's paper http://rsnr.royalsocietypublishing.org/content/52/1/39.full.pdf you gave derives the "shape of the ideal dome" without, however, DEFINING what it MEANS to be an "ideal dome." So I have a certain amount of skepticism Heyman knows what he is doing. If he knew, then the paper should contain a definition and a theorem? I would propose that the definition should involve uniform stress, which means the meridian-compression stress is everywhere equal to a constant C, and the latitude-tension stress is everywhere equal to a constant T. Assume constant-weight-density material. The solution will involve both specifying the shape (radius vs height curve) of the dome, and also the thickness (assumed thin everywhere, but it varies) as a function of height. There will be a system of 2 differential equations. I claim it is not possible to do it with uniform-thickness; if you tried you would succeed in uniformizing one (say C), but the other would do whatever it wanted, which unless a miracle occurs will not be to be uniform. To uniformize both quantities you need two functions worth of control, i.e. both thickness and shape. Why is uniform-stress <==> optimal shape? Because if nonuniform stress you can remove a little material from a low-stress place and relocate it to a high-stress place, thus getting a better dome (can survive using a weaker-strength material) involving same amount material. But one must admit this argument makes more sense if "stress" is a scalar-valued quantity like in a thin-beam arch (compression only). For a dome, stress is 2-dimensional quantity. [Also there could be a third kind of stress, shear, but due to symmetry the shear will always be zero for a thin dome. For 3D bulk elasticity as opposed to 2D membrane elasticity stress would be 6-dimensional.] So really, I think the real truth is that there is a magic function F(C,T) of the two (or more) kinds of stress, such that the material self-destructs when F>threshold. and really we want to uniformize this FUNCTION, which IS scalar-valued. So if you knew F, then you could derive an optimum shape for a uniform-thickness dome. However, this function is unknown -- the engineers as far as I know have basically never characterized it despite centuries -- and it presumably depends on the material-type. [To see that, consider a liquid "material" which self-destructs if tension>0 but no problem with compression. Obviously this has a different F than a typical solid.] Therefore, to avoid having to worry about that, I'm suggesting uniformizing both C and T individually rather than F(C,T). Notice that this F-issue is not discussed at all by Heyman, again suggesting at a fundamental level he does not know what he is doing. Frankly, I believe this entire area is dominated by people who do not know what they are doing at a fundamental level. [I am not sure what Heyman had in mind, but perhaps it was: uniform compressional stress, uniform thickness, and variable uncontrolled tensional stress? Or perhaps it was: zero tension, and uncontrolled variable compressional stress? Apparently the latter: Heyman in his EQ9 giving his main result, has the property that y, the nega-height, gets infinitely large while x, the radius, grows very slowly. How can this be true while still having uniform stress?? Think it cannot. So I claim Heyman's "optimal" dome, which is infinitely high, would in practice if you built it too high in a constant g-field, self-destruct due to high compressive stress near the base.] Now let me try to (begin to) formulate the math. r(y) = radius as function of height y. q(y) = thickness ditto. The compressional stress at any given y equals the weight above that y (expressible as integral), divided by 2*pi*r*q. This is to be set equal to C to enforce uniform stress. The tensional stress due to weight above a thin band of dome at any given y is proportional to the weight above that y, times the curvature of the meridian-curve there (assume always convex so positive sign), times r/q. There is also a countervailing term arising from the weight within that thin band itself. It is proportional to r*q*density*r'(s) where s is arclength along meridian. [I said "proportional" not "equal" since there will some factors of density, pi, and stuff I'm too lazy to figure out now.] The difference of these has to be set equal to T. You then have a system of 2 integro-differential equations in 2 unknown functions -- which can be converted to pure-differential form by using fundamental theorem of calculus. You then attempt to solve the system, which will probably be impossible in closed form, but let's see. I think Heyman actually did have a cute idea here, which if I understand aright was to set T=0 thus modeling a "liquid" material aka "oversimplified masonry", but he failed to recognize the need for BOTH thickness-curve and shape-curve control to try to uniformize the compressive stress C, therefore obtaining a manifestly-ridiculous result.