From: "Cordwell, William R" <wrcordw@sandia.gov> Isn't there a shape that has purely compressional tension?

--"purely compressional tension"??

The St. Louis arch? It does not have a vertical tangent, as you note.

--(thin) arches have only one direction of interest; domes being membranes have two. Hence there is no issue for arches with multi-dimensional stress. For membrane and bulk elasticity, there is such an issue. If you want to control both kinds of stress, you need two functions -- thickness and shape -- to control it with. Heyman's solution (his EQ9) has no vertical tangent, which I presume is related to him getting zero tension. But actually, I guess maybe you could allow a vertical tangent if it occurs exactly at dome base assuming dome base anchored. (If dome base free to slide,then there would be tension in the base-hoop.) Heyman has no base. Heyman's solution goes to infinite height rapidly which is ridiculous. The true solution, I claim, with zero tension imposed, should have a finite maximum allowable height; or if infinite height then this would have to be accomplished by having roughly exponentially-growing thickness (soon violating the thin-membrane assumption) as we go downward.