Stresses in a clamped circular plate
All of the solutions in Roark's are closed form differentials with straightforward solutions, although some are tougher than others. Given some time for research and headscratching, I could come up with something for you, but I'm not exactly inclined. I have, however, created a finite element model that very closely followed the results of the annular plate formulas and I assumed "traverse" to mean the outer diameter. I've used these formulas for everything from washer-type pipe expansion joints to concrete caps on Mississippi River sheet pile cells with manholes in them (that's the one I did the FEA on). Besides, the connotation of the word "traverse" indicates all the way across, not some minimum distance.
If you're curious, try it out yourself. I, for one, firmly believe it to be so from practice...