In my message yesterday, Mnemonic 3 (for the case of no corners) incorrectly
gave sqrt(3) as the critical thickness constant in the absence of corners
(circles and ellipses).  The problem arose from my inadvertently dividing by
2/sqrt(3) when I should have multiplied by that amount.  (I was thinking
resistance instead of displacement.)  It should read:

Mnemonic 3:
The absence of corners increases the critical thickness factor C from C = 2
to C = 2.309... = 4/sqrt(3) (an increase of 15%).

Sorry about that.

One other point is that I punted on the thick-wall case of hollow cantilevers
by simply passing the buck back to the standard formula z = F*L^3/(3*E*I) in
its unseparated form.  While this may well be the most practical solution,
for the sake of completing the separated mnemonic system to cater for
thick walls here is one more mnemonic.

Mnemonic 4:
For thick-walled hollow cantilevers, move the thickness term from
equation (1) to (2) so as to unite it with A.  This gives

 *************
 z = x*(C*L)^2         (1')
 *************

 *****************
 x = F*L/(E*(A*T^2))   (2')
 *****************

The unified term A*T^2 denotes the area moment of inertia I to within a
constant factor that (1) and (2) continue to take care of automatically.
(A*T^2 overstates I by 3*C^2 but one doesn't need to keep track of that
when working just with cantilevers.)

Where previously one would have replaced A by A1 - A0 where A1 is the total
enclosed area and A0 the area of the inside hollow, instead replace A*T^2
by A1*T1^2 - A0*T0^2 where T1 is the outside diameter or thickness and T0
the inside diameter or thickness.  (This is the same as one does with the
standard formula, while retaining the mnemonic system's benefit of automatic
constant management.)  Mnemonic 4 thus corrects (2') to

 x = F*L/(E*(A1*T1^2 - A0*T0^2))   (2")

This remains correct when T0 approaches either zero (the solid case) or T1
(the thin-walled case).  However in both those limits the difference between
the exact term A1*T1^2 - A0*T0^2 and its approximation by (A1-A0)*T1^2
tends to zero, making Mnemonic 4 unnecessary for either of those limits.

The range of applicability of this approximation is independent of whether
one uses the standard formulas or this mnemonic approach.  The principle
benefit of the mnemonic approach is the absorption of all constants into
the uniform notion of critical thickness factor modulo corners, namely 2
with corners and 4/sqrt(3) without.

Vaughan Pratt