This concerns a way of memorizing a whole litany of cantilever displacement
formulas at one time, generalizing the bar case I described in my previous
message on this topic to a number of other shapes of cross-section.  If all
this is already in the literature my apologies for the duplication.

Given a horizontal cantilever held fixed at the base:

 BASE*=====CANTILEVER========>TIP

we can either tug the tip to the right, thereby slightly increasing the
cantilever's length, or press down on the tip, thus bending the cantilever
into a curve, with the portion nearest the base remaining horizontal,
and with the portion nearest the tip having vanishing curvature.

For a given force F applied to the tip in either of these directions, let
x denote the rightwards displacement under tugging, and z the downwards
movement under pressing.

Now suppose we can vary the thickness of the cantilever.  Define the
*critical thickness* to be that thickness for which the magnitude of the
x and z displacements of the tip are the same for the same magnitude of
force, according to the standard formulas for elongation and cantilever
displacement respectively.

The interest in the critical thickness is (i) it is a simple concept, and
(ii) x displacement (stretching) is simpler than z (bending) by virtue of
not involving cantilever dynamics, making it more natural to obtain the
formula for z from that for x than vice versa.

Mnemonic 1:
A cantilever with rectangular cross-section has critical thickness twice
its length L, i.e. C*L where C = 2 is the critical thickness factor.

Now consider varying the length L and thickness T of the cantilever
independently.

Mnemonic 2:
z/x depends quadratically on increasing L and decreasing T.

These two mnemonics combine to give

 ****************
 z = x*(C*L/T)^2  (1)
 ****************

The definition of Young's modulus of elasticity E as stress/elongation,
namely E = (F/A)/(x/L), gives

 *************
 x = F*L/(E*A)   (2)
 *************

Plugging (2) into (1) and taking C = 2 gives z = 4*F*L^3/(E*A*T^2).  This is
the formula given explicitly by Isa Kiyat for the bar case where A = W*T,
and implicitly by Venkat Indy in terms of z = F*L^3/(3*E*I) where I, the
area or geometric moment of inertia, is A*T^2/12 = W*T^3/12 for a bar.

The point of keeping (1) and (2) separate in this way is that they are more
readily remembered separately than in combination.


What makes these mnemonics really useful is that they apply not only to
bar-shaped cross-sections but to many other shapes such as box, circular
rod, circular tube, elliptical rod, and elliptical tube.  For the latter
four cases however we need one further mnemonic.

Mnemonic 3:
The absence of corners reduces the critical thickness factor C from C = 2
to C = 1.732... = sqrt(3) (a reduction of 13%).

Thus whereas the factor is C = 2 for bars and boxes, it is C = sqrt(3)
for circular rods, circular tubes, elliptical rods, and elliptical tubes.
All other details remain unchanged, in particular continue to use formulas
(1) and (2).

Notes.

(i)  The cross-section area A of the cantilever is needed in x = F*L/(E*A),
which holds for all cross-section shapes however complex.  For hollow
cantilevers, namely boxes and tubes, obtain A by subtracting the inside
area from what would have been the area had it been solid.

(ii) The area of an ellipse is pi*D*d/4 where D and d are its major and
minor diameters respectively.  This generalizes the formula pi*D^2/4 for
area of a circle of diameter D.

(iii) T is the vertical outside thickness or outside diameter.  In the
hollow case with thin walls this is a fine approximation; with relatively
thick walls, T can be reduced a little, but not all the way to the average
of inside and outside diameter due to the quadratic dependency.  When high
accuracy is needed in the thick-walled case, use instead the appropriate area
moment of inertia I in conjunction with the single formula z = F*L^3/(3*E*I),
as Venkat indicated.  The above separation of this formula into (1) and (2)
depends on expressions for I that are valid primarily for solid cantilevers
and relatively thin-walled hollow ones.

(iv) The formulas start to become unreliable for either T or z greater
than around L/3.  Software based on these formulas should check for this
requirement and warn as appropriate.  It also follows that the definition
of "critical thickness" is based on a white lie and should not be taken
literally!

(v) Cantilevers with nonuniform cross-section are more challenging.

Vaughan Pratt