My day job is in machine design for a contract R&D company so I have access to some very clever people. I have thought a lot about the problem of bending necks and, as I am making a mandolin for my own use and don't want to have to do it twice, I collared a clever analytical engineer to talk over the mechanics with.
In the course of the discussions I have got some insights into the workings of truss rods and the possibilities of neck stiffening that I feel should be shared.
I also now understand why it is much easier to straighten a neck by clamping it straight and tightening the truss rod than by just tightening the truss rod. This will be tricky to explain without a whiteboard and much hand waving but here goes!

Truss rods and why they work

So, imagine a neck made of wood - no stiffening or truss rod. When a bending force is applied to the neck, the wood will see the force in 2 ways. On the inside of the curve (close to the strings) the wood will be in compression and on the outside of the curve the wood will be in tension.

At a point somewhere inside the neck there will be a line where the wood is in neither compression or tension. This is called the "Neutral Axis" (NA) (for the serious this axis runs through the centre of the 2nd moment of area Ixx).
Simply put, the neck will remain straight providing the forces on each side of the Neutral Axis are in balance.
You can ignore the curve in the truss rod as, contrary to popular belief, it plays little or no part in controlling the straightness of the neck.
The important thing is the distance that the line connecting the centre of the ends of the truss rod is below(away from the strings)the Neutral Axis(NA).

On a mandolin there is likely to be a ratio of about 5:1 between the distance from the NA to the strings and the distance from the NA to the centre of the truss rod so, taking the worst case string tension from Donald Lashomb's useful data of 190lb (say 200lb to account for the angle of the string from the nut), we get a requirement of about 1000lb or 1/2 ton tension in the truss rod to completely balance the bending force of the strings.
This is well within the capacity of a 3/16" truss rod.

Why is it so hard to straighten a bent neck with a truss rod?

The primary task of the truss rod is to stop a neck from bending NOT to straighten an already bent neck.
I have had to replace a number of truss rods for players who sheared off the threaded end of the rod trying to straighten a bent neck without reducing or removing the tension of the strings. This has been mostly on guitars but also the occasional banjo.
When a neck starts to bend under the tension of the strings, the line of action of the truss rod moved towards the NA as the strings move away thus reducing the effectiveness of the truss rod and increasing bending force generated by the strings. In extreme situations (never reached on a mandolin due to the shortness of the neck but often the case with thin necked guitars) the truss rod will start to act on the wrong side of the NA thus increasing the bending of the neck. If an attempt is made to straighten a neck in this state without restoring the geometry by loosening the strings and clamping the neck straight, the truss rod will not be able to deliver enough tension to straighten the neck before the rod is broken and could, in the case of a very bent neck, even add to the bending of the neck.

Replacing a broken truss rod is an expensive time consuming and inherently traumatic experience.

How to straighten a bent neck

To straighten a bent neck first slacken or remove the strings - on a mandolin removal would probably be best. If the neck doesn't return to a straight condition once the strings have been removed you will need to force it to a straight condition by the action of a clamp. once the neck is straight the truss rod should be firmly tightened sufficiently to maintain the neck straight once the strings have been replaced.
Don't forget to lubricate the truss rod threads before tightening the truss rod nut.