Well, we seem to have stirred things up a bit here! I'll try and respond to the various points in the order that they have been made.
Ron - you are again quite right in suggesting that as a tubular rod recovers from bending, its power is effectively increased. What it certainly does not do is enhance the innate power of the rod. It's the same principle as a variable potentiometer in an electrical circuit, which cuts off some of the current as you turn it - a volume control on a radio for example. The current is still fixed. A rod can only generate the power it is capable of, no more. Bending a tubular rod reduces the power; unbending gives it back.
The bow analogy is irrelevant in this context; the way a longbow reacts under deflection will be subject to exactly the same principles as a fishing rod and is dependant on the material used in the same way. You cannot compare how the bowstring reacts to how a rod does; the rod and the bow are the same, not the string.
Mark - you say that there is no detectable distortion in a tubular fishing rod. In the tip sections no; they will fracture before any measurable change in the shape of the tube. The lower sections do show some distortion - only slightly, but they do all the same. The reason behind this is the relationship between the diameter of the tube and the wall thickness.
This distortion is actually very tiny; Sean suggests that it cannot be more than 1 or 2 percent. In fact it is more like 0.03 of a percent, or about 1 ten-thousandths of an inch in a typical fly rod butt. Insignificant in practical terms certainly; a tiny change in taper is sufficient to nullify any perceived weakness, which is why modern tubular rods are entirely satisfactory.
The fact remains however, that solid rods do not distort; half of the rod's cross section (that which falls above a line drawn through the centre of the rod in such a way as to leave the same area of material on either side) will be under tension, wheras the half below the line will be under compression; both halves want to move towards this line of neutrality but, as the tension and compression are equal, they are not able to, so no distortion occurs.
Sean - you ask how it is that a bamboo rod can resist bending better than a tubular rod? You are right to point out that such a test requires comparable rods, but it does not mean identical tapers, blank diameter or wall thicknesses; for one thing you don't have a wall thickness in a solid split cane rod. What you can do however is produce a cane blank and a tubular blank with the same curve under load i.e. the same test curve in pounds and the same rate of curvature. The blanks themselves will appear quite different but this doesn't matter - we are not comparing blank diameter or taper here. Having done this, put them under load. There will be no appreciable difference until, with carbon fibre anyway, the tubular rod breaks; this will happen before the cane one does.
Beyond a certain point the tubular rod begins to distort but, due to the bittleness of the material, fracture occurs before the distortion becomes so much that the rod's resistance to deflection becomes noticeable. To stop the tube fracturing it would be necessary to make it out of a less brittle material such as mild steel, which would make the decrease in resistance due to ovality of the cross section quite noticeable - but this would result in a heavy floppy rod quite useless for fishing. The principle however is the same.
The solid rod will keep going on resisting until the fibres on the outside of the curve seperate by being pulled apart; what governs this is tensile strength of the longitudinal fibres and good quality bamboo has plenty of it.
To be continued...