Mathematics and Fly Fishing

The Rod Tip

It follows from the calculations of the moments that the moment at the tip is zero. The moment increases as we move down the rod.
The implication of the small diameter near the tip is that the theoretical diameters required to match the stress level for the middle and the lower parts of the rod are so small that the tip becomes mechanical fragile.
So we have to increase the diameter to make the tip mechanical manageable. But this is not always enough.
The tip may be flimsy and give unwanted waves in the line. So we have to increase further to get rid if we want to get rid of the waves.

The waves are not too decremental to the action and it is more a matter of learning to cast with a very soft tip.

Fighting even a small fish on a rod with a very soft tip is a lot of fun - until it hits into the weeds.

The very soft tip does not offer any advantage as far as protecting the leader tip. It is the overall stiffness of the rod that determines the buffer action. When a hooked fish loads the rod, the angle between the first 10" of the tip and the line is small, and there is not much buffer action to be had from the tip.

Garrisons approach was to calculate the rods for a constant stress, and then make corrections to the tip section, if needed.

We may wonder why Garrison bothered to calculate the stress at the point 1". A better approach would be to define the diameter at the top eye. It has the additional advantage, that the rod mass between the top eye and the 1" point will be included in the calculation.

Calculating for uniform stress at a high load will therefore give very small diameters at the tip. Resulting in a flimsy tip. The section area is adequate with regard to the bending stress; but mechanical will be very fragile.

The stress reduction must be found by experience, but we may get some clues by looking at Garrison's rods. See About the programs for a suggestion.
When using other stress profiles, you are on your own.

Conclusion:
There is a theoretical reason for the reduction of stresses near the tip.
A short rod may not need the stress reduction, unless it is very soft, whereas a long rod always will.

Warning.
On the Garrison stress curves drawn with three tips, the light tip may be too soft and give some waves on the line.

The Modulus of Elasticity