Mathematics and Fly Fishing

Elementary Solution

The simplified differential equation for the deflection curve - neglecting the square of first derivative is:

A quote from: Large Deflections... By K. E. Bisshopp and D. C. Drycker

"Specifically the elementary theory neglects the square of the first derivative in the curvature formula and provides no correction for the shortening of the moment arm as the loaded end of the beam deflects. For large finite loads, it gives deflections greater than the length of the beam!"

Neglecting the square of the first derivative is not that disastrous.
But it is really ridiculous to accept that the tip of the beam remains at x=L.
The length of the beam do not change, and the length of the beam is measured along the curved beam and not along the x-axis.
So the elementary solution is valid only for x<x(tip).

There is no easy way of calculating the length of the beam along the curve - but there is shortcut to solve that problem.

An extensive study of the path of a the tips of tapered cantilever beams have been done by Parkingson et al.

The result can for a beam with constant cross section be expressed as

which with the text book solution for the curve

enables us to calculate the x(tip).

The diagram below shows curves for a beam deflected to 30% and 50% of its length. The green curve at the right is the path of the tip.
The red curves are calculated using the iteration procedure and the blue curves are calculated from the elementary equation.

For the 50% deflection the the difference between the two curves range from 20% at the base to 3% at the tip.

Conclusion: The elementary equation gives a fair approximation even for large deflections - when calculated from 0 to x(tip).

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