Mathematics and Fly Fishing

Proof of Convergence

Consider a cantilever beam with constant cross section, constant modulus of elasticity and a load P applied at the free end.
In order to start the calculation we have to assume an initial shape of the deflection curve.
Let the initial curve be a straight line.
The equation for the curvature is:

Using the procedure described in The Deflection Curve we can calculate a sequence of curves. Each curve is defined by its curveture K and results in a new value of x(tip). Assigning an index to each curve we have:

K₀ = 0 is the smallest curvature and x₀(tip) is the largest x-value for the tip.
K₁ = 0 is the largest curvature and x₁(tip) is the smallest x-value for the tip.
Now for curve₂ we must have:

The curvature K₃ calculated using x₂(tip ) must be larger than K₂ which was calculated using the smaller smaller x₁(tip). That places the curve₃ above curve₂ making x₃ lager than x₂.
Continuing the line of thought we end up with:

Which leads to the conclusion:

So after a certain number of iterations the curvature will approach a definite value K_true, which will never be reached, but we can get as close as we want to.

Simple Supported Beam