As illustrated in the diagram below, the beam curvature, κ, is approximately equal to the second derivative (curvature) of the neutral axis line (dotted line in diagram)
It follows that
Since the moment at the section concerned can also be written, for a cantilever beam, as
it follows that
This second order differential equation can be integrated (twice), with appropriate boundary conditions, to find the deflection of the beam at different points along its length. For a cantilever beam, this operation is shown below.
which can be rearranged to give
For example, at the loaded end ( x = L ), this gives
The corresponding operation for symmetrical 3-point bending can be seen by clicking here .
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