This problem is being discussed on the AAMT email discussion list. I have a meter long metal ruler. I push the ends together so that they're only 99cm apart, which means the ruler will bow a bit. How tall is that arc?
The problem we haven't been able to solve is 'what is the shape of the arc'?
Elastic materials assume shapes that are the graphs of polynomials of degree at most 3. Knowing the length of the ruler, the distance between the endpoints, and making an assumtion about symmetry with respect to the middle may give you enough data to get the equation.
Alternatively, if all you want is a bound, you could probably assume some extreme shapes, like the equal sides of an isosceles triangle, and get an answer that way.