I've been reading through my mechanics of materials textbook recently, notably in regard to the section on the deflection of beams. The well regarded Euler-Bernoulli beam theory relates the radius of curvature for the beam to the internal bending moment and flexural rigidity. However the theory approximates curvature by defining the derivative of the deflection function as much less than one causing the curvature function to approximate to the second derivative of the deflection. This essentially means the derived function is only valid for small angular deflections of the beam which is great for real world application because large angular deflections are not desirable in structures obviously. My book mentions that there are a small number of problems that the full relation of curvature can be used to solve because it produces a second order non linear degree one ODE. Out of curiosity, does anyone know of an example showing this put into use? I have been playing around with the ODE for a few hours but I am unsure of how to deal with the squared derivative term in the curvature equation as I have never encountered differential equations in my studies where the derivative terms are ever raised to a power. Thanks in advance!