Simply supported beam—energy minimization with Fourier series

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our trip through various methods to derive the equation for the center deflection of a uniformly loaded simply supported beam, today we’re going to do the first of two solutions using the Rayleigh-Ritz method. Of all the possible shapes a beam can deform into, the shape it will deform into is the one that minimizes the potential energy of the system, the system being the beam and the load. The equation for the potential energy for our beam, Π, is Π=∫0L12EI(y″)2dx−∫0Lwydx where the first term comes from the bending of the beam (note its similarity to the formula 12kx2 for a spring) and the second term comes from the load acting through the deflection. The first term is positive because the potential energy of the beam increases as the beam bends; the second term is negative because the potential energy of the uniform load decreases as…