Simply supported beam—second-order ODE

[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.] Here’s the first of the derivations for the center deflection of a simply supported beam with a uniform load. We start with the differential relationship between the bending moment, M, and the deflection, y: M=−EId2ydx2 The second derivative of y is the curvature of the beam (for small deflections, which is one of the fundamental assumptions of beam theory), and the negative sign is there to account for the usual sign conventions for moment and displacement. M is a parabola that passes through 0 at each end of the beam and peaks at wL2/8 at the center. Its formula is M=wL2x−w2x2 Therefore, y″=wEI(12x2−L2x) where I’ve started using primes for differentiation. Integrating once gives y′=wEI(16x3−14Lx2)+C1 Symmetry tells us the slope at the center of the beam is zero, so y′(L2)=wEI(148L3−116L3)+C1=0 Which means C1=wL324EI Plugging this result in and…