Simply supported beam—slope-deflection equation

[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.] The next technique we’ll use to derive the formula for the center deflection of a simply supported beam with a uniform load is the slope-deflection equation: MA=2EIL(2θA+θB−3ψ)−FEMA Let’s start by explaining where all the terms come from. Here’s a beam of length L with arbitrary end supports (could be simple, fixed, free, or sprung) and an arbitrary applied load. We’ll call the left end A and the right end B. The moment at A is the sum of five terms, which come from the superposition1 of five conditions. First is the fixed-end moment (FEM), which is the moment that would exist at A if the beam had both ends fixed against vertical displacement and rotation: The other four terms come from analysis of the unloaded beam when specific geometric end conditions are applied. The end conditions are specified by the clockwise rotation at each end, θA, and…