At any cut along a loaded beam, the loads and reactions on one side of the cut exert a net moment that the section must resist internally — that is the bending moment at that point. Solving a single-span beam starts from static equilibrium: the sum of vertical forces and the sum of moments about a support both equal zero, which yields the reactions, and integrating the loading from the left end then gives the moment at every station along the member.
A handful of closed-form maxima cover the everyday cases. A simply supported span with a central point load peaks at M = PL/4 at midspan; a full-span uniformly distributed load peaks at wL²/8; a cantilever develops its largest moment at the fixed end — PL for a tip load, wL²/2 for a full UDL. A 10 m simply supported beam under a 100 kN midspan load therefore carries a maximum moment of 100 × 10 / 4 = 250 kN·m.
The bending-moment diagram (BMD) plots this quantity along the beam so the critical section can be read directly — a triangle peaking under a central point load, a parabola under a UDL. The usual sign convention is sagging-positive, with applied couples taken counter-clockwise positive, and superposition lets any mix of point loads, UDLs, and applied moments be summed into one diagram.