Fixed-End Moment — Fixed-Fixed Beam, Uniform Load

Clamp both ends of a uniformly loaded beam so they cannot rotate and the moment diagram redistributes: $wL^2/12$ hogging at each support, $wL^2/24$ sagging at midspan, and inflection points at 0.211L from each end. Compare that with the simply supported $wL^2/8$ all concentrated at midspan. A 12 kN/m load on a 6 m span gives 36 kN·m at the supports against 54 kN·m for the same beam on pins, so fixity cuts the peak moment by a third and cuts the deflection to $wL^4/384EI$, a fifth of the simply supported value.

This is the flagship number in Hardy Cross's moment distribution method, published in 1930 and the tool that made multi-storey frame analysis possible with a slide rule. You start every member at its fixed-end moment, then release joints one at a time and let the unbalance distribute according to stiffness. Computers made the arithmetic obsolete; the fixed-end moment table did not, because it is still how an engineer estimates a continuous beam in their head. A useful mental note: the interior span of a long continuous beam behaves close to fixed-fixed, so $wL^2/12$ is the right first guess for it, while an end span with a pinned outer support sits closer to $wL^2/10$.

The catch is that true fixity is a fiction. Real supports rotate, and the actual moment lands somewhere between $wL^2/12$ and $wL^2/8$ depending on the relative stiffness of the beam and whatever restrains it. Designing the midspan section for $wL^2/24$ on the assumption of perfect fixity is genuinely unsafe: if the support flexes, moment migrates back to midspan. Support settlement does the same thing, and unlike the simply supported case, a fixed-fixed beam is statically indeterminate, so a support that drops even slightly generates real moments the load never asked for.