Moment distribution

Hardy Cross's moment distribution method solves continuous beams and frames by successive corrections, without ever writing simultaneous equations. Imagine every joint first locked against rotation, so each span carries its fixed-end moments. Then release the joints one at a time. The unbalanced moment at a released joint is shared among the members meeting there according to their relative stiffness, and half of each share is carried over to the far end of the member. Repeat until the corrections are small.

The member stiffness factor is 4EI/L for a far end fixed and 3EI/L for a far end pinned (with no carry-over to the pin). Distribution factors at a joint are each member's stiffness divided by the total, and they add to one. The carry-over factor is one half for a prismatic member with a fixed far end. Sway in frames is handled by a second distribution with an arbitrary sway, then scaling to satisfy storey shear.

The method converges quickly for ordinary beams, gives designers a feel for how moments flow towards stiffer members, and is still useful for checking software output. It also explains why continuous slabs and beams have hogging moments over supports and smaller sagging moments in spans than simply supported ones.