Slope-deflection method

The slope-deflection method was one of the first systematic ways to analyse continuous beams and rigid frames by hand. Instead of choosing redundant forces, it takes the unknown joint rotations (and sway displacements) as the main unknowns. Each member's end moments are written in terms of those rotations plus the fixed-end moments that the loads would cause if both ends were clamped. Joint equilibrium then gives as many equations as unknowns.

For a member AB of length L and stiffness E I, the end moment M_AB = M_F,AB + (2EI/L)(2θA + θB − 3ψ), where ψ is the chord rotation caused by relative sideways movement of the ends. A clockwise moment on the member end is usually taken as positive. At every rigid joint the sum of member end moments must be zero (or equal any external moment applied there). For a frame that sways, an extra shear equation for each storey closes the set.

Its strengths are clarity and a direct link to the stiffness method: the coefficients 4EI/L and 2EI/L are exactly the terms of a beam element stiffness matrix. It becomes tedious beyond a few joints, which is why moment distribution and then computers took over, but it remains the cleanest way to understand how stiffness, carry-over and sway interact.