Frames by the Displacement Method

In frames, the displacement method uses rigid-joint rotations and, for sway frames, independent translations of joints or storeys as unknowns. A kinematic analysis is therefore required before the equilibrium equations are assembled.

Non-sway frames

If geometry, supports, or bracing prevent joint translation, the unknowns are primarily joint rotations. Each rigid joint receives a moment-equilibrium equation after the end moments of all connected members have been expressed in terms of those rotations.

Sway frames

In a sway frame, one or more translational coordinates are added. A unit horizontal translation bends the columns and creates reactions in the temporary translational restraint. The canonical equation for that coordinate expresses equilibrium of the corresponding generalized horizontal force.

Member end actions

Member end moments depend on external loading, rotations at both ends, and relative transverse translation of the member ends. In classical hand analysis these relations are represented by tabulated primary-system reactions or slope-deflection-type equations.

Equation assembly

For each rotational coordinate, write moment equilibrium at the associated joint. For each translational coordinate, write force or generalized-reaction equilibrium in the displacement direction. Together these conditions form the system for $Z_i$.

Recovering results

After solving for joint displacements, calculate member end moments, then shears and axial forces, and finally support reactions. The completed solution must satisfy equilibrium of every joint and of the frame as a whole.

Connection to the matrix method

The classical displacement method already contains the central stiffness-method idea: joint displacements are unknowns and joint forces are linear functions of them. Matrix structural analysis systematizes this procedure for large numbers of elements and degrees of freedom.