Continuous Beams by the Displacement Method

For a continuous beam, the displacement method is especially convenient because, in the absence of support settlement, the primary unknowns are often only rotations of the rigid intermediate joints. Ordinary support translations are prescribed or restrained.

Primary system

Artificial rotational restraints are introduced at the rigid intermediate joints. Each span of the primary system is then treated as a beam element with restrained or prescribed end displacements.

Load state

External loading acting while the joint rotations are restrained produces member end moments. These moments generate the load reactions of the artificial rotational restraints.

Unit rotations

Each independent joint is given a unit rotation in turn. Adjacent spans resist that rotation according to their flexural stiffness $EI$ and length. Reactions produced by the unit states form the coefficients of the canonical equations.

Joint equilibrium

After the actual rotations have been found, the sum of member end moments meeting at a joint, together with any applied joint moment, must satisfy moment equilibrium. This is the physical basis of the corresponding canonical equation.

Final moment diagram

Member end moments and $M(x)$ in each span are recovered by superposing the load state and unit-displacement states. Span equilibrium then gives shears and support reactions.

Support settlement

Prescribed vertical support movements produce additional member end moments and enter the equations as known kinematic effects.