Elementary Indeterminate Beams

One redundant support is solved by removing it, finding the deflection there under the loads, and finding the reaction that pushes the beam back to zero deflection. That is superposition plus one compatibility equation.

Worked: a propped cantilever

Without the roller, the tip deflects wL⁴/(8EI) down. The roller force R deflects it RL³/(3EI) up. Setting them equal: R = 3wL/8 = 3 × 1.5 × 20/8 = 11.25 kips. The wall takes the rest, 30 − 11.25 = 18.75 kips, and a moment of 11.25 × 20 − 1.5 × 20²/2 = −75 ft-kips (wL²/8, hogging). The largest sagging moment is 9 × 1.5 × 400/128 = 42.2 ft-kips, 7.5 ft from the roller.