Beam Deflection by Superposition

Example: The cantilever beam ABC is of length L and flexural rigidity EI. It is subjected

to a concentrated moment, M, at midspan (B) and a concentrated load, P, at its end, C.

Find the deflection and slope of the beam at midspan, B, and at the end, C.

Strategy: Replace the beam under two loads with two beams each with one load as shown.

Note in the tables the concentrated moment acts at the end of the beam (top figure below.)

From table: y_C = − ML² / 2EI and slope θ_C = − ML / EI ** if M acted at C

Also y₁(x) = − (M/2EI) x² for 0 ≤ x ≤ L/2

From table: y_C = − PL^{3 /} 3EI and slope , θ_C = − PL²/2EI Also y₂(x) = (Px²/6EI)(3L – x)

** Since the concentrated moment, M, actually acts midspan, direct superposition will not

work. Adjustment must be made to account for its acting at midspan prior to superposition.

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