ILD: reaction at B, propped cantilever 12 m

Sign convention: sagging moment (+), hogging moment (-); shear positive when the left part tends to go up; reaction positive upward. Ordinates are for a unit load moving along the beam.

Method (Muller-Breslau)

Propped cantilever AB of span $L = 12$ m: A fixed, B roller, constant $EI$. Remove the support at B and give B a unit upward displacement. The ILD for $R_B$ is the deflected shape of the cantilever AB (fixed at A) scaled so that the ordinate at B equals 1.

By Maxwell's reciprocal theorem, the deflection at B due to a unit load at $x$ from A equals the deflection at $x$ due to a unit load at B. So

$$R_B(x) = \frac{\delta_{Bx}}{f_{BB}} = \frac{x^2(3L-x)/6EI}{L^3/3EI} = \frac{x^2(3L-x)}{2L^3}$$

Sample: $x = 9$ m: $R_B = \dfrac{9^2(3\times12 - 9)}{2\times12^3} = 0.6328$.

Ordinates (3 m interval)

ILD shape

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The ordinate is 0 at A (a load at the fixed support is carried by A), increases with zero slope at A and reaches 1 at B. It is positive throughout, so a downward load always increases $R_B$. A UDL over the whole span gives $R_B = w\times$ (area of the ILD) $= \dfrac{3wL}{8}$.