ILD: fixed-end reaction, propped cantilever 5 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

Propped cantilever AB, $L = 5$ m, A fixed, B roller, constant $EI$. The structure is indeterminate to the first degree; take $R_B$ as the redundant. For a unit load at $x$ from A, the deflection at B of the released cantilever is $\delta_{Bx} = \dfrac{x^2(3L-x)}{6EI}$, and $f_{BB} = \dfrac{L^3}{3EI}$, so

$$R_B = \frac{x^2(3L-x)}{2L^3},\qquad R_A = 1 - R_B,\qquad M_A = x - R_BL\ \text{(hogging)} = \frac{x(L-x)(2L-x)}{2L^2}$$

Sample: $x = 2$: $R_B = \dfrac{4\times13}{250} = 0.208$, so $R_A = 0.792$; $M_A = 2 - 0.208\times5 = 0.96$ (hogging).

ILD for the vertical reaction $R_A$ at the fixed end (1 m interval)

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Muller-Breslau: release the vertical restraint at A (keeping the fixed rotation) and displace A upward by 1; the shape equals $1 - R_B(x)$.

ILD for the reaction moment $M_A$ at the fixed end (1 m interval; hogging shown negative)

The moment ILD is negative (hogging) throughout, with a maximum of about $-0.96$ at $x \approx 0.42L = 2.1$ m, and it is zero at A and B.