ILD: fixed-end moment of propped cantilever

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, $L = 10$ m, A fixed, B roller. The reaction moment at A is the moment at the fixed support (hogging, so negative in the above convention). Muller-Breslau: replace the fixed support by a hinge and give it a unit rotation; the deflected shape of the beam gives the ILD.

Analytically, with a unit load at $x$ from A, $a = x$, $b = L - x$: fixed-end moment at A is $ab^2/L^2$ and the fixed-end moment at B is $a^2b/L^2$. Releasing B (roller) carries half of the latter to A:

$$M_A = \frac{ab^2}{L^2} + \frac{a^2 b}{2L^2} = \frac{x(L-x)(2L-x)}{2L^2}\quad(\text{hogging})$$

Sample: $x = 4$: $M_A = \dfrac{4\times6\times16}{2\times100} = 1.92$ (hogging).

Ordinates (2 m interval; values shown as moments in the sagging-positive convention)

ILD shape

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The ILD is entirely hogging, zero at A and B, with a maximum $-1.92$ at about $x = 4.2$ m ($0.42L$).