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ILD: vertical reaction at fixed support

Short answer

For a propped cantilever, R_A = 1 - x^2(3L-x)/(2L^3). Ordinates at 0.25L intervals are 1, 0.9141, 0.6875, 0.3672 and 0, all positive upward, falling smoothly from the fixed support to zero at the prop.

Draw influence diagram for the vertical reaction at the fixed support of a propped cantilever beam. Plot ordinates at 0.25 times span length.

Method

Propped cantilever AB (A fixed, B roller) of span LL. Muller-Breslau: remove the vertical restraint at A (slide it vertically, keeping the rotation at A fixed) and give a unit upward displacement; the deflected shape is the ILD of RAR_A. Equivalent statement: RA=1−RBR_A = 1 - R_B, with RB=x2(3L−x)2L3R_B = \dfrac{x^2(3L-x)}{2L^3} (cantilever deflection at B by a unit load at xx, divided by the deflection L3/3EIL^3/3EI due to a unit load at B). Therefore

RA=1−x2(3L−x)2L3=2L3−3Lx2+x32L3R_A = 1 - \frac{x^2(3L-x)}{2L^3} = \frac{2L^3 - 3Lx^2 + x^3}{2L^3}

Sample: x=0.5Lx = 0.5L: RA=1−0.25×2.52=0.6875R_A = 1 - \dfrac{0.25\times2.5}{2} = 0.6875.

Ordinates (at 0.25L intervals; positive upward)

Load at (x/L)00.250.500.751.00
RAR_A10.91410.68750.36720

ILD shape

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The ordinate is 1 under the support A, falls smoothly (with zero slope at A) to 0 at B. It is always positive: any downward load produces an upward reaction at A.

✓ Answer checked by Claude Opus 5.5 (high effort)

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