--- title: "ILD: reaction at B, propped cantilever 12 m" subject: "Theory of Structures II" chapter: 4 key: "semester_5/theory_of_structures_ii:ild-reaction-b-propped-12m-3m" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-reaction-b-propped-12m-3m site: BCT Engineering --- # ILD: reaction at B, propped cantilever 12 m Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams - Asked in: 2070 Asar - Video: [Influence Line Diagrams for Propped Cantilever Beams - Problem No 1 (Reaction at A & B and Moment A)](https://www.youtube.com/watch?v=s920Z5PGLkw) — Stan Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-reaction-b-propped-12m-3m - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4 ## Question Draw influence line diagram for reaction at support B of the propped cantilever beam shown in figure below. Determine ordinates at 3 m interval. [Figure: beam AB, A fixed, B roller, span 12 m. Printed as the alternative (OR) to the focal point question above.] ## Short answer Use Muller-Breslau: RB = x^2(3L-x)/(2L^3) for a propped cantilever of span 12 m. Ordinates at 3 m intervals are 0, 0.0859, 0.3125, 0.6328, 1, positive throughout and zero at A. ## Answer 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) | x from A (m) | 0 | 3 | 6 | 9 | 12 | |---|---|---|---|---|---| | $R_B$ | 0 | 0.0859 | 0.3125 | 0.6328 | 1 | #### ILD shape ```text ***** ******** ********* ************ ***************---------------------------------- ^ ^ ``` 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) $= \dfrac{3wL}{8}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. ILD for R_B is A. Deflected shape of the cantilever AB scaled to 1 at B B. Linear from 0 at A to 1 at B C. Parabola with maximum mid-span D. Same as ILD for R_A Answer: A. Muller-Breslau with unit upward displacement at B. (Key point: Method) 2. R_B = A. x(L-x)/L B. x^2(3L-x)/(2L^3) C. x/L D. x^2(L-x)/L^3 Answer: B. Maxwell: delta_Bx / f_BB. (Key point: Formula) 3. Full-span UDL gives R_B = A. wL/2 B. 5wL/8 C. 3wL/8 D. wL/3 Answer: C. w x area of ILD = 3wL/8. 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