--- title: "ILD: reaction at B, propped cantilever 10 m" subject: "Theory of Structures II" chapter: 4 key: "semester_5/theory_of_structures_ii:ild-reaction-b-propped-10m-2m" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-reaction-b-propped-10m-2m site: BCT Engineering --- # ILD: reaction at B, propped cantilever 10 m Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams - Asked in: 2069 Asar, 5 marks - 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-10m-2m - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4 ## Question Draw influence line diagram for the reaction at B of the propped cantilever beam shown in figure below. Find the ordinates at 2 m interval. [Figure: beam AB, A fixed, B roller, span 10 m.] ## Short answer Use Muller-Breslau: RB = x^2(3L-x)/(2L^3) for a propped cantilever of span 10 m. Ordinates at 2 m intervals are 0, 0.056, 0.208, 0.432, 0.704, 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 $L = 10$ 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: $x = 8$ m: $R_B = \dfrac{8^2(3\times10 - 8)}{2\times10^3} = 0.704$. #### Ordinates (2 m interval) | x from A (m) | 0 | 2 | 4 | 6 | 8 | 10 | |---|---|---|---|---|---|---| | $R_B$ | 0 | 0.056 | 0.208 | 0.432 | 0.704 | 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. The R_B ILD at A has A. Zero ordinate and zero slope B. Ordinate 1 C. Maximum slope D. Negative ordinate Answer: A. Increases from 0 with zero slope at fixed A. (Key point: Slope at A) 2. R_B at x = 8 m, L = 10 m is A. 0.64 B. 0.704 C. 0.432 D. 0.8 Answer: B. 64 x 22 / 2000 = 0.704. (Key point: Value) 3. The ILD for R_B is A. Negative throughout B. Positive then negative C. Positive throughout D. Zero except at B Answer: C. Downward load always increases R_B. (Key point: Sign) ## Close variants set in other exams - [ILD: shear at section 2-2, propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-shear-section-2-2-propped) — Theory of Structures II, chapter 4, asked 3× ## Related in Theory of Structures II - [ILD: reaction at B, propped cantilever 12 m](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-reaction-b-propped-12m-3m) — Theory of Structures II, chapter 4 - [ILD: fixed-end reaction, propped cantilever 5 m](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-fixed-end-reaction-propped-5m) — Theory of Structures II, chapter 4, asked 2× - [ILD: moment at support B, fixed-propped beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-moment-b-propped-2l-l) — Theory of Structures II, chapter 4 - [ILD: fixed-end moment of propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-reaction-moment-fixed-propped-10m) — Theory of Structures II, chapter 4, asked 2× - [ILD: bending moment at fixed support, 10 m beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-bm-fixed-support-10m-1) — Theory of Structures II, chapter 4 - [ILD: shear at A, propped cantilever 7 m](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-shear-a-propped-7m-1m) — Theory of Structures II, chapter 4, asked 2× ## Related in other subjects - [Propped cantilever with UDL, two elements](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Apropped-cantilever-two-elements) — Computational Techniques in Civil Engineering, chapter 4 - [Cantilever ILD for SF and BM at C](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aild-cantilever-sf-bm-c) — Theory of Structures I, chapter 5 - [Cantilever 3 m: bending stress 25 mm below top](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acantilever-3m-udl-20kn-t-section) — Strength of Materials, chapter 7 - [Propped cantilever with UDL by integration](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Apropped-cantilever-udl-prop-reaction) — Strength of Materials, chapter 7