--- title: "ILD: fixed-end reaction, propped cantilever 5 m" subject: "Theory of Structures II" chapter: 4 key: "semester_5/theory_of_structures_ii:ild-fixed-end-reaction-propped-5m" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-fixed-end-reaction-propped-5m site: BCT Engineering --- # ILD: fixed-end reaction, propped cantilever 5 m Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams - Asked in: 2068 Chaitra, 8 marks - Video: [EX 06 | INFLUENCE LINE DIAGRAM | PROPPED CANTILEVER BEAM | INDETERMINATE STRUCTURE | ILD](https://www.youtube.com/watch?v=6zSJMZb3m_c) — Structural Funda by:- KheNiSiR - 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-fixed-end-reaction-propped-5m - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4 ## Question Draw ILD at 1 m interval for the support reaction at the fixed end of a propped cantilever beam of span 5 m. Take EI is constant. ## Short answer For a 5 m propped cantilever, RB = x^2(3L-x)/(2L^3) and RA = 1 - RB. Ordinates for RA at 1 m intervals are 1, 0.944, 0.792, 0.568, 0.296, 0. ## 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 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) | x from A (m) | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | $R_A$ | 1 | 0.944 | 0.792 | 0.568 | 0.296 | 0 | ```text *************** ************ ********* ******** --------------------------------------------***** ^ ^ ``` 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) | x from A (m) | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---| | $M_A$ | 0 | -0.72 | -0.96 | -0.84 | -0.48 | 0 | 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. R_A = A. 1 - R_B B. R_B C. 1 + R_B D. x - R_B Answer: A. Vertical equilibrium. (Key point: Redundant) 2. R_A at x = 2 m, L = 5 m A. 0.208 B. 0.792 C. 0.568 D. 0.96 Answer: B. 1 - 0.208. (Key point: Value) 3. M_A at x = 2 m (hogging) A. 0.792 B. 0.84 C. 0.96 D. 0.72 Answer: C. 2 - 0.208 x 5 = 0.96. (Key point: Moment) 4. Maximum of M_A ILD is near A. 0.42L B. 0.5L C. 0.25L D. 0.75L Answer: A. About 2.1 m at 0.42L. (Key point: Max) ## Close variants set in other exams - [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× ## Related in Theory of Structures II - [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: 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: 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: reaction at B, propped cantilever 10 m](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-reaction-b-propped-10m-2m) — Theory of Structures II, chapter 4, asked 2× - [ILD: vertical reaction at fixed support](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-vertical-reaction-fixed-propped-025) — Theory of Structures II, chapter 4 - [ILD: fixed-end moment, 12 m beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-support-moment-fixed-12m-3m) — Theory of Structures II, chapter 4 ## Related in other subjects - [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 - [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 - [Propped cantilever by strain energy method](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Apropped-cantilever-redundant-reaction) — Mechanics of Solids, chapter 5 - [ILD for SF and BM at C of beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aild-sf-bm-c-40-60-5knm) — Theory of Structures I, chapter 5