---
title: "ILD: fixed-end moment of propped cantilever"
subject: "Theory of Structures II"
chapter: 4
key: "semester_5/theory_of_structures_ii:ild-reaction-moment-fixed-propped-10m"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-reaction-moment-fixed-propped-10m
site: BCT Engineering
---

# ILD: fixed-end moment of propped cantilever

Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams

- Asked in: 2076 Chaitra, 6 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-reaction-moment-fixed-propped-10m
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4

## Question

Draw ILD for reaction moment at the fixed support of the propped cantilever beam of span 10 m. Take ordinate interval as 2 m.

## Short answer

The ILD is entirely hogging, zero at ends. Ordinates at 2 m intervals: 0, -1.44, -1.92, -1.68, -0.96, 0 at x=0,2,4,6,8,10 m. Peak -1.92 at 4 m.

## 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, $L = 10$ 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)
| x from A (m) | 0 | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|---|
| $M_A$ | 0 | -1.44 | -1.92 | -1.68 | -0.96 | 0 |

#### ILD shape
```text




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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is the ILD for the reaction moment at a fixed support obtained?
   A. Remove the support and give a unit vertical displacement
   B. Replace the fixed support by a hinge and give a unit rotation
   C. Replace it by a roller and apply a unit load
   D. Cut the beam at mid-span and rotate
   Answer: B. Moment-type quantity: release the corresponding rotational restraint and impose a unit rotation. (Key point: Muller-Breslau)

2. For load at x from A, M_A (hogging) equals
   A. x(L-x)/L
   B. x(L-x)^2/L^2 only
   C. x(L-x)(2L-x)/(2L^2)
   D. x^2(3L-x)/(2L^3)
   Answer: C. Fixed-end moment ab^2/L^2 plus half of a^2b/L^2 carried over gives x(L-x)(2L-x)/(2L^2). (Key point: Formula)

3. What is M_A for a load at x = 4 m on the 10 m span?
   A. 2.40
   B. 1.44
   C. 1.68
   D. 1.92 (hogging)
   Answer: D. 4 x 6 x 16 / (2 x 100) = 1.92. (Key point: Value)

## Close variants set in other exams

- [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×

## 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: 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: 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
- [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: 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

## 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