---
title: "Cantilever ILD for SF and BM at C"
subject: "Theory of Structures I"
chapter: 5
key: "semester_4/theory_of_structures_i:ild-cantilever-sf-bm-c"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/ild-cantilever-sf-bm-c
site: BCT Engineering
---

# Cantilever ILD for SF and BM at C

Theory of Structures I · Chapter 5: Influence Lines for Simple Structures

- Asked in: 2080 Chaitra, 6 marks
- Video: [I L D  for Cantilever Beam - Rolling Loads and Influence Line Diagrams  - Structural analysis 1](https://www.youtube.com/watch?v=ghBiK6sDfTM) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/ild-cantilever-sf-bm-c
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-5

## Question

Draw ILD for SF and BM at C and calculate S.F and B.M at section C of the given cantilever beam using these influence lines. [Figure: cantilever AB fixed at A; 50 kN load at C, 2 m from A; D is 2 m beyond C; 10 kN/m UDL over DB of 4 m.]

## Short answer

Shear force at C is 40 kN just right of the section and 90 kN just left. Bending moment is -160 kN m (hogging).

## Answer

**Data:** cantilever $AB$ fixed at $A$ (free end $B$). $C$ is 2 m from $A$ with a 50 kN load; $D$ is 2 m beyond $C$; 10 kN/m UDL on $DB$ (4 m). So $B$ is 8 m from $A$. Let $s$ be the distance of the unit load from $A$ and the section at $C$ ($s_C = 2$ m).

#### ILD for shear at C
A load to the left of $C$ is carried by the fixed support only, so $V_C = 0$. A load to the right of $C$ is transmitted to the section, so $V_C = +1$.

#### ILD for bending moment at C
For $s<2$: $M_C = 0$. For $s>2$: $M_C = -(s-2)$, a straight line from 0 at $C$ to $-6$ at $B$ ($B$ is 6 m from $C$).

```text
 A (fixed)  C     D           B
 |==========|=====|===========|
 ILD SF:  0  | 1   1   1   1   1
 ILD BM:  0  | 0  -2          -6
```

#### Values
The 50 kN load is at the section. For shear take the section just right of $C$ (the load then is on the left of the section); the value just left of $C$ is given as well.

**Shear force at C:**

- Just right of $C$: $V_C = 10\times 4\times 1 = 40$ kN (UDL on $DB$, ordinate 1).
- Just left of $C$ (50 kN load taken to the right of the section): $V_C = 50\times 1 + 40 = 90$ kN.

**Bending moment at C:** ordinate at $D$ is $-2$, at $B$ it is $-6$, so the average ordinate under the UDL is $-4$.

$$
M_C = 50\times 0 + 10\times 4\times(-4) = -160\ \text{kN m}
$$

(Check by statics: UDL resultant $40$ kN acts 4 m from $C$, so $M_C = -160$ kN m, hogging.)

**Answer: $V_C = 40$ kN (90 kN just left of the 50 kN load); $M_C = -160$ kN m (hogging).**

## Check yourself

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

1. For a cantilever fixed at A, what is the shear ILD at C?
   A. 0 for load left of C and +1 for load right of C
   B. +1 everywhere
   C. 0 for load right of C
   D. Linear from 0 to 1
   Answer: A. Loads to the left go directly to the support, those to the right pass through the section. (Key point: Shear ILD)

2. What is the BM ILD ordinate at the free end B, 6 m from C?
   A. +6
   B. -6
   C. -2
   D. 0
   Answer: B. M_C = -(s - 2), so at B (s = 8) it is -6. (Key point: BM ILD)

3. What is M_C for the 10 kN/m UDL over DB (4 m)?
   A. -80 kN m
   B. -240 kN m
   C. -160 kN m
   D. +160 kN m
   Answer: C. Average ordinate -4 gives 10 x 4 x (-4) = -160 kN m, hogging. (Key point: UDL moment)

4. What is V_C just right of C?
   A. 90 kN
   B. 50 kN
   C. 0
   D. 40 kN
   Answer: D. Only the 40 kN UDL lies to the right, with ordinate 1. (Key point: Shear value)

## Related in Theory of Structures I

- [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
- [ILD: RA, RB, SF and BM at C](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aild-ra-rb-sf-bm-c-15knm-1-5kn) — Theory of Structures I, chapter 5
- [ILD at mid-span: BM and SF under given loads](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aild-mid-span-bm-sf-50-10-30) — Theory of Structures I, chapter 5
- [ILDs for BM and SF at 10 m, 25 m beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aild-beam-bm-sf-10m-25m) — Theory of Structures I, chapter 5, asked 2×
- [Suspension 150 m: UDL right half, SF and BM at 50 m](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Asuspension-150m-two-cables-udl-right-half) — Theory of Structures I, chapter 7
- [Suspension 100 m: UDL on right half, SF and BM at 20 m](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Asuspension-100m-5knm2-10knm2-right-half) — Theory of Structures I, chapter 7, asked 2×

## Related in other subjects

- [ILD: shear at section of propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-shear-propped-10m-sec4) — 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
- [Slope deflection: beam, 360 kN load, fixed end](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aslope-deflection-beam-10kn-20knm-360kn) — Theory of Structures II, chapter 3
- [Castigliano: deflection at 50 kN load, cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-deflection-50kn-cantilever-100kn) — Theory of Structures II, chapter 2