---
title: "Overhang beam: ILD, max positive/negative moment at D"
subject: "Theory of Structures I"
chapter: 5
key: "semester_4/theory_of_structures_i:overhang-ild-moment-d-three-loads"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/overhang-ild-moment-d-three-loads
site: BCT Engineering
---

# Overhang beam: ILD, max positive/negative moment at D

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

- Asked in: 2076 Baisakh, 10 marks
- Video: [Influence Line Diagrams for Overhanging beams - Problem No 2](https://www.youtube.com/watch?v=9D4uuWhbTJs) — Stan Academy
- 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/overhang-ild-moment-d-three-loads
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-5

## Question

For the overhang beam, draw the Influence Line Diagram for moment at D and then using that ILD to determine the maximum positive and negative moment at the D due to the three concentrated loads as shown in figure below which moves from left to right. [Figure: overhanging beam C-A-D-B-E; CA = 2 m, AD = 3 m, DB = 5 m, BE = 2 m; supports at A and B; loads 20 kN, 25 kN, 30 kN with spacings 2 m and 3 m.]

## Short answer

ILD for M_D: negative on overhangs (−1.25 at C, −0.75 at E), positive between supports (+1.875 at D). Maximum positive moment is 81.875 kN m with 25 kN over D; maximum negative is −37.50 kN m with 30 kN at C.

## Answer

**Data:** overhanging beam $C$–$A$–$D$–$B$–$E$: $CA = 2$ m, $AD = 3$ m, $DB = 5$ m, $BE = 2$ m; supports at $A$ and $B$ (span 8 m); $D$ is 3 m from $A$. Loads 20, 25 and 30 kN, 2 m and 3 m apart as drawn, moving left to right, so the 30 kN load leads. Let $x$ be the position of the unit load measured from $C$.

#### ILD for $M_D$
Reactions: $R_B = \dfrac{x-2}{8}$, $R_A = 1 - R_B$.

- Load between $C$ and $D$ ($x\le 5$): take moments of the right part: $M_D = R_B\times 5 = \dfrac{5(x-2)}{8}$.
- Load between $D$ and $E$ ($x\ge 5$): take moments of the left part: $M_D = R_A\times 3 = \dfrac{3(10-x)}{8}$.

| Position | $C$ ($x=0$) | $A$ (2) | $D$ (5) | $B$ (10) | $E$ (12) |
|---|---|---|---|---|---|
| Ordinate of $M_D$ (m) | −1.25 | 0 | +1.875 | 0 | −0.75 |

```text
 C   A       D        B   E
 -1.25 \  0  /\1.875/ 0  \ -0.75
```

(The ILD is negative on the overhangs and positive between $A$ and $B$.)

#### Maximum positive moment at D
Put the heavy loads where the positive ordinates are largest. Try the train with 25 kN at $D$: then 30 kN is at $x = 8$ m, 25 kN at $x = 5$ m (over $D$), 20 kN at $x = 3$ m.

Ordinates: at $x = 8$: $\dfrac{3(10-8)}{8} = 0.75$; at $x=5$: $1.875$; at $x=3$: $1.875\times\dfrac13 = 0.625$.

$$
M_{D,max}^{+} = 30(0.75) + 25(1.875) + 20(0.625) = 22.5 + 46.875 + 12.5 = 81.875\ \text{kN m}
$$

(Other positions, checked by shifting the train, give smaller values.)

#### Maximum negative moment at D
The largest negative ordinate is $-1.25$ at $C$, so place the leading 30 kN load at $C$ ($x = 0$) as the train enters the beam; the other loads are still off the beam:

$$
M_{D,max}^{-} = 30\times(-1.25) = -37.50\ \text{kN m}
$$

**Answer: maximum positive moment at $D$ = 81.875 kN m (25 kN load over $D$); maximum negative moment at $D$ = -37.50 kN m (30 kN load at the end $C$).**

## Check yourself

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

1. What is the peak ILD ordinate for M_D?
   A. +1.875 m
   B. +3 m
   C. +1.25 m
   D. +0.75 m
   Answer: A. Section at D has peak ab/L = 3 x 5/8 = 1.875. (Key point: Ordinate at D)

2. What is the sign of the ILD on the overhangs?
   A. Positive
   B. Negative
   C. Zero
   D. Positive on C, negative on E
   Answer: B. The ordinate is -1.25 at C and -0.75 at E. (Key point: Overhang ordinates)

3. What is the maximum positive moment at D?
   A. 37.5 kN m
   B. 46.875 kN m
   C. 81.875 kN m
   D. 60 kN m
   Answer: C. 30(0.75) + 25(1.875) + 20(0.625) = 81.875 kN m. (Key point: Max positive)

4. What is the maximum negative moment at D?
   A. -25 kN m
   B. -20 kN m
   C. -81.875 kN m
   D. -37.5 kN m, with the 30 kN load at C
   Answer: D. 30 x (-1.25) = -37.5 kN m. (Key point: Max negative)

## Related in Theory of Structures I

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