---
title: "Four loads: maximum BM at C"
subject: "Theory of Structures I"
chapter: 5
key: "semester_4/theory_of_structures_i:beam-maxbm-c-50-70-80-60"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/beam-maxbm-c-50-70-80-60
site: BCT Engineering
---

# Four loads: maximum BM at C

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

- Asked in: 2065 Chaitra, 8 marks
- Video: [Influence Line Diagrams for Simply Supported Beams - Problem No 6 (with 4 wheel loads)](https://www.youtube.com/watch?v=bbsIf1bZvVg) — 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/beam-maxbm-c-50-70-80-60
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-5

## Question

Find the maximum bending moment at C for the beam and loading as shown in fig-5. [Figure: loads 50 kN, 70 kN, 80 kN, 60 kN with spacings 2 m, 2 m, 3 m moving over a beam AB with A hinge, B roller; AC = 6 m and CB = 9 m (span 15 m).]

## Short answer

Place the 70 kN load at C as the critical load. With the influence line ordinates 2.40, 3.60, 2.80, 1.60 m for the train positions, the maximum bending moment at C is 692.0 kN m.

## Answer

**Data:** simply supported beam $AB$, $L = 15$ m; $C$ at $a = 6$ m from $A$, $b = 9$ m. Loads 50, 70, 80, 60 kN with spacings 2 m, 2 m, 3 m (50 kN at one end). Total $W = 260$ kN.

#### ILD for $M_C$
Peak ordinate $\dfrac{ab}{L} = \dfrac{6\times 9}{15} = 3.6$ m. Ordinate $0.6x$ for $x\le 6$ and $0.4\,(15-x)$ for $x\ge 6$.

#### Critical load
Condition: $\dfrac{W_L + P_k}{a} \ge \dfrac{W}{L} \ge \dfrac{W_L}{a}$, with $\dfrac{W}{L} = \dfrac{260}{15} = 17.33$ kN/m.

For the **70 kN load at $C$** (train moving left to right, 60 kN leading): load on the left without it $=50$ kN, with it $=120$ kN.

$$
\frac{120}{6} = 20 \ge 17.33 \ge \frac{50}{6} = 8.33
$$

The condition holds, so the 70 kN load must be at $C$. Positions from $A$: 50 kN at 4 m, 70 kN at 6 m, 80 kN at 8 m, 60 kN at 11 m.

| Load (kN) | Position (m) | Ordinate (m) | P x y (kN m) |
|---|---|---|---|
| 50 | 4 | 2.40 | 120.0 |
| 70 | 6 | 3.60 | 252.0 |
| 80 | 8 | 2.80 | 224.0 |
| 60 | 11 | 1.60 | 96.0 |

$$
M_{C,max} = 692.0\ \text{kN m}
$$

If the train moves in the opposite direction (50 kN leading), the 80 kN load goes to $C$ (positions 60 kN at 3 m, 80 kN at 6 m, 70 kN at 8 m, 50 kN at 10 m), and the same calculation gives 692.0 kN m, the same value.

**Answer: maximum bending moment at $C$ = 692.0 kN m.**

## Check yourself

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

1. Which numbers show that 70 kN is the critical load at C (W/L = 17.33 kN/m, a = 6 m)?
   A. 120/6 = 20 >= 17.33 >= 50/6 = 8.33
   B. 50/6 >= 17.33
   C. 120/6 <= 17.33
   D. 260/6 = 43.3
   Answer: A. With 70 kN at C the left load is 120 kN (20 per m) and without it 50 kN (8.33 per m); 17.33 lies between them. (Key point: Checking the 70 kN load)

2. What is the ILD ordinate under C for this beam?
   A. 3.6 m
   B. 2.4 m
   C. 6.0 m
   D. 7.5 m
   Answer: A. ab/L = 6 x 9/15 = 3.6 m. (Key point: Peak ILD ordinate)

3. Summing P x y for 50, 70, 80, 60 kN with ordinates 2.4, 3.6, 2.8, 1.6, what is M_C max?
   A. 572 kN m
   B. 692 kN m
   C. 936 kN m
   D. 260 kN m
   Answer: B. 120 + 252 + 224 + 96 = 692 kN m. (Key point: Result)

## Related in Theory of Structures I

- [Four wheel loads on 20 m beam: absolute max BM](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Afour-wheel-30-80-60-30-20m) — Theory of Structures I, chapter 5
- [Four loads on 60 m girder: BM at C, absolute](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atrain-170-215-150-120-60m) — Theory of Structures I, chapter 5, asked 2×
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- [Four loads on 50 m girder: BM at 20 m, absolute](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Afour-loads-100-150-200-150-50m) — Theory of Structures I, chapter 5
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- [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

## Related in other subjects

- [Flexural rigidity; T-beam bending stresses (60 kN loads)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aflexural-rigidity-t-beam-60kn) — Strength of Materials, chapter 7
- [Moment distribution: frame with 80 kN, 60 kN](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-frame-80kn-10knm-60kn) — Theory of Structures II, chapter 3
- [Beam AFD, SFD, BMD with hinge (80 kN)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Abeam-afd-sfd-bmd-20knm-80kn) — Strength of Materials, chapter 1
- [ILD: bending moment at B, two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-bm-b-two-span-8) — Theory of Structures II, chapter 4