---
title: "Principle of superposition with example"
subject: "Strength of Materials"
chapter: 1
key: "semester_3/strength_of_materials:principle-of-superposition"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/principle-of-superposition
site: BCT Engineering
---

# Principle of superposition with example

Strength of Materials · Chapter 1: Axial Forces, Shearing Forces and Bending Moments

- Asked in: 2081 Baisakh, 4 marks; 2080 Baisakh, 4 marks (asked 2 times)
- Video: [Strength of Materials - Principle of Superposition](https://www.youtube.com/watch?v=PnEqtO4iYh4) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/principle-of-superposition
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-1

## Question

State and explain the principle of superposition with a suitable example.

## Short answer

For a linear elastic structure, the total effect of multiple loads equals the sum of effects of each load acting alone. Example: a simply supported beam with UDL and point load; mid-span moment from UDL = 9 kNm, from point load = 9 kNm, combined = 18 kNm.

## Answer

**Principle of superposition:** for a linear elastic structure (small deformations, Hooke's law valid), the total effect (reaction, shear force, bending moment, stress or deflection) produced by several loads acting together is equal to the algebraic sum of the effects produced by each load acting separately.

#### Explanation
- Each load is applied alone on the original, undeformed structure and its effect is found.
- The individual effects are then added, taking care of signs.
- It works because the equations of equilibrium and the stress–strain relation are linear: doubling a load doubles every effect, and the effect of one load does not change the effect of another.

#### Example
A simply supported beam of span $L = 6$ m carries a UDL $w = 2$ kN/m over the full span and a point load $P = 6$ kN at mid-span.

```text
        P = 6 kN
 w = 2 kN/m  |
 ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓
 A▲------- 3 m -------●------- 3 m -------▲B
```

Case 1 – UDL alone: $M_{mid,1} = \dfrac{wL^2}{8} = \dfrac{2\times6^2}{8} = 9$ kN·m.

Case 2 – point load alone: $M_{mid,2} = \dfrac{PL}{4} = \dfrac{6\times6}{4} = 9$ kN·m.

Superposing: $M_{mid} = 9 + 9 = 18$ kN·m. Similarly the reactions are $R_A = R_B = 6 + 3 = 9$ kN, which equals the sum of the reactions of the two cases ($6$ kN from the UDL and $3$ kN from the point load).

Hence a complicated loading is split into simple standard cases whose results are tabulated, and the answers are added.

## Check yourself

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

1. According to the principle of superposition, the total effect of several loads on a structure is:
   A. The algebraic sum of the effects of each load acting separately
   B. The effect of the largest load only
   C. The product of the individual effects
   D. The average of the individual effects
   Answer: A. The total reaction, moment, stress or deflection equals the algebraic sum of the effects of each load applied alone. (Key point: Statement of principle)

2. Which condition must hold for superposition to be valid?
   A. Large plastic deformation is allowed
   B. The structure is linear elastic with small deformations
   C. Only point loads may be used
   D. The structure must be statically indeterminate
   Answer: B. Superposition works because equilibrium and the stress-strain relation are linear, which requires Hooke's law and small deformations. (Key point: Conditions for validity)

3. A beam of span 6 m carries w = 2 kN/m and a mid-span P = 6 kN. What is the mid-span moment by superposition?
   A. 9 kN·m
   B. 27 kN·m
   C. 18 kN·m
   D. 36 kN·m
   Answer: C. The UDL gives wL^2/8 = 9 kN·m and the point load gives PL/4 = 9 kN·m. Their sum is 18 kN·m. (Key point: Worked example)

## Related in Strength of Materials

- [Principle of superposition, example and limitations](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprinciple-of-superposition-limitations) — Strength of Materials, chapter 1
- [Superposition: stepwise procedure for bending moment](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asuperposition-stepwise-bending-moment) — Strength of Materials, chapter 1

## Related in other subjects

- [Principle of superposition with examples](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aprinciple-of-superposition) — Theory of Structures I, chapter 1, asked 7×
- [Superposition: 90, 50, 130 ohm circuit](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Asuperposition-90-50-130) — Basic Electronics Engineering, chapter 1, asked 3×
- [Cantilever deflection by integration and superposition](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Acantilever-udl-point-load-deflection) — Strength of Materials, chapter 7