---
title: "What is strain energy; real work method for beam"
subject: "Theory of Structures I"
chapter: 3
key: "semester_4/theory_of_structures_i:real-work-beam-explain-strain-energy"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/real-work-beam-explain-strain-energy
site: BCT Engineering
---

# What is strain energy; real work method for beam

Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method

- Asked in: 2071 Magh (old course), 1+3 marks
- Video: [Structural Analysis - Strain energy Method  - Introduction](https://www.youtube.com/watch?v=9zYcxqOT_ho) — Dr.M.Purushothaman MP
- 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/real-work-beam-explain-strain-energy
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3

## Question

What is strain energy? Explain with an example the real work method to calculate the deflection of a beam.

## Short answer

Strain energy is the energy stored in an elastic body, equal to external work ½PΔ. Real work method for a beam with a single load: compute U = ∫ M²/(2EI)dx, equate to ½PΔ, solve for Δ. Example: cantilever with tip load P gives Δ = PL³/(3EI).

## Answer

**Strain energy** is the energy stored in a deformed elastic body, equal to the work done by the external loads when they are applied gradually: $U = \tfrac12 P\Delta$. It is recovered on unloading within the elastic limit.

#### Real work method for beam deflection
The method uses the principle of conservation of energy: the **external work** done by the load equals the **internal strain energy** of the beam.

1. Find the bending moment $M$ along the beam due to the single load $P$.
2. Strain energy: $U = \displaystyle\int \frac{M^2}{2EI}\,dx$ (shear energy is usually neglected).
3. External work: $W = \tfrac12 P\,\Delta$, where $\Delta$ is the deflection at the load point.
4. Equate $W = U$ and solve for $\Delta$.

**Example.** Cantilever of length $L$ with a load $P$ at the free end.

$$
M = Px,\qquad U = \int_0^L \frac{P^2x^2}{2EI}dx = \frac{P^2L^3}{6EI}
$$

$$
\tfrac12 P\Delta = \frac{P^2L^3}{6EI} \;\Rightarrow\; \Delta = \frac{PL^3}{3EI}
$$

The method gives the deflection **only under a single load, at its point of application**, in the direction of the load.

## Check yourself

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

1. Strain energy is:
   A. Energy stored in a deformed elastic body, equal to the work done by gradually applied loads
   B. Energy lost as heat during loading
   C. Energy of the load only
   D. Energy needed to reach failure
   Answer: A. It is the stored energy U = (1/2)P Delta, recovered on unloading within the elastic limit. (Key point: Strain energy definition)

2. In the real work method for a beam, the deflection is found by:
   A. Equating (1/2) P Delta to the integral of M^2/(2EI) dx
   B. Integrating M twice
   C. Equating P Delta to M
   D. Applying a unit load
   Answer: A. External work (1/2) P Delta equals bending strain energy; solving gives Delta. (Key point: Real work method steps)

3. For a cantilever with end load P, U equals:
   A. P^2 L^3 / (6EI)
   B. P L^3 / (3EI)
   C. P^2 L^2 / (2EI)
   D. P^2 L^3 / (3EI)
   Answer: A. U = integral of P^2 x^2/(2EI) dx = P^2 L^3/(6EI), which gives Delta = PL^3/(3EI). (Key point: Cantilever result)

4. The method gives deflection:
   A. At any point of the beam
   B. Only under a single load, at its point and direction
   C. Only for multiple loads
   D. Only for rotations
   Answer: B. Because one equation has one unknown, it works for a single load at its own point. (Key point: Method limitation)

## Related in Theory of Structures I

- [Strain energy and real work deflection, overhang beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Areal-work-strain-energy-overhang-5m) — Theory of Structures I, chapter 3
- [Strain energy deflection at A, spring support](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Astrain-energy-beam-spring-deflection-a) — Theory of Structures I, chapter 2
- [Overhang beam free end deflection by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-vw-free-end-d-2065) — Theory of Structures I, chapter 3
- [Strain energy in rectangular beam, central load](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Ass-beam-strain-energy-rect) — Theory of Structures I, chapter 2
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