---
title: "Define strain energy; relation for bending"
subject: "Theory of Structures I"
chapter: 2
key: "semester_4/theory_of_structures_i:define-strain-energy-complementary-bending"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/define-strain-energy-complementary-bending
site: BCT Engineering
---

# Define strain energy; relation for bending

Theory of Structures I · Chapter 2: Analysis by the Strain Energy Method

- Asked in: 2069 Bhadra, 6 marks
- Video: [Strain energy (part1) - Axial and Bending loads](https://www.youtube.com/watch?v=yNASIxckvzg) — Bawa Kulkarni
- 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/define-strain-energy-complementary-bending
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-2

## Question

Define strain energy and complementary strain energy. Also derive the relationship of strain energy due to bending.

## Short answer

Strain energy is energy stored in an elastic body equal to work done by external forces, area under load-displacement curve. Complementary strain energy is area between curve and load axis. For linear systems both equal ½PΔ. Strain energy due to bending: dU = M²dx/(2EI), integrated over length gives U_b = ∫ M²/(2EI) dx.

## Answer

#### Strain energy
**Strain energy** $U$ is the energy stored in an elastic body by the work done by external forces in deforming it. It equals the area under the load-displacement curve (up to the displacement):

$$
U = \int_0^\Delta P\,d\Delta
$$

For linear behaviour, $U = \tfrac12 P\Delta$.

#### Complementary strain energy
**Complementary strain energy** $U^*$ is the area between the load-displacement curve and the load axis:

$$
U^* = \int_0^P \Delta\,dP
$$

```text
 P |        /
   |  U*  /|
   |    /  |      U + U* = P*Delta
   |  /  U |
   +-------+---- Delta
```

For a linear structure, $U^* = U = \tfrac12 P\Delta$; for a non-linear one they differ. Castigliano's theorems use $U$ (first) and $U^*$ (second).

#### Strain energy due to bending
An element $dx$ under moment $M$ rotates by $d\theta = \dfrac{M\,dx}{EI}$. The moment grows gradually, so

$$
dU = \tfrac12 M\,d\theta = \frac{M^2\,dx}{2EI}
$$

$$
U_b = \int_0^L \frac{M^2}{2EI}\,dx
$$

Using the stress method as a check: $\sigma = \dfrac{My}{I}$, $dU = \displaystyle\int\frac{\sigma^2}{2E}dA\,dx = \frac{M^2}{2EI^2}\Big(\int y^2dA\Big)dx = \frac{M^2dx}{2EI}$.

Example: cantilever with end load $P$: $M = Px$, so $U_b = \displaystyle\int_0^L \frac{P^2x^2}{2EI}dx = \frac{P^2L^3}{6EI}$.

## Check yourself

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

1. Strain energy is represented by the:
   A. Area between curve and load axis
   B. Slope of the curve
   C. Area under the load-displacement curve, integral P dΔ
   D. Product P x Δ always
   Answer: C. U is the integral of P dΔ. (Key point: Strain energy)

2. Complementary strain energy U* is:
   A. Area under the curve
   B. Equal to U always
   C. Slope of the curve
   D. Area between curve and load axis, integral Δ dP
   Answer: D. U + U* = PΔ. (Key point: Complementary energy)

3. For a linear structure:
   A. U* = U = (1/2) P Δ
   B. U* = 2U
   C. U* = 0
   D. U* = PΔ
   Answer: A. The two areas are equal triangles. (Key point: Linear case)

4. Bending strain energy of a beam is:
   A. integral of M/(EI) dx
   B. integral of M^2/(2EI) dx
   C. integral of M^2/(EI) dx
   D. integral of M^2/(2GJ) dx
   Answer: B. dU = (1/2) M dθ. (Key point: Bending)

## Related in Theory of Structures I

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