---
title: "Strain energy due to bending, rectangular beam"
subject: "Theory of Structures I"
chapter: 2
key: "semester_4/theory_of_structures_i:strain-energy-bending-derivation"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/strain-energy-bending-derivation
site: BCT Engineering
---

# Strain energy due to bending, rectangular beam

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

- Asked in: 2079 Chaitra, 4 marks; 2076 Baisakh, 4 marks (asked 2 times)
- Video: [Structural analysis -I  # Derive strain energy U deu to bending of beam || old question solution ||](https://www.youtube.com/watch?v=nuAj7FC1C78) — Study with Jyoti
- 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/strain-energy-bending-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-2

## Question

Derive an expression for the strain energy stored in a rectangular beam due to bending moment.

## Short answer

Bending strain energy derived from work of moment on rotation: dθ = M dx/EI, dU = ½ M dθ = M^2 dx/(2EI). For whole beam, Ub = ∫₀ᴸ M^2/(2EI) dx. This applies to any section; for rectangle, I = bd^3/12.

## Answer

Strain energy due to bending is the energy stored in a beam because of bending (flexural) stresses.

#### Derivation
Consider an elemental length $dx$ of a beam under a bending moment $M$. The ends of the element rotate relative to each other by

$$
d\theta = \frac{M\,dx}{EI}
$$

The moment rises gradually from 0 to $M$ while the rotation rises from 0 to $d\theta$, so the work done (strain energy stored) in the element is

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

**Alternative (stress method):** the bending stress at a distance $y$ from the neutral axis is $\sigma = \dfrac{My}{I}$, and the energy per unit volume is $\dfrac{\sigma^2}{2E}$:

$$
dU = \int \frac{M^2y^2}{2EI^2}\,dA\,dx = \frac{M^2}{2EI^2}\left(\int y^2\,dA\right)dx = \frac{M^2\,dx}{2EI}
$$

For the whole beam of length $L$:

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

#### For a rectangular beam
With $I = \dfrac{bd^3}{12}$, a simply supported beam of span $L$ carrying a central load $P$ has $M = \dfrac{P x}{2}$ for $0 \le x \le L/2$:

$$
U_b = 2\int_0^{L/2}\frac{P^2x^2}{8EI}\,dx = \frac{P^2L^3}{96EI} = \frac{P^2L^3}{8Ebd^3}
$$

For a cantilever with an end load $P$: $M = Px$, so $U_b = \dfrac{P^2L^3}{6EI} = \dfrac{2P^2L^3}{Ebd^3}$.

**Result:** $U_b = \displaystyle\int_0^L \frac{M^2}{2EI}\,dx$

## Check yourself

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

1. Relative rotation of the ends of an element dx under M is:
   A. M dx/(GJ)
   B. EI dx/M
   C. M dx/(EI)
   D. M dx/(AE)
   Answer: C. dθ = M dx/EI. (Key point: Rotation)

2. The energy stored in the element is:
   A. M dθ
   B. 2 M dθ
   C. (1/2) M^2 dθ
   D. (1/2) M dθ because the moment rises gradually
   Answer: D. Gradually applied moment does half product work. (Key point: Work done)

3. Total bending strain energy is:
   A. integral of M^2/(2EI) dx
   B. integral of M/(2EI) dx
   C. integral of M^2/(EI^2) dx
   D. integral of M^2/(2GA) dx
   Answer: A. Sum over length. (Key point: Result)

4. Cantilever with end load P has bending energy:
   A. P^2L^3/(96EI)
   B. P^2L^3/(6EI)
   C. P^2L^3/(3EI)
   D. P^2L^3/(48EI)
   Answer: B. Integrate (Px)^2/2EI from 0 to L. (Key point: Cantilever)

## Related in Theory of Structures I

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