---
title: "Strain energy, gradual vs sudden load, shear"
subject: "Theory of Structures I"
chapter: 2
key: "semester_4/theory_of_structures_i:strain-energy-gradual-sudden-shear"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/strain-energy-gradual-sudden-shear
site: BCT Engineering
---

# Strain energy, gradual vs sudden load, shear

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

- Asked in: 2071 Bhadra, 6 marks
- Video: [Derivation on strain energy ||gradually applied load || suddenly applied load || Impact load](https://www.youtube.com/watch?v=D2QkeeUl08k) — Engineering visions
- 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-gradual-sudden-shear
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-2

## Question

Define strain energy and explain with examples the difference between gradually and suddenly applied direct loads. Derive the expression for strain energy due to shear force in a beam in bending.

## Short answer

Strain energy is stored elastic energy. Gradually applied load: stress = P/A, deflection = PL/AE, work = ½Pδ. Suddenly applied load: full P acts instantly, stress = 2P/A, deflection = 2PL/AE. Shear strain energy: Us = ∫ (α V^2/(2GA)) dx, α=1.2 for rectangle.

## Answer

**Strain energy** is the energy stored in a deformed elastic body due to the work done by external loads in deforming it. It is recovered when the load is removed (within the elastic limit). For a direct load, $U = \dfrac{\sigma^2}{2E}\times\text{Volume}$.

#### Gradually and suddenly applied direct loads
| Point | Gradual load | Sudden load |
|---|---|---|
| Application | Load increases slowly from 0 to $P$ | Full load $P$ acts at once, from zero deflection |
| Work done | $\tfrac12 P\delta$ | $P\delta$ (force is constant throughout) |
| Stress | $\sigma = P/A$ | $\sigma = 2P/A$ |
| Deflection | $\delta = PL/AE$ | $\delta = 2PL/AE$ |
| Example | Slowly lifting a weight with a hoist | A weight released suddenly on a collar from zero height |

For sudden loading, the work done $P\delta_{max}$ equals the strain energy $\tfrac12 k\delta_{max}^2$, so $\delta_{max} = 2P/k$: the stress and deflection are twice those for a gradually applied load.

#### Strain energy due to shear force in a beam
For an element of length $dx$ with shear force $V$, the shear stress at a distance $y$ from the neutral axis is $\tau = \dfrac{VA'\bar y}{Ib}$, and the energy per unit volume is $\dfrac{\tau^2}{2G}$. Integrating over the area, for a rectangular section:

$$
dU = \int \frac{\tau^2}{2G}\,dA\,dx,\qquad \tau = \frac{V}{2I}\left(\frac{d^2}{4}-y^2\right)
$$

$$
dU = \frac{V^2 b}{8GI^2}\cdot\frac{d^5}{30}\,dx = \frac{6}{5}\cdot\frac{V^2}{2GA}\,dx
$$

For any section:

$$
U_s = \int_0^L \frac{\alpha V^2}{2GA}\,dx
$$

where $\alpha$ is the shear form factor ($\alpha = 1.2$ for a rectangle, $10/9$ for a solid circle).

## Check yourself

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

1. Strain energy is:
   A. Heat lost in deformation
   B. Energy lost to friction
   C. Energy stored in a deformed elastic body, recovered on unloading
   D. Energy needed to break a body
   Answer: C. Stored elastic energy equals work done. (Key point: Definition)

2. A suddenly applied load produces stress:
   A. P/A
   B. P/2A
   C. 4P/A
   D. 2P/A, twice the gradual value
   Answer: D. Work P δ = (1/2) k δ^2 gives δ = 2P/k. (Key point: Sudden load)

3. Work done by a gradually applied load is:
   A. (1/2) P δ
   B. P δ
   C. 2 P δ
   D. P δ^2
   Answer: A. Force grows from 0 to P. (Key point: Work done)

4. For a rectangular section shear strain energy is:
   A. V^2/(2GA)
   B. (6/5) V^2/(2GA) per unit length
   C. (10/9) V^2/(2GA)
   D. V^2/(2EI)
   Answer: B. Form factor 1.2. (Key point: Shear energy)

## Related in Theory of Structures I

- [Strain energy due to shear in rectangular beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Astrain-energy-shear-rect-beam) — Theory of Structures I, chapter 2, asked 3×
- [Strain energy due to bending, rectangular beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Astrain-energy-bending-derivation) — Theory of Structures I, chapter 2, asked 2×
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