---
title: "Total strain energy formulae; derive shear energy"
subject: "Theory of Structures I"
chapter: 2
key: "semester_4/theory_of_structures_i:total-strain-energy-formulae-shear"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/total-strain-energy-formulae-shear
site: BCT Engineering
---

# Total strain energy formulae; derive shear energy

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

- Asked in: 2070 Bhadra, 6 marks; 2066 Kartik, 6 marks (asked 2 times)
- Video: [Strain energy and strain energy density introduced](https://www.youtube.com/watch?v=m14sqLGg4BQ) — Engineer4Free
- 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/total-strain-energy-formulae-shear
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-2

## Question

Write down the formula for determination of total strain energy due to axial force, bending moment, shear force and torsion in a structural system. Derive the expression for energy due to shear force in an element of a structural system.

## Short answer

Total strain energy = sum of integrals of N^2/(2AE), M^2/(2EI), αV^2/(2GA), T^2/(2GJ) over length. Shear energy per element dx: from shear stress τ = V A' ȳ/(Ib), integrating τ^2/(2G) over area gives dU = α V^2/(2GA) dx, with α = (A/I^2) ∫ (A' ȳ/b)^2 dA. For rectangle α=6/5.

## Answer

#### Total strain energy of a structural system
For a member of length $L$ with axial force $N$, bending moment $M$, shear force $V$ and torque $T$, the total strain energy is the sum:

$$
U = \int_0^L \frac{N^2}{2AE}\,dx + \int_0^L \frac{M^2}{2EI}\,dx + \int_0^L \frac{\alpha\,V^2}{2GA}\,dx + \int_0^L \frac{T^2}{2GJ}\,dx
$$

Here $A$ = area, $I$ = moment of inertia, $J$ = polar moment of inertia, $E$ and $G$ = Young's and shear moduli, $\alpha$ = shear form factor ($6/5$ for rectangle, $10/9$ for solid circle). For a structure, the sum is taken over all members.

#### Derivation of energy due to shear force
Consider an element of length $dx$ with shear force $V$. Shear stress $\tau$ at a level $y$ of the section (area element $dA$):

$$
\tau = \frac{V A'\bar y}{Ib}
$$

Energy per unit volume $= \dfrac{\tau^2}{2G}$, so for the element

$$
dU = \int_A \frac{\tau^2}{2G}\,dA\,dx
$$

If the shear stress were uniform, $\tau = V/A$, giving $dU = \dfrac{V^2}{2GA}dx$. To account for the actual non-uniform distribution a **shear form factor** is defined as

$$
\alpha = \frac{A}{I^2}\int_A \left(\frac{A'\bar y}{b}\right)^2 dA
$$

Then

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

For a rectangular section $\tau = \dfrac{V}{2I}\left(\dfrac{d^2}{4}-y^2\right)$, which gives $\alpha = 1.2$.

## Check yourself

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

1. Strain energy due to axial force N is:
   A. integral of N^2/(2GJ) dx
   B. integral of N/(AE) dx
   C. integral of N^2/(2AE) dx
   D. integral of N^2/(2EI) dx
   Answer: C. Axial uses AE. (Key point: Axial term)

2. Torsional strain energy is:
   A. integral of T^2/(2EI) dx
   B. integral of T^2/(2GA) dx
   C. integral of T/(GJ) dx
   D. integral of T^2/(2GJ) dx
   Answer: D. Torsion uses G and polar moment J. (Key point: Torsion term)

3. Why is a shear form factor alpha introduced?
   A. Shear stress is not uniform, so V/A underestimates the energy
   B. To convert E to G
   C. To account for torsion
   D. To include axial effect
   Answer: A. alpha corrects for actual distribution. (Key point: Form factor definition)

4. Shear form factors are:
   A. 10/9 for rectangle and 6/5 for circle
   B. 6/5 for rectangle and 10/9 for solid circle
   C. 1 for both
   D. 3/2 for rectangle and 4/3 for circle
   Answer: B. Standard values. (Key point: Values)

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