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

# Define strain energy and complementary

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

- Asked in: 2077 Chaitra, 2 marks
- Video: [Strain and complimentary  energy - derivations](https://www.youtube.com/watch?v=0O5PBpJl-zE) — Anup Patil
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-2

## Question

Define strain energy and complementary strain energy.

## Short answer

Strain energy is energy stored in an elastic body during deformation, equal to work done by loads, area under load-displacement curve. Complementary strain energy is area between load-displacement curve and load axis. For linear elastic materials both equal ½PΔ and sum to PΔ; they differ only for non-linear materials.

## Answer

**Strain energy** is the energy stored in an elastic body when it is deformed by external loads. It equals the work done by the loads and is recovered fully on unloading. It is the area under the load-displacement curve:

$$
U = \int_0^\Delta P\,d\Delta \quad(= \tfrac12 P\Delta \text{ for a linear body})
$$

**Complementary strain energy** is the area between the load-displacement curve and the load axis:

$$
U^* = \int_0^P \Delta\,dP \quad(= \tfrac12 P\Delta \text{ for a linear body})
$$

The sum $U + U^* = P\Delta$. For linearly elastic materials $U^* = U$; they differ only for non-linear materials.

## Check yourself

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

1. Strain energy is:
   A. Energy permanently lost
   B. Heat generated
   C. Energy stored in an elastic body, recovered fully on unloading
   D. Energy that equals PΔ always
   Answer: C. Work done equals stored energy. (Key point: Strain energy)

2. Complementary energy is the area:
   A. Under the curve
   B. Above the displacement axis only
   C. Of the whole rectangle
   D. Between the curve and the load axis
   Answer: D. U* = integral Δ dP. (Key point: Complementary)

3. The relation between U and U* is:
   A. U + U* = PΔ, and U* = U for linear materials
   B. U = 2U* always
   C. U* = PΔ
   D. They differ even for linear materials
   Answer: A. They differ only for non-linear materials. (Key point: Relation)

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