Strain Energy & Energy Theorems Notes

Last reviewed 16 Sept 2026 · 6 min read

Strain energy and resilience

When a load deforms an elastic body, the work done by the load is stored inside the body as strain energy $U$ . On removing the load the energy is released — that is how springs and bows work.

DefinitionTerms

Resilience — total strain energy stored in a body (N·mm, J).
Proof resilience — maximum strain energy that can be stored up to the elastic limit.
Modulus of resilience — proof resilience per unit volume $= \frac{σ _{e l}^{2}}{2 E}$ (area under the elastic part of the stress–strain curve).
Modulus of toughness — energy per unit volume up to fracture (area under the whole curve).

Strain energy in direct stress

A load increased gradually from zero to $P$ stretches a bar by $δ$ . Work done $= \frac{1}{2} P δ$ .

U = \frac{1}{2} P δ = \frac{σ ^{2}}{2 E} \times A L = \frac{P ^{2} L}{2 A E}

For a member with varying force or section: $U = \int \frac{P ^{2} d x}{2 A E}$ .

Gradual, sudden and impact loading

Loading	Maximum stress	Remark
Gradual	$σ = P / A$	Load increases slowly from zero
Sudden (applied all at once, no drop height)	$σ = 2 P / A$	Twice the gradual stress
Impact (falls from height $h$ )	$σ = \frac{P}{A} [1 + 1 + \frac{2 A E h}{P L}]$	Reduces to $2 P / A$ when $h = 0$

Derivation (impact). Loss of potential energy of the falling load = strain energy stored:

P (h + δ) = \frac{σ ^{2}}{2 E} A L, δ = \frac{σ L}{E}

which is a quadratic in $σ$ with the positive root above. If $δ$ is negligible compared with $h$ : $σ \approx \frac{2 E P h}{A L}$ .

A load $P$ moving with velocity $v$ on impact is treated with $h = v^{2} /2 g$ .

Exam TipDesigning against shock

For a given impact energy, stress $\propto 1/ A L$ — a longer bar or a bar with more volume absorbs the same energy at a lower stress. Bolts for shock loading are sometimes turned down to the root diameter over the shank so the whole length stretches uniformly and stores more energy.

Strain energy in shear

Under shear stress $τ$ on volume $V$ :

U = \frac{τ ^{2}}{2 G} V

Strain energy in bending

A beam element $d x$ under moment $M$ rotates by $d θ = \frac{M}{E I} d x$ ; energy $= \frac{1}{2} M d θ$ :

U = \int_{0}^{L} \frac{M ^{2}}{2 E I} d x

(Shear-deformation energy is usually negligible for slender beams.)

Beam	Strain energy
Cantilever, end load $W$	$\frac{W ^{2} L ^{3}}{6 E I}$
Cantilever, UDL $w$	$\frac{w ^{2} L ^{5}}{40 E I}$
Simply supported, central load $W$	$\frac{W ^{2} L ^{3}}{96 E I}$
Simply supported, UDL $w$	$\frac{w ^{2} L ^{5}}{240 E I}$

Strain energy in torsion

U = \int \frac{T ^{2}}{2 G J} d x = \frac{T ^{2} L}{2 G J}

Solid shaft: $U = \frac{τ _{ma x}^{2}}{4 G} V$ ; hollow shaft: $U = \frac{τ _{ma x}^{2}}{4 G} \cdot \frac{D ^{2} + d ^{2}}{D ^{2}} V$ .

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Strain Energy & Energy Theorems