Learning topic

Work of External Forces and Strain Energy

Learn strain-energy formulas for axial loading, shear, torsion, and bending, including energy density and the link to Castigliano’s theorem.

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Strain energy $U$ is the energy stored in an elastic body as a result of deformation. In a linearly elastic system under gradual static loading, it equals the work performed by the external forces.

For a linearly elastic system loaded statically from zero to the final force values, the work of external forces equals the stored strain energy:

$$A=U=\frac{1}{2}\sum_i P_i\delta_i+\frac{1}{2}\sum_j M_j\varphi_j.$$

For one generalized force $P$ and its corresponding displacement $\delta$:

$$U=\frac{1}{2}P\delta.$$

$A$ — work of external forces;
$U$ — strain energy;
$P_i$ — generalized force;
$\delta_i$ — corresponding linear displacement;
$M_j$ — generalized moment;
$\varphi_j$ — corresponding angular displacement.

The factor $1/2$ appears because, in a linear system, the force and its corresponding displacement increase proportionally from zero to their final values.

Energy expressed through internal force resultants

For a linearly elastic member, strain energy can be calculated by integrating contributions from internal force resultants along its length:

$$U_N=\int_0^L\frac{N^2(x)}{2EA}\,dx,$$

$$U_M=\int_0^L\frac{M^2(x)}{2EI}\,dx,$$

$$U_T=\int_0^L\frac{T^2(x)}{2GJ_p}\,dx.$$

If shear deformation is included, its contribution is written consistently with the adopted beam model, for example:

$$U_Q=\int_0^L\frac{Q^2(x)}{2\kappa GA}\,dx,$$

$U_N$, $U_M$, $U_T$, $U_Q$ — strain-energy contributions;
$N(x)$ — axial force;
$M(x)$ — bending moment;
$T(x)$ — torque;
$Q(x)$ — shear force;
$E$, $G$ — Young's and shear moduli;
$A$ — cross-sectional area;
$I$ — second moment of area;
$J_p$ — polar second moment of area;
$\kappa$ — shear correction factor.

In a linear system, the total strain energy is the sum of the applicable independent quadratic contributions.

For a member subjected simultaneously to several deformation modes, the total energy within the applicable linear model is obtained by adding contributions from axial loading, bending, torsion, and, where necessary, transverse shear.

Axial loading

For a prismatic bar with constant $N$, $A$, and $E$:

$$U_N=\frac{N^2L}{2EA}.$$

Since $\Delta L=NL/(EA)$, the same result can be written as $U_N=N\Delta L/2$.

Torsion

For a circular prismatic shaft with constant $T$, $G$, and $J_p$:

$$U_T=\frac{T^2L}{2GJ_p}=\frac{1}{2}T\varphi,$$

where $\varphi$ is the total angle of twist.

Bending

For a beam, the principal strain-energy contribution in the classical model is often associated with the bending moment:

$$U_M=\int_0^L\frac{M^2(x)}{2EI}\,dx.$$

If transverse-shear deformation is significant, the corresponding shear contribution is added.

Strain-energy density

For a uniaxial linearly elastic state:

$$u=\frac{U}{V}=\frac{1}{2}\sigma\varepsilon=\frac{\sigma^2}{2E}.$$

For pure shear:

$$u=\frac{1}{2}\tau\gamma=\frac{\tau^2}{2G}.$$

Volumetric and distortional energy

For an isotropic linearly elastic material, the total strain-energy density can be separated into volumetric and deviatoric parts. The distortional-energy component is related to differences between principal stresses and forms the basis of the von Mises yield criterion.

For ductile isotropic materials under a multiaxial stress state, the Tresca and von Mises criteria are widely used.

Tresca:

$$\sigma_{\mathrm{eq,T}}=\max\left(|\sigma_1-\sigma_2|,|\sigma_2-\sigma_3|,|\sigma_3-\sigma_1|\right).$$

von Mises:

$$\sigma_{\mathrm{eq,VM}}=\sqrt{\frac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}.$$

$\sigma_1$, $\sigma_2$, $\sigma_3$ — principal stresses;
$\sigma_{\mathrm{eq,T}}$ — Tresca equivalent stress;
$\sigma_{\mathrm{eq,VM}}$ — von Mises equivalent stress.

In allowable-stress design, the corresponding equivalent stress is compared with an allowable value consistent with the material properties and the adopted design method.

Practical significance

The energy formulation is particularly useful for determining displacements by Castigliano's theorem or the unit-load method and for analyzing systems subjected to several simultaneous internal force resultants.

About this topic

During elastic deformation, external loads perform work stored inside the body as strain energy U. This page provides formulas for strain energy under axial loading, pure shear, torsion, and bending. It also details strain energy density divided into volumetric change and shape distortion components.