Strain energy is the energy stored inside a material when it deforms under load. You calculate it by finding the work done by internal forces as the material stretches, compresses, bends, or twists. For a simple linear elastic material, the core formula is U = ½ × stress × strain × volume, and it simplifies into specific equations depending on the type of loading.
The General Principle
When you apply a force to a solid object and it deforms, energy transfers into the material. That stored energy is strain energy. If the material behaves elastically (meaning it returns to its original shape when unloaded), this energy equals the work done by the applied force during deformation.
At the most fundamental level, the strain energy in a tiny piece of material equals one-half times the stress times the strain times the volume of that piece:
dU = ½ × σ × ε × dV
To get the total strain energy in an entire structure, you integrate (add up) this expression across the full volume. In practice, engineers rarely use this general form directly. Instead, they use simplified versions tailored to specific loading types: axial, bending, shear, or torsion.
Strain Energy Under Axial Loading
Axial loading means a force pulls or pushes straight along the length of a bar or rod. This is the simplest and most common case you’ll encounter. For a bar with a constant cross-sectional area A, length L, elastic modulus E, and an internal axial force N that stays constant along the length, the strain energy is:
U = N²L / (2EA)
You can also write this in terms of the total elongation Δ instead of the force:
U = EA Δ² / (2L)
Both formulas give the same result. Use the force version when you know the load, and the displacement version when you know how much the bar stretched. If the force or cross-section changes along the length (for example, a tapered rod or a bar with multiple segments carrying different loads), you need to integrate along the length or calculate each segment separately and add them up.
Example: Multi-Segment Rod
Imagine a rod made of three segments: steel (E = 200 GPa), brass (E = 101 GPa), and aluminum (E = 73.1 GPa). Each segment carries its own internal force depending on how loads are applied. You would calculate U = N²L / (2EA) for each segment individually, using that segment’s force, length, area, and elastic modulus, then sum all three values to get the total strain energy in the assembly.
Strain Energy in Bending
When a beam bends under a transverse load (like a floor joist supporting weight), the strain energy comes from the internal bending moment M along the beam’s length. The formula is:
U = ∫ M² / (2EI) dx
Here, E is the elastic modulus, I is the moment of inertia of the cross-section (a geometric property that measures how resistant the shape is to bending), and the integral runs along the beam’s length. For a beam with constant EI, you only need to figure out how M varies with position x, square it, and integrate.
If the bending moment is simple, like from a single point load or a uniform distributed load, the integral often works out to a clean expression. For more complex loading, you may need to split the beam into segments where M has different expressions and integrate each one separately.
Strain Energy From Shear
Shear loading occurs when forces act parallel to a surface rather than perpendicular to it. The strain energy due to a shear force Q in a member of length L, cross-sectional area A, and shear modulus G is:
U = Q²L / (2AG)
When the shear force varies along the length, this becomes an integral:
U = ∫ Q² / (2AG) ds
In most beam problems, shear strain energy is small compared to bending strain energy and is often neglected. It becomes significant in short, deep beams or in connections where shear dominates.
Strain Energy From Torsion
Torsion means twisting, like a drive shaft transmitting power. For a circular shaft with polar moment of inertia J, shear modulus G, length L, and internal torque T, the strain energy is:
U = T²L / (2GJ)
This follows the same pattern as the axial formula: force squared times length, divided by twice the stiffness. For a steel shaft with a 40 mm radius and G = 75 GPa, you would calculate J from the geometry (J = π r⁴ / 2), determine the internal torque at each section, then plug into the formula. If the torque changes along the shaft, integrate T² / (2GJ) over the length.
Strain Energy Density
Sometimes you need energy per unit volume rather than total energy. This quantity, called strain energy density, equals the area under the stress-strain curve. For a material loaded in one direction within the elastic range:
u = ½ × σ × ε
The units work out to force per area (like N/mm²), which is equivalent to energy per volume (N·mm/mm³). Strain energy density is useful for comparing how much energy different materials can absorb regardless of their size.
Two related concepts come from the stress-strain curve. Resilience is the strain energy density the material can absorb while still returning to its original shape. It equals the area under the curve up to the yield point. Toughness is the total area under the entire stress-strain curve up to fracture, representing the maximum energy the material can absorb before breaking. A tough material like mild steel has a large area under its curve because it deforms extensively before failure. A brittle material like glass has very little.
Using Strain Energy To Find Deflections
One of the most powerful uses of strain energy is calculating how much a structure deflects. Castigliano’s theorem states that if you take the partial derivative of the total strain energy with respect to a particular force, the result equals the displacement at the point where that force is applied, in the direction of that force:
δ = ∂U / ∂F
For axial loading, this recovers the familiar result. If U = F²L / (2AE), then differentiating with respect to F gives δ = FL / (AE). For torsion, differentiating U = T²L / (2GJ) with respect to T gives the angle of twist θ = TL / (GJ).
For beams in bending, the theorem becomes:
δ = ∫ (M / EI) × (∂M / ∂F) dx
You express the bending moment M in terms of the applied loads, differentiate with respect to the load at the point where you want the deflection, then integrate along the beam. If no actual load exists at the point where you need the deflection, you add a fictitious (dummy) force there, carry it through the calculation, and set it to zero at the end.
Castigliano’s theorem is especially useful for statically indeterminate structures and curved members where standard deflection formulas don’t apply.
Nonlinear Materials
All the formulas above assume the material follows Hooke’s law, where stress is proportional to strain. For materials that don’t behave linearly (rubber, biological tissue, some polymers), strain energy is described by a strain energy density function that depends on the full deformation state rather than just stress and strain. These functions are defined so that the stored energy is zero when the material is unstressed. The energy for small deformations can be approximated by expanding the function as a polynomial in strain, but for large deformations you need specific material models fitted to experimental data. The key difference is that you can no longer use the simple ½ × stress × strain relationship and must instead integrate the actual stress-strain curve of the material.