What this calculator does
Shear strain measures how much a material deforms sideways under a shearing force, relative to its original size. It is one of the basic quantities in materials science and structural engineering, alongside normal strain, and it describes distortion rather than a simple change in length.
The shear strain formula has two equivalent forms. Given a lateral displacement and the original height or length over which it occurs, shear strain is that displacement divided by the length: γ = Δx / L. For small deformations this is also equal to the tangent of the angle the material has skewed through, so the same quantity can be worked out directly from an angle instead, without needing the raw displacement figure.
The formula
Choose whichever pair of values you actually have. From a displacement and a length, shear strain is the displacement divided by the length. From an angle of deformation, shear strain is the tangent of that angle. The two are the same underlying quantity, just entered a different way, and this calculator converts between them either way.
| Term | Meaning |
|---|---|
| γ (gamma) | Shear strain: the ratio of lateral displacement to original length, a dimensionless number (or expressed as a percentage). |
| Δx | The lateral displacement, how far the top face has moved sideways relative to the bottom face. |
| L | The original length or height over which that displacement occurs. |
| θ (theta) | The angle of deformation, for small angles equal to the arctangent of the shear strain. |
The inputs explained
| Field | What to enter |
|---|---|
| Known value | Pick "Displacement and length" if you know how far the material moved sideways, or "Angle of deformation" if you know the skew angle directly. |
| Lateral displacement (Δx) (mm) | The lateral (sideways) displacement of one face relative to the other. |
| Original length (L) (mm) | The original length or height between the two faces, measured before any deformation. |
| Angle of deformation (θ) (°) | The angle the material has sheared through, in degrees. |
When to use it
Testing a material sample
A shear test measures how far a sample deforms sideways under a known load; converting that displacement and the sample height into shear strain lets it be compared against published material properties or a shear modulus.
Checking a bonded joint or adhesive layer
Adhesive and bonded joints are often rated by how much shear strain they can sustain before failing, so a measured displacement across the bond line converts directly into a comparable strain figure.
Working from a measured tilt angle
When a structure or sample has visibly skewed rather than been measured for displacement, the angle of deformation converts straight to shear strain using the shear strain formula in its tangent form.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How shear strain changes with lateral displacement at a fixed length
A fixed 100 mm length, across a range of lateral displacements.
| Lateral displacement | Shear strain (γ) | Equivalent angle of deformation |
|---|---|---|
| 0.5 mm | 0.005000 | 0.2865 ° |
| 1 mm | 0.010000 | 0.5729 ° |
| 2 mm | 0.020000 | 1.146 ° |
| 5 mm | 0.050000 | 2.862 ° |
| 10 mm | 0.100000 | 5.711 ° |
| 20 mm | 0.200000 | 11.310 ° |
How shear strain changes with the angle of deformation
A range of deformation angles, converted straight to shear strain.
| Angle of deformation | Shear strain (γ) | As a percentage |
|---|---|---|
| 0.5° | 0.008727 | 0.8727% |
| 1° | 0.017455 | 1.746% |
| 2° | 0.034921 | 3.492% |
| 5° | 0.087489 | 8.749% |
| 10° | 0.176327 | 17.633% |
| 20° | 0.363970 | 36.397% |
Questions
What is the shear strain formula?
Shear strain is γ = Δx / L, the lateral displacement divided by the original length over which it occurs. It is also equal to tan θ, where θ is the angle the material has skewed through, which is the same quantity expressed as an angle rather than a ratio.
Is shear strain the same as normal strain?
No. Normal strain measures a change in length along the same direction as the applied force (stretching or compressing). Shear strain measures a sideways, angular distortion at right angles to the applied force, with no change in the material's overall length.
Does shear strain have units?
Shear strain is dimensionless, a ratio of two lengths, so it is often written as a plain number or a percentage rather than with a unit. Radians are sometimes used informally since it approximates an angle at small deformations, but strictly it carries no unit.
How is shear strain used with the shear modulus?
Shear stress divided by shear strain gives the shear modulus (G) of a material, a measure of its stiffness against shearing. Once shear strain is known from a test, and the corresponding shear stress is also known, the two together characterise how rigid the material is.
For a related structural calculation on beams under load, see the beam deflection calculator. To work out the second moment of area of a beam cross-section, see the second moment of area calculator.