Shear Strain Calculator
Choose a calculation method, enter your values, and get shear strain instantly with the full working shown. The displacement method uses lateral movement and element height. The stress-modulus method applies Hooke's law for shear. The torsion method handles circular shafts under twist, giving both the strain at any radius and the maximum surface strain.
Formula
Worked example
A steel block 50 mm tall is subjected to a lateral shear displacement of 2 mm: gamma = 2 / 50 = 0.04 rad (about 2.29 degrees). That same block, if made of structural steel (G = 80,000 MPa), carries a shear stress of 0.04 x 80,000 = 3,200 MPa - well above yield, so the small-displacement scenario would be unusual for steel. More realistically, at tau = 40 MPa: gamma = 40 / 80,000 = 0.0005 rad.
What is shear strain?
Shear strain (symbol gamma, unit: radians) measures the angular distortion a material element undergoes when shearing forces are applied parallel to one of its faces. Unlike normal strain, which describes elongation or compression along an axis, shear strain describes the change in angle between two originally perpendicular lines in the material. It is dimensionless: a shear strain of 0.01 means the right angle between two originally perpendicular planes has changed by 0.01 radians (about 0.57 degrees). Shear strain is central to structural and mechanical engineering. Beams resist transverse loads partly through shear, shafts under torque experience shear strain across their cross-section, adhesive joints fail in shear, and soils in slope stability problems are assessed in terms of shear. Understanding shear strain is a prerequisite for computing shear stress, predicting yielding, and designing against fatigue.
Three ways to calculate shear strain
This calculator supports all three standard methods used in practice: 1. Displacement method (gamma = Δx / h): Use when you can directly measure how far one face of an element moves laterally relative to the opposite face. Δx is the lateral displacement and h is the perpendicular dimension of the element. This is often the most direct method for testing or field measurement. 2. Stress-modulus method (gamma = tau / G): Hooke's law for shear states that, within the elastic range, shear stress tau is proportional to shear strain gamma through the shear modulus G. Given the applied shear stress and the material's G, this is the quickest route to strain. This method is only valid for isotropic, linearly elastic materials below their yield point. 3. Torsion method (gamma = rho * phi / L): When a circular shaft of length L is twisted by an angle phi, every longitudinal line on the shaft spirals. The shear strain at any radius rho equals rho * phi / L, and the maximum strain occurs at the outer surface (rho = c): gamma_max = c * phi / L. The linear distribution of torsional shear strain with radius is a key assumption in the classical (Saint-Venant) torsion theory.
Shear modulus, shear stress, and Hooke's law for shear
The shear modulus G (also called the modulus of rigidity) is a material constant that relates shear stress to shear strain in the elastic regime: tau = G * gamma. It has the same units as stress (pascals or MPa). For isotropic materials, G is related to Young's modulus E and Poisson's ratio nu by G = E / (2 * (1 + nu)). For steel, E is about 200,000 MPa and nu is about 0.3, giving G = 200,000 / (2 * 1.3) = 76,923 MPa, commonly rounded to 80,000 MPa in calculations. Below the elastic limit, the stress-strain relationship is linear and reversible: remove the load and the deformation recovers. Above the shear yield stress (approximately 0.577 * tensile yield stress for metals, from the von Mises criterion), the response becomes non-linear and the simple gamma = tau / G formula no longer applies without plasticity corrections.
Small-angle approximation and when it breaks down
The definition gamma = tan(theta) is exact, where theta is the actual angle of distortion. For small angles (theta less than about 10 degrees or 0.175 radians), tan(theta) is approximately equal to theta in radians, so gamma is numerically equal to the angle. This calculator shows the converted angle in degrees alongside the dimensionless strain value so you can judge whether the small-angle assumption holds. For rubber, elastomers, and soft biological tissues, shear strains above 0.5 are common and the large-strain (finite deformation) framework is required. The formulas in this calculator assume small strains (gamma << 1), which is appropriate for metals, ceramics, stiff polymers, and most structural engineering applications. When gamma exceeds roughly 0.1, results should be treated as approximate.
Typical shear modulus values for common engineering materials
| Material | Shear modulus G (GPa) | Approx. shear yield strain |
|---|---|---|
| Structural steel (A36) | 79 - 81 | ~0.001 |
| Stainless steel (304) | 73 - 75 | ~0.001 |
| Aluminium alloy (6061) | 26 - 27 | ~0.002 |
| Copper (pure) | 44 - 48 | ~0.001 |
| Titanium (Grade 5) | 41 - 44 | ~0.005 |
| Cast iron (grey) | 40 - 45 | ~0.0005 |
| Concrete (compression) | ~10 - 20 | N/A (brittle) |
| Natural rubber | 0.0003 - 0.001 | ~0.5 - 5 |
| HDPE polymer | ~0.25 - 0.40 | ~0.02 |
| Glass (borosilicate) | ~26 - 30 | N/A (brittle) |
Use these G values in the shear stress / modulus mode. Values are approximate and vary by alloy, temper, and temperature.
Frequently asked questions
What are the units of shear strain?
Shear strain is dimensionless. It is the ratio of a displacement to a length (Δx/h), which cancels units, or equivalently an angle measured in radians. Radians are themselves dimensionless (arc length divided by radius). In practice, shear strain is quoted as a plain number or in rad/rad to indicate it is an angular deformation. Some sources express it in microstrains (1 microstrain = 1 x 10^-6).
What is the difference between shear strain and shear stress?
Shear stress (tau) is a force per unit area (units: Pa or MPa). Shear strain (gamma) is the resulting angular deformation (dimensionless). They are related by the shear modulus G: tau = G x gamma. Shear stress tells you how hard the material is being loaded; shear strain tells you how much it has actually deformed in response.
How do I find shear strain in a circular shaft under torsion?
Use the torsion formula: gamma = rho x phi / L, where rho is the radial distance from the shaft axis, phi is the total angle of twist (in radians) over the shaft length L. The maximum shear strain occurs at the outer surface (rho = c, the outer radius): gamma_max = c x phi / L. This distribution is linear: strain is zero at the axis and maximum at the surface, which is why solid shafts are sometimes replaced by hollow ones for weight saving without much strength penalty.
At what shear strain value does a material yield?
The shear yield strain is tau_y / G, where tau_y is the shear yield stress. For structural steel with a tensile yield stress of 250 MPa, the shear yield stress is approximately 0.577 x 250 = 144 MPa (von Mises criterion), giving a shear yield strain of 144 / 80,000 = 0.0018 rad. The reference table above lists approximate yield strain values for common materials. These are approximate because actual yield depends on alloy, heat treatment, strain rate and temperature.
What is the relationship between shear strain and engineering shear strain?
The engineering shear strain (used in most structural analysis) is exactly gamma = Δx / h, which equals 2 x epsilon_xy where epsilon_xy is the tensorial (mathematical) shear strain. The factor of two arises from the symmetric definition used in tensor notation. This calculator uses engineering shear strain throughout, which is the standard in most textbooks and codes.
Can I use this calculator for soil mechanics?
Yes, the displacement and stress/modulus modes both apply to soils. Soil shear modulus G is commonly determined from laboratory tests (triaxial, simple shear, or resonant column) or estimated from SPT/CPT in-situ data. Typical G values for medium-dense sand range from 20 MPa to 100 MPa. For clays and soft soils, G is much lower. Enter the appropriate G for your soil type in the stress/modulus mode.