Shear Strain Calculator

Calculation mode: The geometry you are solving: simple shear for sideways sliding, or shaft torsion for twisting.

Engineering shear strain, gamma: The standard classroom shear strain measure. It is dimensionless, meaning it has no unit.

Tensor shear strain, epsilon_xy: A strain-component form used in tensor notation. In this calculator it is one-half of engineering shear strain for the same case.

Equivalent shear angle (rad): The deformation angle written in radians, which is often the preferred angle unit in mechanics formulas.

Equivalent shear angle (deg): The same deformation angle written in degrees for easier intuition.

Radius from center to point of interest: In shaft torsion, the distance from the shaft center to the exact point where you want the strain.

Angle of twist: The total shaft rotation over the entered shaft length.

Interpretation note: A short message that explains how to read the result in the chosen mode.

Mistake: Entering Sideways displacement (selected length unit) in mm but Original height (selected length unit) in cm.
Fix: Set one Length unit and enter all length values in that same unit before you calculate.

Mistake: Typing a degree value into Angle of twist while Angle unit for twist input is set to radians.
Fix: Change Angle unit for twist input so it matches your twist number exactly, then recalculate.

Mistake: Treating Engineering shear strain, gamma and Tensor shear strain, epsilon_xy as the same number.
Fix: Use Strain definition to emphasize to check which one your class, book, or software expects. Tensor shear strain is half of engineering shear strain.

Mistake: Using zero for Original height (selected length unit) or Shaft length (selected length unit) .
Fix: Enter a value greater than 0 for the active mode. Those inputs are divisors, so zero is not allowed.

Mistake: Expecting the shaft result to be the same everywhere in the shaft cross section.
Fix: Enter the specific Radius from center to point of interest (selected length unit) you care about. A larger radius gives a larger surface strain, and radius 0 gives zero strain at the center.

Mistake: Assuming a negative answer is automatically wrong when Engineering shear strain, gamma or Equivalent shear angle is below zero.
Fix: Keep the sign if your Sideways displacement or Angle of twist is negative. The sign shows shear direction, not a math error.

• The calculator uses the listed engineering formulas for simple shear and circular shaft torsion. It does not model complex shapes, noncircular shafts, or detailed stress distributions.
• For Simple shear, the engineering relation uses displacement divided by height. When deformation is not small, the geometric angle from arctangent can differ noticeably from the engineering strain value.
• For Shaft torsion, the result is the engineering shear strain at the entered Radius from center to point of interest (selected length unit), not automatically the whole cross section.
• The tensor result is reported as one-half of the engineering shear strain. If your course uses a different notation convention, match your class definition before comparing answers.
• All entered lengths must use the same selected unit. Mixed units will give wrong results even though the strain itself is dimensionless.
• Original height (selected length unit) and Shaft length (selected length unit) must be greater than 0. Radius may be 0, which gives zero strain at the shaft center in torsion mode.
• Negative displacement or negative twist angle is allowed and kept as a signed result so direction information is not lost.

Core equations

This calculator uses one formula set for simple shear and another for circular shaft torsion. It then reports both engineering shear strain and tensor shear strain so you can compare the two common definitions.

γ = x / h

For simple shear, engineering shear strain γ equals sideways displacement x divided by original height h [1].

ψ = atan(x / h)

The equivalent geometric shear angle ψ is found with arctangent. For small deformations, ψ in radians is very close to engineering shear strain.

γ = rφ / L

For a circular shaft in torsion, engineering shear strain at radius r equals radius times twist angle φ in radians divided by shaft length L [2][3].

εxy = γ / 2

Tensor shear strain ε_xy is one-half of engineering shear strain in this small-strain convention [4].

deg = rad x 180 / π

The calculator also converts the equivalent angle between radians and degrees for display.

How the calculator handles units

Shear strain is dimensionless, so the length unit cancels out as long as all active length inputs use the same selected unit. In shaft torsion mode, the twist input is first interpreted using your chosen angle unit. If you enter degrees, the calculator converts degrees to radians before using the torsion equation.

Worked mini-example

Simple shear example: if Sideways displacement is 2 mm and Original height is 50 mm, then engineering shear strain is 2 / 50 = 0.04.

γ = 2 / 50 = 0.04

The tensor shear strain is half of that, so 0.02.

εxy = 0.04 / 2 = 0.02

The equivalent geometric angle is atan(0.04), which is about 0.03998 rad or about 2.2906 deg.

ψ = atan(0.04) = 0.03998 rad = 2.2906 deg

Interpretation tips

A larger magnitude of engineering shear strain means more angular distortion. In shaft torsion mode, strain increases linearly with radius, so the outer surface has the largest strain and the center has zero strain.

Assumptions used here

This tool follows the standard introductory mechanics relations above. It is most useful for small-strain interpretation and for circular shafts in torsion. If deformation is large, geometry is unusual, or material behavior is beyond the basic model, real results can differ.