Shear Strain Calculator

This shear strain calculator finds the strain due to shearing stresses in different situations (shear force or torque).

Shear stresses act in almost every mechanical situation, and the strain energy of these shear stresses is a powerful tool for solving deflection problems using energy methods. For this reason, it's essential to be aware of shear strain and the equations behind it.

If you're interested in knowing where things come from, in the following sections, we present the shear strain equations behind this calculator and the max shear strain formula for a shaft under torsion.

Suppose you apply two tangent forces on the opposite faces of a cubic or rectangular element. If that occurs, one of the faces will displace a distance $x$ relative to the other.

The shear strain is no more than the angle $\gamma$ (shown in the image) caused by the displacement $x$. In practice, displacement $x$ is much smaller than $h$, and $\gamma$ is a small angle.

Depending on the situation and the information available, you'll need a different equation to calculate the shear strain.

Cubic object with known displacement

If we know the displacement $x$, we can calculate the shear strain with the following formula:

, where:

• $\gamma$ — Shear strain;
• $x$ — Displacement due to shear stress caused by forces $F_{||}$; and
• $h$ — Transverse dimension of the element, as shown in the previous image.

Strictly speaking, $x/h$ is not the shear strain ($\gamma$) but the tangent of the shear strain. Even so, we can say $x/h$ and ($\gamma$) are practically the same, as for small $\gamma$ angles, $\tan(\gamma) \approx \gamma$

Element with known shear modulus subjected to shear stress

We usually don't know the displacement $x$ due to shear stress, and when that occurs, we can calculate the shear strain this way:

, where:

• $\tau$ — Shear stress applied to the element; and
• $G$ — Shear modulus, also known as modulus of rigidity, of the material.

Contrary to the previous case, this formula provides the angle $\gamma$ instead of its tangent.

Circular shaft subjected to torsion

When we apply a torque to a circular shaft, every element parallel and perpendicular to the shaft axis suffers shear strain. We can calculate this shear strain with the formula:

where:

• $\rho$ — Distance from the axis to the point at which we want to know the shear strain;
• $\phi$ — Angle of twist of the transverse section (obtain it with the angle of twist calculator); and
• $L$ — Length of the shaft.

As you can note from the previous formula, a greater $\rho$ implies a higher strain. At the shaft surface, where $ρ$ equals the shaft radius ($c$), the shear strain is maximum:

The shear strain unit is radian, a dimensionless unit. For this reason, the shear strain unit is often omitted. Besides, the normal strain unit is also radians or dimensionless.

To calculate the shear strain γ, divide the shear stress 𝜏 by the shear modulus G. For example, in a brass element (G = 3.5×10¹⁰ Pa) subjected to a shear stress of 𝜏 = 8.0×10⁶ Pa, shear strain equals γ = (8.0×10⁶ Pa)/(3.5×10¹⁰ Pa) = 2.3×10⁻⁴.

That's how to calculate shear strain! You can also use our shear strain calculator.

The shear strain symbol is the Greek letter γ, gamma. Depending on the direction and plane of actuation of the stress, γ can have different subscripts:

• The shear strain symbol γ_xy represents the strain in the XY plane caused by the shearing stresses 𝜏_xy and 𝜏_yx.
• γ_yz represents the same in the YZ plane, caused by 𝜏_yz and 𝜏_zy, the stresses in that plane.
• γ_zx represents the same in the ZX plane, caused by the 𝜏_yz and 𝜏_zy stresses.

To calculate the modulus of rigidity G in a torsion test:

1. Organize the list of angles of twist (in radians) corresponding to each torque applied.
2. Plot each torque (vertical axis) against its corresponding angle (horizontal axis).
3. Get the slope of the part of the graph that follows a linear relationship.
4. As torque T and angle of twist ϕ relate each other linearly by the equation: T = (JG/L)ϕ, to get G, equal the slope of the graph to JG/L, and solve for G. The result would be G = slope × L/J.

• To calculate the strain energy density for a material under plane shearing stress, use the equation u = 𝜏²/(2G), where:

  • 𝜏 — Shear stress applied to the material;
  • G — Shear modulus of the material; and
  • u — Strain energy density (energy per unit volume).

• To calculate the total strain energy of shear stress U, integrate the strain energy density through the entire volume V:

  U = ∫[𝜏²/(2G)] dV