Shear Modulus Calculator (G = τ/γ, or from E and ν)

Understanding the shear modulus calculator — concept and principle

This calculator finds the shear modulus (modulus of rigidity) G from a shear test, using the force, sheared area, height and sideways displacement, or from Young's modulus and Poisson's ratio. It also gives the shear stress, shear strain and shear angle. Materials and mechanical engineering students use it.

G is shear stress per radian of tilt; a tiny 0.08° lean already takes 100 MPa in steel-like metal.

How it is calculated

τ = F ÷ A; γ = Δx ÷ L₀

G = τ ÷ γ

From E and ν: G = E ÷ (2(1 + ν))

F in kN, A in mm², heights in mm; G in GPa and MPa.

The principle

Push the top face of a block sideways while holding the bottom, and the block leans over. The shear stress is the force per area of the face, and the shear strain is how far the face moves relative to the block's height, which is also the angle of lean in radians for small angles. Their ratio is the shear modulus, the material's resistance to change of shape. For isotropic materials it is tied to Young's modulus through Poisson's ratio.

Worked example

Default inputs: shear test, F = 200 kN, A = 2,000 mm², L₀ = 50 mm, Δx = 0.0667 mm.

τ = 200,000 ÷ 2,000 = 100 MPa.
γ = 0.0667 ÷ 50 = 0.001334 rad, a shear angle of 0.07643°.
G = 100 ÷ 0.001334 = 74,963 MPa = 74.96 GPa.

Each point is the calculator re-run with only “Sheared area A (mm²)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Sheared area A (mm²)	Shear modulus G (GPa)
500	299.9
1,000	149.9
1,500	99.95
2,000	74.96
2,500	59.97
3,000	49.98
4,000	37.48

How to read the result

OpenStax lists typical shear moduli of about 75 GPa for steel, 44 GPa for copper and 25 GPa for aluminium, so this result is typical of steel. With E = 200 GPa and ν = 0.30 the formula gives 76.9 GPa, close to the test value. Large differences suggest measurement error or that the material is not isotropic.

Common mistakes

Using the height L₀ in the wrong direction.
Mixing kN with N or mm² with m².
Testing beyond the elastic range.
Using the E–ν formula for wood or composites.

Keep in mind: Real designs use certified material data and applicable codes; consult a qualified engineer.

Frequently asked questions

What is the shear modulus of steel?

About 75–80 GPa.

How is G related to E?

G = E/(2(1 + ν)) for isotropic materials.

Why measure in radians?

Shear strain is defined as the angle of deformation, Δx/L₀.

Do fluids have a shear modulus?

No; fluids flow under any shear stress.

Sources: OpenStax, University Physics Volume 1 (2016), section 12.3 "Stress, Strain, and Elastic Modulus" · MIT OpenCourseWare 2.001 Mechanics & Materials I (Fall 2006), lecture 13 notes

Checked 9 Nov 2026. OpenStax, University Physics Volume 1 (2016), §12.3 “Stress, Strain, and Elastic Modulus” (S = (F∥/A)/(Δx/L₀), Table 12.1); G = E/(2(1 + ν)): MIT OpenCourseWare 2.001 Mechanics & Materials I (Fall 2006), lecture 13 notes.

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