Learn: Shear modulus
Young’s modulus describes resistance to stretching; shear modulus describes resistance to a change of shape. This relationship lets you find the shear stiffness of an isotropic material from two familiar elastic properties.
What the formula is saying
The factor 2(1 + ν) connects normal and shear deformation in isotropic elasticity. Divide E by this complete factor, rather than dividing by 2 and then adding ν.
Read the symbols in plain language
- E
- Young’s modulus
Elastic stiffness: the stress change needed for a unit strain in the stated material model. It is not a strength limit.
PaOne pascal is one newton per square metre. 1 MPa = 10⁶ Pa.
- ν
- Poisson's ratio
The negative transverse-to-axial strain ratio of the isotropic material; this describes lateral contraction or expansion.
ratio / no unitA dimensionless ratio has no physical unit; 0.01 as a ratio is 1% when the percent option is selected.
- G
- Result to find
Shear modulus. Divide the tensile stiffness by the isotropic conversion factor.
Pa
Sort out the units first
Enter E in pascals, or choose a supported stress unit. G has the same stress dimension. For example, 200 GPa equals 200,000,000,000 Pa, not 200 Pa.
Assumptions before calculating
Treat the material as homogeneous and linearly elastic, with small strains. Use properties for the actual temperature and loading direction; the calculation is a model of behaviour before yielding, not a failure test.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find G and explain the result in the stated output unit.
- E · Young’s modulus
- 200000000000 Pa
- ν · Poisson's ratio
- 0.3
Build the elastic conversion factor
Add one to Poisson’s ratio before multiplying the whole bracket by two.
2 × (1 + (0.3)) = 2.6Find shear modulus
Divide the tensile stiffness by the isotropic conversion factor.
(200000000000) ÷ (2.6) ≈ 76923076920 Pa
Does this worked answer make sense?
At ν = 0.30, G is E/2.6 and is smaller than E. Doubling E at fixed ν doubles G.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- E · Young’s modulus
- 70000000000 Pa
- ν · Poisson's ratio
- 0.33
Build the elastic conversion factor
Add one to Poisson’s ratio before multiplying the whole bracket by two.
2 × (1 + (0.33)) = 2.66Find shear modulus
Divide the tensile stiffness by the isotropic conversion factor.
(70000000000) ÷ (2.66) ≈ 26315789470 Pa
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- E · Young’s modulus
- 30000000000 Pa
- ν · Poisson's ratio
- 0.2
Find: Learn: Shear modulus
A hint, not the answer
The factor 2(1 + ν) connects normal and shear deformation in isotropic elasticity. Divide E by this complete factor, rather than dividing by 2 and then adding ν.
Enter E in pascals, or choose a supported stress unit. G has the same stress dimension. For example, 200 GPa equals 200,000,000,000 Pa, not 200 Pa.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Build the elastic conversion factor
Add one to Poisson’s ratio before multiplying the whole bracket by two.
2 × (1 + (0.2)) = 2.4Find shear modulus
Divide the tensile stiffness by the isotropic conversion factor.
(30000000000) ÷ (2.4) = 12500000000 Pa
Avoid the common trap
Keep the brackets around 2(1 + ν). Poisson’s ratio is a decimal, so 0.30 must not be entered as 30. Young’s modulus and shear modulus are not interchangeable.
When this method applies — and when it does not
The relationship requires isotropic elasticity, E > 0 and −1 < ν < 0.5. Do not use it to replace a measured directional shear modulus for timber, laminates or other anisotropic materials.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Shear modulus. Use the relevant elastic-response, equilibrium, constitutive-relations or shear-and-torsion module; the lesson states its particular sign convention.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.