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Generated on: July 20, 2026 at 09:33 PM

Shear Modulus Calculator

Inputs

Shear force500 N
Area10 cm²
Displacement0.1 mm
Original length50 mm

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Physics

Shear Modulus Calculator

Calculate the shear modulus (modulus of rigidity) G from a shear test. Enter the shear force, the area it acts over, the sideways displacement and the original length to get G = τ/γ, with the shear stress and shear strain shown alongside.

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G = \dfrac{\tau}{\gamma} = \dfrac{?\,\text{MPa}}{?} = ?\,\text{GPa}

Details

\tau = \dfrac{F}{A} = \dfrac{500\,\text{N}}{10\,\text{cm²}} = ?\,\text{MPa}
\gamma = \dfrac{\Delta x}{L} = \dfrac{0.1\,\text{mm}}{50\,\text{mm}} = ?

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Shear Modulus

The shear modulus, also called the modulus of rigidity, is a single number that captures how strongly a material resists a change of shape — a twisting or sliding load that skews it sideways without pulling it apart. Steel resists shear about three times more strongly than aluminium and thousands of times more than rubber, and that difference is exactly what the modulus measures. This calculator works it out from a simple shear test: a force pushing parallel to a face of known area, over a sample of known length, that slides sideways by a measured amount.

Where the formula comes from

A sideways force produces two things. The shear stress is the force shared over the face it acts on, τ=F/A\tau = F/A, measured in pascals. The shear strain is how far the surface slides relative to the length between the sheared faces, γ=Δx/L\gamma = \Delta x / L, a pure ratio. In the elastic region — before any permanent distortion — stress and strain stay proportional, and the constant that links them is the shear modulus, G=τ/γG = \tau/\gamma. Substituting the two definitions gives G=(F L)/(A Δx)G = (F\,L)/(A\,\Delta x).

Formula

QuantitySymbolMeaning
Shear modulusGGG=τγ=F LA ΔxG = \dfrac{\tau}{\gamma} = \dfrac{F\,L}{A\,\Delta x}
Shear stressτ\tauτ=FA\tau = \dfrac{F}{A}
Shear strainγ\gammaγ=ΔxL\gamma = \dfrac{\Delta x}{L}
Shear forceFFLoad applied parallel to the face
AreaAAArea the shear force acts across

Because strain has no units, GG carries the units of stress — pascals — and is most often quoted in gigapascals (GPa).

Worked example

A rubber-like block with a 10 cm² face (A=1×10−3 m2A = 1\times10^{-3}\ \text{m}^2) and length L=50 mm=0.05 mL = 50\ \text{mm} = 0.05\ \text{m} is pushed sideways with F=500 NF = 500\ \text{N}, and the top face slides by Δx=0.1 mm\Delta x = 0.1\ \text{mm}:

τ=FA=5001×10−3=5×105 Pa=0.5 MPaγ=ΔxL=0.00010.05=2×10−3G=τγ=5×1052×10−3=2.5×108 Pa=0.25 GPa\begin{aligned} \tau &= \frac{F}{A} = \frac{500}{1\times10^{-3}} = 5\times10^{5}\ \text{Pa} = 0.5\ \text{MPa} \\ \gamma &= \frac{\Delta x}{L} = \frac{0.0001}{0.05} = 2\times10^{-3} \\ G &= \frac{\tau}{\gamma} = \frac{5\times10^{5}}{2\times10^{-3}} = 2.5\times10^{8}\ \text{Pa} = 0.25\ \text{GPa} \end{aligned}

A quarter of a gigapascal places the sample among soft plastics, far below the tens of gigapascals of a structural metal.

A block with a 1.5 in² face is pushed sideways with F=110 lbfF = 110\ \text{lbf} over a length of L=2 inL = 2\ \text{in}, and the top face slides by Δx=0.004 in\Delta x = 0.004\ \text{in}. Keeping every length in inches makes the strain a clean ratio:

τ=FA=1101.5=73.3 psiγ=ΔxL=0.0042=2×10−3G=τγ=73.32×10−3=36,700 psi\begin{aligned} \tau &= \frac{F}{A} = \frac{110}{1.5} = 73.3\ \text{psi} \\ \gamma &= \frac{\Delta x}{L} = \frac{0.004}{2} = 2\times10^{-3} \\ G &= \frac{\tau}{\gamma} = \frac{73.3}{2\times10^{-3}} = 36{,}700\ \text{psi} \end{aligned}

That 36,700 psi is the imperial counterpart of 0.25 GPa.

Shear modulus versus Young's modulus

Young's modulus EE measures resistance to stretching — a pull along the axis that makes a sample longer. Shear modulus GG measures resistance to shape change — a force parallel to a face that skews the sample. For an isotropic material the two are tied together through Poisson's ratio ν\nu:

G=E2 (1+ν)G = \frac{E}{2\,(1 + \nu)}

With a typical ν≈0.3\nu \approx 0.3, the shear modulus comes out at roughly 0.38 E0.38\,E, which is why GG is usually about a third of EE for common metals. Steel, for example, has E≈200 GPaE \approx 200\ \text{GPa} and G≈79 GPaG \approx 79\ \text{GPa}.

Typical values

The range across materials is enormous: rubber sits near 0.0003 GPa, aluminium around 26 GPa, glass near 26 GPa, copper about 45 GPa, and steel near 79 GPa. A higher shear modulus means the material barely distorts under a twisting or sliding load — useful for a drive shaft that must not wind up under torque, less relevant for a flexible gasket.

Limitations

The modulus only describes the elastic region, where the material springs back to its original shape once the load is removed. Past the yield point the relationship stops being linear, the sample deforms permanently, and the shear modulus no longer applies. It also measures rigidity, not strength: a stiff material can still fail at a low load. The test assumes a uniform sample loaded in pure shear, with the force spread evenly over the face and no bending or twisting mixed in.

Frequently Asked Questions (FAQ)

What is the formula for shear modulus?

The shear modulus is G = τ/γ, the ratio of shear stress to shear strain. Shear stress is the force spread over the area it acts on, τ = F/A, and shear strain is the sideways slide as a fraction of the length, γ = Δx/L. Combining them gives G = (F·L)/(A·Δx). Because strain is dimensionless, G has the units of stress — pascals — and is usually quoted in gigapascals (GPa).

How is shear modulus different from Young's modulus?

Young's modulus E describes stretching: a pull along the axis that makes a sample longer. Shear modulus G describes twisting or sliding: a force parallel to a face that skews the shape without changing its volume much. For an isotropic material the two are linked through Poisson's ratio ν by G = E / (2(1 + ν)), so G is roughly a third of E for typical metals.

What are typical shear modulus values?

Rough figures: rubber near 0.0003 GPa, aluminium about 26 GPa, glass around 26 GPa, copper about 45 GPa, and steel near 79 GPa. A larger shear modulus means the material barely distorts under a twisting or sliding load — steel resists shear roughly three times more strongly than aluminium.

Is shear modulus the same as rigidity?

Yes. "Modulus of rigidity" is another name for the shear modulus, because it measures how strongly a material resists a change of shape at constant volume. It governs how much a shaft twists under torque or how a bolted joint slips, which is why it appears throughout mechanical and structural design.

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