Shear Modulus Calculator

The shear modulus $G$, also called the modulus of rigidity, measures a material's resistance to being skewed — one face sliding parallel to the opposite face, without any change in volume.

$G = \frac{\tau}{\gamma} = \frac{F/A}{\Delta x / L}$

Symbols and SI units:

• $G$ — shear modulus, pascals (Pa), normally quoted in GPa
• $\tau$ — shear stress, Pa; the force $F$ in newtons divided by the area $A$ in m² parallel to the force
• $\gamma$ — shear strain, dimensionless (radians); the sideways displacement $\Delta x$ in m divided by the separation $L$ in m
• $\Delta x$ — lateral displacement of the sheared face, m

Typical values: structural steel $\approx 79$ GPa, aluminium $\approx 26$ GPa, copper $\approx 45$ GPa, rubber $\approx 0.5$ MPa.

When it applies: small elastic strains, below the proportional limit, in an isotropic material.

The assumption people forget: $A$ is the area parallel to the applied force, not the cross-section perpendicular to it as in tension. That single difference separates $G$ from Young's modulus $E$.

For an isotropic material the three elastic constants are not independent:

$G = \frac{E}{2(1 + \nu)}$

with $E$ Young's modulus in Pa and $\nu$ Poisson's ratio (dimensionless, about $0.30$ for metals). Since $\nu$ is near $0.3$, $G$ lands near $0.385E$ — steel's $200$ GPa gives roughly $77$ GPa. It follows that $G < E$ for every ordinary material.

$G$ is the constant that governs torsion of a circular shaft:

$\theta = \frac{TL}{GJ}, \qquad J = \frac{\pi d^4}{32}\ \text{(solid shaft)}$

• $\theta$ — angle of twist, radians
• $T$ — applied torque, N·m
• $L$ — shaft length, m
• $J$ — polar second moment of area, m⁴

A stiffer $G$, or a fatter shaft, means less twist — and because $J$ goes as $d^4$, diameter dominates.

The assumption people forget: $J = \pi d^4/32$ is for a solid circular shaft. Hollow and non-circular sections need their own $J$.

• Using the perpendicular cross-section for $A$ — shear stress divides by the area the force slides along.
• Confusing $G$ with $E$ — they answer different questions, and $G$ is roughly $0.4E$ for metals.
• Treating shear strain as a length — $\gamma = \Delta x/L$ is dimensionless, an angle in radians for small distortions.
• Quoting $G$ in newtons — it is a stress-like quantity, so Pa or GPa.
• Using the radius in $J = \pi d^4/32$ — that formula takes the diameter; with the radius the constant becomes $\pi r^4/2$.
• Leaving $\theta$ in radians when degrees were asked for — multiply by $180/\pi$.
• Applying $G = E/[2(1+\nu)]$ to an anisotropic material — wood and composites do not obey it.