Shear Modulus Calculator - G from Stress, Strain, or E and nu

A shear modulus calculator computes G, also called the modulus of rigidity, a material's resistance to being deformed by a shear force. It turns a shear force, the area it acts on, the original length, and the lateral deformation into G, or derives the same number from Young's modulus and Poisson's ratio. Because G quantifies stiffness against sliding, it is the quantity you need when a shaft twists, a bolt shears, or a rubber pad deflects.

• • Shaft and torsion design: Read G to predict how much a shaft twists for a given torque, since the angle of twist depends on the modulus of rigidity.
• • Bolt and fastener shear checks: Estimate whether a bolt yields in shear by comparing the applied shear stress to G-linked elastic limits.
• • Spring and rubber pad deflection: Compute sideways deflection of elastomeric mounts and springs, where G governs shear stiffness.
• • Materials data lookup: Recover G from tabulated E and nu when a supplier lists only those for a metal or polymer.

Shear is different from stretching. A tensile load pulls a body longer and thinner, but a shear load slides one face parallel to the opposite face. The modulus of rigidity measures exactly that sliding stiffness, which is why it shows up in torsion rather than tension.

The same material has three stiffness numbers: Young's modulus E for pulling, the bulk modulus K for squeezing, and shear modulus G for sliding. This calculator produces G and shows how it connects to E and nu.

Shear modulus measures resistance to sliding, while the bulk modulus calculator measures resistance to uniform compression, so they describe two different stiffness responses of the same material.

The calculator uses the definition of shear modulus as shear stress divided by shear strain. It computes shear stress from the force and area, then shear strain from the deformation and length, and divides the two. A second route applies G = E / (2(1+nu)) when Young's modulus and Poisson's ratio are known.

• F: Shear force applied parallel to the face in newtons.
• A: Cross-sectional area the shear force acts over in square metres.
• L: Original length of the body along the shear direction in metres.
• delta_x: Lateral deformation, the slide of the top face relative to the bottom, in metres.
• E: Young's modulus (axial stiffness) in pascals, used in the elastic-constants route.
• nu: Poisson's ratio, the lateral-to-axial strain ratio, dimensionless.

Shear stress tau is force per unit area, so doubling the force doubles tau while doubling the area halves it. Shear strain gamma is a ratio of two lengths with no units; a gamma of 0.001 means the top slid one-thousandth of the body's height.

When you provide E and nu instead of a measured force, this shear modulus calculator switches to G = E / (2(1+nu)). For steel with E = 200 GPa and nu = 0.30 this gives about 76.9 GPa, matching published tables.

According to Wikipedia (Shear modulus), the shear modulus G equals shear stress divided by shear strain, and for isotropic materials G = E / (2(1+nu)).

The tau = F / A step is exactly the output of the shear stress calculator, which you can use to confirm the shear stress before dividing by shear strain.

Four ideas sit under the output. Read them once and the worked example falls into place.

The link to E and nu is why the calculator offers a second route: with a material data sheet, you skip the force measurement entirely.

Because G, E, and K describe the same elastic solid, knowing two lets you recover the third using the elastic-constants relations.

The link G = E / (2(1+nu)) is one of the relations solved by the elastic constants calculator, which recovers any constant from the other two.

Two input routes are available in this shear modulus calculator. Use the force-and-geometry route when you measured a load, or the material-data route when you have E and nu from a table.

1. 1 Enter the shear force: Type the parallel force in newtons. Multiply kilonewtons by 1000 before typing.
2. 2 Enter the cross-sectional area: Type the area in square metres. A 100 mm by 100 mm face is 0.01 m2; a 10 mm by 10 mm face is 0.0001 m2.
3. 3 Enter the length and deformation: Type the original length and lateral slide in metres. Use 0.001 m for a 1 mm slide on a 1 m block.
4. 4 Or enter E and nu instead: If you have a material card, type Young's modulus in pascals and Poisson's ratio. This route is used when E is greater than zero.
5. 5 Read the four outputs: The panel shows shear stress in Pa, shear strain, G in Pa, and G in GPa for table comparison.
6. 6 Compare to a tabulated value: Check G in GPa against the published shear modulus. Steel reads about 77 GPa and aluminum about 26 GPa.

A steel block with E = 200 GPa and nu = 0.30 returns G = 76.92 GPa. The same block loaded with 1000 N over 0.01 m2 and sheared 1 mm on a 1 m height returns G = 100 MPa, because the load was light. This shear modulus calculator reaches both values from the same inputs.

The gamma = delta_x / L step matches the output of the shear strain calculator, which returns the dimensionless strain from the deformation and the original length.

Reasons to use this shear modulus calculator instead of evaluating tau over gamma by hand or hunting through material tables.

• • Both measurement and data-sheet routes: The same panel returns G from an applied force or from E and nu, so it works in the lab and at the desk.
• • Stress and strain shown separately: Shear stress and shear strain appear as their own outputs, so you can verify each step before reading G.
• • GPa output for table comparison: G is shown in both pascals and gigapascals, matching the units in most materials handbooks.
• • Validates impossible inputs: Zero or negative area, length, deformation, or nu below -1 return a clear message instead of a wrong number.
• • Closed-form and repeatable: Every result is a direct evaluation of G = tau / gamma or G = E / (2(1+nu)), repeatable by hand.
• • Connects to related mechanics tools: The shear stress and shear strain outputs feed into the dedicated shear stress and shear strain calculators.

Because G = tau / gamma is Hooke's law for shear, the Hooke's law calculator provides the axial version that pairs with this sliding-stiffness result.

What changes the answer, and what the simple elastic model cannot capture.

• • It assumes linear elastic behaviour. Once the material yields, the stress-strain line is no longer straight and G from tau / gamma no longer represents the stiffness.
• • It assumes an isotropic solid. Anisotropic materials such as wood or fibre composites need a full stiffness tensor, not a single G.

Use the result as the elastic shear stiffness; real parts near their yield point will deviate from the computed G.

For a measured load the two routes agree only if the strain is small and the material is genuinely isotropic and elastic.

According to Engineering Toolbox (Young's and shear modulus of materials), typical shear modulus values are about 77 GPa for steel and about 26 GPa for aluminum at room temperature.

According to Wikipedia (Shear modulus), the shear modulus of most materials decreases with increasing temperature and changes abruptly at phase transitions.

Loading errors in the force and area feed straight into shear stress, so the stress calculator is useful when you need normal and shear stress from the same load set.