# Shear Modulus Calculator

Created by Luis Hoyos
Reviewed by Steven Wooding
Last updated: Dec 21, 2022

Welcome to our shear modulus calculator, where you'll be able to calculate the shear modulus of a cubic element subjected to a force tangent to its area.

The shear modulus, also known as the modulus of rigidity, is a material property used in many applications. For example:

Read on to learn the modulus of rigidity equation used by this calculator and the modulus of rigidity of steel and many other materials.

## Formula for shear modulus calculation

Although an idealization, Hooke's law is a powerful tool for studying the behavior of materials that follow a linear stress-strain relationship. In a material subjected to shear stress, this law takes the form:

$\tau = G\gamma$

, where:

• $\tau$ — Shear stress;
• $G$ — Shear modulus; and
• $\gamma$ — Shear strain.

In this linear equation, the modulus of rigidity is the proportionality constant. A higher shear modulus implies we'll need to apply higher shear stress to get the same deformation.

A typical example of shear stress is the cubic element in the following image, subjected to a force tangent to its surface.

• Shear stress is the quotient of the force and the surface area $A$ over which that force $F$ is applied: $\tau = \frac{F}{A}$.
• Shear strain is the angle $\gamma$ owing to the deformation $\Delta x$. This angle is usually very tiny, and we can approximate it to $\gamma = \frac{\Delta x}{l}$.

Inputting the previous expressions into $\tau = G\gamma$ and solving for $G$, we finally get the shear modulus formula used by this calculator:

$G =\frac{FL}{A \Delta x}$

## Shear modulus of steel and other common materials

Sometimes, we know the material to use in advance and don't need a modulus of rigidity equation. In that case, here are the shear modulus values of common materials:

• Aluminum: 25 GPa (3.6 × 10⁶ psi).
• Brass: 35 GPa (5.1 × 10⁶ psi).
• Copper: 44 GPa (6.4 × 10⁶ psi).
• Iron: 77 GPa (3.8 × 10⁶ psi).
• Lead: 6 GPa (0.87 × 10⁶ psi).
• Silicone rubber: 0.002 GPa (0.00029 × 10⁶ psi).
• Steel: 75 GPa (10.9 × 10⁶ psi).

For specific materials used in engineering, look at the following table:

Source: Hibbeler RC Pearson Educación (2017).

Material

Shear modulus G

Aluminum wrought alloys

2014-T6

27 GPa (3.9 × 10⁶ psi)

6061-T6

26 GPa (3.8 × 10⁶ psi)

Cast iron alloys

Gray ASTM 20

27 GPa (3.9 × 10⁶ psi)

Malleable ASTM A-197

68 GPa (9.9 × 10⁶ psi)

Copper alloys

Red Brass C83400

37 GPa (5.4 × 10⁶ psi)

Bronze C86100

38 GPa (5.5 × 10⁶ psi)

Magnesium alloy

Am 1004-T61

18 GPa (2.6 × 10⁶ psi)

Steel alloys

Structural A-36

75 GPa (10.9 × 10⁶ psi)

Structural A992

Stainless 304

Tool L2

Titanium alloy

Ti-6Al-4V

44 GPa (6.4 × 10⁶ psi)

## What are the units of modulus of rigidity?

The modulus of rigidity unit is the pascal (Pa) in the International System of Units and pound per square inch (psi) in the United States customary units system. The modulus of rigidity of materials is typically so large that we express it in gigapascals (GPa) instead of Pascals.

## FAQ

### What is the shear modulus of steel?

77 GPa or 11×106 psi. Please consider that steel's shear modulus (modulus of rigidity) can vary slightly with temperature, composition, heat treatment, and mechanical working. The modulus of rigidity of steel materials used in industry can range from 75 to 77.2 GPa.

### How do I calculate shear modulus from Young's modulus?

To calculate shear modulus from Young's modulus:

1. Define Young's modulus and Poisson's ratio of the material.

2. Use the shear modulus formula:

G = E / [2(1 + ν)]

where:

• E — Modulus of elasticity in tension or compression, also known as Young's modulus;
• ν — Poisson's ratio, another material constant; and
• G — shear modulus (also known as modulus of rigidity).

You can use this relationship between the modulus of elasticity and the modulus of rigidity as long as it's a homogenous isotropic material.

### What is the modulus of rigidity of aluminum?

26 GPa or 3.8×106 psi. Please consider that the shear modulus of aluminum can vary slightly with temperature, composition, heat treatment, and mechanical working. The modulus of rigidity of aluminum materials used in industry can range from 26 to 28 GPa.

### Is modulus of rigidity a material property?

Yes, the modulus of rigidity is a material property, as it doesn't depend on the amount of material. Like many properties, it can vary as a function of other properties. For example, the shear modulus of metals slightly decreases with temperature.

### What is the modulus of rigidity of 6061-T6 aluminum?

26 GPa, or 3.7×106 psi. The modulus of rigidity of 6061-T6 aluminum doesn't differ much from other aluminum alloys, whose modulus of rigidity range from 24 to 28 GPa, or 3.7×106 to 4×106 psi.

Luis Hoyos
Input F, A, l, and Δx
Force magnitude (F)
lbf
Area over which the force acts (A)
ft²
Transverse length (l)
ft
Displacement (Δx)
ft
or input 𝜏 and γ
Shear stress (𝜏 = F/A)
psi
Shear strain (γ ≈ Δx/L)
Shear modulus (G)
psi
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