Shear Modulus Calculator
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:
- In shear strain analysis, the shear modulus is necessary to find the shear strain due to shear stress.
- It's also needed to calculate the angle of twist in shafts subjected to torsion.
- From the theoretical point of view, it has great relevance, as shear modulus, Young's modulus, and Poisson's ratio are the constants that form the generalized Hooke's law for homogeneous isotropic materials.
Read on to learn the modulus of rigidity equation used by this calculator and the modulus of rigidity of steel and many other materials. Pick your unit system — American (Imperial) or Metric (SI) — and the calculator handles the conversions for you.
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:
τ = G · γ
- τ (tau) — Shear stress;
- G — Shear modulus; and
- γ (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 a cubic element subjected to a force tangent to its surface. Shear stress is the quotient of the force F and the surface area A over which that force is applied: τ = F / A.
Shear strain is the angle γ owing to the deformation Δx. This angle is usually very tiny, and we can approximate it to γ = Δx / L, where L is the height of the element.
Inputting the previous expressions into τ = G·γ and solving for G, we finally get the shear modulus formula used by this calculator:
G = F · L / (A · Δx)
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, switch the calculator to Material lookup mode, or use the shear modulus values of common materials below:
| Material | Shear modulus G (GPa) | Shear modulus G (×10⁶ psi) |
|---|---|---|
| Aluminum | 25 | 3.6 |
| Brass | 35 | 5.1 |
| Copper | 44 | 6.4 |
| Iron | 77 | 11.2 |
| Lead | 6 | 0.87 |
| Silicone rubber | 0.002 | 0.00029 |
| Steel | 75 | 10.9 |
| Titanium | 41 | 5.9 |
| Glass | 26 | 3.8 |
| Tungsten | 161 | 23.3 |
These are typical reference values; the exact modulus of rigidity of a specific alloy or grade can vary with composition, temperature, and processing.
What are the units of modulus of rigidity?
Because the shear modulus is the ratio of shear stress (force per unit area) to shear strain (a dimensionless angle), it has the same units as stress or pressure:
- In the Metric (SI) system, shear modulus is expressed in pascals (Pa). Engineering values are usually quoted in megapascals (MPa) or gigapascals (GPa), since 1 GPa = 10⁹ Pa.
- In the American (Imperial) system, shear modulus is expressed in pounds per square inch (psi), most often as ×10⁶ psi (Mpsi) or ksi (kips per square inch).
For reference: 1 GPa ≈ 145,038 psi ≈ 0.145 ×10⁶ psi. This calculator reports your result in GPa, MPa, ksi, and ×10⁶ psi at the same time, so you can use whichever unit your project requires.
How to use this shear modulus calculator
- From force & deformation — Enter the tangential force F, the area A it acts on, the element height L, and the resulting deflection Δx. The calculator returns G = F·L/(A·Δx), plus the shear stress and shear strain.
- From shear stress & strain — If you already know the shear stress τ and shear strain γ, the calculator returns G = τ/γ directly.
- Material lookup — Pick a material from the list to instantly read its typical shear modulus in every unit.
Frequently Asked Questions
What is the shear modulus (modulus of rigidity)?
The shear modulus G, or modulus of rigidity, is a material property that measures a material's resistance to shear deformation. It is defined as the ratio of shear stress to shear strain (G = τ/γ) within the elastic range, where the material follows Hooke's law.
What is the shear modulus of steel?
The shear modulus of steel is approximately 75 GPa (10.9 ×10⁶ psi). Values for specific steel grades typically range from about 75 to 80 GPa.
How do I calculate the shear modulus from a force?
Use G = F·L/(A·Δx). Divide the tangential force F by the area A to get the shear stress τ, divide the deformation Δx by the height L to get the shear strain γ, and then divide the shear stress by the shear strain. The "From force & deformation" mode of this calculator does all of that for you.
How are shear modulus, Young's modulus, and Poisson's ratio related?
For a homogeneous isotropic material, the three are linked by E = 2·G·(1 + ν), where E is Young's modulus, G is the shear modulus, and ν is Poisson's ratio. Knowing any two of them lets you compute the third.
Does this calculator support American (Imperial) and Metric units?
Yes. Use the Unit System selector to switch between the SI metric system (N, mm, MPa/GPa) and the US customary / Imperial system (lbf, in, psi). The result is always shown in GPa, MPa, ksi, and ×10⁶ psi so you can read it in any unit.