Free Shear Modulus Calculator

Understanding Shear Modulus (Modulus of Rigidity)

Shear modulus, also called the modulus of rigidity, quantifies how a material resists deformation when a force is applied parallel to one of its faces. This property is essential in structural and mechanical engineering—it controls the angle of twist in shafts under torsion, the behaviour of bolted joints under lateral loads, and the response of seismic dampers. An online Shear Modulus Calculator (or Modulus of Rigidity Calculator) lets engineers and students quickly determine $G$ from simple input quantities—tangential force, cross‑sectional area, original length, and lateral displacement—or, conversely, to compute shear stress or shear strain if the shear modulus is already known.

The Shear Modulus Formula

For a linear‑elastic isotropic material, the relationship between shear stress and shear strain follows Hooke’s law in its shear form:

$$\tau = G \, \gamma$$

where $\tau$ is the shear stress (force per unit area), $G$ the shear modulus, and $\gamma$ the engineering shear strain. Shear stress is defined as:

$$\tau = \frac{F}{A}$$

with $F$ the applied tangential force and $A$ the area of the face on which it acts. When the deformation is small, the shear strain can be approximated by the angle of distortion:

$$\gamma \approx \frac{\Delta x}{l}$$

Here, $\Delta x$ is the lateral deflection of the top surface relative to the bottom, and $l$ is the original height (or thickness) of the element. Substituting these expressions into Hooke’s law and solving for $G$ gives the shear modulus formula used by this calculator:

$$G = \frac{F \, l}{A \, \Delta x}$$

A higher modulus of rigidity indicates that a larger stress must be applied to achieve the same angular deformation—the material is stiffer in shear.

Relationship with Young’s Modulus and Poisson’s Ratio

For homogeneous isotropic materials, the three fundamental elastic constants are interrelated. The shear modulus $G$ is linked to Young’s modulus $E$ and Poisson’s ratio $\nu$ by:

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

This equation provides a convenient way to estimate the shear modulus when direct test data are not available, since $E$ and $\nu$ are commonly tabulated for most engineering alloys.

Typical Values of Shear Modulus for Common Materials

The shear modulus spans a wide range across different material classes. The following table lists representative values for some common substances:

Material	Shear Modulus $G$ (GPa)	Equivalent (psi)
Aluminum	25	$3.6 \times 10^{6}$
Brass	35	$5.1 \times 10^{6}$
Copper	44	$6.4 \times 10^{6}$
Iron	77	$3.8 \times 10^{6}$
Lead	6	$0.87 \times 10^{6}$
Silicone rubber	0.002	$0.00029 \times 10^{6}$
Steel	75	$10.9 \times 10^{6}$

For specific engineering alloys, the modulus of rigidity can vary slightly with composition and heat treatment. Some typical examples (based on standard references such as Hibbeler’s Mechanics of Materials) include:

• Wrought aluminum 2014‑T6: ≈ 27 GPa
• Wrought aluminum 6061‑T6: ≈ 26 GPa
• Gray cast iron (ASTM 20): ≈ 27 GPa
• Malleable cast iron (ASTM A‑197): ≈ 68 GPa
• Red brass (C83400): ≈ 37 GPa
• Bronze (C86100): ≈ 38 GPa
• Magnesium alloy AM 1004‑T61: ≈ 18 GPa
• Structural steel A‑36: ≈ 75 GPa
• Titanium alloy Ti‑6Al‑4V: ≈ 44 GPa

Units of Measurement

In the International System of Units (SI), the shear modulus is expressed in pascals (Pa). Because most materials are extremely stiff in shear, values are normally reported in gigapascals (GPa). In the United States Customary System, the unit is pounds per square inch (psi). This calculator accepts inputs in either system and returns the result accordingly, simplifying unit conversions.

Whether you are designing a shaft, verifying a material specification, or studying shear deformation, this tool functions as both a Shear Stress Calculator and a Shear Strain Calculator by leveraging the fundamental relation $\tau = G \gamma$.