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Modulus of Rigidity Calculator

modulus of rigidity calculator shear modulus G from Young's modulus E and Poisson's ratio nu. Accurate engineering formulas and unit conversions for engineers, students & technicians.

Calculate Modulus of Rigidity Calculator

Enter your engineering parameters below to compute verified physical and mathematical metrics.

Young's Modulus of elasticity in Gigapascals (e.g. Steel = 200 GPa).
Isotropic Poisson's ratio (e.g. Steel = 0.30, Aluminum = 0.33, Rubber = 0.49).

Calculation Results

Primary Metric Output --
Metric Breakdown 1 --
Metric Breakdown 2 --
Metric Breakdown 3 --
Metric Breakdown 4 --
Metric Breakdown 5 --
Mathematical Standard --

Calculated using verified physical methodology: Modulus of Rigidity (Shear Modulus): G = \E / 2(1 + \nu)
Bulk Modulus: K = \E / 3(1 - 2\nu)

*Note: Results represent standard engineering estimates. Validate with structural codes (AISC, Eurocode) or laboratory test measurements for mission-critical applications.

Quick Summary

The Modulus of Rigidity Calculator evaluates the Shear Modulus ($G = \E / 2(1 + \nu)$) for isotropic materials, determining shear stiffness and torsional resistance from Young's modulus and Poisson's ratio.

Formula Explanation

Modulus of Rigidity (Shear Modulus): G = \E / 2(1 + \nu)
Bulk Modulus: K = \E / 3(1 - 2\nu)

How It Works

The Modulus of Rigidity Calculator evaluates an isotropic solid's shear stiffness ($G$) by dividing Young's Modulus ($E$) by $2(1 + \nu)$. It outputs Modulus of Rigidity in GPa, MPa, and Imperial Mpsi ($10^6\text{ psi}$), also deriving Bulk Modulus ($K$).

Step-by-Step Worked Example

Practical Problem: Calculate the Modulus of Rigidity ($G$) and Bulk Modulus ($K$) for structural steel ($E = 200\text{ GPa}$, Poisson's ratio $\nu = 0.30$).

  1. Step 1: Identify Input Parameters: Young's Modulus $E = 200\text{ GPa}$, Poisson's Ratio $\nu = 0.30$.
  2. Step 2: Apply the Isotropic Shear Modulus Relation: $G = \E / 2(1 + \nu)$.
  3. Step 3: Execute Numeric Calculation: $G = \200 / 2 \times (1 + 0.30) = \200 / 2.60 = 76.923\text{ GPa}$.
  4. Step 4: Calculate Bulk Modulus ($K$): $K = \E / 3(1 - 2\nu) = \200 / 3 \times (1 - 0.60) = \200 / 1.20 = 166.667\text{ GPa}$.
  5. Step 5: Convert and Interpret Imperial Metric Outputs: Modulus of Rigidity $G = 76.92\text{ GPa}$ ($76,923\text{ MPa}$). Imperial Mpsi: $76.923 \times 0.145038 = 11.16\text{ Mpsi}$. Shear-to-Elastic Ratio: $G/E = 0.3846$ ($38.5\%$).

Real-World Calculation Examples

Scenario 1: Structural Carbon Steel Drive Shaft

Parameters: $E = 200\text{ GPa}$, $\nu = 0.30$

Result: $G = 76.92\text{ GPa}$ (11.16 Mpsi). Torsional shaft rigidity.

Scenario 2: Aircraft Aluminum 6061-T6 Spar

Parameters: $E = 68.9\text{ GPa}$, $\nu = 0.33$

Result: $G = 25.90\text{ GPa}$ (3.76 Mpsi). Aluminum wing spar shear modulus.

Scenario 3: Brass Torsional Spring Element

Parameters: $E = 110\text{ GPa}$, $\nu = 0.34$

Result: $G = 41.04\text{ GPa}$ (5.95 Mpsi). Brass spring shear rigidity.

Scenario 4: Structural Concrete Pier

Parameters: $E = 30\text{ GPa}$, $\nu = 0.20$

Result: $G = 12.50\text{ GPa}$ (1.81 Mpsi). Structural concrete shear modulus.

Key Benefits of Using This Calculator

Torsional Shaft Sizing

Sizes power transmission shafts, drive axles, and helical springs against torsional deflection.

Bulk Modulus Derivation

Derives volumetric Bulk Modulus ($K = \E / 3(1-2\nu)$) simultaneously.

Multi-Unit Conversion

Provides readouts in GPa, MPa, and Imperial Mpsi ($10^6\text{ psi}$).

100% Client-Side Engine

Fast interactive calculations running locally in your browser with zero data transmission.

Frequently Asked Questions (FAQ)

What is Modulus of Rigidity?

Modulus of Rigidity (also called Shear Modulus G) is an elastic material property measuring resistance to shear deformation and angular distortion.

What is the formula for Modulus of Rigidity?

G = tau / gamma = E / (2(1 + nu)), where tau is shear stress, gamma is shear strain, E is Young's modulus, and nu is Poisson's ratio.

What are standard Shear Modulus values for common metals?

Steel: ~77 to 80 GPa (11.2 to 11.6 Mpsi); Aluminum: ~26 GPa (3.8 Mpsi); Titanium: ~44 GPa (6.4 Mpsi); Copper: ~45 GPa (6.5 Mpsi).

Why is Shear Modulus G always smaller than Young's Modulus E?

For stable isotropic materials, Poisson's ratio nu ranges between 0.0 and 0.5, making 2(1 + nu) range between 2.0 and 3.0. Thus G = E / 2.6 ~ 0.38 * E.

How converts GPa to Mpsi?

Multiply GPa by 0.145038 to obtain Mpsi (e.g. 76.92 GPa = 11.16 Mpsi).

How does Modulus of Rigidity relate to shaft torsion?

Angle of twist theta = (T * L) / (G * J), where T is torque, L is length, G is Modulus of Rigidity, and J is polar moment of inertia.

What is the theoretical maximum ratio of G to E?

When Poisson's ratio nu = 0, G = E / 2 = 0.5 * E (Shear modulus can be at most 50% of Young's modulus).

Does temperature affect Modulus of Rigidity?

Yes, as temperature increases, atomic thermal kinetic energy weakens shear bond resistance, causing G to decrease.

What is the relationship between G and acoustic transverse shear wave speed?

Transverse shear wave speed Vs = sqrt(G / rho), where G is shear modulus and rho is material mass density.

How does anisotropic material behavior affect G?

Anisotropic materials (like wood or carbon fiber composites) possess different shear moduli along different crystallographic planes (G12, G23, G13).

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