Modulus Of Rigidity Of Shaft Given Shear Strain Energy In Ring Of Radius 'r' Calculator

Formula Used:

\[ G = \frac{2\pi (\tau^2) L (r_{center}^3) \delta x}{2U (r_{shaft}^2)} \]

Shear Stress on Surface of Shaft (τ):

Pascal

Length of Shaft (L):

Meter

Radius 'r' from Center Of Shaft (r_center):

Meter

Length of Small Element (δx):

Meter

Strain Energy in Body (U):

Joule

Radius of Shaft (r_shaft):

Meter

Modulus of Rigidity of Shaft (G):

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1. What is Modulus of Rigidity?

The modulus of rigidity, also known as shear modulus, is the elastic coefficient when a shear force is applied resulting in lateral deformation. It gives us a measure of how rigid a body is and its resistance to shear deformation.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ G = \frac{2\pi (\tau^2) L (r_{center}^3) \delta x}{2U (r_{shaft}^2)} \]

Where:

\( G \) — Modulus of rigidity of shaft (Pascal)
\( \tau \) — Shear stress on surface of shaft (Pascal)
\( L \) — Length of shaft (Meter)
\( r_{center} \) — Radius from center of shaft (Meter)
\( \delta x \) — Length of small element (Meter)
\( U \) — Strain energy in body (Joule)
\( r_{shaft} \) — Radius of shaft (Meter)

Explanation: This formula calculates the modulus of rigidity based on the shear strain energy stored in a ring of radius 'r' from the center of the shaft.

3. Importance of Modulus of Rigidity Calculation

Details: Accurate calculation of modulus of rigidity is crucial for designing mechanical components, analyzing structural integrity, and predicting material behavior under shear stress conditions.

4. Using the Calculator

Tips: Enter all values in appropriate units. All input values must be positive numbers. The calculator will compute the modulus of rigidity in Pascals.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of modulus of rigidity?
A: Modulus of rigidity measures a material's resistance to shearing deformation. Higher values indicate greater stiffness and resistance to shear stress.

Q2: How does modulus of rigidity differ from Young's modulus?
A: Young's modulus measures resistance to linear deformation (tension/compression), while modulus of rigidity measures resistance to shear deformation.

Q3: What are typical values for modulus of rigidity?
A: Values vary by material. Steel typically has G ≈ 79.3 GPa, aluminum ≈ 26 GPa, rubber ≈ 0.0003 GPa.

Q4: Why is this specific formula used?
A: This formula specifically calculates modulus of rigidity based on the strain energy stored in a ring element of the shaft, providing accurate results for torsional applications.

Q5: What are the limitations of this calculation?
A: The calculation assumes homogeneous material properties, linear elastic behavior, and uniform stress distribution throughout the shaft.