Radius Of Disc Given Circumferential Stress In Solid Disc Calculator

Formula Used:

\[ r_{disc} = \sqrt{\frac{\left(\frac{C_1}{2} - \sigma_c\right) \times 8}{\rho \times \omega^2 \times \left((3 \times \nu) + 1\right)}} \]

Constant at boundary condition (C₁):

Circumferential Stress (σc):

Pascal

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Disc Radius (r_disc):

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1. What is the Disc Radius Calculation?

This calculator determines the radius of a solid disc given circumferential stress and other material properties. It's based on the mechanical properties of rotating discs under stress conditions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{disc} = \sqrt{\frac{\left(\frac{C_1}{2} - \sigma_c\right) \times 8}{\rho \times \omega^2 \times \left((3 \times \nu) + 1\right)}} \]

Where:

\( r_{disc} \) — Disc radius (m)
\( C_1 \) — Constant at boundary condition
\( \sigma_c \) — Circumferential stress (Pa)
\( \rho \) — Density of disc (kg/m³)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio

Explanation: This formula calculates the radius of a solid disc based on stress distribution and material properties under rotational conditions.

3. Importance of Disc Radius Calculation

Details: Accurate disc radius calculation is crucial for mechanical design, stress analysis, and ensuring structural integrity in rotating machinery components.

4. Using the Calculator

Tips: Enter all required parameters with appropriate units. Ensure values are physically meaningful (positive values, valid Poisson's ratio between 0-0.5).

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the constant C₁?
A: C₁ is a boundary condition constant that depends on the specific stress conditions and material properties at the disc's boundaries.

Q2: How does angular velocity affect disc radius?
A: Higher angular velocities generally require smaller radii to maintain the same stress levels, as the relationship is inversely proportional to ω².

Q3: What is the typical range for Poisson's ratio?
A: For most engineering materials, Poisson's ratio ranges between 0.1 and 0.5, with 0.3 being common for many metals.

Q4: When is this calculation most applicable?
A: This calculation is particularly useful for solid rotating discs in mechanical systems where circumferential stress is a critical design parameter.

Q5: Are there limitations to this formula?
A: This formula assumes homogeneous material properties and may have limitations for complex geometries or non-uniform stress distributions.