Radius Of Circular Disc Given Radial Stress In Solid Disc Calculator

Formula Used:

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

Constant at boundary condition (C₁):

Radial Stress (σᵣ):

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Disc Radius (r):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Disc Radius Calculation?

The disc radius calculation determines the radius of a circular solid disc based on radial stress, material properties, and rotational characteristics. This formula is essential in mechanical engineering for designing rotating discs under stress.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The equation calculates the disc radius by considering the stress distribution, material density, rotational speed, and material's Poisson's ratio.

3. Importance of Disc Radius Calculation

Details: Accurate disc radius calculation is crucial for designing rotating machinery components, ensuring structural integrity, and preventing failure due to excessive stress in rotating discs.

4. Using the Calculator

Tips: Enter all required values with appropriate units. Ensure constant value, density, and angular velocity are positive. Poisson's ratio should be between 0 and 0.5 for most materials.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the constant at boundary condition?
A: The constant represents the integration constant obtained from solving the differential equations of stress distribution in rotating discs.

Q2: How does angular velocity affect disc radius?
A: Higher angular velocities generally require smaller disc radii to maintain the same stress levels, as centrifugal forces increase with rotational speed.

Q3: What are typical Poisson's ratio values for disc materials?
A: For metals, Poisson's ratio typically ranges from 0.25 to 0.35. For example, steel is around 0.3, aluminum about 0.33.

Q4: When is this formula applicable?
A: This formula applies to solid circular discs with uniform thickness rotating about their central axis under plane stress conditions.

Q5: What are the limitations of this calculation?
A: The formula assumes homogeneous material properties, constant thickness, and may not account for temperature effects or complex loading conditions.