Outer Radius Of Disc Given Max Circumferential Stress In Solid Disc Calculator

Formula Used:

\[ r_{outer} = \sqrt{\frac{8 \cdot \sigma_c}{\rho \cdot \omega^2 \cdot (3 + \nu)}} \]

Circumferential Stress (σc):

Pascal

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

(dimensionless)

Outer Radius Disc (r_outer):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Outer Radius Disc Formula?

The formula calculates the outer radius of a solid disc given the maximum circumferential stress, density, angular velocity, and Poisson's ratio. It's derived from the stress analysis of rotating discs under centrifugal forces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{outer} = \sqrt{\frac{8 \cdot \sigma_c}{\rho \cdot \omega^2 \cdot (3 + \nu)}} \]

Where:

\( r_{outer} \) — Outer radius of the disc (m)
\( \sigma_c \) — Circumferential stress (Pascal)
\( \rho \) — Density of the disc material (kg/m³)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio (dimensionless)

Explanation: This formula calculates the maximum outer radius a solid disc can have without exceeding the specified circumferential stress when rotating at a given angular velocity.

3. Importance of Outer Radius Calculation

Details: Accurate calculation of outer radius is crucial for designing rotating discs in mechanical systems such as turbines, flywheels, and gears to ensure they operate within safe stress limits.

4. Using the Calculator

Tips: Enter circumferential stress in Pascals, density in kg/m³, angular velocity in rad/s, and Poisson's ratio (typically between 0.1-0.5 for metals). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is circumferential stress in a rotating disc?
A: Circumferential stress (hoop stress) is the tensile stress acting tangentially to the circumference of the disc, caused by centrifugal forces during rotation.

Q2: Why is Poisson's ratio important in this calculation?
A: Poisson's ratio accounts for the material's tendency to expand laterally when stretched, affecting the stress distribution in the disc.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.25-0.35. Rubber has about 0.5, while cork is close to 0.

Q4: Can this formula be used for hollow discs?
A: No, this specific formula is derived for solid discs. Hollow discs have different stress distributions and require different formulas.

Q5: What safety factors should be considered?
A: Always apply appropriate safety factors to the calculated stress values, typically 1.5-4.0 depending on the application and material properties.