Disc Outer Radius Given Radial Stress In Solid Disc Calculator

Formula Used:

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

Radial Stress (σr):

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Radius of Element (r):

Outer Radius Disc (router):

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1. What is the Disc Outer Radius Formula?

The formula calculates the outer radius of a solid disc given radial stress, density, angular velocity, Poisson's ratio, and element radius. It's derived from the stress distribution equations in rotating discs under centrifugal forces.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_{outer} \) — Outer radius of the disc (m)
\( \sigma_r \) — Radial stress induced by bending (Pa)
\( \rho \) — Density of the disc material (kg/m³)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio (dimensionless)
\( r \) — Radius of the element considered (m)

Explanation: This formula accounts for the stress distribution in a rotating solid disc, considering material properties and rotational dynamics.

3. Importance of Outer Radius Calculation

Details: Calculating the outer radius is crucial for designing rotating discs in mechanical systems, ensuring structural integrity under centrifugal forces, and preventing failure due to excessive stress.

4. Using the Calculator

Tips: Enter radial stress in Pascals, density in kg/m³, angular velocity in rad/s, Poisson's ratio (typically 0.1-0.5), and element radius in meters. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is radial stress in a rotating disc?
A: Radial stress is the stress component acting perpendicular to the radius 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 or contract in directions perpendicular to the applied stress, affecting stress distribution.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.1 and 0.5, with common values around 0.3 for steel and aluminum.

Q4: When is this formula applicable?
A: This formula applies to solid, homogeneous discs rotating at constant angular velocity with uniform material properties.

Q5: What are the limitations of this formula?
A: The formula assumes linear elastic material behavior, constant angular velocity, and may not account for temperature effects or material imperfections.