Circumferential Stress In Solid Disc Given Outer Radius Calculator

Circumferential Stress Formula:

\[ \sigma_c = \frac{\rho \cdot \omega^2 \cdot ((3+\nu) \cdot r_{outer}^2 - (1+3\nu) \cdot r^2)}{8} \]

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Outer Radius Disc (r_outer):

Radius of Element (r):

Circumferential Stress (σ_c):

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1. What is Circumferential Stress in Solid Disc?

Circumferential stress, also known as hoop stress, is the force per unit area exerted circumferentially perpendicular to the axis and the radius in a rotating disc. It's a critical parameter in mechanical engineering for designing rotating components like turbine discs, flywheels, and brake rotors.

2. How Does the Calculator Work?

The calculator uses the circumferential stress formula:

\[ \sigma_c = \frac{\rho \cdot \omega^2 \cdot ((3+\nu) \cdot r_{outer}^2 - (1+3\nu) \cdot r^2)}{8} \]

Where:

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

Explanation: The formula calculates the stress distribution in a rotating solid disc, accounting for material properties and rotational dynamics.

3. Importance of Circumferential Stress Calculation

Details: Accurate calculation of circumferential stress is crucial for designing rotating machinery components to prevent failure due to excessive stress. It helps engineers determine appropriate dimensions, materials, and operating limits for rotating discs.

4. Using the Calculator

Tips: Enter all values in appropriate units. Density should be in kg/m³, angular velocity in rad/s, Poisson's ratio between 0.1-0.5, and all radii in meters. Ensure all values are positive and within reasonable physical limits.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of Poisson's ratio in this calculation?
A: Poisson's ratio represents the material's tendency to expand or contract in directions perpendicular to the direction of loading, affecting how stress distributes in the disc.

Q2: How does angular velocity affect circumferential stress?
A: Circumferential stress increases with the square of angular velocity, meaning doubling the speed quadruples the stress.

Q3: Where is the maximum stress typically located in a rotating disc?
A: For a solid disc, the maximum circumferential stress typically occurs at the center (r = 0).

Q4: What materials are commonly used for rotating discs?
A: Common materials include various steel alloys, aluminum alloys, titanium alloys, and composite materials, chosen based on strength, density, and application requirements.

Q5: How does disc thickness affect circumferential stress?
A: For a thin disc (where thickness is much smaller than diameter), thickness doesn't significantly affect the circumferential stress distribution in the plane of the disc.