Circumferential Stress In Solid Disc Calculator

Circumferential Stress Formula:

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

Constant at boundary condition (C₁):

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Disc Radius (r_disc):

Poisson's Ratio (ν):

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 stress exerted circumferentially in a rotating disc. It represents the force per unit area acting tangentially to the circumference of the disc and is perpendicular to both the axis of rotation and the radial direction.

2. How Does the Calculator Work?

The calculator uses the circumferential stress formula for solid discs:

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

Where:

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

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

3. Importance of Circumferential Stress Calculation

Details: Calculating circumferential stress is crucial for designing rotating mechanical components like turbine discs, flywheels, and gears. It helps determine if the disc can withstand the rotational forces without failure and ensures structural integrity under operating conditions.

4. Using the Calculator

Tips: Enter all required values with appropriate units. The constant at boundary condition, density, angular velocity, and disc radius must be positive values. Poisson's ratio should be between 0 and 0.5 for most engineering materials.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the constant at boundary condition?
A: The constant C₁ is determined from boundary conditions and represents the integration constant obtained when solving the stress equilibrium equations for rotating discs.

Q2: How does angular velocity affect circumferential stress?
A: Circumferential stress increases with the square of angular velocity, meaning higher rotational speeds generate significantly higher stresses in the disc.

Q3: What is the typical range for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.1 and 0.5. Common values are around 0.3 for steel and 0.33 for aluminum.

Q4: Where is circumferential stress maximum in a solid disc?
A: In a solid rotating disc, circumferential stress is typically maximum at the center of the disc and decreases toward the outer edge.

Q5: What are the practical applications of this calculation?
A: This calculation is essential for designing rotating machinery components, analyzing disc brake systems, designing flywheel energy storage systems, and ensuring the structural integrity of rotating parts in various mechanical applications.