Maximum Circumferential Stress In Solid Disc Calculator

Formula Used:

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

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Outer Radius Disc (r_outer):

Circumferential Stress (σ_c):

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1. What is Maximum Circumferential Stress In Solid Disc?

Maximum circumferential stress in a solid disc refers to the highest stress value occurring at the circumference of a rotating solid disc. This stress is caused by centrifugal forces acting on the disc material during rotation and is an important factor in mechanical design and failure analysis.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

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)

Explanation: This formula calculates the maximum circumferential stress that occurs at the outer edge of a solid rotating disc, considering material properties and rotational dynamics.

3. Importance of Circumferential Stress Calculation

Details: Calculating maximum circumferential stress is crucial for designing rotating machinery components such as flywheels, turbine discs, and gears. It helps engineers ensure that the disc can withstand the centrifugal forces without failure or excessive deformation.

4. Using the Calculator

Tips: Enter density in kg/m³, angular velocity in rad/s, Poisson's ratio (typically between 0.1-0.5 for metals), and outer radius in meters. All values must be positive with Poisson's ratio within the valid range.

5. Frequently Asked Questions (FAQ)

Q1: Why is circumferential stress maximum at the outer radius?
A: The circumferential stress increases with radius and reaches its maximum value at the outer edge of the disc due to the cumulative effect of centrifugal forces acting on all mass elements.

Q2: What materials are typically used for rotating discs?
A: Common materials include various steel alloys, aluminum alloys, titanium alloys, and composite materials, chosen based on strength-to-weight ratio and fatigue resistance.

Q3: How does angular velocity affect circumferential stress?
A: Circumferential stress increases with the square of angular velocity, meaning doubling the rotational speed quadruples the stress, making high-speed applications particularly critical.

Q4: What safety factors are typically used in disc design?
A: Safety factors typically range from 1.5 to 3.0 depending on the application, material properties, operating conditions, and consequences of failure.

Q5: Can this formula be used for discs with central holes?
A: No, this formula is specifically for solid discs. Discs with central holes have different stress distributions and require different formulas for accurate calculation.