Outer Radius Of Disc Given Circumferential Stress Calculator

Formula Used:

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

Circumferential Stress (σc):

Pascal

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 Outer Radius Of Disc Given Circumferential Stress Formula?

The formula calculates the outer radius of a rotating disc based on circumferential stress, material density, angular velocity, Poisson's ratio, and element radius. This is essential in mechanical engineering for designing rotating components that can withstand specific stress conditions.

2. How Does The Calculator Work?

The calculator uses the following formula:

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

Where:

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

Explanation: This formula accounts for the stress distribution in a rotating disc and calculates the required outer radius to maintain specified circumferential stress conditions.

3. Importance Of Outer Radius Calculation

Details: Accurate calculation of outer radius is crucial for designing rotating discs in various applications such as turbines, flywheels, and mechanical drives. It ensures structural integrity and prevents failure due to excessive stress.

4. Using The Calculator

Tips: Enter circumferential stress in Pascals, density in kg/m³, angular velocity in rad/s, Poisson's ratio (typically between 0.1-0.5), and element radius in meters. 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 stress acting tangentially to the circumference of the disc, caused by rotational forces.

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 the stress distribution.

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-like materials can have values close to 0.5.

Q4: How does angular velocity affect the outer radius?
A: Higher angular velocities generally require larger outer radii to maintain the same circumferential stress level, as centrifugal forces increase with rotational speed.

Q5: Can this formula be used for non-uniform discs?
A: This formula assumes a uniform, homogeneous disc. For discs with varying thickness or material properties, more complex analysis is required.