Constant At Boundary Condition Given Circumferential Stress In Solid Disc Calculator

Formula Used:

\[ C1 = 2 \times \left( \sigma_c + \frac{\rho \times \omega^2 \times r_{disc}^2 \times ((3 \times \nu) + 1)}{8} \right) \]

Circumferential Stress (σc):

Pascal

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Disc Radius (rdisc):

Poisson's Ratio (ν):

Constant At Boundary Condition (C1):

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1. What is Constant At Boundary Condition?

Constant at boundary condition is value obtained for stress in solid disc. It represents the constant term in the stress distribution equation for a rotating solid disc under specific boundary conditions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ C1 = 2 \times \left( \sigma_c + \frac{\rho \times \omega^2 \times r_{disc}^2 \times ((3 \times \nu) + 1)}{8} \right) \]

Where:

\( C1 \) — Constant at boundary condition
\( \sigma_c \) — Circumferential Stress (Pascal)
\( \rho \) — Density Of Disc (kg/m³)
\( \omega \) — Angular Velocity (rad/s)
\( r_{disc} \) — Disc Radius (m)
\( \nu \) — Poisson's Ratio

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

3. Importance of Constant Calculation

Details: Accurate calculation of the boundary condition constant is crucial for determining stress distribution in rotating solid discs, which is essential for mechanical design, structural integrity analysis, and failure prevention in rotating machinery.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of this constant?
A: The constant represents the stress value at the boundary condition of a solid disc and is used to determine the complete stress distribution throughout the disc.

Q2: How does angular velocity affect the constant?
A: Higher angular velocities significantly increase the constant value due to the ω² term, reflecting increased centrifugal forces.

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, with common values around 0.3 for many engineering materials.

Q4: Can this formula be used for hollow discs?
A: No, this specific formula is derived for solid discs. Hollow discs have different boundary conditions and require modified equations.

Q5: What units should be used for accurate results?
A: Consistent SI units must be used: Pascals for stress, kg/m³ for density, radians/second for angular velocity, and meters for disc radius.