Radius Value 'X' For Inner Cylinder Given Hoop Stress At Radius X Calculator

Formula Used:

\[ r_{\text{cylindrical shell}} = \sqrt{\frac{b_2}{\sigma_\theta - a_2}} \]

Radius Of Cylindrical Shell:

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1. What is Radius Value 'X' For Inner Cylinder?

The radius value 'x' for inner cylinder given hoop stress at radius x is a calculation used in mechanical engineering to determine the radial position in a thick-walled cylinder where a specific hoop stress occurs, using Lame's equations for stress distribution.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{\text{cylindrical shell}} = \sqrt{\frac{b_2}{\sigma_\theta - a_2}} \]

Where:

\( r_{\text{cylindrical shell}} \) — Radius of cylindrical shell (m)
\( b_2 \) — Constant 'b' for inner cylinder
\( \sigma_\theta \) — Hoop stress on thick shell (Pa)
\( a_2 \) — Constant 'a' for inner cylinder

Explanation: This formula calculates the radial position where a specific hoop stress occurs in a thick-walled cylinder, derived from Lame's equations for stress distribution in pressurized cylinders.

3. Importance of Radius Calculation

Details: Accurate calculation of radius for specific stress values is crucial for designing pressure vessels, piping systems, and other cylindrical structures to ensure they operate within safe stress limits and prevent failure.

4. Using the Calculator

Tips: Enter constant 'b' for inner cylinder, hoop stress value, and constant 'a' for inner cylinder. Ensure that (σθ - a₂) > 0 for valid calculation. All values must be valid numerical inputs.

5. Frequently Asked Questions (FAQ)

Q1: What are Lame's constants a and b?
A: Lame's constants are parameters derived from boundary conditions that describe the stress distribution in thick-walled cylinders under internal and/or external pressure.

Q2: When is this calculation typically used?
A: This calculation is used in mechanical engineering for designing pressure vessels, hydraulic cylinders, gun barrels, and other thick-walled cylindrical components.

Q3: What units should be used for input values?
A: Hoop stress should be in Pascals (Pa), while constants a and b should be in consistent units. The radius result will be in meters (m).

Q4: Are there limitations to this formula?
A: This formula assumes homogeneous, isotropic material, linear elastic behavior, and applies specifically to thick-walled cylinders with internal pressure.

Q5: How does this relate to thin-walled cylinder theory?
A: Thin-walled theory assumes uniform stress distribution, while this thick-walled approach accounts for stress variation through the wall thickness, providing more accurate results for cylinders with high diameter-to-thickness ratios.