Circumferential Stress Given Circumferential Strain In Thick Cylindrical Shell Calculator

Formula Used:

\[ \sigma_{\theta} = (e_1 \times E) + (\nu \times (\sigma_l - \sigma_c)) \]

Circumferential Strain (e₁):

unitless

Modulus of Elasticity (E):

Poisson's Ratio (ν):

unitless

Longitudinal Stress (σₗ):

Compressive Stress (σ꜀):

Circumferential Stress (σθ):

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1. What is the Circumferential Stress Formula?

The formula calculates the circumferential (hoop) stress in a thick cylindrical shell based on circumferential strain, modulus of elasticity, Poisson's ratio, longitudinal stress, and compressive stress. This is essential for pressure vessel design and structural analysis.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \sigma_{\theta} = (e_1 \times E) + (\nu \times (\sigma_l - \sigma_c)) \]

Where:

\( \sigma_{\theta} \) — Circumferential stress (Pa)
\( e_1 \) — Circumferential strain (unitless)
\( E \) — Modulus of elasticity (Pa)
\( \nu \) — Poisson's ratio (unitless)
\( \sigma_l \) — Longitudinal stress (Pa)
\( \sigma_c \) — Compressive stress (Pa)

Explanation: The formula accounts for both the elastic deformation and the stress interactions in different directions within the thick cylindrical shell.

3. Importance of Circumferential Stress Calculation

Details: Accurate circumferential stress calculation is crucial for designing pressure vessels, pipelines, and cylindrical structures to ensure they can withstand internal and external pressures without failure.

4. Using the Calculator

Tips: Enter all required values with appropriate units. Ensure Poisson's ratio is between 0.1 and 0.5 for most metals and alloys. All input values must be valid numerical values.

5. Frequently Asked Questions (FAQ)

Q1: What is circumferential stress?
A: Circumferential stress (hoop stress) is the stress exerted circumferentially in both directions on every particle in the cylinder wall when pressure is applied.

Q2: How does this differ from thin wall cylinder formulas?
A: Thick wall cylinder formulas account for radial stress variations through the wall thickness, while thin wall formulas assume uniform stress distribution.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.25 and 0.35. Rubber-like materials can approach 0.5.

Q4: When is this formula applicable?
A: This formula is used for thick cylindrical shells where the wall thickness is significant compared to the diameter, typically when the ratio of inner diameter to wall thickness is less than 20.

Q5: What are the limitations of this calculation?
A: This calculation assumes isotropic material properties, linear elastic behavior, and uniform material properties throughout the cylinder wall.