Circumferential Stress Given Change In Radius Of Thick Cylindrical Shell Calculator

Formula Used:

\[ \sigma_\theta = \frac{\Delta r \times E}{r_{\text{cylindrical shell}}} + \mu \times (\sigma_l - \sigma_c) \]

Change in Radius (Δr):

Modulus of Elasticity (E):

Radius of Cylindrical Shell (r):

Poisson's Ratio (μ):

Longitudinal Stress (σl):

Compressive Stress (σc):

Circumferential Stress (σθ):

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1. What is Circumferential Stress?

Circumferential stress (hoop stress) is the stress exerted circumferentially in both directions on particles within a cylindrical shell when pressure is applied. It's a critical parameter in pressure vessel design and structural analysis.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \sigma_\theta = \frac{\Delta r \times E}{r} + \mu \times (\sigma_l - \sigma_c) \]

Where:

\( \sigma_\theta \) — Circumferential stress (Pa)
\( \Delta r \) — Change in radius (m)
\( E \) — Modulus of elasticity (Pa)
\( r \) — Radius of cylindrical shell (m)
\( \mu \) — Poisson's ratio
\( \sigma_l \) — Longitudinal stress (Pa)
\( \sigma_c \) — Compressive stress (Pa)

Explanation: The formula calculates the circumferential stress by considering the elastic deformation and the effect of other stress components through Poisson's ratio.

3. Importance of Circumferential Stress Calculation

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

4. Using the Calculator

Tips: Enter all values in appropriate units. Radius must be positive. Poisson's ratio typically ranges between 0.1 and 0.5 for most materials.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between hoop stress and circumferential stress?
A: Hoop stress and circumferential stress are essentially the same - both refer to the stress acting tangentially to the circumference of a cylindrical object.

Q2: When is this formula applicable?
A: This formula is applicable for thick-walled cylindrical shells where the wall thickness is significant compared to the radius.

Q3: What materials is this formula valid for?
A: The formula is valid for isotropic, homogeneous, and linearly elastic materials.

Q4: How does Poisson's ratio affect the result?
A: Poisson's ratio accounts for the lateral strain effect when the material is under stress, influencing the overall stress distribution.

Q5: What are typical values for modulus of elasticity?
A: Modulus of elasticity varies by material: steel ~200 GPa, aluminum ~70 GPa, concrete ~20-30 GPa, rubber ~0.01-0.1 GPa.