Diameter Of Spherical Shell Given Change In Diameter Of Thin Spherical Shells Calculator

Formula Used:

\[ D = \sqrt{\frac{\Delta d \cdot (4 \cdot t \cdot E)}{(1 - \mu) \cdot P_i}} \]

Change In Diameter (Δd):

Thickness Of Thin Spherical Shell (t):

Modulus of Elasticity (E):

Poisson's Ratio (μ):

Internal Pressure (P_i):

Diameter of Sphere (D):

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1. What is the Diameter Calculation Formula?

The formula calculates the diameter of a spherical shell based on the change in diameter, thickness, modulus of elasticity, Poisson's ratio, and internal pressure. It is derived from the principles of thin-walled pressure vessel theory and elasticity.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ D = \sqrt{\frac{\Delta d \cdot (4 \cdot t \cdot E)}{(1 - \mu) \cdot P_i}} \]

Where:

\( D \) — Diameter of Sphere (m)
\( \Delta d \) — Change in Diameter (m)
\( t \) — Thickness Of Thin Spherical Shell (m)
\( E \) — Modulus of Elasticity (Pa)
\( \mu \) — Poisson's Ratio
\( P_i \) — Internal Pressure (Pa)

Explanation: The formula relates the dimensional changes in a thin spherical shell under internal pressure to its material properties and geometry.

3. Importance of Diameter Calculation

Details: Accurate diameter calculation is crucial for designing pressure vessels, storage tanks, and other spherical containers to ensure structural integrity and safety under internal pressure conditions.

4. Using the Calculator

Tips: Enter all values in appropriate units. Change in diameter and thickness in meters, modulus of elasticity and internal pressure in Pascals. Poisson's ratio is dimensionless (typically between 0-0.5).

5. Frequently Asked Questions (FAQ)

Q1: What is the range of validity for this formula?
A: This formula is valid for thin spherical shells where the thickness is much smaller than the radius (typically t/R < 0.1).

Q2: How does Poisson's ratio affect the result?
A: Poisson's ratio accounts for the lateral contraction/expansion of the material when subjected to axial stress. Higher values indicate more lateral deformation.

Q3: What are typical values for modulus of elasticity?
A: For steel: 200 GPa, aluminum: 70 GPa, rubber: 0.01-0.1 GPa. The value depends on the material.

Q4: Can this formula be used for thick-walled spheres?
A: No, this formula is specifically derived for thin-walled spherical shells. Thick-walled vessels require more complex equations.

Q5: What safety factors should be considered?
A: Engineering designs typically include safety factors of 2-4 depending on the application, material properties, and operating conditions.