Modulus Of Elasticity Given Change In Diameter Of Thin Spherical Shells Calculator

Formula Used:

\[ E = \frac{P_i \times D^2}{4 \times t \times \Delta d} \times (1 - \mu) \]

Internal Pressure (P_i):

Diameter of Sphere (D):

Thickness of Thin Spherical Shell (t):

Change in Diameter (Δd):

Poisson's Ratio (μ):

(unitless)

Modulus of Elasticity (E):

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1. What is the Modulus of Elasticity Formula?

The formula calculates the modulus of elasticity for thin spherical shells based on internal pressure, diameter, thickness, change in diameter, and Poisson's ratio. It provides a measure of the material's stiffness and resistance to deformation under stress.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ E = \frac{P_i \times D^2}{4 \times t \times \Delta d} \times (1 - \mu) \]

Where:

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

Explanation: The formula relates the elastic properties of a thin spherical shell to the applied internal pressure and resulting deformation, accounting for the material's Poisson's ratio.

3. Importance of Modulus of Elasticity Calculation

Details: Accurate calculation of modulus of elasticity is crucial for designing pressure vessels, understanding material behavior under stress, and ensuring structural integrity in engineering applications.

4. Using the Calculator

Tips: Enter all values in appropriate units (Pa for pressure, meters for dimensions). Ensure Poisson's ratio is between 0 and 0.5. All input values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is modulus of elasticity?
A: Modulus of elasticity measures a material's resistance to elastic deformation under stress. It quantifies the stiffness of a material.

Q2: What is Poisson's ratio?
A: Poisson's ratio is the ratio of transverse strain to axial strain when a material is stretched. For most materials, it ranges between 0 and 0.5.

Q3: When is this formula applicable?
A: This formula is specifically for thin spherical shells under internal pressure where the thickness is small compared to the diameter.

Q4: What are typical values for modulus of elasticity?
A: Values vary by material: steel ~200 GPa, aluminum ~70 GPa, rubber ~0.01-0.1 GPa, concrete ~30 GPa.

Q5: How does thickness affect the calculation?
A: Thinner shells will experience greater deformation for the same internal pressure, resulting in a different calculated modulus of elasticity.