Modulus Of Elasticity Given Original Difference Of Radii At Junction Calculator

Formula Used:

\[ E = \frac{2 \times r \times (a_1 - a_2)}{\Delta r_{\text{original}}} \]

Radius at Junction (r):

Constant 'a' for outer cylinder (a₁):

Constant 'a' for inner cylinder (a₂):

Original difference of radii (Δr):

Modulus of Elasticity (E):

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1. What is the Modulus of Elasticity Formula?

The formula calculates the modulus of elasticity for thick shells using the radius at junction, constants for inner and outer cylinders, and the original difference of radii. It provides a measure of a material's resistance to deformation under stress.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ E = \frac{2 \times r \times (a_1 - a_2)}{\Delta r_{\text{original}}} \]

Where:

\( E \) — Modulus of Elasticity (Pa)
\( r \) — Radius at Junction (m)
\( a_1 \) — Constant for outer cylinder
\( a_2 \) — Constant for inner cylinder
\( \Delta r_{\text{original}} \) — Original difference of radii (m)

Explanation: The formula calculates the elastic modulus based on geometric parameters and material constants derived from Lame's equations for thick-walled cylinders.

3. Importance of Modulus of Elasticity Calculation

Details: Accurate calculation of modulus of elasticity is crucial for designing pressure vessels, compound cylinders, and other thick-walled structures to ensure they can withstand internal pressures without excessive deformation.

4. Using the Calculator

Tips: Enter all values in appropriate units. Radius and difference of radii must be positive values. The constants a₁ and a₂ are dimensionless parameters from Lame's equations.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of modulus of elasticity?
A: Modulus of elasticity measures a material's stiffness - its resistance to elastic deformation under applied stress.

Q2: How are the constants a₁ and a₂ determined?
A: These constants are derived from Lame's equations based on internal and external pressures and radii of the compound cylinders.

Q3: What are typical values for modulus of elasticity?
A: For steel: 200 GPa, aluminum: 70 GPa, concrete: 20-30 GPa, rubber: 0.01-0.1 GPa.

Q4: When is this formula particularly useful?
A: This formula is essential for analyzing compound cylinders and pressure vessels where thick-walled theory applies.

Q5: What are the limitations of this calculation?
A: The formula assumes linear elastic material behavior and may not be accurate for materials with non-linear elasticity or at very high stresses.