Moment of Inertia given Deflection at Free End of Column with Eccentric Load Calculator

Moment of Inertia Formula:

\[ I = \frac{P}{\varepsilon_{column} \times \left(\left(\frac{\text{arcsec}\left(\frac{a_{crippling}}{e_{load}} + 1\right)}{l_{column}}\right)^2\right)} \]

Eccentric Load on Column:

Modulus of Elasticity:

Deflection of Free End:

Eccentricity of Load:

Column Length:

Tolerance:

Moment of Inertia (kg·m²):

Tolerance Range (±5%):

1. What is Moment of Inertia given Deflection at Free End?

Definition: This calculator determines the moment of inertia required for a column to resist a specified deflection under an eccentric load.

Purpose: It helps structural engineers design columns that can withstand eccentric loads without excessive deflection.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ I = \frac{P}{\varepsilon_{column} \times \left(\left(\frac{\text{arcsec}\left(\frac{a_{crippling}}{e_{load}} + 1\right)}{l_{column}}\right)^2\right)} \]

Where:

\( I \) — Moment of Inertia (kg·m²)
\( P \) — Eccentric load on column (N)
\( \varepsilon_{column} \) — Modulus of elasticity of column (Pa)
\( a_{crippling} \) — Deflection of free end (m)
\( e_{load} \) — Eccentricity of load (m)
\( l_{column} \) — Column length (m)

3. Importance of Moment of Inertia Calculation

Details: Proper moment of inertia calculation ensures structural stability, prevents excessive deflection, and maintains safety under eccentric loads.

4. Using the Calculator

Tips: Enter all required values in consistent units. The tolerance field (default ±5%) allows for manufacturing and material variations.

5. Frequently Asked Questions (FAQ)

Q1: What is eccentric load on a column?
A: An eccentric load is a force applied off-center from the column's central axis, creating both compressive and bending stresses.

Q2: Why include a tolerance percentage?
A: The tolerance accounts for material variations, manufacturing imperfections, and safety factors in real-world applications.

Q3: What's a typical modulus of elasticity for columns?
A: For steel columns, about 200 GPa; for concrete, typically 20-30 GPa depending on mix and strength.

Q4: How does column length affect the moment of inertia?
A: Longer columns generally require higher moments of inertia to resist the same deflection under load.

Q5: What if my deflection is zero?
A: The formula becomes undefined at zero deflection. Small deflections are normal in structural design.