Distance of Section from Fixed End given Deflection at Section of Column with Eccentric Load Calculator

Formula:

\[ X_d = \frac{\cos^{-1}\left(1-\frac{\delta_c}{a_{crippling}+e_{load}}\right)}{\sqrt{\frac{F}{\varepsilon_{column} \times I}}} \]

Deflection of Column:

Deflection of Free End:

Eccentricity of Load:

Eccentric load on column:

Modulus of elasticity:

Moment of Inertia:

kg·m²

Distance b/w fixed end and deflection point:

1. What is Distance of Section from Fixed End given Deflection?

Definition: This calculator determines the distance from the fixed end to a section point where deflection occurs in a column under eccentric load.

Purpose: It helps structural engineers analyze column behavior under eccentric loading conditions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ X_d = \frac{\cos^{-1}\left(1-\frac{\delta_c}{a_{crippling}+e_{load}}\right)}{\sqrt{\frac{F}{\varepsilon_{column} \times I}}} \]

Where:

\( X_d \) — Distance between fixed end and deflection point (meters)
\( \delta_c \) — Deflection of column at free end (meters)
\( a_{crippling} \) — Deflection of free end (meters)
\( e_{load} \) — Eccentricity of load (meters)
\( F \) — Eccentric load on column (Newtons)
\( \varepsilon_{column} \) — Modulus of elasticity of column (Pascals)
\( I \) — Moment of inertia (kg·m²)

3. Importance of This Calculation

Details: Accurate calculation helps in designing columns that can withstand eccentric loads without excessive deflection.

4. Using the Calculator

Tips: Enter all required parameters with appropriate units. Values must be positive numbers. Results have ±5% tolerance.

5. Frequently Asked Questions (FAQ)

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

Q2: Why is the cosine inverse function used?
A: The acos function helps solve the trigonometric relationship between deflection and column properties in the derived formula.

Q3: What affects the modulus of elasticity?
A: Material properties and temperature primarily affect the modulus of elasticity.

Q4: How precise are the results?
A: Results have ±5% tolerance to account for real-world material variations and approximations in the formula.

Q5: When would this calculation be needed?
A: When designing columns for buildings, bridges, or other structures where off-center loads are expected.