Euler Column Formula

Overview

Engineering reference data for Euler Column Formula in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

Symbol	Description	Unit
$F$	Force	N
$m$	Mass	kg
$a$	Acceleration	m/s²
$v$	Velocity	m/s

End Conditions

The factor $n$ or $k$ accounts for how the column is supported at its ends. The relationship between them is $k = (1/n)^{1/2}$.

Slenderness Ratio

The slenderness ratio $L/r$ is a key parameter in column design, where $L$ is the column length and $r$ is the radius of gyration.

$r = \sqrt{\frac{I}{A}}$

where $I$ is the moment of inertia and $A$ is the cross-sectional area.

General behavior:

• Higher slenderness ratio → lower critical stress to cause buckling
• Lower slenderness ratio → higher critical stress to cause buckling

Example: Column Fixed at Both Ends

A column with length $L = 5$ m is fixed at both ends. The column is made of an aluminium I-beam 7 × 4½ × 5.80 with a moment of inertia $I_y = 5.78 \text{ in}^4$.

Given:

• $E = 69 \text{ GPa} = 69 \times 10^9 \text{ Pa}$
• $n = 4$ (both ends fixed)
• $I_y = 5.78 \text{ in}^4 = 241 \times 10^{-8} \text{ m}^4$

Converting moment of inertia:

$I_y = 5.78 \text{ in}^4 \times (0.0254 \text{ m/in})^4 = 241 \times 10^{-8} \text{ m}^4$

Calculating the Euler buckling load:

$F = \frac{n \pi^2 E I}{L^2} = \frac{4 \times \pi^2 \times (69 \times 10^9) \times (241 \times 10^{-8})}{5^2}$

$\boxed{F = 262{,}594 \text{ N} \approx 263 \text{ kN}}$

Interactive Charts

References

• Original Source