Columns: Buckling Failure

STRUT: A structural member which carries an axial compressive load.

COLUMN: A vertical strut is known as column.

A long column becomes unstable when its axial compressive load reaches a limit called critical buckling load. Its lateral deflection called buckling.

Load carrying capacity of columns in depend upon -

• Material
• End connections
• dimension or slenderness ratio

Euler's Theory

Assumptions:

• Column is perfectly straight and uniform cross section.
• Applied compressive load is perfectly axially.
• Stresses are within elastic limit.
• The material is homogenous and isotropic.

Maximum allowable buckling load or Euler's critical load is given by:

\[\boxed{P_e = \frac{{\pi}^2EI_{min}}{L_e^2}}\]

where, $I_{min}$ = Moment of inertia about centroids axis

Le = Effective length                            

NOTE: This formula doesn't take account the axial stress. It is applicable for long columns where effect of crushing is neglected. 

Slenderness ratio (S)

Slenderness ratio is defined as the ratio of its effective length to least radius of gyration. 

\[S = \frac{L_e}{k}\]

\[ K = \sqrt{\frac{I_{min}}{A}}\]

\[Buckling \quad Stress \quad \sigma_b = \frac{P_e}{A} = \frac{{\pi}^2EI_{min}}{A L_e^2} = \frac{{\pi}^2E}{S^2}\]

Rankine's Theory

\[\frac{1}{P_R} = \frac{1}{P_C} + \frac{1}{P_e}\]

$P_R$ = Rankine Load or Crippling load

$P_C$ = crushing load = $\sigma_C$

$P_e$ = Buckling Load

\[P_R = \frac{\sigma_c A}{1 + K'(\frac{L_e}{k})^2}\]

K' = Rankkine's constant = $\frac{\sigma_c}{\pi^2E}$

• Effect of crushing and buckling considered in this formula.
• This formula is applicable to any column.