Buckling of Columns - Mechanics of Materials

Column: A long and slender beam that is subjected to a compressive(axial) force.
Buckling: Deflection in a column. Will occur about the centroidal principle axes of the cross section where the moment of inertia is the least.

A columns critical load is the maximum compressive load that it can support without buckling. Any additional load will cause buckling.

Ideal Column With Pin Supports: straight column before load applied and made of a homogeneous material where the compressive load is applied to the centroid of the cross section of the column.

As the moment of inertia for a column increases, so does the capacity for loading. Therefore, efficient columns are built so that most of the area (in the cross section) is located a max distance from the centroidal axes.

Good column designs are circular tubes. Also, other shapes with Ix = Iy

For a pin supported column, the buckling equation is...
Pcr = (pi)^2 EI / L^2

• Pcr is the critical axial load before buckling
• E is the modulus of elasticity
• I is the least moment of inertia
• L is the unsupported length of the column

This equation can also be rewritten expressing I = Ar^2
σcr = (pi)^2 E / (L/r)^2

• σcr is the critical stress
• E is the modulus of elasticity
• L is the unsupported length
• r is the smallest radius of gyration. Determined from r = sqrt(Imin / A)

L/r is the slenderness ratio and measures the flexibility of the column



For columns having various types of supports

Pcr = (pi)^2 EI / Le^2

σcr = (pi)^2 E / (Le/r)^2

• Le = KL and is the effective length. It is the distance between the zero-moment points (inflection points)
  
• K is the effective length factor

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