Euler Buckling Explained
In 1757, the mathematician Leonhard Euler created an equation to calculate the maximum axial force that a long, slender, ideal column can carry without buckling. The equation works for homogeneous materials free from stress. Today Euler Buckling is still one of the governing limits of compression in steel columns.
The Euler Buckling equations are: P_e = \frac{\pi^2 E I}{K L^2}
F_e = \frac{P_e}{A} = \frac{\pi^2 E}{\left(\frac{KL}{r}\right)^2}

where:
 P_{e} = Maximum Euler Buckling Force (lbs, kips, etc.)
 F_{e} = Maximum Euler Buckling Stress (psi, ksi, etc.)
 E = modulus of elasticity (29,000 ksi for carbon steel)
 I = moment of inertia
 K = effective length factor = 1.0 for pinnedpinned member & 0.65 for fixedfixed member
 L = length
 A = cross sectional area
 r = radius of gyration (in)
Note: The term \frac{L}{r} or \frac{KL}{r} is often known as the slenderness ratio. Most of the AISC compression strength tables are based off of these ratios.