Boundary Conditions and Euler Buckling

The end conditions of a deforming rod determine its effective buckling length, and hence the Critical Buckling force. This is shown for two cases: Fixed Ends for which the effective length (KL) is 50% of the total length, and a rod with one end fixed and the other pinned. For this last case the effective length is 70% of the total length. The buckling force, P_cr , and stress, s_cr , are then given by:

P_cr = (p²EI)/(KL)², s_cr = (p²E)/(KL/r)²

where r is the radius of gyration of the rod.

For Fixed Ends:
P_cr^fixed = 4 P_cr

For Pinned-Fixed conditions:
P_cr^pin-fix = 2 P_cr

From: Hibbeler
"Statics and Mechanics of Materials"
Macmillan, (1993)