Euler Buckling Load of a Column
Euler Buckling Load of a Slender Column
Slender columns fail by buckling — a sudden sideways instability — long before the material reaches its crushing stress. This sheet computes the critical (Euler) load at which a pin-ended column buckles, and the corresponding stress.
The Euler critical load is P_cr = π²EI/(KL)², where K is the effective-length factor (1.0 for pinned–pinned ends).
Inputs
A solid circular column of diameter d and length L, pinned at both ends (K = 1).
E := 200 GPa=200000 MPa
d := 50 mm=50 mm
L := 3 m=3 m
K := 1=1
Section property
For a solid circle the second moment of area is I = πd⁴/64.
I := pi·d^4/64=3.07e-07 m^4
Critical load and stress
P_cr := pi^2·E·I/(K·L)^2=67287.93 N
sigma_cr := P_cr/(pi·d^2/4)=34.27 MPa
Results
If the critical stress comes out above the material yield strength, the column is not actually slender enough to buckle elastically and an inelastic (Johnson) or code column-curve check governs instead. The Euler load is an upper bound — real columns with imperfections carry less.
Assumptions & limitations
- Perfectly straight, concentrically loaded, pin-ended column.
- Elastic material; valid only for high slenderness ratios.
- No initial crookedness or load eccentricity.