How this calculation works
A slender column can lose lateral stability before its average compressive stress reaches material yield. Euler theory connects that instability to bending stiffness and effective length. This calculation keeps critical load separate from an entered operating load.
Pcr = π²EI/(KL)²; r = √(I/A); slenderness = KL/rA worked example
At E = 200 GPa, I = 1000000 mm⁴, L = 3000 mm and K = 1, the ideal critical load is about 219324.54 N. With area 1000 mm², the radius of gyration is 31.6228 mm and slenderness is 94.8683.
Good to know
The model is an ideal elastic bifurcation calculation, not design resistance. Real columns need assessment of imperfections, eccentricity, material yielding, local buckling and applicable standards. An entered load below Pcr alone does not establish safety.
Reference: MIT — elastic column buckling (opens in a new tab) ↗
What this model answers
A slender column can lose lateral stability before its average compressive stress reaches material yield.
Euler theory connects that instability to bending stiffness and effective length. This calculation keeps critical load separate from an entered operating load.
The Beam point load deflection and support comparison calculator shows how a temperature change can create axial compression under rigid restraint; such a force still needs a separate column-stability assessment.
Keep the input conventions consistent
Use the weakest relevant bending axis and its corresponding second moment of area.
The effective length factor represents assumed end restraint in the buckling plane. Enter K explicitly rather than inferring perfect fixity from a support label.
Read the result carefully
Critical load varies inversely with effective length squared.
A small change in unsupported length can therefore have a large effect. Radius of gyration and slenderness make the geometry behind that change visible without implying that every column is slender enough for Euler theory.
End restraint changes effective length rather than the physical amount of material. Keep that distinction when comparing support cases. A shorter effective length raises the ideal Euler load quadratically, but uncertain fixity should not be represented by choosing a favorable factor merely because it produces a larger answer.
Work through the numbers
At E = 200 GPa, I = 1000000 mm⁴, L = 3000 mm and K = 1, the ideal critical load is about 219324.54 N.
With area 1000 mm², the radius of gyration is 31.6228 mm and slenderness is 94.8683.
Where the calculation stops
The model is an ideal elastic bifurcation calculation, not design resistance.
Real columns need assessment of imperfections, eccentricity, material yielding, local buckling and applicable standards. An entered load below Pcr alone does not establish safety.
A few questions, answered
Why does area not change Pcr directly?
Euler load uses EI and effective length; area enters slenderness and average stress. Changing a real section usually changes both A and I.
What should I change in the comparison?
Double effective length while keeping physical geometry fixed; critical load falls to one quarter.