How to use this calculator
- Choose end condition. Pick pinned, fixed-free, fixed-pinned or fixed-fixed to set the effective length factor K.
- Enter length and modulus. Enter the unsupported length L and Young’s modulus E.
- Define the section. Use a solid round diameter or enter the least I and area for another shape.
- Enter load. Enter the compressive load you want to compare with Pcr.
- Read utilization. Keep applied load below the Euler critical load with your required design margin.
How it works
Euler buckling estimates the elastic critical load of a slender column:
Pcr = pi^2 · E · I / (K · L)^2
The effective length K·L converts the real end restraints into an
equivalent pinned column. More fixity lowers K; a cantilever raises it.
The calculator also computes the radius of gyration
r = sqrt(I/A) and the slenderness ratio K·L/r. For a
solid round section, it derives I = pi·d^4/64 and
A = pi·d^2/4. For any other section, enter the weak-axis
I and cross-section area directly.
Worked example
Verified against the live calculator
Take the default 25 mm steel round column, L = 1000 mm,
E = 200 GPa and pinned-pinned ends (K = 1).
The round section has I = pi x 25^4 / 64 = 19,174.76 mm^4 and
A = 490.87 mm^2. Euler gives
Pcr = pi^2 x 200000 x 19174.76 / 1000^2 = 37.85 kN.
The radius of gyration is r = sqrt(I/A) = 6.25 mm, so the
slenderness ratio is 1000 / 6.25 = 160. Against the default
10 kN load, utilization is 10 / 37.85 = 0.264.
Frequently asked questions
What is Euler buckling load?
Euler buckling load is the elastic critical compressive load of a slender column: Pcr = pi^2*E*I/(K*L)^2. E is Young’s modulus, I is the least area moment of inertia, L is unsupported length and K is the effective length factor for end restraint.
What effective length factor should I use?
Pinned-pinned uses K = 1.0, fixed-free cantilever uses K = 2.0, fixed-pinned is often approximated as K = 0.7, and ideal fixed-fixed uses K = 0.5. Real connections are rarely perfectly fixed, so use conservative restraint assumptions unless you have frame analysis.
When is Euler buckling valid?
Use Euler for long, slender columns that buckle elastically before material yield. Stocky columns can yield or crush before Euler buckling, so they need a Johnson/parabolic, code-based or material-yield check.
Which moment of inertia should I enter?
Use the least area moment of inertia about the weak buckling axis. A column buckles in the direction where it is least stiff, so using the strong-axis I can overstate the critical load.
Can this check a hydraulic cylinder rod?
It can do the first-pass Euler rod buckling math if you enter the unsupported rod length, diameter, modulus and end condition. Final cylinder selection should also check mounting geometry, side load, stroke, rod material, column charts and manufacturer ratings.
Does this work in imperial units?
Yes. Toggle to imperial to enter length in inches, modulus in Mpsi, force in lbf and inertia in in^4. The calculator converts to fixed internal units before applying Euler’s formula.
Method & assumptions
- Euler elastic buckling only; use long, slender columns that buckle before material yielding.
- End conditions are idealized. Real joints and frames usually fall between the listed K factors.
- Use the weak-axis moment of inertia. Local wall buckling, holes, welds, residual stress and connections are not included.
- The load is concentric compression. Eccentricity, side load, initial curvature and imperfection reduce real buckling strength.
- Codes and manufacturers may require additional reduction factors, yield checks or column curves.