Column Buckling Calculator

Euler critical buckling load and stress for columns with pinned, fixed, or free ends, with slenderness ratio and a warning when Euler theory does not apply.

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

Unsupported Length L

Length between supports or bracing points.

Length Unit

Unit for the value entered above.

End Condition

Material (Elastic Modulus E)

Custom Elastic Modulus E

Used only when the material is set to Custom.

Custom E Unit

Unit for the value entered above.

Second Moment of Area I

About the axis the column buckles about, normally the weaker axis.

I Unit

Unit for the value entered above.

Cross-Section Area A

Used for the critical stress and the radius of gyration.

Area Unit

Unit for the value entered above.

Yield Strength σy (optional)

Enter 0 to skip the Euler validity check.

Yield Strength Unit

Unit for the value entered above.

Validity Threshold (fraction of σy)

Warn when σcr exceeds this fraction of σy. Use 1 for yield, or about 0.5 to approximate the proportional limit.

What This Calculator Does

A slender column under compression fails by bending sideways long before the material crushes. Euler's formula gives the load at which that happens for an ideal, straight, elastic column: it depends on stiffness $EI$, length, and how the ends are held, and it does not depend on material strength at all. This calculator returns the critical load $P_{cr}$, the critical stress, the radius of gyration, and the slenderness ratio $KL/r$. If you give a yield strength, it flags the case where the critical stress exceeds yield, which means the column yields before it buckles elastically and Euler's formula overestimates its capacity.

It combines Unsupported Length L, Length Unit, End Condition, Material (Elastic Modulus E) to estimate Critical buckling load Pcr, Critical stress σcr, Slenderness ratio KL/r.

Formula & Method

Euler critical load:

P_{cr} = \frac{\pi^2 E I}{(KL)^2}

Critical stress:

\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2 E}{(KL/r)^2}

Radius of gyration and slenderness ratio:

r = \sqrt{\frac{I}{A}}, \qquad \lambda = \frac{KL}{r}

where

E

is the elastic modulus,

I

is the smallest second moment of area of the cross-section,

A

is the cross-section area,

L

is the unsupported length, and

K

is the effective length factor set by the end conditions: pinned–pinned 1.0, fixed–free 2.0, fixed–pinned 0.699, fixed–fixed 0.5. The Euler result is valid only while

\sigma_{cr}

stays at or below the yield strength (or the proportional limit).

Notation used in the formulas: $R$ = Critical buckling load Pcr; $x_{1}$ = Unsupported Length L; $x_{2}$ = Length Unit; $x_{3}$ = End Condition; $x_{4}$ = Material (Elastic Modulus E); $x_{5}$ = Custom Elastic Modulus E; $x_{6}$ = Custom E Unit.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it for long, slender columns, struts, and push rods to find the load at which elastic buckling begins, and to see how much the end conditions matter. Fixing both ends of a pinned column quadruples its critical load.

Worked Examples

Pinned steel column, 3 m (metric)

Given: L = 3 m, E = 200 GPa, I = 8.0×10⁻⁶ m⁴, A = 4000 mm², pinned at both ends, σy = 250 MPa Method: Euler's formula with K = 1.0, then compare the critical stress with yield. Pcr = π² × 200×10⁹ × 8.0×10⁻⁶ / (1.0 × 3)² = 1754.6 kN r = √(8.0×10⁻⁶ / 4.0×10⁻³) = 44.7 mm, so KL/r = 3000 / 44.7 = 67.1 σcr = 1754.6×10³ / 4.0×10⁻³ = 438.6 MPa σcr is above σy = 250 MPa, so this column yields before it buckles elastically. The Euler load of 1754.6 kN is not achievable. Use the Johnson parabola or a design code instead.

Steel column fixed at the base, free at the top (imperial)

Given: L = 10 ft = 120 in, E = 29×10⁶ psi, I = 19.2 in⁴, A = 6.2 in² Method: cantilever column, so K = 2.0 and the effective length is 240 in. Pcr = π² × 29×10⁶ × 19.2 / 240² = 95,400 lbf (95.4 kip) r = √(19.2 / 6.2) = 1.76 in, so KL/r = 240 / 1.76 = 136 σcr = 95,400 / 6.2 = 15.4 ksi, well below the yield of a typical structural steel, so Euler's formula applies here.

Effective Length Factors

The theoretical values come from the exact solution for each ideal end condition. Design codes round some of them up because real supports are never perfectly fixed.

End condition	Theoretical K	Recommended design K (AISC)
Pinned–pinned	1.0	1.0
Fixed–pinned	0.699	0.80
Fixed–fixed	0.5	0.65
Fixed–free (cantilever)	2.0	2.1

This calculator uses the theoretical values. Check the AISC Steel Construction Manual commentary (Table C-A-7.1 in the editions that include it) for the current recommended values.

Reference

Euler's column formula and its limits are covered in Beer and Johnston, Mechanics of Materials, and in Gere and Goodno, Mechanics of Materials. Design practice for steel columns is set out in the AISC Steel Construction Manual.

Common Mistakes

Using the wrong axis for $I$. A column buckles about its weaker axis, so use the smaller of $I_x$ and $I_y$ unless bracing forces it about the other.
Applying Euler's formula to a short, stocky column. When $\sigma_{cr}$ exceeds yield the formula overpredicts capacity, sometimes by a factor of two or more.
Treating $P_{cr}$ as an allowable load. It is the ideal failure load. Real design applies safety factors and accounts for initial crookedness and load eccentricity.
Using theoretical $K$ values with real connections. A base plate on a footing is not perfectly fixed, so $K = 0.5$ is optimistic.
Using the full length when intermediate bracing exists. $L$ is the unsupported length between braces.

When Not To Use This Calculator

Short or intermediate columns, where $\sigma_{cr}$ exceeds yield or the proportional limit. Use the Johnson parabola or a design code such as AISC, Eurocode 3, or the NDS for timber.
Columns with significant eccentric loading or initial curvature. These need a secant formula or second-order analysis.
Local buckling of thin plates and flanges, torsional buckling, and lateral-torsional buckling of beams.
Frames where sway changes the effective length. Use the frame alignment charts or a stability analysis.
Tapered columns, or columns whose cross-section varies along the length.

Frequently Asked Questions

What is the Euler critical load?: It is the axial compressive load at which an ideal, straight, elastically behaving column becomes unstable and bends sideways: Pcr = π²EI/(KL)². It depends on the stiffness EI and the effective length KL, not on the material's strength.
How does the end condition change the buckling load?: It changes the effective length factor K. Fixed–fixed (K = 0.5) carries four times the load of pinned–pinned (K = 1.0), and fixed–free (K = 2.0) carries one quarter of it. Because K is squared in the formula, small changes in end restraint matter.
What is the slenderness ratio, and when does Euler's formula apply?: The slenderness ratio is KL/r, where r = √(I/A). Euler's formula applies to slender columns where the critical stress stays below the yield strength or the proportional limit. For stockier columns the column yields first, and the Johnson parabola or a design code gives a better estimate.
Which axis should I use for I?: The weaker one. A column buckles about the axis with the smaller second moment of area, unless bracing prevents that. For an I-beam used as a column this is normally the minor axis, Iy.
Is the critical load the same as the safe load?: No. Pcr is the failure load for a perfect column. Real columns have initial crookedness, load eccentricity, and residual stress, and design codes apply safety factors and reduction curves. Treat Pcr as an upper bound.

Inputs Used

Unsupported Length L: Length between supports or bracing points.
Length Unit: Unit for the value entered above.
End Condition: Used directly in the calculation.
Material (Elastic Modulus E): Used directly in the calculation.
Custom Elastic Modulus E: Used only when the material is set to Custom.
Custom E Unit: Unit for the value entered above.
Second Moment of Area I: About the axis the column buckles about, normally the weaker axis.
I Unit: Unit for the value entered above.
Cross-Section Area A: Used for the critical stress and the radius of gyration.
Area Unit: Unit for the value entered above.
Yield Strength σy (optional): Enter 0 to skip the Euler validity check.
Yield Strength Unit: Unit for the value entered above.
Validity Threshold (fraction of σy): Warn when σcr exceeds this fraction of σy. Use 1 for yield, or about 0.5 to approximate the proportional limit.

See Also

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