Euler Column Buckling Calculator

Euler column buckling

UnitsSIUS customary

End conditions

Modulus of elasticity E (GPa)Steel is about 200 GPa (29 000 ksi).

Minimum moment of inertia I (mm⁴)The smaller of Ix and Iy. Scientific notation works, e.g. 1e6.

Unbraced length L (m)Between end restraints.

Area A (optional) (mm²)For slenderness ratio and critical stress.

Yield strength Sy (optional) (MPa)Needs A. Checks whether Euler applies.

Euler critical load Pcr

219.3 kN

Effective length K·L

3 m (K = 1)

Slenderness ratio K·L/r

134.2

Euler critical stress

109.7 MPa

Governing critical load

219.3 kN

λ = 134.2 is at or above the transition slenderness λc = 125.7, so the column is long and Euler governs.

First buckling mode, Pinned–pinned (K = 1)

Intermediate steps
Step	Formula	Value
Effective length	Le = K·L = 1 × 3 m	3 m
Euler critical load	Pcr = π²·E·I / Le² = π² × 200 GPa × 1.000e+6 mm⁴ / Le²	219.3 kN
Radius of gyration	r = √(I / A)	22.36 mm
Slenderness ratio	λ = K·L / r	134.2
Euler critical stress	σcr = Pcr / A = π²E / λ²	109.7 MPa
Transition slenderness	λc = √(2π²E / Sy)	125.7

Inputs

End conditions — Pinned–pinned (K = 1), fixed–free (K = 2), fixed–pinned (K = 0.699) or fixed–fixed (K = 0.5).
Modulus of elasticity E (GPa or ksi) — From the material data sheet.
Moment of inertia I (mm⁴ or in⁴) — The smaller of Ix and Iy, unless bracing stops buckling about the weak axis.
Length L (m or in) — Unbraced length between the end restraints.
Area A (optional) (mm² or in²) — Gives the radius of gyration, slenderness ratio and critical stress.
Yield strength Sy (optional) (MPa or ksi) — Needs A. Gives the transition slenderness and the Johnson check.

Key formulas

Formulas used on this page, with the textbook each is cited from
Formula	Expression	Where	Cited from
Euler critical load	`Pcr = π²·E·I / (K·L)²`	K·L = effective length	Gere & Goodno, Mechanics of Materials
Effective-length factors (ideal ends)	`K = 1 (pinned–pinned), 2 (fixed–free), 0.699 (fixed–pinned), 0.5 (fixed–fixed)`	theoretical values from the buckled mode shapes	Timoshenko & Gere, Theory of Elastic Stability
Slenderness and critical stress	`r = √(I / A), λ = K·L / r, σcr = π²·E / λ²`	r = radius of gyration	Hibbeler, Mechanics of Materials
Transition slenderness	`λc = √(2π²·E / Sy)`	Euler applies for λ ≥ λc	Shigley’s Mechanical Engineering Design
J. B. Johnson parabola	`σcr = Sy − (Sy·λ / 2π)² / E`	for intermediate columns, λ < λc	Shigley’s Mechanical Engineering Design

Assumptions

An ideal column: perfectly straight, loaded exactly through the centroid, homogeneous and linear-elastic up to buckling.
Real columns have initial crookedness, load eccentricity and residual stresses, so they fail below Pcr. Apply the factor of safety or design code your job requires; this is not a code check.
K values are the theoretical ones for ideal restraints. Real fixed ends are never perfectly rigid, and design codes recommend larger K values for them.
Buckling is about the axis with the smaller I (and r) unless the column is braced about that axis.
Local buckling of thin walls, torsional buckling and lateral–torsional buckling are not checked.

Worked example

Example: A pinned steel strut

Given

A steel strut 3 m long, pinned at both ends (K = 1).
E = 200 GPa (typical for carbon steel), minimum I = 1.0 × 10⁶ mm⁴, A = 2000 mm².
Yield strength Sy = 250 MPa, an assumed value for a mild steel grade.

Find

The Euler critical load.
Whether Euler’s formula is valid for this strut.

Solution

Convert I to m⁴ and apply Euler’s formula with an effective length of 1 × 3 m.Pcr = π² × 200 × 10⁹ × 1.0 × 10⁻⁶ / 3² = 219 300 N = 219.3 kN
Find the radius of gyration and the slenderness ratio.r = √(1.0 × 10⁶ / 2000) = 22.36 mm, λ = 3000 / 22.36 = 134.2
Find the transition slenderness for this steel.λc = √(2π² × 200 000 / 250) = 125.7
λ = 134.2 is above λc = 125.7, so the strut is long and Euler governs. Check the critical stress against yield.σcr = 219 300 / 2000 = 109.7 MPa < Sy = 250 MPa
For comparison, fixing one end and freeing the other (K = 2) divides the load by K² = 4.Pcr = 219.3 / 4 = 54.8 kN

Answer: Pcr = 219.3 kN. λ = 134.2 exceeds λc = 125.7, so Euler’s formula applies, with σcr = 109.7 MPa.

Common mistakes

Using the strong-axis I. An unbraced column buckles about its weak axis. Use the smaller of Ix and Iy.
Applying Euler to short columns. Below the transition slenderness, Euler overpredicts the load. Use the Johnson parabola or your design code there.
Treating Pcr as an allowable load. Pcr is the theoretical buckling load of a perfect column. Divide by a factor of safety, or use a code curve that accounts for imperfections.
Assuming a welded or bolted base is fully fixed. Real connections rotate. Using K = 0.5 for a joint that is not rigid can overestimate capacity by a large margin.
Mixing mm and m. Keep E·I and L² in the same units. The calculator converts for you; hand checks are where this goes wrong.

Frequently asked questions

What is the effective length factor K?

K scales the actual length to the length of an equivalent pinned–pinned column that buckles at the same load. It comes from the shape of the buckled column for each pair of end restraints.

When is Euler’s formula valid?

When the column is slender enough to buckle elastically, before any part of it yields. In practice that means a slenderness ratio above the transition value √(2π²E/Sy). Shorter columns fail by a mix of yielding and buckling.

Why does doubling the length cut the load to a quarter?

Pcr is proportional to 1/(K·L)². Twice the effective length gives one quarter of the critical load.

Does a stronger steel buckle at a higher load?

Not for a slender column. Euler’s load depends on E and I, not on strength, and E is nearly the same for all steels. A higher yield strength only helps short and intermediate columns.

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Sources

J. M. Gere, B. J. Goodno, Mechanics of Materials, Cengage
R. C. Hibbeler, Mechanics of Materials, Pearson
S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, McGraw Hill (reprinted by Dover)
R. G. Budynas, J. K. Nisbett, Shigley’s Mechanical Engineering Design, McGraw Hill
NIST Special Publication 811, Guide for the Use of the International System of Units (conversion factors), NIST

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Euler Column Buckling Calculator