What is Euler’s critical buckling load formula?

Euler’s critical buckling load for a slender, perfectly straight column with pinned ends is given by Pcr = π² E I / L², where Pcr is the critical load, E is Young’s modulus, I is the minimum second moment of area, and L is the unsupported length of the column. For other end conditions, an effective length factor K is used: Pcr = π² E I / (K L)².

When is Euler’s buckling formula valid?

Euler’s buckling formula is valid for long, slender columns that fail by elastic buckling before yielding. The column should be straight, centrally loaded in compression, with constant cross-section and material properties, and stresses should remain below the material’s proportional limit. For stocky columns or inelastic behavior, other design formulas (e.g., Johnson or code-based curves) must be used.

How do end conditions affect buckling load?

End conditions are represented by an effective length factor K. A lower K means a stiffer column and higher buckling load. Typical values: pinned–pinned K = 1.0, fixed–fixed K = 0.5, fixed–pinned K ≈ 0.7, fixed–free (cantilever) K = 2.0. The Euler load is Pcr = π² E I / (K L)², so halving K increases Pcr by a factor of four.

What safety factor should I use for buckling?

Safety factors for buckling depend on design codes, uncertainty in loads, imperfections, and consequences of failure. In practice, safety factors between 1.5 and 3 are common for Euler buckling, but structural design should follow the applicable standard (e.g., AISC, Eurocode) which uses resistance factors or partial safety factors instead of a simple global factor.

Buckling Load Calculator (Euler Critical Load)

Compute the Euler critical buckling load for columns with different end conditions, materials, and cross-sections. Get design load, safety factor, and slenderness ratio in one place.

Buckling Load Calculator

This tool uses Euler’s elastic buckling formula. It is intended for long, slender columns that fail by elastic buckling. Always verify results against the applicable design code.

Material (E)

Unsupported length L

End condition (effective length factor K)

Cross-section & moment of inertia I

Section type

Width b

Height h (buckling axis)

Buckling is assumed about the weak axis (use the smaller I).

Safety factor (optional)

If provided, design buckling load = P_cr / SF.

Applied axial load (optional)

If provided, the calculator reports the buckling safety factor.

Euler buckling load formula

For a long, slender column loaded in axial compression, Euler’s elastic buckling load is:

General form with effective length factor K

\[ P_\text{cr} = \frac{\pi^2 E I}{(K L)^2} \]

\(P_\text{cr}\): critical buckling load
\(E\): Young’s modulus of the material
\(I\): minimum second moment of area of the cross-section
\(L\): unsupported length of the column
\(K\): effective length factor (depends on end conditions)

Typical effective length factors K

End condition	Description	K
Pinned–Pinned	Both ends free to rotate, no moment restraint	1.0
Fixed–Fixed	Both ends fully restrained against rotation	0.5
Fixed–Pinned	One end fixed, one pinned	≈ 0.7
Fixed–Free (Cantilever)	One end fixed, other free	2.0

Section properties used by this calculator

The calculator can compute the second moment of area \(I\) for common shapes, or you can enter a custom value.

Solid rectangular section

For a solid rectangle of width \(b\) and height \(h\), buckling about the axis through the centroid and parallel to \(b\):

\[ I = \frac{b h^3}{12} \]

For buckling about the weaker axis, use the smaller of the two principal moments of inertia.

Solid circular section

\[ I = \frac{\pi d^4}{64} \]

\(d\): diameter

Hollow circular (tube)

\[ I = \frac{\pi}{64} \left(d_o^4 - d_i^4\right) \]

\(d_o\): outer diameter
\(d_i\): inner diameter

Radius of gyration and slenderness ratio

The radius of gyration \(r\) and slenderness ratio \(\lambda\) are:

\[ r = \sqrt{\frac{I}{A}}, \qquad \lambda = \frac{L_e}{r} = \frac{K L}{r} \]

Large slenderness ratios (e.g. > 100) indicate columns that are more likely to fail by elastic buckling rather than material yielding.

When can you use Euler’s buckling formula?

Column is long and slender (high slenderness ratio).
Material behaves elastically up to buckling (stresses below yield).
Load is concentric and purely axial.
Column is initially straight with small imperfections.
Cross-section and material properties are uniform along the length.

For stocky columns or inelastic buckling, design codes use more complex interaction formulas and reduction factors instead of pure Euler theory.

Practical design notes

Always check both principal axes and use the smaller buckling load.
Consider imperfections, residual stresses, and eccentricity of loading.
Apply appropriate safety factors or resistance factors as required by your code.
For built-up or braced columns, use the correct effective length and boundary conditions.

Frequently asked questions

What is Euler’s critical buckling load?

Euler’s critical buckling load is the smallest axial compressive load at which a perfectly straight, slender column becomes unstable and buckles laterally. Below this load, the column remains straight; at or above it, any small lateral disturbance grows and the column deflects sideways.

How do I choose the correct end condition?

Look at how the column is connected to the structure. If both ends are pinned (hinged) with no moment restraint, use pinned–pinned. If both ends are rigidly connected to beams or slabs, fixed–fixed may be appropriate. Real conditions are often between these ideal cases; design codes provide guidance and alignment charts to select an effective K.

Why is my buckling load much higher than the applied load?

This can happen if the column is short and stiff, or if the assumed end conditions are too favorable (e.g. fixed–fixed instead of pinned–pinned). It may also indicate that material yielding or local buckling will govern before global Euler buckling. Always check material strength and follow code provisions.

Can this calculator replace code-based design?

No. This tool is intended for quick estimates, education, and preliminary sizing. Final design of structural members must follow the relevant standard (AISC, Eurocode, etc.), which includes additional safety factors, interaction equations, and checks for local buckling, lateral–torsional buckling, and other failure modes.