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Euler Buckling Formula and Critical Buckling Load

Euler buckling formula for the critical buckling load of a slender column: Pcr = π²EI/(KL)². Learn effective length, end conditions, assumptions and see a worked example.

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The Euler buckling formula gives the critical load of an ideal slender column that buckles while the material remains linearly elastic:

\[P_{cr}=\frac{\pi^2EI}{(KL)^2}.\]

For a straight, slender elastic column, Euler's critical load is:

$$N_{cr}=\frac{\pi^2 E I_{min}}{l_{eff}^2}=\frac{\pi^2 E I_{min}}{(\mu L)^2}.$$

$N_{cr}$ — critical compressive load;
$E$ — Young's modulus;
$I_{min}$ — minimum centroidal second moment of area;
$L$ — actual column length;
$\mu$ — effective-length factor;
$l_{eff}=\mu L$ — effective length.

The minimum $I$ identifies the weakest buckling plane. The formula applies to elastic buckling under conditions close to Euler's idealized column model.

The calculator does not treat $\mu$ as an independent unknown; use the already determined effective length $l_{eff}$.

Derivation for a pinned column

For a pinned-pinned column of length $L$ under centric compression $P$, a small lateral deflection $y(x)$ produces the bending moment $M(x)=-Py(x)$. The elastic-curve equation becomes

\[EI\frac{d^2y}{dx^2}+Py=0.\]

With $k^2=P/(EI)$, the solution is

\[y(x)=C_1\sin(kx)+C_2\cos(kx).\]

The boundary conditions $y(0)=0$ and $y(L)=0$ require $C_2=0$ and $\sin(kL)=0$. The first mode has $kL=\pi$, giving

\[P_{cr}=\frac{\pi^2EI}{L^2}.\]

General form

Other end conditions are represented by the effective length $L_{eff}=KL$:

\[P_{cr}=\frac{\pi^2EI_{min}}{L_{eff}^2}=\frac{\pi^2EI_{min}}{(KL)^2}.\]

Physical meaning

The Euler load is proportional to Young's modulus and the relevant second moment of area, and inversely proportional to the square of effective length. It is a stability limit of an ideal model, not necessarily the failure load of a real column.

Worked example

For a pinned steel column with $E=200$ GPa, $L=2$ m and $I_{min}=2\cdot10^{-6}$ m⁴:

\[P_{cr}=\frac{\pi^2(200\cdot10^9)(2\cdot10^{-6})}{2^2}\approx987\ \text{kN}.\]

Assumptions

The classical model assumes elastic material, centric compression, an initially straight prismatic member, ideal end restraints and small deflections up to the critical state. Its applicability must be checked using slenderness and critical stress.

Euler buckling formula FAQ

What is the Euler buckling formula?

For an ideal slender elastic column, $P_{cr}=\pi^2EI/(KL)^2$.

Is the Euler load the actual failure load?

Not necessarily. Imperfections, residual stresses, local buckling and non-ideal restraints reduce the resistance of real columns.

About this topic

Use the Euler buckling formula to calculate the critical buckling load of a slender column. This guide explains Pcr = π²EI/(KL)², Euler critical load, effective length and end conditions, the weakest buckling axis, slenderness ratio, assumptions, units, and a worked numerical example.