What is the Euler buckling formula and what are the effective length factor K values?

The Euler critical buckling load is P_cr = π²EI / (KL)², where E is Young's modulus, I is the second moment of area, L is the column length, and K is the effective length factor. K values by end condition: pinned-pinned K=1.0, fixed-free (cantilever) K=2.0, fixed-fixed K=0.5, fixed-pinned K≈0.7. A fixed-fixed column carries 4 times the buckling load of an equivalent pinned-pinned column.

What slenderness ratio is required for Euler's buckling formula to be valid?

Euler's elastic buckling formula is valid for slender columns where the slenderness ratio λ = KL/r is approximately 100 or greater (r = √(I/A) is the radius of gyration). For intermediate columns with λ < 100, inelastic buckling occurs before the elastic limit is reached, and the Johnson parabola formula applies: σ_cr = σ_y − σ_y²(KL/r)²/(4π²E). This calculator displays a warning when λ < 100.

Why are hollow tube sections preferred for column buckling resistance?

A hollow circular tube (outer diameter D, inner diameter d) has second moment of area I = π(D⁴ − d⁴)/64. Concentrating material at the outer perimeter maximises the radius of gyration r = √(I/A) for a given cross-sectional area, directly increasing P_cr = π²EI/(KL)². However, if the D/t wall-thickness ratio is too large, local shell buckling may govern before global Euler buckling.

How does column length affect the buckling load?

From P_cr = π²EI/(KL)², the critical load is inversely proportional to the square of the effective length KL. Doubling the column length reduces P_cr by a factor of 4. On the log-log P_cr vs L chart this appears as a straight line with slope −2. For practical design, improving end fixity from pinned-pinned (K=1.0) to fixed-fixed (K=0.5) quadruples P_cr without any change in column length.

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Buckling Analysis Tool

Euler Buckling Load Calculator

Vary cross-section shape, end conditions, and column length to calculate critical buckling load in real time. Simultaneously visualize buckling mode shapes and the P_cr–L relationship.

$$P_{cr} = \frac{\pi^2 EI}{(KL)^2}$$

Parameters

Young's Modulus E 200 GPa

Cross-Section Shape

Second Moment of Area I = —

Column Length L 2.0 m

End Conditions

Critical Load P_cr

—

Effective Length KL

—

Moment of Inertia I

—

m⁴

Slenderness Ratio KL/r

—

(Euler valid when > 100)

Buckling Mode Shape

P_cr vs L (Log Scale)

Theory — Euler Buckling

Fundamental Equation

$$P_{cr}=\frac{\pi^2 EI}{(KL)^2}$$

K is the effective length factor depending on end conditions

Effective Length Factor K

Pinned-Pinned: K=1.0
Fixed-Free: K=2.0
Fixed-Fixed: K=0.5
Fixed-Pinned: K≈0.7

Circular Section I

$$I=\frac{\pi d^4}{64},\quad I_{tube}=\frac{\pi(D^4-d^4)}{64}$$

Range of Applicability

The Euler formula is valid for slender columns with KL/r > 100. For short columns, compressive failure governed by yield stress σ_y occurs first.

KL/r < 100 → Use Johnson's Formula