ℹ About This Calculator
A slender column under axial compression can fail by buckling — bending sideways suddenly — at a load far below its material crushing strength. This calculator computes the Euler critical buckling load, the critical stress and the slenderness ratio for a pin- or fixed-ended column from its unbraced length, moment of inertia, cross-sectional area, modulus of elasticity and end-condition factor, in US units. It is the fundamental check for steel and other compression members, columns, struts and bracing.
Euler's formula gives the critical axial load at which an ideal slender column buckles elastically. The load depends on the flexural stiffness EI, inversely on the square of the effective length KL, where K accounts for the end restraints (0.5 fixed-fixed, 0.7 fixed-pinned, 1.0 pinned-pinned, 2.0 fixed-free cantilever). The slenderness ratio KL/r determines whether buckling is elastic (Euler applies, high slenderness) or inelastic (short columns, where material yielding governs and the AISC inelastic column curve is used instead).
Worked Example
A 10 ft (120 in) pinned-pinned steel column (K=1.0) with I = 10 in⁴, A = 5 in², and E = 29,000,000 psi: Pcr = π² × 29e6 × 10 / 120² = 198,750 lb (199 kips). Critical stress = 198,750 / 5 = 39,750 psi; slenderness ratio = 120 / √(10/5) = 85.
Frequently Asked Questions
What is the Euler buckling formula?
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The Euler critical buckling load is Pcr = π²·E·I / (K·L)², where E is the modulus of elasticity, I the least moment of inertia of the cross-section, L the unbraced length and K the effective length factor for the end conditions. It gives the axial load at which a slender column buckles elastically. Any load above Pcr causes the column to bend sideways and fail, even if the material is nowhere near its crushing strength.
What is the effective length factor K?
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K accounts for how the column's ends are restrained against rotation and translation. Common values are K = 1.0 for pinned-pinned ends, 0.5 for fixed-fixed, 0.7 for fixed-pinned, and 2.0 for a fixed-free cantilever. Because the effective length KL is squared in the denominator, better end restraint (lower K) dramatically increases the buckling capacity — a fixed-fixed column carries four times the load of the same pinned-pinned column.
What is the slenderness ratio?
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The slenderness ratio is KL/r, where r = √(I/A) is the radius of gyration. It measures how slender a column is: high values mean a slender column that buckles elastically (Euler's formula applies), while low values mean a stocky column that fails by material yielding or inelastic buckling. The transition is around KL/r of 100-120 for steel; below that, the AISC inelastic column curve governs rather than pure Euler.
When does a column buckle instead of crush?
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A column buckles when it is slender enough that the critical buckling stress (Pcr/A) is below the material's yield strength. Slender columns (high KL/r) buckle elastically at low stress; stocky columns (low KL/r) reach yield and crush or fail inelastically. The critical slenderness where the two modes meet depends on the material — for A36 steel it is around KL/r of 120-130. This is why the slenderness ratio is the key design parameter.
How do I increase a column's buckling capacity?
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Increase the moment of inertia (use a larger or more efficient cross-section like a wide-flange or tube), reduce the unbraced length by adding intermediate bracing, or improve the end restraint (lower K). Because Pcr depends on I directly and on length squared, adding bracing to halve the unbraced length quadruples the buckling capacity — often the most economical fix. Orient the section so the weak axis (least I) is braced.
Does Euler's formula apply to all columns?
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No — Euler's formula assumes elastic buckling, which applies only to slender columns above the critical slenderness ratio. For shorter, stocky columns the stress reaches the inelastic range before Euler buckling, and the actual capacity is lower than Euler predicts; design codes like AISC use a separate inelastic column curve for these. The slenderness ratio tells you which regime governs, which is why this calculator reports it.
Is this column buckling calculator accurate?
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It computes the exact Euler critical load, stress and slenderness ratio, which are accurate for slender (elastic) columns. For real design, verify the slenderness ratio is high enough for elastic buckling, apply the appropriate safety factor or LRFD/ASD provisions, check both axes for the least moment of inertia, and use the AISC inelastic column curve for stocky members. Have final structural designs stamped by a licensed structural engineer.
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