Column Buckling Calculator

🏛️ Column Buckling

Field: Structural

Written and maintained by the PhDino author · Last reviewed 21 September 2026 · Checked against 5 independent reference calculations · how PhDino checks its numbers

Limits of this calculator: This is the column's buckling CAPACITY, not the load it carries. The load on a column comes from the tributary floor or roof area above it; check that load against the design capacity, and never size a footing from the capacity.

Related standards: AISC 360

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How much axial load a steel column can carry before it buckles: Euler's elastic formula, with the AISC reduction for stocky columns.

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A short, stocky column fails by crushing — the material simply reaches its yield stress. A long, slender column can fail a completely different way: it suddenly bows sideways (buckles) at a load far below what the material could otherwise carry. Leonhard Euler worked out the critical stress at which this instability sets in, as a function of the column's slenderness — how long and skinny it is relative to its stiffness.

The end conditions matter enormously: a column pinned at both ends buckles at a much lower load than the same column fixed rigidly at both ends, because fixed ends resist the rotation that buckling needs. Between the two extremes sits a wide middle range of columns that are neither slender enough for Euler's formula nor stocky enough to simply crush; the AISC curve used here covers that range.

Key formula

Slenderness ratio: λ = KL/r
Euler stress: F_e = π²E / λ²
If λ ≤ 4.71·√(E/F_y):  F_cr = [0.658^(F_y/F_e)] · F_y
Otherwise:  F_cr = 0.877 · F_e
Nominal capacity: P_n = F_cr × A;  design capacity φP_n with φ = 0.90 (LRFD)

Variables

K: effective length factor (0.5–2.0 depending on end restraint)
L: unbraced column length
r: radius of gyration of the cross-section about the axis it buckles around (use the weaker axis unless it is braced)
A: cross-sectional area
E, F_y: Young's modulus and yield strength of the steel

How to use the Column Buckling calculator

Use this to find how much axial load a straight steel column can carry before it buckles sideways, and how much of that a designer may count on (the LRFD design capacity). It suits a post, a strut, a brace or a column in a frame, where the question is not whether the steel can be crushed but whether a slender member will bow out first.

The answer is a capacity, a ceiling, and it is not the load the column actually carries. That load comes from the floor or roof area the column supports; compare the two, and never use this figure to size anything downstream of the column.

Column Length (ft): The unbraced length: the distance between points that stop the column moving sideways, such as floors, beams or braces. It is not always the full height; a brace at mid-height can halve it for the direction it restrains.
Cross-Section Area (A) (in²): The cross-sectional area of the steel from the section table (or π d² ÷ 4 for a solid round bar).
Youngs Modulus (E) (ksi): Young's modulus of the steel, 29,000 ksi for every structural grade. Aluminum is about 10,000 ksi but its buckling curve is different, so this calculator is for steel.
Yield Strength (Fy) (ksi): The steel's yield strength: 36 ksi for A36 plate and angles, 50 ksi for the A992 wide-flange shapes in common use, and roughly 42 to 50 ksi for hollow structural sections depending on grade and shape.
Radius of Gyration (r, weaker axis) (in): The radius of gyration about the axis the column will buckle around, from the section table. A column buckles about its weaker axis, so enter the smaller of r_x and r_y unless that axis is braced.
End Condition: How the two ends are held. Pinned-pinned (K = 1.0) is the safe default for a column in a braced frame; fixed-free (K = 2.0) is a flagpole. Ideal fixity rarely exists in real connections, so the fixed choices assume more than a bolted joint delivers.

Worked example: a W8×31 column, 10 feet tall

A steel column in a small building is a W8×31 in A992 steel, 10 ft between floors, pinned at both ends. The section tables give A = 9.13 in², r_x = 3.47 in and r_y = 2.02 in. What can it carry?

You enter	Value
Column Length	10 ft
Cross-Section Area (A)	9.13 in²
Youngs Modulus (E)	29,000 ksi
Yield Strength (Fy)	50 ksi
Radius of Gyration (r, weaker axis)	2.02 in
End Condition	Pinned-Pinned (K=1.0)

The calculator returns	Value
Slenderness Ratio (KL/r)	59.4
Euler Buckling Stress (Fe)	81.1 ksi
Critical Stress (Fcr)	38.6 ksi
Nominal Capacity (Pn = Fcr × A)	352.7 kips
Design Capacity (φPn, φ = 0.90)	317.4 kips

Worked by hand:

Pick the axis. About the strong axis the slenderness would be 10 × 12 ÷ 3.47 = 34.6; about the weak axis it is larger, so the weak axis governs. Use r_y = 2.02 in.
Slenderness. KL/r = 1.0 × 10 ft × 12 ÷ 2.02 in = 59.4.
Euler stress. F_e = π² × 29,000 ÷ 59.4² = 81.1 ksi.
Which branch applies. The boundary is 4.71 × √(E ÷ F_y) = 113.4. The column's 59.4 is below it, so part of the section yields before it buckles and the inelastic curve governs: F_cr = 0.658^(F_y ÷ F_e) × F_y = 0.658^(50 ÷ 81.1) × 50 = 38.6 ksi.
Capacity. P_n = F_cr × A = 38.6 × 9.13 = 353 kips, and the LRFD design capacity is 0.90 × 353 = 317 kips.
A check on the method. Euler alone would give F_e × A = 740 kips, about 110% too high, which is exactly why the inelastic reduction exists.

This column can be counted on for about 317 kips of factored axial load. That is the number to compare with the factored load from the floors above it, worked out from the area it supports. If it falls short, the cheapest cure is usually bracing the weak axis: a brace at mid-height halves the unbraced length to 29.7 in slenderness and lifts the design capacity to about 385 kips.

Reading the result: slenderness decides everything

The slenderness ratio KL/r is the number to look at first. It says how likely the column is to buckle rather than crush, and it selects which formula applies. The dividing line is 4.71 × √(E ÷ F_y), about 113 for 50 ksi steel and 134 for 36 ksi steel.

Below about 40 the column is stocky and its capacity is close to yield stress times area; buckling barely matters.
From about 40 to the boundary (most building columns) capacity falls steadily below yield because of partial yielding and initial crookedness, which is the inelastic branch this calculator uses.
Above the boundary Euler's elastic formula governs and capacity drops quickly with length. Design standards prefer KL/r no greater than 200 for compression members.
End conditions move KL a great deal. The ideal factors are 0.5, 0.7, 1.0 and 2.0, but design practice recommends larger values (commonly 0.65, 0.80 and 2.1) for the cases that assume fixity, because real joints are not perfectly rigid.

Notes & limitations

The result is the column's CAPACITY — the most it can carry — not the load it does carry. The load on a column comes from the floor or roof area it supports; check that load against the design capacity, and never size a footing from the capacity.
A column buckles about its weaker axis, so use the smaller radius of gyration (for a W-shape, r_y) unless bracing stops that mode. Using the strong-axis radius can overstate the capacity several times over.
K=1.0 (pinned-pinned) is the traditional textbook baseline; K values of 0.5, 0.7 and 2.0 assume ideal end conditions. Real connections are rarely perfectly fixed or pinned, so design codes recommend larger K values than the ideal ones for the fixed cases.

Common mistakes

Entering the strong-axis radius of gyration. A column buckles about its weaker axis, and using r_x can overstate the capacity several times over for a wide-flange shape.
Treating the capacity as the load. The load comes from tributary area and design load combinations. Passing this figure to a footing or a connection design produces a wildly oversized result.
Using Euler's formula for a stocky column. Below the slenderness boundary it predicts a capacity far above what the steel can deliver, which is why the calculator applies the inelastic curve.
Ignoring sway. In a frame that is free to move sideways the effective length factor exceeds 1.0 for every end condition, so the pinned-pinned default is unconservative there.
Applying a steel curve to timber or aluminum. Wood posts follow a different column stability method and aluminum has its own strength curve; use the standard for the material.
Forgetting the other limit states. This covers flexural buckling of a concentrically loaded column only; local buckling of thin flanges, twisting modes, load eccentricity, and the connections all need their own checks.

Frequently asked questions

What is the difference between nominal and design capacity?

Nominal capacity P_n is the predicted strength. The LRFD design capacity multiplies it by a resistance factor of 0.90 to leave a margin for variability, and design loads are factored up on the other side of the comparison. Allowable-strength design instead divides P_n by a safety factor of 1.67.

Where do I find A and r for my section?

In the manufacturer's or standards body's section tables. Do it by hand for simple shapes: a solid round bar has r = d ÷ 4 and A = π d² ÷ 4, and a solid rectangle has r = depth ÷ √12 about the axis across its depth.

Can I use it for a wood post or an aluminum column?

No. The curve here is for structural steel. Timber columns are checked under the wood design standard with adjusted design values, and aluminum columns under the aluminum design specification.

Does bracing really help that much?

Yes, when the braced axis is the weak one. Buckling capacity depends on the square of the unbraced length in the elastic range, so a brace at mid-height can raise it dramatically, as the worked example shows.

Papers worth reading

Inelastic column theory Shanley, F. R. (1947), Journal of the Aeronautical Sciences. The classic explanation of how a column buckles once part of it is stressed past the elastic range, which is where the Euler formula overstates strength.