Column Buckling Calculator — Euler & Johnson Critical Load, Slenderness & LTB

Governing standard: Euler · Johnson (Shigley §4.12)· Column check: classical Euler critical load + J.B. Johnson parabola (Shigley §4.12) — NOT the EN 1993-1-1 §6.3.1 buckling-curve method, and less conservative than it. The lateral-torsional panel is a screening after EN 1993-1-1:2005 §6.3.2 (χ_LT per the §6.3.2.3 form; §6.3.3 interaction is not implemented), and the thin-wall panel follows EN 1993-1-5:2006 Table 4.1–4.2.

Page last updated 16 October 2026

Video walkthrough (3:21)

Read the transcript

Here's a steel column, 3 m long, carrying 150 kN. The tool says it's SAFE, with a safety factor of 3.09.

Now make it 1 m longer. Same bar, same load, and the safety factor drops to 1.74.

This is the MechanixCalc column buckling calculator. It works out the critical load, the slenderness and the eccentric first yield. Let's go through it, step by step.

A slender column under compression doesn't crush. It bows sideways, at a load far below its yield strength.

That load is the critical load. It falls with the square of the length, and it rises with the fourth power of the diameter.

Step one is the end conditions. Both ends are pinned, so K is 1, and the column is 3000 mm long.

Step two is the cross-section. A solid round bar, 80 mm in diameter. Its slenderness is 150.

Step three is the material, steel S355. For an 80 mm bar the standard's minimum yield is 325 MPa, so the tool designs at 325.

Step four is the load. 150 kN on the axis, and a target safety factor of 2.5.

Step five shows the checks. A slenderness of 150 is above the Johnson limit of 112.9, so this is an Euler column.

The critical load is 463 kN, against 150 applied. That's a safety factor of 3.09, above the 2.5 target, so the verdict is SAFE.

It's classical Euler and Johnson theory, not a Eurocode check, and the tool says so on screen.

Now make the column 4000 mm long. The slenderness rises to 200. The verdict changes to CHECK.

The critical load drops to 260.5 kN. A third more length costs 44 % of the capacity, and the safety factor is 1.74.

The fix is a thicker bar. At 90 mm the safety factor is 2.78, and the verdict turns green.

Now put the load 20 mm off the axis. The buckling safety factor is still 2.78, but the verdict changes to CHECK.

The offset load bends the bar as it compresses. The outer fibre yields at 1.85 times the load, below the target, so eccentric yield governs.

The fix is a 100 mm bar. The first yield factor rises to 2.70, and the verdict turns green.

The ends matter just as much. Make it a flagpole, fixed at the base and free at the top. The design K is 2.1, so the effective length is 8400 mm.

The buckling safety factor falls to 0.96. The load is above the critical load, and the verdict is FAIL.

What the tool does not check is listed too: the code resistance with imperfections, the real end restraint, and fire. So silence never reads as a pass.

Every input, every formula and every check goes into a PDF report. A report your reviewer can follow.

That's a column, checked. Try it on your own design, on MechanixCalc.com.

The MechanixCalc column buckling calculator checks structural columns and struts against the Euler elastic buckling load and the Johnson intermediate parabola (classical column theory, not the EN 1993-1-1 §6.3.1 buckling curves), with a slender-wall (local buckling) check and a torsional screen for tees and angles; a lateral-torsional buckling screen after EN 1993-1-1 §6.3.2 is offered for beams. Enter the cross-section geometry, effective length factor, applied axial load, and material, and the tool returns the critical load, governing slenderness ratio, buckling regime (Euler / Johnson / stub column), and a safety factor — all in a single pass across nine cross-section types and four end conditions. Every section is checked about its axis of minimum stiffness, including the minor principal axis for angles.

Structural, civil, and mechanical engineers use the calculator to verify steel columns, compression struts, and beam-columns where the load may cause lateral instability rather than simple yielding. The tool covers the thin-wall local buckling check per EN 1993-1-5 Table 4.1–4.2 (plate slenderness and effective width), the eccentricity secant formula for off-centre loads, and a simplified first-yield beam-column P-M interaction check — and it generates a branded PDF engineering report with the governing standard and method clearly shown.

What this calculator does

Euler and Johnson critical load for nine cross-section types — solid & hollow circular, square (SHS) & rectangular (RHS) hollow, rectangular, I-section, channel, tee and angle — each about its axis of minimum stiffness
Four end conditions with recommended design effective-length factors (pinned-pinned K=1.0, fixed-fixed K=0.65, fixed-free K=2.1, fixed-pinned K=0.80; theoretical 0.5 / 2.0 / 0.7)
EN 1993-1-1 §6.3.2 lateral-torsional buckling screening (χ_LT per the §6.3.2.3 form) with fabrication-dependent imperfection curves (Table 6.5)
Thin-wall local buckling and effective-width reduction factor ρ per EN 1993-1-5 Tables 4.1 and 4.2
Beam-column combined axial + bending P-M interaction with moment-amplification factor
Slenderness sensitivity chart comparing σ_cr vs λ across the standard section types simultaneously
Branded PDF engineering report showing the governing formula, regime, and safety factor

Method & formulas

Euler elastic and Johnson intermediate buckling

Long columns buckle elastically; the Euler formula gives the critical stress as π²E/λ², where λ = Le/r is the slenderness ratio. Below the Euler–Johnson tangent slenderness λ₁ = π·√(2E/Sy) the Johnson parabola governs, blending the yield stress down quadratically with slenderness. Very short columns stay on that same parabola, which converges to the yield stress as λ → 0 — a stub column is the short end of the Johnson curve, not a separate case, so MechanixCalc does not substitute σ_cr = Sy below any cut-off. Euler and Johnson are tangent at λ₁ (both equal Sy/2 there), giving one continuous curve and no step where a column could be assigned the less conservative of two loads.

The effective length Le = K·L is computed from the buckling length L and the recommended design effective-length factor K (1.0 for pin-pin, 0.65 for fixed-fixed, 2.1 for fixed-free, 0.80 for fixed-pin — above the theoretical 0.5 / 2.0 / 0.7 because real fixed ends are not rigid). The section radius of gyration r = √(I/A) is derived from the cross-section geometry so slenderness and critical load are fully coupled to the section choice.

Euler critical stress

σ_cr = π² · E / λ² [Euler regime, λ > λ₁]

where σ_cr = critical buckling stress (MPa); E = elastic modulus (MPa); λ = Le/r = slenderness ratio; Le = K·L = effective length (mm); r = radius of gyration (mm)

Johnson parabolic critical stress

σ_cr = Sy · [1 − Sy · λ² / (4π²E)] [Johnson regime, λ ≤ λ₁]

where Sy = yield strength (MPa); λ₁ = π·√(2E/Sy) = tangent slenderness at which σ_euler = Sy/2

Lateral-torsional buckling (EN 1993-1-1 §6.3.2)

An I-section column or beam-column can also fail by twisting sideways out of the plane of bending — lateral-torsional buckling (LTB). The EN 1993-1-1 method computes the elastic critical moment M_cr from the weak-axis flexural stiffness, the Saint-Venant torsion constant J, and the warping constant C_w. The LTB slenderness λ_LT = √(Wy·fy / M_cr) is then reduced by an imperfection factor α_LT that depends on the fabrication method (rolled vs. welded) and the depth-to-width ratio h/b — curves b, c, or d from EN 1993-1-1 Table 6.5. The reduction factor χ_LT gives the LTB design moment M_b,Rd = χ_LT · Wy · fy.

Elastic critical moment (uniform bending, doubly symmetric I-section)

M_cr = (π/L_b) · √(E·Iy·G·J) · √[1 + (π²·E·C_w) / (G·J·L_b²)]

where L_b = unbraced length (mm); Iy = weak-axis second moment of area (mm⁴); G = shear modulus (MPa); J = Saint-Venant torsion constant (mm⁴); C_w = warping constant (mm⁶). Multiply by the load-case factor C_b for non-uniform moment. By default a transverse load is taken on the top flange (z_g = h/2), which lowers M_cr: 1/M_cr = 1/M_cr,sc + z_g/(k·GJ_eff), with M_cr,sc the shear-centre value above (including C_b), GJ_eff = G·J + π²·E·C_w/L_b², k = 1 on the I-section panel and 0.5 on the plate-girder panel (moment entered by hand); select 'shear centre' only for end moments or a load at or below the shear centre.

LTB reduction factor χ_LT (EN 1993-1-1 §6.3.2.3)

χ_LT = 1 / [Φ_LT + √(Φ_LT² − λ_LT²)], Φ_LT = 0.5·[1 + α_LT·(λ_LT − 0.4) + λ_LT²]

where λ_LT = √(Wy·fy / M_cr); λ_LT,0 = 0.4 and β = 1.0 (stricter than the recommended 0.75); χ_LT = 1 for λ_LT ≤ 0.4; α_LT = imperfection factor from EN 1993-1-1 Table 6.3 (curve b=0.34, c=0.49, d=0.76 per Table 6.5); χ_LT ≤ 1.0

Thin-wall local buckling and beam-column interaction

Slender cross-section elements — outstanding flanges, internal compression plates, webs — can buckle locally before the column reaches its global critical load. The tool checks the plate slenderness λ_p = √(fy / σ_cr,plate) against the EN 1993-1-5 limit and computes the effective-width reduction factor ρ (Table 4.1 for internal elements, Table 4.2 for outstanding elements), giving the effective width b_eff = ρ·b. For a solid circular member under a combined axial force P and bending moment M, the beam-column panel's unity check UC = P/P_cr + (M/M_pl)·1/(1 − P/P_e) — the moment term amplified for the destabilising effect of the axial load, uniform moment (C_m = 1) — must not exceed 1.0.

Worked example

Find the Euler critical buckling load of a pinned-pinned S355 steel column with a solid circular cross-section: diameter d = 50 mm, length L = 2000 mm, E = 210 000 MPa, Sy = 355 MPa.

Given

Diameter d50 mm
Column length L2000 mm
End conditionPinned-pinned (K = 1.0)
Elastic modulus E210 000 MPa
Yield strength Sy355 MPa

Result

Slenderness ratio λ160 (Euler regime)
Euler critical stress σ_cr80.96 MPa
Critical buckling load P_cr≈ 159 kN

Compute cross-section properties: A = π·d²/4 = π·50²/4 = 1963.5 mm²; I = π·d⁴/64 = π·50⁴/64 = 306 796 mm⁴; r = √(I/A) = √(306 796 / 1963.5) = √156.25 = 12.5 mm.
Effective length: Le = K·L = 1.0 × 2000 = 2000 mm. Slenderness ratio: λ = Le/r = 2000/12.5 = 160.
Check the regime: λ₁ = π·√(2E/Sy) = π·√(2 × 210 000 / 355) = π·√1183.1 = π·34.40 = 108.1. Since λ = 160 > λ₁ = 108.1, the Euler formula governs.
Euler critical stress: σ_cr = π²·E / λ² = 9.870 × 210 000 / 25 600 = 80.96 MPa (well below Sy = 355 MPa, confirming elastic buckling).
Critical buckling load: P_cr = σ_cr · A = 80.96 × 1963.5 = 158 940 N ≈ 159 kN.

Illustrative example — verify against your own geometry, material and end conditions. Apply an appropriate safety factor (typically SF ≥ 3 for columns in service) and check lateral-torsional buckling and thin-wall local buckling where applicable.

Frequently asked questions

Which standard does this column buckling calculator use?

The primary buckling check uses the classical Euler elastic formula and the Johnson intermediate parabola (regime selected automatically by slenderness). Lateral-torsional buckling (LTB) for I-sections is a screening after EN 1993-1-1:2005 §6.3.2 (χ_LT per the §6.3.2.3 form; the §6.3.3 interaction is not implemented), including the fabrication-dependent imperfection factors from Table 6.5 (curves b, c, d). The thin-wall local buckling check uses EN 1993-1-5:2006 Tables 4.1–4.2 for the effective-width reduction factor ρ. The governing standard and formula are shown in the generated PDF report.

What is the difference between Euler, Johnson and short-column buckling?

Long, slender columns fail elastically by Euler buckling before the material yields. Below the tangent slenderness λ₁ = π·√(2E/Sy) the Johnson parabola governs, accounting for yielding that begins before full elastic buckling. Very short, stocky columns yield in compression rather than buckling laterally — and the parabola already says so, converging to Sy·A as λ → 0. The calculator stays on that one curve rather than substituting Sy·A below a cut-off, so the reported load has no step in it as a column gets shorter; λ < 20 is reported as the stub-column band for information.

What is lateral-torsional buckling and when does it govern?

Lateral-torsional buckling (LTB) occurs when an I-section column or beam-column buckles sideways out of the bending plane by twisting and deflecting simultaneously. It governs for deep I-sections with long unbraced lengths or unfavourable loading (e.g. uniform moment). The EN 1993-1-1 LTB check computes the elastic critical moment M_cr, derives the LTB slenderness λ_LT, and applies an imperfection-based reduction factor χ_LT to obtain the design moment resistance M_b,Rd.

Can it check columns with combined axial load and bending?

Yes. Two panels check it. The P-M interaction panel on the main column uses a first-yield interaction P/P_cr + (M/M_y)·1/(1 − P/P_e); the circular-section beam-column panel checks UC = P/P_cr + (M/M_pl)·1/(1 − P/P_e) with a uniform-moment factor C_m = 1. In both, the moment-amplification factor 1/(1 − P/P_e) accounts for second-order effects, and the ratio must be ≤ 1.0 for the column to remain stable. Interaction charts are generated.

Is the column buckling calculator free?

You can run the full calculation during a free 30-minute preview with no sign-up required. A free 14-day account trial unlocks every calculator with no credit card. The branded PDF engineering report with the method and safety-factor narrative, and the ability to save and reload calculations, are included in the free 14-day trial and in every paid plan.

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