CTR K

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 does implement EN 1993-1-1:2005 §6.3.2–§6.3.3, and the thin-wall panel EN 1993-1-5:2006 Table 4.1–4.2.

The MechanixCalc column buckling calculator checks structural columns and struts against the Euler elastic buckling load, the Johnson intermediate parabola, and the EN 1993-1-1 lateral-torsional buckling (LTB) criterion. 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 full beam-column P-M interaction unity 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 (pinned-pinned K=1.0, fixed-fixed K=0.5, fixed-free K=2.0, fixed-pinned K=0.7)
  • EN 1993-1-1 §6.3.2–§6.3.3 lateral-torsional buckling check 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 theoretical effective-length factor K (1.0 for pin-pin, 0.5 for fixed-fixed, 2.0 for fixed-free, 0.7 for fixed-pin). 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, 20 < 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.

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); 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. When a combined axial force P and bending moment M are present, the unity check UC = P/P_cr + M/M_pl (amplified for the destabilising effect of axial load via a moment-amplification factor) 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
  1. 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.
  2. Effective length: Le = K·L = 1.0 × 2000 = 2000 mm. Slenderness ratio: λ = Le/r = 2000/12.5 = 160.
  3. 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.
  4. 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).
  5. 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 follows EN 1993-1-1:2005 §6.3.2–§6.3.3, 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. The beam-column P-M interaction panel checks the unity condition UC = P/P_cr + M/M_pl (amplified for the destabilising effect of axial load). A moment-amplification factor 1/(1 − P/P_e) accounts for second-order effects. UC must be ≤ 1.0 for the column to remain stable. A full interaction envelope chart is 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.

Related calculators

Run the Column Buckling on your own numbers

Free 30-minute preview — no sign-up. A free 14-day account trial unlocks every tool and the branded PDF report, no credit card required.

Start free

Using MechanixCalc at work? See plans & pricing — one subscription unlocks all 50 calculators, PDF reports and saved projects.