Column Buckling Calculator computes slenderness, critical stress, and critical load using Euler (elastic) and Johnson (inelastic) methods. Use it to assess capacity under axial compression while switching end conditions (K) and unit systems.
About end condition factor K. The effective length factor transforms the clear length into an effective buckling length, reflecting boundary conditions: Pinned–Pinned (K = 1.0), Fixed–Pinned (≈0.7), Fixed–Fixed (0.5), and Fixed–Free (2.0). Choosing K correctly is critical because it directly scales the slenderness (KL/r).
About slenderness and radius of gyration. Slenderness is the non-dimensional ratio KL/r, where r = √(I/A). Large KL/r indicates a slender member that tends to buckle elastically; smaller KL/r indicates a stockier member where yielding can start before global instability. I is the second moment of area and A is the cross-section area about the buckling axis of interest.
About material properties E and Fy. E (Young’s modulus) controls stiffness (≈ 200–210 GPa for structural steel). Fy (yield stress) bounds the onset of inelastic behavior. Euler’s elastic formula depends on E and geometry only, while Johnson’s inelastic formula uses both E and Fy to account for early yielding in intermediate-slenderness columns.
About units. The calculator keeps a single SI state internally and renders inputs/outputs in Metric or Imperial on demand. Stresses are shown in MPa and ksi; loads in kN and kip. Length units switch between meters and feet, while section properties accept common engineering units (cm², cm⁴ or in², in⁴).
About modes (Euler vs Johnson). Euler (Elastic) is appropriate for slender columns where the member remains elastic at buckling; Johnson (Inelastic) is appropriate for intermediate columns where yielding begins before buckling. The Auto mode compares slenderness to a transition boundary to pick the appropriate formula.
Calculation. The procedure evaluates the radius of gyration and slenderness, checks the transition slenderness to determine whether elastic or inelastic behavior governs, computes the corresponding critical stress, and multiplies by the area to obtain the critical load. Safety factors, partial factors, and code interaction checks must be applied separately according to your design standard.
Formulas:
- r = √(I/A) (radius of gyration)
- Slenderness = KL/r (non-dimensional slenderness)
- σe = π²E / (KL/r)² (Euler elastic buckling stress)
- σJ = Fy[1 − (Fy/(4π²E))(KL/r)²] (Johnson inelastic buckling stress)
- (KL/r)c = π√(2E/Fy) (transition slenderness: Johnson ↔ Euler)
- Pcr = σcr·A (critical load from controlling stress)
Examples:
- Intermediate column (Johnson). L = 3.0 m, K = 0.7, E = 200 GPa, I = 8500 cm⁴, A = 58.0 cm², Fy = 275 MPa.
r ≈ 121.1 mm, KL/r ≈ 17.3 → method: Johnson.
σcr ≈ 272.1 MPa; Pcr ≈ 1578.3 kN. - Very slender column (Euler). L = 12.0 m, K = 2.0, E = 200 GPa, I = 8500 cm⁴, A = 58.0 cm², Fy = 275 MPa.
r ≈ 121.1 mm, KL/r ≈ 198.3 → method: Euler.
σcr ≈ 50.2 MPa; Pcr ≈ 291.3 kN. - Pinned–Pinned, typical steel (Johnson). L = 3.0 m, K = 1.0, E = 200 GPa, I = 8500 cm⁴, A = 58.0 cm², Fy = 275 MPa.
r ≈ 121.1 mm, KL/r ≈ 24.8 → method: Johnson.
σcr ≈ 269.1 MPa; Pcr ≈ 1560.9 kN.
Corresponding tools. For companion checks and inputs, see:
Structural Steel Section Properties Calculator,
Bending Stress Calculator,
Steel Weight Calculator,
Plate Weight Calculator.
Design responsibility note: The calculator provides idealized elastic/inelastic buckling estimates. Always apply the relevant code (e.g., Eurocode 3, AISC 360), load combinations, safety/partial factors, and interaction checks for final design.
