Buckling per Euler and Tetmajer
Calculate the buckling safety of straight compression members: from the Euler case, member length and cross-section follow the slenderness ratio, buckling stress, critical buckling load and the available safety factor. The calculator automatically distinguishes elastic buckling (Euler), inelastic buckling (Tetmajer) and the crushing range, plots the working point in the σ-λ chart and additionally checks the compressive stress.
Buckling calculator (Euler/Tetmajer)
Model: ideal straight member with constant cross-section under centric static compression; imperfections are covered globally by the buckling safety factor. No torsional or lateral-torsional buckling of open thin-walled sections, no transverse load. Sizing tool for machine and fixture design – structural members must be verified per EN 1993-1-1 (buckling curves); do not mix the two concepts. Use the spring calculator for compression springs and the beam calculator for members under transverse load.
Results
Calculating …
Formulas and fundamentals
A centrically loaded ideal column becomes unstable once the compressive force reaches the critical buckling load. In the elastic range Euler's formula applies: F_k = π²·E·I_min/l_k². The smallest second moment of area I_min always governs, because the member buckles about its weak axis. The end conditions enter through the effective (buckling) length l_k = β·L: the four Euler cases give β = 2 (fixed at the base, free at the top – the classic piston rod), β = 1 (pinned at both ends, the basic case), β = 0.699 (fixed / pinned) and β = 0.5 (fixed at both ends).
Whether Euler applies at all is decided by the slenderness ratio λ = l_k/i with the radius of gyration i = √(I_min/A). Above the material-dependent limit slenderness λ_g (S235: 105, E295/S355: 89, grey cast iron: 80) the buckling stress σ_k = π²·E/λ² stays below the proportional limit – the member buckles elastically. Below it the material partially yields before buckling; here the empirical Tetmajer equations σ_k = a + b·λ + c·λ² apply, for S235 for instance σ_k = 310 − 1.14·λ. Below the crushing limit λ_0 (S235: 65.8) the straight line reaches the yield strength: buckling is no longer the issue and the check becomes a plain compressive stress check with σ_k = R_e.
The critical buckling load follows from F_k = σ_k·A, the available safety factor from S_avail = F_k/F. In mechanical engineering, buckling safety factors of 3 to 6 are customary because of the high sensitivity to imperfections (initial curvature, load eccentricity, flexible end fixity), tending towards the upper end in the slender Euler range – the calculator defaults to 4. In addition the compressive stress σ_d = F/A is checked against the yield strength. The method applies to straight, prismatic members under centric static load; structural members governed by building codes must instead be verified per Eurocode 3 with buckling curves.
Worked example
A round bar of S235 with 30 mm diameter and 1500 mm free length is pinned at both ends (Euler case 2, β = 1) and carries a centric compressive force of 7 kN. Section properties: A = 706.9 mm², I_min = 39,761 mm⁴, radius of gyration i = √(I_min/A) = 7.5 mm.
With the effective length l_k = 1·1500 mm = 1500 mm the slenderness ratio is λ = 1500/7.5 = 200. Since λ = 200 ≥ λ_g = 105 the member buckles elastically – the Euler range governs. The buckling stress is σ_k = π²·210,000/200² = 51.8 N/mm², the critical buckling load F_k = σ_k·A = 36.6 kN.
The available safety factor is S_avail = F_k/F = 36,626/7000 = 5.23 and exceeds the required buckling safety S_req = 4 – the check passes. The compressive stress check is uncritical with σ_d = F/A = 9.9 N/mm² against R_e = 235 N/mm², as is typical for slender members: stability, not strength, limits the load capacity.
Frequently asked questions
When does Euler apply, when Tetmajer?
Above the limit slenderness λ_g the member buckles elastically per Euler (σ_k = π²·E/λ²), below it inelastically per the empirical Tetmajer equations. Below the crushing limit λ_0 there is no stability problem at all, just yielding under compression. The calculator assigns the range automatically and shows it as a badge.
Which Euler case should I choose?
Always by the actual end conditions, not by the case number – the numbering is not consistent across the literature. An extended piston rod corresponds to case 1 (β = 2), a pin-ended strut to case 2 (β = 1). Real fixed ends are rarely ideally rigid: when in doubt, conservatively use the larger β value, since the buckling load drops with 1/β².
Why does the smallest second moment of area I_min govern?
The member buckles about the axis with the least resistance. For a rectangular section that is the axis parallel to the long side (I_min = h·b³/12 with b ≤ h). Mixing up I_min and I_max is the most common user error in buckling checks – the calculator derives I_min automatically from the dimensions.
Why is the required buckling safety of 3 to 6 so high?
Because the buckling load is extremely sensitive to imperfections: initial curvature, eccentric load introduction and flexible end fixity reduce the real capacity well below the ideal value. These effects are covered globally by the safety factor. The more slender the member, the higher it should be chosen; cylinder manufacturers often specify their own values for piston rods.
What does the badge "crushing range – compression check governs" mean?
For very stocky members (λ < λ_0) the Tetmajer line would yield stresses above the yield strength – physically meaningless, the material yields first. The calculator therefore caps the buckling stress at R_e; the verification becomes a plain compressive stress check. For grey cast iron the compressive strength σ_dB takes the place of the yield strength.
Does the calculator replace a Eurocode 3 verification?
No. Euler/Tetmajer with a global safety factor is the classic sizing practice in machine and fixture design. Structures governed by building regulations must be verified per EN 1993-1-1 (flexural buckling with buckling curves and partial safety factors) – a different concept that must not be mixed with the buckling safety S = 3…6. The historical ω-method per DIN 4114, still found in old calculations, was superseded in the 1970s.