Calculate the slenderness ratio λ = lk/i of a compression member from buckling length lk, second moment of area I and cross-sectional area A, with the radius of gyration i = √(I/A) as the intermediate value. Slenderness is the one figure that condenses length, end conditions and section shape of a compression member into a single dimensionless number.
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Calculate the slenderness ratio λ = lk/i of a compression member from buckling length lk, second moment of area I and cross-sectional area A, with the radius of gyration i = √(I/A) as the intermediate value. Slenderness is the one figure that condenses length, end conditions and section shape of a compression member into a single dimensionless number.
A pinned-pinned member (β = 1) with l = 2,000 mm, I = 100 cm⁴ and A = 20 cm² has radius of gyration i = √(10⁶ mm⁴/2,000 mm²) = 22.36 mm and therefore λ = 2,000 mm/22.36 mm ≈ 89.4. For structural steel S235 with λ₀ ≈ 104 this member lies below the limit slenderness, so the Euler formula would already be inadmissible here.
Prismatic member with a constant cross-section along its length; I and A must come from the same section. Slenderness is a purely geometric figure and contains no material property – deciding whether λ is large or small always requires the material-dependent limit slenderness λ₀. Tapered sections, built-up and multi-part compression members, torsional buckling and the local plate buckling that additionally governs thin-walled profiles are excluded.
The slenderness ratio λ answers whether a compression member is a buckling problem at all. This calculator determines λ = lk/i from buckling length, second moment of area and cross-sectional area, and rearranges the relationship for each of them.
The radius of gyration i = √(I/A) is a substitute measure for section shape: it states at what distance from the centroidal axis the entire area A would have to sit to produce the same second moment of area I. The slenderness ratio λ = lk/i relates the buckling length to it and is therefore dimensionless. λ condenses length, end conditions and section shape into one number and is the input to every buckling curve and code buckling check.
Two broom handles of equal length, one thin and one thick. Both are equally long, but the thin one is more slender and flicks away much earlier. λ measures exactly this ratio of length to cross-sectional extent – independently of absolute size, which is why a model and its scaled-up original share the same slenderness ratio.
i = √(I/A); λ = lk / i
Radius of gyration: i = √(I/A)Slenderness ratio: λ = lk/i = lk·√(A/I)Buckling length: lk = β·lValidity limit of the Euler formula: λ > λ₀ = π·√(E/|σP|)| Symbol / input | Meaning |
|---|---|
| Slenderness ratio λ | The dimensionless ratio of buckling length to radius of gyration. Large values (λ ≳ 100 for steel) mean a slender member that buckles elastically according to Euler; small values mean a stocky member in which the material yields first and the Euler formula no longer applies. Always compare λ with the material's limit slenderness λ₀. |
| Buckling length lk | The reduced buckling length lk = β·l from member length and the end-condition factor of the Euler case: β = 2 (fixed/free), β = 1 (pinned/pinned), β ≈ 0.699 (fixed/pinned), β = 0.5 (fixed/fixed). Do not enter the member length l itself. |
| Smallest second moment of area I | Second moment of area about the buckling axis. If the member can deflect in any plane, use the smallest principal value I₂, because a member buckles about its weak axis. Source: a section table or the principal-moments-of-inertia calculator for arbitrary cross-sections. |
| Cross-sectional area A | The load-bearing area of the very same cross-section that I was taken from. I and A must belong to the same profile, otherwise the radius of gyration is meaningless. |
lk is the reduced buckling length lk = β·l with the Euler-case end-condition factor β, not the member length. I is the second moment of area about the buckling axis; if the member can deflect in any plane, the smallest principal value I₂ governs. A is the cross-sectional area of the same profile. I and A must belong to the identical section.
Take the section values I and A from a profile table or a section calculator, making sure both refer to the same profile. Determine β from the end conditions and form lk = β·l. Then compare the calculated slenderness with the material's limit slenderness λ₀: for λ > λ₀ the elastic Euler check applies; for λ < λ₀ an inelastic method or the governing code procedure must be used.
A pinned-pinned member (Euler case II, β = 1) with l = 2,000 mm, I = 100 cm⁴ = 10⁶ mm⁴ and A = 20 cm² = 2,000 mm² has radius of gyration i = √(10⁶/2,000) mm = 22.36 mm, hence λ = 2,000/22.36 ≈ 89.4. Structural steel S235 has λ₀ ≈ 104, so this member is below the limit slenderness and is a stocky member for which the Euler formula would overestimate the load. Extending the same section to l = 3,000 mm raises λ to about 134, making the elastic Euler check admissible.
λ grows in proportion to buckling length and inversely with the radius of gyration. For the same area a tube has a far larger radius of gyration than a solid bar and therefore a smaller slenderness ratio – which is why tubes are the most material-efficient compression members. To reduce λ, shorten the buckling length or distribute the existing area further outwards; simply enlarging the section while keeping its shape helps only through the length dependence, not through λ itself.
The calculation runs in coherent SI units: lk in metres, I in m⁴, A in m². You may enter lk in mm, cm, m or inches, I in cm⁴, mm⁴ or in⁴, and A in cm², mm² or in². Because I and A enter only as the ratio I/A, λ itself is unit-free – in hand calculations with mm and mm⁴, i comes out directly in mm.
Deciding up front whether a compression member buckles elastically or inelastically; input to buckling curves and code buckling checks; comparing profile options (tube, solid bar, angle) at equal cross-sectional area; placing intermediate supports on columns, spindles, connecting rods and scaffold members.
Valid for prismatic members with a constant cross-section along the length. λ is a purely geometric figure containing no material property; without the material-dependent limit slenderness λ₀ no statement about capacity follows from it. Tapered and built-up sections, multi-part compression members with flexible connections, torsional buckling and the local plate buckling of thin-walled profiles – which can govern before buckling in very thin tubes and sheets – are not covered.
Common mistake: The classic error is entering the member length l instead of the buckling length lk. Mixing I and A from different profiles, or using the large instead of the small principal second moment of area, is nearly as common. Another misconception is reading λ as a measure of safety: a small slenderness ratio does not mean “safe”, only that the Euler formula no longer applies and a different verification method is required.
Deciding up front whether a compression member buckles elastically or inelastically; input to buckling curves and code buckling checks; comparing profile options (tube, solid bar, angle) at equal cross-sectional area; placing intermediate supports on columns, spindles, connecting rods and scaffold members.
lk is the reduced buckling length lk = β·l with the Euler-case end-condition factor β, not the member length. I is the second moment of area about the buckling axis; if the member can deflect in any plane, the smallest principal value I₂ governs. A is the cross-sectional area of the same profile. I and A must belong to the identical section.
Valid for prismatic members with a constant cross-section along the length. λ is a purely geometric figure containing no material property; without the material-dependent limit slenderness λ₀ no statement about capacity follows from it. Tapered and built-up sections, multi-part compression members with flexible connections, torsional buckling and the local plate buckling of thin-walled profiles – which can govern before buckling in very thin tubes and sheets – are not covered.