Steel Column Design to DBN V.2.6-198:2014: Effective Length, Slenderness, Buckling

What is checked in a column

A centrally compressed column needs four checks:

• section strength (cl. 8.1.1, formula 8.1);
• buckling of the member under central compression (cl. 8.1.3, formula 8.3): for columns this is usually the governing check;
• limiting slenderness (cl. 13.4, Table 13.9);
• local stability of webs and flanges (cl. 8.3): cl. 8.1.3 directly requires compliance with 8.3.2–8.3.8.

If the column also carries a bending moment (eccentric compression), the check is made under Section 10 of DBN. That is a separate subject and is not covered here.

Input data

The calculation needs six quantities. The table shows where each comes from and what the worked example at the end of the article assumes. The section data of the I-beam come from the I-beam sortament.

About γn and γc in this example

Consequence class CC2 and γc = 1.0 are assumed for the example. For a real project the designer determines them.

Two different “categories” are easy to confuse. The “responsibility category” (A, B, C) that drives γn under DBN V.1.2-14 is not the same as the “purpose category” (also A, B, C) under cl. 5.3.5 and Table A.1 of DBN V.2.6-198.

Where the design codes do not sort structures into responsibility categories, Note 1 to Table 5 of DBN V.1.2-14 says to take category B (for CC2, γn = 1.050). For columns of public buildings, Table 5.1 of DBN V.2.6-198 (item 2) gives γc = 0.95; for columns of single-storey industrial buildings with overhead cranes (item 3) it gives 1.05.

Step 1. Effective length

Effective length of a constant-section column (cl. 13.3.1, formula 13.2):

lef = μ · lc

Here μ is the effective length factor, which depends on the end restraint. Other arrangements (uneven loading, elastic end restraint, frame columns) are in Table 13.7, cl. 13.3.4–13.3.7 and Annex R; effective lengths of stepped columns are in cl. 13.3.8 and Annex S.

The effective length can differ between planes: in the plane of the frame the column may be fixed, and out of plane it may be pinned. Out of plane, lef is recommended to be taken equal to the distance between nodes that hold the column against horizontal movement out of plane: column supports, crane beams, bracing and girder connections and so on (cl. 13.3.10). In the example below, bracing or wall girts halve lef out of plane.

For the main arrangements, Table 13.7 gives these values of μ:

Step 2. Slenderness and relative slenderness

Slenderness of the member in each principal plane, and the relative slenderness that enters the formulas for φ:

λ = lef / i; λ̄ = λ · √( Ry / E )

The modulus of elasticity of rolled steel is E = 2.06·10⁵ N/mm² (Table B.1). Both planes are checked. The governing one is where φ is smaller, which as a rule is the plane about the axis with the smaller radius of gyration.

Step 3. Buckling curve

Depending on the section shape, buckling curve a, b or c is used (Table 8.1). The coefficients α and β for each curve are given below.

Note to Table 8.1: for rolled I-beams deeper than 500 mm, buckling in the plane of the web is checked with curve a.

Step 4. Buckling coefficient φ

For relative slenderness λ̄ ≥ 0.4, φ is calculated by formula (8.4) with the auxiliary coefficient δ from formula (8.5):

φ = 0.5 / λ̄² · ( δ − √( δ² − 39.48 · λ̄² ) )

δ = 9.87 · ( 1 − α + β·λ̄ ) + λ̄²

The value of φ from (8.4) should not be taken greater than 7.6 / λ̄² for curve a at λ̄ > 3.8, for curve b at λ̄ > 4.4 and for curve c at λ̄ > 5.8. At λ̄ < 0.4, φ = 1.0 may be taken for all curves.

It does not have to be computed by hand: φ can be read from Table Zh.1 of Annex Zh. The tabulated values are multiplied by 1000, and between tabulated values of λ̄ linear interpolation is used.

Step 5. Buckling check

Formula (8.3):

N · γn / ( φ · A · Ry · γc ) ≤ 1

Here A is the gross cross-sectional area and φ is the smaller of the two values found for the two principal axes.

Step 6. Strength check

Formula (8.1) for steel with Ryn ≤ 440 N/mm²:

N · γn / ( An · Ry · γc ) ≤ 1

Here An is the net area. For a solid column without holes An = A. In practice buckling usually governs, but for very stocky short columns (λ̄ < 0.4, φ = 1) the margin on strength is small.

Step 7. Limiting slenderness

Slenderness λ must not exceed the limit λu from Table 13.9. The formula below is for main columns (item 4; other members have other items in the table). The α here is a separate quantity, not the same as α in Table 8.1:

λu = 180 − 60 · α

α = N / ( φ · A · Ry · γc ) ≥ 0.5 (without γn, per the note to Table 13.9)

Under cl. 13.4.2 the limiting slenderness may be increased by 10 % for members of purpose category C (Table A.1) in structures of responsibility classes CC3 and CC2, and for all members of CC1 structures. The main members of industrial-building columns are purpose category A (Table A.1, item 4, a), so this increase does not apply to them at CC2.

Example: I-beam No. 30, 4 m column

Conditions: N = 600 kN, pinned at both ends in both planes (μ = 1.0), lc = 4 m. I-beam No. 30 (DSTU 8768:2018): A = 46.5 cm² = 4650 mm², Ix = 7080 cm⁴, ix = 123 mm, Iz = 337 cm⁴, iz = 26.9 mm. Steel S255, flange thickness 10.2 mm, Ry = 240 N/mm²; E = 2.06·10⁵ N/mm²; curve b (I-beam): α = 0.04, β = 0.09.

Strength (8.1): 600·10³ · 1.05 / ( 4650 · 240 · 1.0 ) = 0.56 ≤ 1.

Strong axis x–x. lef = 4000 mm; λx = 4000 / 123 = 32.5; λ̄x = 32.5 · √(240 / 206 000) = 1.11. From (8.4)–(8.5), φx = 0.937. Table Zh.1 gives 0.948 at λ̄ = 1.0 and 0.927 at 1.2, and interpolation gives the same result.

Weak axis z–z without bracing. lef = 4000 mm; λz = 4000 / 26.9 = 148.7; λ̄z = 148.7 · √(240 / 206 000) = 5.08. Formula (8.4) gives φ = 0.311, but for curve b at λ̄ > 4.4 φ should not exceed 7.6 / λ̄² = 0.295, so φ = 0.295 is used. Table Zh.1 after interpolation gives the same (0.304 at 5.0 and 0.281 at 5.2). Check (8.3): 600·10³ · 1.05 / ( 0.295 · 4650 · 240 ) = 1.91 > 1, so buckling resistance is not sufficient.

Weak axis with bracing at mid-height (bracing or wall girts; lef,z = 2000 mm, the distance between out-of-plane restraint nodes, cl. 13.3.10). This holds for a uniform axial force; in other cases the effective length is found under Annex R. λz = 2000 / 26.9 = 74.3; λ̄z = 2.54; φz = 0.734 (Table Zh.1: 0.760 at 2.4 and 0.722 at 2.6). Check (8.3): 600·10³ · 1.05 / ( 0.734 · 4650 · 240 ) = 0.77 ≤ 1, buckling resistance is sufficient. The strong axis gives 600·10³ · 1.05 / ( 0.937 · 4650 · 240 ) = 0.60, so it is less critical and the weak axis governs.

Limiting slenderness (Table 13.9, item 4): α = 600·10³ / ( 0.734 · 4650 · 240 ) = 0.73, λu = 180 − 60 · 0.73 = 136. Both slenderness values, λz = 74.3 and λx = 32.5, are below the limit, so the condition is met. The 10 % increase under cl. 13.4.2 is not applied to main columns of category A.

Local stability of the web and flanges (cl. 8.3) is not checked in the example; it is done separately to requirements 8.3.2–8.3.8.

Example results

The No. 30 I-beam column, 4 m high under 600 kN, passes on strength with margin, but without bracing out of the plane of lower stiffness buckling resistance is not sufficient. This is a typical trap: area and strength are calculated, and the weak axis is forgotten.

Common mistakes

• checking buckling only about the x–x axis: the weak z–z axis may govern;
• taking μ = 1 for every case: for a cantilever column fixed at the bottom lef = 2.0·lc, and φ drops sharply;
• ignoring intermediate out-of-plane restraints (wall girts, bracing, purlins): they shorten lef in one plane;
• using curve b for round tubes and closed rectangular sections: they take curve a, which gives a larger φ;
• forgetting γc and γn or taking them “by habit” without checking Table 5.1 and DBN V.1.2-14;
• skipping the limiting slenderness: λ must not exceed 180 − 60α (taking cl. 13.4.2 into account).

What about the foundation

The column in the example (pinned ends, central compression) passes only the axial force N to the foundation. If the column ends are fixed, a moment and a shear force are added; all these forces are the input data for anchor bolts and embedded parts.

Codes used for this article

• DBN V.2.6-198:2014 “Steel structures. Design standards” (cl. 8.1, 13.3, 13.4, Tables 5.1, 8.1, 13.7, 13.9, B.1, G.2, Zh.1);
• DBN V.1.2-14:2018 “General principles of reliability and structural safety of buildings and structures” (Table 5, values of γn).

A section can be chosen and its mass estimated with the metal handbook and the metal weight calculator.

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