What Column for a 6 m Storey Height? Axial Plus Moment, Checked End to End

Ask a structural engineer "what column for a 6 m storey height" and the fast answer is a section off a capacity table. Find the shape whose axial capacity beats the load, done. If the column carried nothing but axial force that answer would even be close to right, because a pure column is sized by one number. The trap is that almost no real storey column carries only axial force. It also carries a moment, from an unbalanced floor beam, from frame action, from wind, and the moment does not politely wait its turn. It runs through the same interaction check as the axial load, and on a tall storey it is usually the moment that decides.

This article works one concrete column all the way through and lets the numbers settle the question. A single storey, 6.0 m floor to floor, one column carrying 1400 kN of gravity axial from the floors above and a 150 kN.m moment about its strong axis. We solve it on the CalcSteel finite element engine, cross-check the axial and the moment against hand statics, then race seven HEB sections, from HEB 200 to HEB 320, through AISC 360 buckling, lateral-torsional buckling and the H1.1 interaction. Here is the whole point in advance: the lightest section that passes on axial alone is not the column you build, the 6 m unbraced length quietly knocks both capacities down, and the interaction, not the axial table, names the winner.

Here is the member we carry through the article. A single storey column, 6.0 m from the base connection to the floor above, an HEB hot-rolled wide-flange section in S355 steel (yield 355 MPa, modulus 200 GPa). It sits in a braced frame, so the storey does not sway and the effective-length factor is K = 1: the column is held in position at both floor levels but pinned for rotation, and its buckling length is the full 6.0 m. The floor beams frame into it at the top and bottom only, so between floors the minor axis is unbraced over the whole storey. Hold on to that last fact, because a 6 m unbraced length is what turns this from a lookup into a real check.

Two things load this column. The floors above stack gravity down it as axial compression, and a floor beam plus the lateral system deliver a strong-axis moment at the storey. A pure axial column would be sized by buckling alone. A column with a moment is a beam-column, and a beam-column is checked by an interaction equation that adds the two demands together. The question is not "which section carries 1400 kN", it is "which section carries 1400 kN and 150 kN.m at the same time, over 6 m".

The demand side is a pair of numbers, and neither depends on which section we pick.

• Axial compression, N = 1400 kN, the gravity load from the floors above resolved down this column line at the ultimate (LRFD) combination.
• Strong-axis moment, M = 150 kN.m, the first-order moment at the storey from the unbalanced beam and the lateral system.

Both are checked at the factored load, and both meet the same section at the same time. What the section changes is the capacity that answers them, and the 6 m storey height reaches into both capacities at once:

• In compression, the buckling length is K times the storey, 1.0 x 6.0 = 6.0 m. Divide by the minor-axis radius of gyration and you get the slenderness that sets the buckling stress. The taller the storey, the larger the slenderness, the smaller the capacity.
• In bending, the compression flange is braced only at the floors, so the unbraced length for lateral-torsional buckling is also 6.0 m. On a stocky HEB the plastic moment is reachable only up to a length of about 2.5 to 3.2 m; past that the flexural capacity falls. At 6 m we are well past it.

So the storey height is not a passive dimension in the background. It is the single variable that discounts the compression capacity and the bending capacity together, and it does so before the moment and the axial are even added up. Keep the number 6.0 m in mind: it is why the section that carries the axial cannot carry the pair.

Drop the column into the CalcSteel engine, apply the 1400 kN axial and the transverse load that delivers the 150 kN.m, and read the results. The engine returns exactly what statics predicts, which is the point of running the control first:

• Axial force in the column, N = 1400.0 kN, matching the applied gravity load to 0.000 percent.
• Peak strong-axis moment, M = 150.0 kN.m, matching the closed-form wL squared over 8 for the equivalent transverse load to 0.000 percent.

That agreement is not decoration. It certifies that the model, the supports and the load path are what we think they are, so every capacity we compute next is answering the right demand. The 1400 kN and 150 kN.m are now fixed. They are the same for every candidate. What changes from HEB 200 to HEB 320 is the capacity that meets them, and, decisively, how the 6 m length discounts that capacity before the interaction adds the two ratios up.

Before we race the sections, size a column yourself. The free column buckling calculator below takes a section, a length and an effective-length factor and returns the slenderness and the design compression, the same AISC 360 Chapter E path we use in a moment. Put in a 6.0 m length with K = 1, try an HEB 240 and an HEB 280, and watch the capacity climb as the section gets stockier. It is the axial half of the check, live and with no login, and it is the fastest way to feel how hard the storey height pushes on the number.

HEB sections are the natural column shape: wide flanges, a depth close to the width, and a large minor-axis radius of gyration, which is exactly what a column needs to resist buckling. We line up seven of them and read the properties straight from the catalogue.

The two properties that matter are on show. The minor-axis radius of gyration ry sets the compression capacity, because minor-axis buckling governs a doubly symmetric column at equal unbraced lengths. The plastic modulus Zx sets the bending capacity. Both grow as the section gets heavier, and the whole article is a contest between how fast each one grows and how hard the 6 m length and the 150 kN.m moment push back.

Start the way a lookup does, with the axial load and nothing else. AISC 360 Chapter E gives the design compression from the slenderness. With K = 1 and the 6.0 m storey, the minor-axis slenderness KL over ry runs from 118.5 for the HEB 200 down to 84.7 for the HEB 280. Feed that through the column curve and you get the buckling stress Fcr, then multiply by the area and the 0.90 factor for the design compression phiPn:

• HEB 220: slenderness 107.4, phiPn = 1221 kN. Against the 1400 kN demand that is a utilisation of 1.15, a fail.
• HEB 240: slenderness 98.7, Fcr = 170.6 MPa, phiPn = 1627 kN. Utilisation 1400 / 1627 = 0.86, a comfortable pass.

So on axial alone the answer is HEB 240. It is the lightest section in the list that carries the 1400 kN, and at 0.86 it has real headroom, about 14 percent. A capacity table, or a column calculator run with axial only, would hand you the HEB 240 and you would move on. Notice how much the 6 m did here already: the HEB 240 is a stocky 84 kg/m section, and the storey height has spent most of it, leaving only that 14 percent. That thin margin is all the axial check knows about. It has never seen the moment.

Now let the moment into the room. A beam-column is checked by the AISC 360 H1.1 interaction equation, which adds the axial utilisation and the bending utilisation into one ratio that must stay at or below 1.0. When the axial utilisation is 0.2 or more, the rule is:

IR = Pr/Pc + (8/9)(Mr/Mc)

where Pr/Pc is the axial ratio we already have and Mr/Mc is the moment ratio. The moment capacity Mc = phiMn is where the 6 m bites a second time. The HEB 240 could reach a design plastic moment of 324.6 kN.m if its flange were braced, but over a 6 m unbraced length lateral-torsional buckling drops it to phiMn = 275.9 kN.m, a 15 percent knockdown. The moment demand is not the bare 150 kN.m either: the second-order amplifier lifts it to 195.3 kN.m (the next section explains why). Put the pieces together for the HEB 240:

IR = 0.86 + (8/9)(195.3 / 275.9) = 0.86 + 0.63 = 1.49

The section that passed the axial check at 0.86 fails the combined check at 1.49, nearly 50 percent over. And it is not alone. Step up one size to the HEB 260, and the axial ratio eases to 0.69 but the interaction is still 1.17, a fail. The moment has eliminated the two lightest sections that carry the axial. The lookup answer is gone.

The 150 kN.m is the first-order moment, the one a linear analysis draws. But the axial load does not act on the undeflected column, it acts on the column that has already bent sideways under the moment, and that offset multiplied by the 1400 kN adds moment the first-order diagram never showed. AISC 360 captures this member effect with the B1 amplifier:

B1 = Cm / (1 - Pr/Pe1) ≥ 1

where Pe1 is the elastic buckling load of the member about the axis of bending. For the HEB 240, Pe1 = 6033 kN, so with Cm = 1.0 the amplifier is 1 / (1 - 1400/6033) = 1.30, turning the 150 kN.m into 195.3 kN.m. For the stiffer HEB 280, Pe1 = 10307 kN and B1 = 1.16, a smaller amplification of the same 150 kN.m to 173.6 kN.m, because a stiffer column deflects less and feeds itself less second-order moment.

This is the quiet reason a beam-column is harder than the sum of its parts. The heavier section is not just carrying more, it is amplifying less. The B1 factor rewards stiffness twice: once in the capacity and once in the demand. On this column it accounts for a third of the moment that fails the HEB 240.

Keep stepping up. The HEB 280 brings a design compression of phiPn = 2445 kN and, after the same 6 m lateral-torsional knockdown, a design moment of phiMn = 413.3 kN.m. Its axial ratio is a relaxed 1400 / 2445 = 0.573, its amplified moment is 173.6 kN.m, and the interaction lands at:

IR = 0.573 + (8/9)(173.6 / 413.3) = 0.573 + 0.373 = 0.946

That is a pass, with a little room. The HEB 280 is the lightest section in the list that carries the axial and the moment together over 6 m. Here is the whole contest on one card:

The number that decides is the last column, and it is nothing like the axial ratio. Sizing on axial alone stopped at the HEB 240, 83.2 kg/m. The interaction forces the HEB 280, 103.1 kg/m. That is 24 percent more steel down every column in the grid, bought not by the axial load a table would look up but by the moment and the 6 m length that a table never sees.

The tempting shortcut, when a section fails, is to reach for a higher grade. If S355 will not do it, surely S460 will. Run the counterfactual: take the failing HEB 240, keep the geometry, and lift the yield from 355 to 460 MPa. The results are a lesson in what actually controls a 6 m column.

• The design compression rises from 1627 kN to only 1696 kN, a 4 percent gain. At a slenderness of 98.7 the column is buckling, not yielding, and buckling depends on stiffness and length, not on yield strength. A stronger steel barely moves the compression capacity, because the column fails by going unstable long before the steel reaches 460 MPa.
• The design moment rises more, from 275.9 to 331.1 kN.m, because bending capacity does scale with yield. But it is not enough.
• The interaction ratio falls from 1.49 to 1.35. Still a fail.

So the S460 HEB 240 still does not carry the pair, and you have paid for a premium steel to move from 1.49 to 1.35. This mirrors the deflection lesson on long-span beams, where a stronger steel does nothing for stiffness. Here a stronger steel does almost nothing for buckling. The levers that work are geometric: a bigger section (more area and more ry, which is what the HEB 280 buys), a shorter unbraced length (brace the minor axis at mid-height and the slenderness halves), or a smaller moment (detail the beam connection to shed the unbalanced moment). Steel grade is not on that list.

The worked check above is AISC 360, but every major code frames a beam-column the same way: a design compression from a buckling curve, a design moment reduced for lateral-torsional buckling, a second-order moment, and one interaction inequality that must not exceed unity.

• AISC 360 (United States), the H1.1 equations used here, with the compression from Chapter E, the flexure from Chapter F, and the B1/B2 amplifiers or a direct second-order analysis for the moment.
• Eurocode 3, EN 1993-1-1 (Europe), clause 6.3.3, with the two interaction expressions and the interaction factors kyy and kzz. The buckling reduction factor chi replaces the AISC column curve, and the moment is reduced by chiLT for lateral-torsional buckling. The bones are identical: axial term plus moment term, at or below 1.0.
• ABNT NBR 8800 (Brazil), the interaction of clause 5.5, closely aligned with the AISC form: an axial ratio plus a moment ratio, with the split at 0.2 and the same 8/9 weighting on the bending term.

The numbers shift a few percent between codes, but the verdict does not. On any of the three, the HEB 240 fails the pair over a 6 m storey and the section climbs. The interaction equation is the universal language of the beam-column, and the axial-only lookup is not.

"The column carries axial, so I sized it for axial." This is the mistake the whole article is built to catch. A storey column almost always carries a moment too, from an unbalanced beam, from frame action, from a notional load, and the moment runs through the interaction with the axial. Size for the pair, not the larger of the two.

"I used the plastic moment for bending capacity." Only valid if the compression flange is braced within the length where the plastic moment is reachable, about 2.5 to 3.2 m for these HEB sections. Over a 6 m storey braced only at the floors, lateral-torsional buckling reduces the moment capacity, here by 15 percent on the HEB 240. Forgetting it is unconservative on exactly the check that governs.

"I used the first-order moment." The axial load riding on the deflected shape adds second-order moment. On this column the B1 amplifier lifts the 150 kN.m by 16 to 30 percent depending on the section. Either apply the amplifier or run a proper second-order analysis; do not check a beam-column with the linear moment.

"Does K = 1 always apply?" Only in a braced frame, where the storey cannot sway. In an unbraced (moment) frame K exceeds 1, often well above 2, the buckling length grows, and the compression capacity falls further. Modern practice sidesteps the K debate with the direct analysis method, which uses K = 1 with reduced stiffnesses and notional loads instead.

"Which axis buckles?" For a doubly symmetric HEB with equal unbraced lengths, the minor axis, because ry is smaller than rx. That is why ry sets the compression capacity in the table above. Brace the minor axis and the game changes.

The question was "what column for a 6 m storey height", and the honest answer is that the height alone cannot answer it. Feed the column its real demand, 1400 kN of axial and 150 kN.m of moment, and the 6 m storey turns out to be the variable that discounts both capacities at once: the buckling length in compression and the unbraced length in bending. The section that carries the axial, the HEB 240, fails the interaction at 1.49. The section you build, the HEB 280, is 24 percent heavier, and a stronger steel cannot close the gap.

That is the difference between a lookup and a check. A table reads the axial load and hands you a section. The interaction reads the axial, the moment, the storey height and the second-order amplification, and hands you the column that actually stands up. You can run the axial half yourself in the calculator above, and the full beam-column check, section by section, in the CalcSteel editor, free and with no login, and free for students. Put in your own storey height and your own moment, and watch which check governs.