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Corner Columns and Biaxial Bending: Why Passing Each Axis Is Not Enough

By 8 min readUpdated

Many of us design a column one direction at a time. We check the moment about x, then the moment about y. If both pass (pasok), we call it safe.

For a corner column, that habit can hide how close the column really is. Here is a sample column run through our RC Column Section Designer (NSCP 2015).

Why a corner column is different

A corner column has beams framing into it from two sides, at right angles. Each beam bends it in its own direction, so it gets a moment about x and a moment about y at the same time.

It carries the least floor area, so it is tempting to make it small. But it is the one bent from two sides.

For columns that belong to two frames, like corner columns, NSCP Section 208 asks for the earthquake in both directions at once: the full force one way plus 30% the other. Overturning also gives corner columns the largest swings in axial force, so the combination with the least axial load needs a check too.

What happens inside the column

Bending about one axis puts one face in compression. Bending about both axes tilts the neutral axis, and the compression zone becomes a slanted wedge. Its shape depends on the axial load. It needs about the same concrete to carry Pu, but on a slant that concrete gives less moment in each direction, and the corner reaches its limit first.

Compression zone: bending about one axis and about both axes Concept sketch, no numbers. Left: with one moment, about the x-axis, the neutral axis is parallel to a face and the compression zone is a rectangular block along one 400 mm face. Right: with moments about both axes the neutral axis tilts and the compression zone becomes a slanted wedge of about the same area, deepest at one corner; its shape depends on the axial load, and that corner reaches its limit first. One moment (about x) One face in compression compression tension side neutral axis block along one 400 mm face Two moments together (about x and y) The neutral axis tilts compression tension side neutral axis, tilted this corner reaches its limit first slanted wedge: shape depends on the axial load About the same concrete in compression, but on a slant it gives less moment in each direction.
Figure 1. Concept sketch, no numbers. With one moment the compression zone is a block along one face. With two moments the neutral axis tilts and the zone becomes a slanted wedge, deepest at one corner.

You cannot spend the same concrete twice. That is why two separate "pass" results do not, by themselves, show that the column is safe.

The test column

I used the sample column C-1 that comes with the sheet, and changed only the moments. It is deliberately heavily loaded:

  • 400 × 500 mm: b = 400 mm along x, h = 500 mm along y. Mux bends the column about the x-axis and is resisted by the 500 mm depth
  • 10-Ø20 (ρg = 1.57%): 3 bars along each 400 mm face, 4 along each 500 mm face. 10 mm ties @ 150 mm plus one crosstie, 40 mm clear cover
  • f'c = 28 MPa, fy = 420 MPa
  • Braced (non-sway), ℓu = 3.0 m both ways: kℓu/r = 20.0 (x) and 25.0 (y), under the limit of 28.0 (406.2.5: 34 + 12 M1/M2, with M1/M2 = −0.5 for this sample). Short, so δns = 1.0 and Mcx = Mux, Mcy = Muy
  • Pu = 1,800 kN (the sample's value)
  • Mux = 170 kN·m and Muy = 140 kN·m, acting together
Corner column C-1: plan at the corner and column section Left: plan of a corner column, 400 by 500 mm, with one beam framing in along y and one along x. The beam that spans in y bends the column about the x-axis (Mux = 170 kN.m, double-headed vector along x); the beam that spans in x bends it about the y-axis (Muy = 140 kN.m, vector along y). Pu = 1,800 kN. Right: the column section to scale, b = 400 mm along x and h = 500 mm along y, with 10 bars of 20 mm: 3 along each 400 mm face and 4 along each 500 mm face, a 10 mm perimeter tie at 150 mm with 40 mm clear cover, and one crosstie parallel to x that holds one middle bar on each 500 mm face. Plan at the corner Pu = 1,800 kN, with Mux and Muy together x y Mux 170 kN·m Muy 140 kN·m Beam spanning in y bends the column about x: Mux Beam spanning in x bends the column about y: Muy Double arrows: moment vectors (right-hand rule) Column section, to scale b = 400 mm along x, h = 500 mm along y x y b = 400 mm h = 500 mm 10-Ø20 bars: 3 along each 400 mm face, 4 along each 500 mm face 10 mm ties @ 150 mm, 40 mm clear cover Crosstie parallel to x: holds one middle bar on each 500 mm face Mux: about the x-axis, resisted by the 500 mm depth Muy: about the y-axis, resisted by the 400 mm depth
Figure 2. The corner column in plan, and its section drawn to scale: 400 × 500 mm with 10-Ø20 bars, a 10 mm tie and one crosstie. The beam spanning in y gives Mux = 170 kN·m and the beam spanning in x gives Muy = 140 kN·m, both acting together with Pu = 1,800 kN.

First, each moment alone. With only Mux, the sheet says SECTION ADEQUATE, with a moment ratio of 0.62. With only Muy, again SECTION ADEQUATE at 0.62. So far, the one-axis habit says the column is fine.

Case A: both moments together

Check Demand Capacity / limit Ratio Result
Axial load1,800 kNφPn,max = 3,122.4 kN0.58PASS
Bending about x170 kN·mφMnx = 274.1 kN·m0.62PASS
Bending about y140 kN·mφMny = 225.4 kN·m0.62PASS
Biaxial, Rx + Ry0.62 + 0.621.001.24FAIL

Bending capacities are at Pu = 1,800 kN; Rx = Mcx/φMnx, Ry = Mcy/φMny. Overall: REVISE SECTION. Every other check passes or does not apply. Only the combined check fails, by the sheet's straight-line rule.

Case A in the RC Column Section Designer, NSCP 2015 edition: the first four rows of the sheet's Design Check Summary and the overall result (the rows in between are left out). Each axis passes at 0.62, and the biaxial row fails at 1.24.
Figure 3. Case A in the RC Column Section Designer, NSCP 2015 edition: the first four rows of the sheet's Design Check Summary and the overall result (the rows in between are left out). Each axis passes at 0.62, and the biaxial row fails at 1.24.

An independent strain-compatibility calculation outside the sheet gives the same capacities: c = 310 mm about x and 247 mm about y, both compression-controlled, so φ = 0.65 (Table 421.2.2).

What about bigger bars?

A common first fix is more steel. I changed the bars from 20 mm to 25 mm, with the same section and loads.

10-Ø20 (case A) 10-Ø25
Steel ratio ρg1.57%2.45%
φMnx at Pu274.1 kN·m323.0 kN·m
φMny at Pu225.4 kN·m268.0 kN·m
Biaxial ratio (sheet)1.24, FAIL1.05, FAIL

The steel area went up about 56% ((25/20)² = 1.56), but the moment strength only about 18% about x and 19% about y. At this high axial load, most of the moment strength comes from the concrete in compression, which did not change, and the tension-side bars are below yield.

Case B: a bigger section, 450 × 550 mm

Same bars, ties and loads. Only the section grows, by 50 mm each way.

Check Demand Capacity / limit Ratio Result
Axial load1,800 kNφPn,max = 3,710.3 kN0.49PASS
Bending about x170 kN·mφMnx = 372.2 kN·m0.46PASS
Bending about y140 kN·mφMny = 316.3 kN·m0.44PASS
Biaxial, Rx + Ry0.46 + 0.441.000.90PASS

Overall: SECTION ADEQUATE.

Case B, 450 × 550 mm, in the same sheet: the first four rows of the Design Check Summary and the overall result. The biaxial ratio is 0.90 and the section is adequate.
Figure 4. Case B, 450 × 550 mm, in the same sheet: the first four rows of the Design Check Summary and the overall result. The biaxial ratio is 0.90 and the section is adequate.

The general form is (Mcx/φMnx)^α + (Mcy/φMny)^α ≤ 1.0. The sheet uses α = 1.0, the straight-line rule:

Mcx / φMnx + Mcy / φMny ≤ 1.0

Hand check: 170 / 372.2 + 140 / 316.3 = 0.4567 + 0.4426 = 0.899, shown as 0.90.

Biaxial check: the three runs against the straight 1.0 line Chart of Rx = Mcx over phi Mnx (horizontal, 0 to 1.0) against Ry = Mcy over phi Mny (vertical, 0 to 1.0). The straight line Rx + Ry = 1.0 is the sheet's limit (alpha = 1.0). Case A, 400 by 500 with 10 bars of 20 mm, plots at 0.62 and 0.62: sum 1.24, outside the line, FAIL. With 25 mm bars it plots at 0.53 and 0.52: sum 1.05, still outside, FAIL. Case B, 450 by 550 with 10 bars of 20 mm, plots at 0.46 and 0.44: sum 0.90, inside the line, PASS. Each moment alone gives 0.62 on its own axis, which passes. Biaxial check: Rx + Ry ≤ 1.0 (the sheet's straight-line rule) 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 Rx = Mcx / φMnx (bending about x) Ry = Mcy / φMny (bending about y) Straight line: Rx + Ry = 1.0 outside the line: FAIL inside the line: PASS Mux alone: 0.62 Muy alone: 0.62 Case A: 400 × 500, 10-Ø20 0.62 + 0.62 = 1.24, FAIL Bigger bars: 400 × 500, 10-Ø25 0.53 + 0.52 = 1.05, FAIL Case B: 450 × 550, 10-Ø20 0.46 + 0.44 = 0.90, PASS All three: Pu = 1,800 kN, Mux = 170 kN·m, Muy = 140 kN·m
Figure 5. The three runs on the sheet's straight-line check. Case A plots at 0.62 and 0.62, a sum of 1.24, outside the line. With 25 mm bars the sum is 1.05, still outside. Case B, 450 × 550 mm, gives 0.90, inside the line.

The axial line checks too (NSCP 422.4.2), with Ag = 450 × 550 = 247,500 mm² and Ast = 10 × π × 20² / 4 = 3,141.6 mm²:

Po = 0.85 f'c (Ag − Ast) + fy Ast = 0.85 × 28 × (247,500 − 3,141.6) + 420 × 3,141.6 = 7,135,199 N = 7,135.2 kN
φPn,max = 0.65 × 0.80 × 7,135.2 = 3,710.3 kN

The gross area grew about 24% (247,500 against 200,000 mm²); the moment strength about 36% about x and 40% about y. At this high axial load, the larger section did more than the heavier bars (by the sheet's check). At lower axial loads this can turn around.

Watch the detailing: the steel ratio drops to 1.27%, still above the 1% minimum, and the clear distance from an untied middle bar to the nearest tied bar becomes 145 mm, just inside the 150 mm limit of 425.7.2.3(b).

This sample is a braced column outside the earthquake-resisting frame. In seismic zone 4, most of the Philippines, a corner column of a concrete moment frame is usually part of that frame, often a sway frame. There the slenderness limit is 22 (406.2.5), so kℓu/r of 25.0 or more about y (k ≥ 1.0) would need second-order effects. The special moment frame rules of NSCP Section 418 also change the design (strong column-weak beam, shear), not only the ties; see Why Column Ties Need 135-Degree Hooks. The biaxial check applies in every case.

The lesson

Two ratios of 0.62 do not mean the column is 62% used. They mean 62% used in each direction, when the other moment is zero. When both act together, the sheet adds them: 0.62 + 0.62 = 1.24.

The straight-line rule is the simple, safe-side form. A full biaxial strain-compatibility analysis of this column, done outside the sheet, finds more strength: it puts case A at about the limit, and it would pass the 25 mm option. If we rely on it, we must do it and keep the calculation. Either way, the two results of 0.62 hid a column that is fully used.

What to check

  1. Find the columns with two moments. Corner columns first, then edge columns with unequal beams on two sides.
  2. Use moments that act together: the same load combination and the same end of the column. Check each combination with its own Pu, Mux and Muy, including the one with the least axial load.
  3. Run the combined check, Mcx/φMnx + Mcy/φMny ≤ 1.0 (or a full biaxial P-Mx-My analysis), after any slenderness magnification.
  4. If it fails, try both a bigger section and more or bigger bars, and compare. Which works better depends on the axial load.
  5. Re-check the new section: steel ratio, bar spacing, and which bars need a tie corner or crosstie. If you enlarge the column, re-run the frame analysis. A stiffer column attracts more moment.
  6. On site, build what the column schedule shows: bar count, size and position, corner bars in tie corners, crossties and 135° hooks. Keep the formwork size and orientation: a 400 × 500 column turned 90 degrees puts its strong direction in the wrong place. No bar substitution or "to match" downsizing without the engineer of record's written approval.
  7. Worried about an existing building? Ask the engineer of record or a licensed structural engineer to review it. A failed simplified check does not mean the column will fail. Do not chip or core it.

This is not about any one engineer or crew. Checking the two directions together is a small step that is easy to miss.

General information only. Follow the approved structural plans and the engineer of record.