RC column design

Worked example: design a reinforced concrete column under axial load and biaxial bending using the N-M interaction diagram.

Problem statement

An interior column in a multi-storey office building carries combined axial compression and biaxial bending from frame action. The column is 400 $\times$ 400 mm and 3.5 m tall between floor levels. The exposure classification is A1 (interior, non-aggressive).

Check the column section capacity under the critical ULS load combination to AS 3600:2018.

Given data

Parameter	Value	Units	Source
Column width ( $b$ )	400	mm	Given
Column depth ( $d$ )	400	mm	Given
Column height ( $L$ )	3500	mm	Given
Concrete grade	N50	—	AS 3600
$f'_c$	50	MPa	N50 grade
Rebar grade	D500N	—	AS/NZS 4671
$f_y$	500	MPa	D500N
Cover	35	mm	AS 3600 Table 4.10.3.2, exposure A1
Stirrup size	N10	—	Assumed
Reinforcement	8N24	—	8 bars evenly around perimeter

Design actions (critical ULS combination)

Load	Value	Units
$N^*$	2500	kN
$M^*_x$	120	kN.m
$M^*_y$	80	kN.m

These actions include second-order effects (moment magnification per AS 3600 Cl. 10.4 has been applied externally).

Reinforcement details

8N24 bars: $A_{st} = 8 \times 452 = 3616$ mm $^2$
Reinforcement ratio: $\rho = 3616 / (400 \times 400) = 2.26\%$
Effective depth: $d_{eff} = 400 - 35 - 10 - 24/2 = 343$ mm

Step-by-step solution

Step 1: Define section geometry

Apply the Rectangular template with $b = 400$ mm, $d = 400$ mm.

Step 2: Set materials

Design code: AS 3600
Concrete grade: N50 ( $f'_c = 50$ MPa)
Rebar grade: D500N ( $f_y = 500$ MPa)
Cover: 35 mm all sides

Step 3: Place reinforcement

Use the Perimeter Pattern tool:

Bar diameter: 24 mm (N24)
Number of bars: 8 (3 per face with corner sharing)

This distributes 8 bars evenly around the perimeter at the cover + stirrup inset.

Configure stirrups: N10 at 300 mm spacing, 2 legs.

Step 4: Enter design loads

In the Applied Loads panel:

Member type: Column
ULS 1: $N^* = 2500$ kN, $M^*_x = 120$ kN.m, $M^*_y = 80$ kN.m

Step 5: Review the interaction diagram

Switch to the ULS tab and open the Interaction panel.

Uniaxial check ( $N$ - $M_x$ )

The uniaxial interaction diagram shows the ( $N^*$ , $M^*_x$ ) point relative to the capacity envelope. Key results:

Result	Value	Units
Squash load ( $\phi N_{u0}$ )	~4300	kN
Balanced point	~(2800, 280)	(kN, kN.m)
Pure bending ( $\phi M_u$ )	~260	kN.m
Uniaxial utilisation	~0.55	—

The design point sits well inside the uniaxial envelope.

Biaxial check

With both $M^*_x = 120$ kN.m and $M^*_y = 80$ kN.m, the biaxial check is critical. ACS reports:

Method	Utilisation
Rigorous (3D surface)	~0.65
Bresler reciprocal	~0.68
Bresler contour ( $\alpha = 1.5$ )	~0.70

All three methods give utilisation below 1.0, confirming adequacy. The rigorous method is the most accurate; the Bresler methods are slightly conservative, which is expected for a symmetric section.

Step 6: View the 3D interaction surface

Click the Interaction tab on the canvas to view the 3D $N$ - $M_x$ - $M_y$ surface. The design point (marked in red) sits inside the surface, confirming adequacy visually.

Rotate the surface to see the capacity envelope from different angles. The $M_x$ - $M_y$ contour at the applied axial load level shows the remaining moment capacity in all directions.

Results summary

Check	Demand	Capacity	Utilisation	Status
Uniaxial $N$ - $M_x$	(2500, 120)	Envelope	~0.55	Pass
Biaxial (rigorous)	(2500, 120, 80)	3D surface	~0.65	Pass
Bresler reciprocal	—	—	~0.68	Pass
Bresler contour	—	—	~0.70	Pass

Discussion

The column is adequate under the critical biaxial load combination with a utilisation ratio of approximately 0.65—0.70 depending on the method.

The biaxial utilisation (0.65—0.70) is significantly higher than the uniaxial utilisation (0.55), demonstrating why biaxial checks are essential for columns with moments about both axes. Ignoring the minor-axis moment would significantly underestimate the demand on the section.

Key observations:

The design is compression-dominated ( $N^* = 2500$ kN is above the balanced point), so the $\phi$ factor is lower (0.65 for compression-controlled members)
The reinforcement ratio of 2.26% is within the typical range for columns (1%—4%)
For a symmetric section with symmetric reinforcement, the Bresler methods are reliable; for asymmetric sections, use the rigorous 3D surface check

If the utilisation were too high, you could:

Increase the column size (most effective for compression-dominated members)
Add more reinforcement (up to the maximum ratio of 4%)
Increase the concrete grade (effective because $N_{u0}$ scales directly with $f'_c$ )

Hand calculation verification

Verify the Bresler reciprocal method:

Step 1: Find $N_{ux}$ (uniaxial capacity at $M^*_x = 120$ kN.m, $M^*_y = 0$ ).

From the $N$ - $M_x$ interaction diagram, at $M_x = 120$ kN.m, $\phi N_{ux} \approx 4100$ kN.

Step 2: Find $N_{uy}$ (uniaxial capacity at $M^*_y = 80$ kN.m, $M^*_x = 0$ ).

By symmetry (square section, symmetric reinforcement), at $M_y = 80$ kN.m, $\phi N_{uy} \approx 4200$ kN.

Step 3: Squash load $\phi N_{u0} \approx 4300$ kN.

Step 4: Bresler reciprocal:

\frac{1}{\phi N_u} = \frac{1}{4100} + \frac{1}{4200} - \frac{1}{4300}

\frac{1}{\phi N_u} = 0.000244 + 0.000238 - 0.000233 = 0.000249

\phi N_u = 4016 \text{ kN}

Utilisation:

\frac{N^*}{\phi N_u} = \frac{2500}{4016} = 0.62

This is close to the ACS rigorous result of ~0.65, confirming the analysis.

M-N interaction theory — interaction diagram background
Section analysis — overview of all analysis types
RC beam example — beam design with flexure and shear