---
title: "Biaxial column 400x600, 1500 kN"
subject: "Design of RCC Structure"
chapter: 10
key: "semester_7/design_of_rcc_structure:biaxial-column-400x600-1500-kn"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/biaxial-column-400x600-1500-kn
site: BCT Engineering
---

# Biaxial column 400x600, 1500 kN

Design of RCC Structure · Chapter 10: Design of compression members: Columns

- Asked in: 2068 Baisakh (old course), 15 marks
- Video: [Design of a Rectangular Column subjected to combined axial loading and biaxial bending](https://www.youtube.com/watch?v=VL2MoB0n9Tg) — Stan Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/biaxial-column-400x600-1500-kn
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10

## Question

Determine the reinforcement in a biaxially loaded column with the following parameters: Size of column = 400 mm x 600 mm; Factored load, $P_u$ = 1500 kN; Factored moment, $M_{ux}$ = 300 kNm; Factored moment, $M_{uy}$ = 200 kNm. Assume M25 concrete and Fe 415 steel.

## Short answer

For a 400×600 mm biaxially loaded column with 1500 kN axial load, 300 kNm major-axis and 200 kNm minor-axis moments, use 8-32 mm bars (2.68%) and 8 mm ties at 300 mm c/c.

## Answer

#### Data and assumptions
- Column 400 mm (width $b$) × 600 mm (depth $D$); $P_u = 1500$ kN; $f_{ck} = 25$ MPa, $f_y = 415$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
- Moments: about major axis (x-x) $M_{ux}$ = 300 kN·m; about minor axis (y-y) $M_{uy}$ = 200 kN·m.
- Unsupported length not given; a short column with $L = 3$ m is assumed. $M_{ux}=300$ kN·m acts about the major axis (depth 600 mm), $M_{uy}=200$ kN·m about the minor axis (depth 400 mm).
- Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

#### Slenderness

| Axis | $l_e$ (mm) | Dimension (mm) | $l_e/$dim | Type |
|---|---|---|---|---|
| x-x (about major) | 3000 | 600 | 5.00 | short |
| y-y (about minor) | 3000 | 400 | 7.50 | short |

Both ratios are below 12, so the column is **short**; no additional moment is required.

#### Minimum eccentricity (cl. 25.4)
$$
e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)
$$
- About x-x: $e_x = 3000/500 + 600/30 = 26.0$ mm (min. 20 mm); adopt 26.0 mm; $M_{ux,min} = P_u e = 39.0$ kN·m
- About y-y: $e_y = 3000/500 + 400/30 = 19.3$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 30.0$ kN·m

#### Design moments
- Design moments: $M_{ux} = 300.0$ kN·m, $M_{uy} = 200.0$ kN·m (not less than the minimum-eccentricity values).

#### Longitudinal reinforcement
$\dfrac{P_u}{f_{ck} b D} = \dfrac{1500000}{25 \times 400 \times 600} = 0.250$
Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.

| Trial | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ |
|---|---|---|---|---|---|---|---|
| 12-25 φ | 2.45 | 0.101 | 531.6 | 318.3 | 0.336 | 1.23 | 1.061 ✗ |
| 20-20 φ | 2.62 | 0.097 | 550.9 | 331.4 | 0.327 | 1.21 | 1.021 ✗ |
| 8-32 φ | 2.68 | 0.107 | 570.6 | 338.2 | 0.324 | 1.21 | 0.991 ✓ |

#### Check
$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 233566 + 0.75 \times 415 \times 6434 = 4630$ kN
$\dfrac{P_u}{P_{uz}} = \dfrac{1500}{4630} = 0.324$, so $\alpha_n = 1.21$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).
$$
\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{300.0}{570.6}\right)^{1.21} + \left(\frac{200.0}{338.2}\right)^{1.21} = 0.991 \le 1.0
$$
The interaction condition is satisfied; the section is **safe**.

Provide **8-32 mm** bars ($A_{sc} = 6434$ mm², $p = 2.68\%$), within 0.8% to 4%.

#### Lateral ties
- Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(32/4, 6) = 8$ mm → provide 8 mm.
- Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 512, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

#### Detailing
```text
CROSS-SECTION 400 x 600
    +-------------------------+
    |   o        o        o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o        o        o   |
    +-------------------------+
   o = 32 mm bars, ties 8 mm @ 300 c/c
```

**Answer:** 400 × 600 mm column with 8-32 mm longitudinal bars ($p = 2.68\%$) and 8 mm ties @ 300 mm c/c.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The question gives no length. How does the answer proceed?
   A. It refuses to design
   B. It assumes a short column with L = 3 m, so no slenderness moments are added
   C. It assumes a slender column
   D. It assumes zero minimum eccentricity
   Answer: B. A short column with L = 3 m is assumed, giving le/D = 5 and le/b = 7.5. (Key point: Assumption of length)

2. With Pu/Puz = 0.324, alpha_n is about
   A. 1.21
   B. 1.62
   C. 2.00
   D. 0.32
   Answer: A. alpha_n = 1 + (0.324 - 0.2)/0.6 = 1.21. (Key point: Alpha_n)

3. For the 8-32 mm trial, the sum is 0.991. Meaning?
   A. Unsafe
   B. Safe, since the sum is <= 1.0
   C. The steel exceeds 4%
   D. Slenderness fails
   Answer: B. The interaction sum 0.991 <= 1.0, so the section is adequate; the 12-25 and 20-20 trials (1.061 and 1.021) fail. (Key point: Check of interaction)

4. In Puz = 0.45 fck Ac + 0.75 fy Asc, what is Ac?
   A. Gross area 240000 mm^2
   B. Gross area minus steel area (233566 mm^2)
   C. Steel area only
   D. Half the gross area
   Answer: B. Ac = 400 x 600 - 6434 = 233566 mm^2. (Key point: Puz calculation)

5. For 32 mm bars in a 400 mm wide column, the ties are
   A. 8 mm at 300 mm c/c
   B. 8 mm at 512 mm c/c
   C. 6 mm at 400 mm c/c
   D. 32 mm at 300 mm c/c
   Answer: A. Dia >= 32/4 = 8 mm; pitch <= min(400, 512, 300) = 300 mm. (Key point: Tie spacing)

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