---
title: "Unbraced column 400x350, 6.0 m"
subject: "Design of RCC Structure"
chapter: 10
key: "semester_7/design_of_rcc_structure:unbraced-column-400x350-6-0-m"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/unbraced-column-400x350-6-0-m
site: BCT Engineering
---

# Unbraced column 400x350, 6.0 m

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

- Asked in: 2079 Bhadra, 14 marks
- 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/unbraced-column-400x350-6-0-m
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10

## Question

Design an unbraced rectangular RC column having clear height 6.0 m, with x-sectional dimension 400 mm x 350 mm subjected to design axial load of 600 kN, design bending moments 100 kNm about major axis and 50 kNm about minor axis. Consider M20 concrete and Fe 415 steel.

## Short answer

Provide 6-32 mm bars (3.45%) and 8 mm ties @ 300 mm c/c. Slenderness added moments give 134.3 and 84.8 kN·m; interaction ratio 0.965 ≤ 1.0.

## Answer

#### Data and assumptions
- Column 350 mm (width $b$) × 400 mm (depth $D$); $P_u = 600$ kN; $f_{ck} = 20$ 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) $M_{ux}$ = 100 kN·m; about minor axis (y-y) $M_{uy}$ = 50 kN·m.
- End conditions are not stated. The column is assumed to be held in position and restrained against rotation at one end, and restrained against rotation but free to sway at the other, so $l_e = 1.2L = 7.2$ m (IS 456 Table 28).
- The 100 kN·m moment acts about the major axis and the 50 kN·m moment about the minor axis; being design moments, they are taken as the larger end moments.
- 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) | 7200 | 400 | 18.00 | slender |
| y-y (about minor) | 7200 | 350 | 20.57 | slender |
The limit for a short column is 12 (IS 456 cl. 25.1.2). The column is slender about both axes, so additional moments are added about both (cl. 39.7.1).

#### Minimum eccentricity (cl. 25.4)
$$
e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)
$$
- About x-x: $e_x = 6000/500 + 400/30 = 25.3$ mm (min. 20 mm); adopt 25.3 mm; $M_{ux,min} = P_u e = 15.2$ kN·m
- About y-y: $e_y = 6000/500 + 350/30 = 23.7$ mm (min. 20 mm); adopt 23.7 mm; $M_{uy,min} = 14.2$ kN·m

#### Design moments
Additional moment due to slenderness: $M_a = \dfrac{P_u D}{2000}\left(\dfrac{l_e}{D}\right)^2$ (38.9 kN·m about x-x, 44.4 kN·m about y-y), reduced by $k = \dfrac{P_{uz}-P_u}{P_{uz}-P_b} \le 1$.
- For the selected steel: $P_{uz} = 2719$ kN, $P_{bx} = 316$ kN, $k_x = 0.88$, $M_{ax} = 34.3$ kN·m; $P_{by} = 10$ kN, $k_y = 0.78$, $M_{ay} = 34.8$ kN·m.
- $P_{by}$ is close to zero because, for bending about the minor axis, two-thirds of the heavy steel (the middle and far bars) lies in the tension zone at the balanced strain; its tensile force almost cancels the concrete compression ($0.36 f_{ck} D x_{u,max} \approx 395$ kN).
- In an unbraced column the additional moment is added to the end moment: $M_{ux} = 100 + 34.3 = 134.3$ kN·m, $M_{uy} = 50 + 34.8 = 84.8$ kN·m (not less than the minimum-eccentricity values).

#### Longitudinal reinforcement
$\dfrac{P_u}{f_{ck} b D} = \dfrac{600000}{20 \times 350 \times 400} = 0.214$
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 | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ |
|---|---|---|---|---|---|---|---|
| 20-16 φ | 2.87 | 0.140 | 197.6 | 164.4 | 0.242 | 1.07 | 1.150 ✗ |
| 14-20 φ | 3.14 | 0.145 | 200.4 | 181.0 | 0.232 | 1.05 | 1.102 ✗ |
| 6-32 φ | 3.45 | 0.160 | 275.2 | 169.2 | 0.221 | 1.03 | 0.965 ✓ |

#### Check
$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 135175 + 0.75 \times 415 \times 4825 = 2719$ kN
$\dfrac{P_u}{P_{uz}} = \dfrac{600}{2719} = 0.221$, so $\alpha_n = 1.03$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).
$$
\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{134.3}{275.2}\right)^{1.03} + \left(\frac{84.8}{169.2}\right)^{1.03} = 0.965 \le 1.0
$$
The interaction condition is satisfied; the section is **safe**.

Provide **6-32 mm** bars ($A_{sc} = 4825$ mm², $p = 3.45\%$), 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) $= \min(350, 512, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c. The middle bar on each 350 mm face is more than 75 mm from the corner bars, so it is held by a cross-tie.

#### Detailing
```text
CROSS-SECTION 350 x 400
    +-------------------------+
    |                         |
    |    o       o       o    |
    |                         |
    |                         |
    |                         |
    |                         |
    |                         |
    |                         |
    |                         |
    |    o       o       o    |
    |                         |
    +-------------------------+
   o = 32 mm bars, ties 8 mm @ 300 c/c
```

**Answer:** 350 × 400 mm column with 6-32 mm longitudinal bars ($p = 3.45\%$) 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. How is Mux = 134.3 kN·m obtained for the unbraced column?
   A. 100 + 34.3 (Mux plus reduced additional moment, no reduction of the end moment)
   B. 0.6(100) + 34.3
   C. 34.3 only
   D. 100 - 34.3
   Answer: A. In a sway column the given moment is the larger end moment and is not reduced, so Mux = 100 + Max = 134.3 kN·m. (Key point: Total design moment)

2. Which expression gives the axial capacity Puz used in k and in αn?
   A. 0.67 fck Ac + 0.87 fy Asc
   B. 0.45 fck Ac + 0.75 fy Asc
   C. 0.36 fck b D
   D. fck Ac + fy Asc
   Answer: B. IS 456 gives Puz = 0.45 fck Ac + 0.75 fy Asc, here 2719 kN. (Key point: Squash load)

3. For bars of 32 mm and 350 mm least dimension, which limit governs the tie pitch?
   A. 16 x 32 = 512 mm
   B. 350 mm
   C. 256 mm
   D. The 300 mm limit
   Answer: D. Pitch <= min(350, 512, 300) = 300 mm. Tie diameter is max(32/4, 6) = 8 mm. (Key point: Tie pitch)

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