---
title: "Short column 300x500, 2000 kN, M30"
subject: "Design of RCC Structure"
chapter: 10
key: "semester_7/design_of_rcc_structure:short-column-300x500-2000-kn-m30"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/short-column-300x500-2000-kn-m30
site: BCT Engineering
---

# Short column 300x500, 2000 kN, M30

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

- Asked in: 2072 Kartik, 16 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/short-column-300x500-2000-kn-m30
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10

## Question

Determine the longitudinal and transverse reinforcements in a short rectangular column subjected to a factored axial load of 2000 kN and factored moment $M_{ux}$ about major axis of 190 kN-m and $M_{uy}$ about minor axis of 95 kN-m. The size of the column is 300 mm x 500 mm and the unsupported length of 3 m. Adopt M30 concrete and Fe 500 grade steel.

## Short answer

Column is short. Provide 16-20 mm longitudinal bars (Asc 5027 mm2, 3.35%) and 8 mm ties at 300 mm c/c; biaxial interaction check is safe for M30 concrete and Fe500 steel.

## Answer

#### Data and assumptions
- Column 300 mm (width $b$) × 500 mm (depth $D$); $P_u = 2000$ kN; $f_{ck} = 30$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
- Moments: about major axis (x-x) $M_{ux}$ = 190 kN·m; about minor axis (y-y) $M_{uy}$ = 95 kN·m.
- 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 | 500 | 6.00 | short |
| y-y (about minor) | 3000 | 300 | 10.00 | 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: $e_x = 3000/500 + 500/30 = 22.7$ mm (min. 20 mm); adopt 22.7 mm; $M_{ux,min} = P_u e = 45.3$ kN·m
- About y-y: $e_y = 3000/500 + 300/30 = 16.0$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 40.0$ kN·m

#### Design moments
- Design moments: $M_{ux} = 190.0$ kN·m, $M_{uy} = 95.0$ kN·m (not less than the minimum-eccentricity values).

#### Longitudinal reinforcement
$\dfrac{P_u}{f_{ck} b D} = \dfrac{2000000}{30 \times 300 \times 500} = 0.444$
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 | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ |
|---|---|---|---|---|---|---|---|
| 14-20 φ | 2.93 | 0.116 | 258.7 | 138.3 | 0.553 | 1.59 | 1.163 ✗ |
| 10-25 φ | 3.27 | 0.121 | 282.5 | 151.3 | 0.526 | 1.54 | 1.029 ✗ |
| 16-20 φ | 3.35 | 0.116 | 299.9 | 148.7 | 0.521 | 1.53 | 0.999 ✓ |

#### Check
$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 30 \times 144973 + 0.75 \times 500 \times 5027 = 3842$ kN
$\dfrac{P_u}{P_{uz}} = \dfrac{2000}{3842} = 0.521$, so $\alpha_n = 1.53$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).
$$
\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{190.0}{299.9}\right)^{1.53} + \left(\frac{95.0}{148.7}\right)^{1.53} = 0.999 \le 1.0
$$
The interaction condition is satisfied; the section is **safe**.

Provide **16-20 mm** bars ($A_{sc} = 5027$ mm², $p = 3.35\%$), within 0.8% to 4%.

#### Lateral ties
- Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(20/4, 6) = 6$ mm → provide 8 mm.
- Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(300, 320, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

#### Detailing
```text
CROSS-SECTION 300 x 500
    +-------------------------+
    |    o   o   o   o   o    |
    |                         |
    |                         |
    |    o               o    |
    |                         |
    |    o               o    |
    |                         |
    |    o               o    |
    |                         |
    |                         |
    |    o   o   o   o   o    |
    +-------------------------+
   o = 20 mm bars, ties 8 mm @ 300 c/c
```

**Answer:** 300 × 500 mm column with 16-20 mm longitudinal bars ($p = 3.35\%$) 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. For le = 3000 mm and a 300 x 500 section, which statement is correct?
   A. le/D = 6.0 and le/b = 10.0; both below 12, so short
   B. le/b = 10.0 exceeds the limit, so slender
   C. Only le/D matters, so short
   D. Both ratios are above 12
   Answer: A. A column is short if le/D and le/b are both below 12; here 6.0 and 10.0. (Key point: Slenderness classification)

2. About the minor axis e = 3000/500 + 300/30 = 16 mm. What moment is taken as the minimum?
   A. 32 kN-m using 16 mm
   B. 40 kN-m using 20 mm
   C. 0 kN-m
   D. 95 kN-m
   Answer: B. emin is not less than 20 mm, so Muy,min = 2000 x 0.020 = 40 kN-m (less than the applied 95 kN-m). (Key point: Minimum eccentricity moments)

3. Which expression gives Puz used for alpha_n?
   A. 0.45 fck Ac + 0.75 fy Asc
   B. 0.87 fy Asc only
   C. 0.67 fck Ag + 0.87 fy Asc
   D. fck Ag + fy Asc
   Answer: A. Puz = 0.45 fck Ac + 0.75 fy Asc, with Ac the concrete area net of steel. (Key point: Axial capacity Puz)

4. The 10-25 trial gave an interaction sum of 1.029. What does this mean?
   A. Safe, as it is close to 1
   B. Not safe; sum exceeds 1.0, so more steel is needed
   C. Safe because p exceeds 3%
   D. Safe if the ties are 8 mm
   Answer: B. The sum must be <= 1.0. The 16-20 trial gave 0.999 and was adopted. (Key point: Trial iteration)

5. With 20 mm main bars in a 300 mm wide column, the tie pitch limit is
   A. min(300, 320, 300) = 300 mm
   B. 320 mm
   C. min(500, 320) = 320 mm
   D. 6 mm
   Answer: A. Pitch <= min(least lateral dimension 300, 16 x 20 = 320, 300) = 300 mm. (Key point: Tie pitch)

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