---
title: "Short column 35x45 cm, 1800 kN"
subject: "Design of RCC Structure"
chapter: 10
key: "semester_7/design_of_rcc_structure:short-column-35x45-cm-1800-kn"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/short-column-35x45-cm-1800-kn
site: BCT Engineering
---

# Short column 35x45 cm, 1800 kN

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

- Asked in: 2072 Chaitra, 15 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-35x45-cm-1800-kn
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10

## Question

Design the longitudinal and transverse reinforcements to be provided for a short column of size 35 cm x 45 cm subjected to the following forces: Factored axial load $P_u$ = 1800 kN; Factored moment $M_{ux}$ = 175 kN-m; Factored moment $M_{uy}$ = 105 kN-m. Reinforcements are distributed equally on two sides. Use M25 concrete and Fe500 steel. Unsupported length = 3.1 m

## Short answer

Column is short. Provide 6-32 mm bars on two faces (Asc 4825 mm2, 3.06%) plus one 12 mm nominal bar at mid-depth of each 450 mm face, with 8 mm ties at 300 mm c/c; biaxial check is safe.

## Answer

#### Data and assumptions
- Column 350 mm (width $b$) × 450 mm (depth $D$); $P_u = 1800$ kN; $f_{ck} = 25$ 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) $M_{ux}$ = 175 kN·m; about minor axis (y-y) $M_{uy}$ = 105 kN·m.
- As the question states, the bars are placed equally on two opposite faces: the two 350 mm faces, which are parallel to the major axis x-x and so resist the larger moment $M_{ux}$ most effectively.
- 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) | 3100 | 450 | 6.89 | short |
| y-y (about minor) | 3100 | 350 | 8.86 | 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: $e_x = 3100/500 + 450/30 = 21.2$ mm (min. 20 mm); adopt 21.2 mm; $M_{ux,min} = P_u e = 38.2$ kN·m
- About y-y: $e_y = 3100/500 + 350/30 = 17.9$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 36.0$ kN·m

#### Design moments
- Design moments: $M_{ux} = 175.0$ kN·m, $M_{uy} = 105.0$ kN·m (not less than the minimum-eccentricity values).

#### Longitudinal reinforcement
$\dfrac{P_u}{f_{ck} b D} = \dfrac{1800000}{25 \times 350 \times 450} = 0.457$
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 | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ |
|---|---|---|---|---|---|---|---|
| 12-20 φ | 2.39 | 0.129 | 231.9 | 122.0 | 0.573 | 1.62 | 1.418 ✗ |
| 8-25 φ | 2.49 | 0.134 | 238.2 | 131.9 | 0.562 | 1.60 | 1.303 ✗ |
| 6-32 φ | 3.06 | 0.142 | 286.1 | 161.3 | 0.510 | 1.52 | 0.996 ✓ |

#### Check
$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 152675 + 0.75 \times 500 \times 4825 = 3527$ kN
$\dfrac{P_u}{P_{uz}} = \dfrac{1800}{3527} = 0.510$, so $\alpha_n = 1.52$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).
$$
\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{175.0}{286.1}\right)^{1.52} + \left(\frac{105.0}{161.3}\right)^{1.52} = 0.996 \le 1.0
$$
The interaction condition is satisfied; the section is **safe**.

Provide **6-32 mm** bars ($A_{sc} = 4825$ mm², $p = 3.06\%$), within 0.8% to 4%.

The two layers are $450 - 2 \times 64 = 322$ mm apart, which is more than the 300 mm maximum spacing along the periphery (IS 456 cl. 26.5.3.1), so also provide one 12 mm nominal bar at mid-depth of each 450 mm face (not counted in the design).

#### 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.

#### Detailing
```text
CROSS-SECTION 350 x 450
    +-------------------------+
    |                         |
    |    o       o       o    |
    |                         |
    |                         |
    |                         |
    |    x               x    |
    |                         |
    |                         |
    |                         |
    |    o       o       o    |
    |                         |
    +-------------------------+
   o = 32 mm bars, x = 12 mm nominal side bars, ties 8 mm @ 300 c/c
```

**Answer:** 350 × 450 mm column with 6-32 mm longitudinal bars ($p = 3.06\%$), 3 on each 350 mm face, plus one 12 mm nominal bar on each 450 mm face, 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 the 350 x 450 column with le = 3100 mm, why is no additional moment needed?
   A. Because the axial load is below 2000 kN
   B. Because le/dimension is below 12 on both axes (6.89 and 8.86), so it is short
   C. Because the moments exceed the minimum-eccentricity moments
   D. Because the steel is Fe500
   Answer: B. IS 456 treats a column as short when le/D and le/b are both below 12, so slenderness moments are not added. (Key point: Short column check)

2. About the minor axis, e = L/500 + b/30 = 17.9 mm. What eccentricity is adopted and why?
   A. 17.9 mm, the calculated value
   B. 20 mm, because emin is not less than 20 mm
   C. 35 mm, one-tenth of the width
   D. 0 mm, because the applied moment is larger
   Answer: B. IS 456 cl. 25.4 requires emin to be at least 20 mm, so 20 mm is adopted, giving Muy,min = 1800 x 0.02 = 36 kN-m. (Key point: Minimum eccentricity)

3. Which condition must be satisfied for the biaxially loaded column to be safe?
   A. Mux + Muy <= Mux1 + Muy1
   B. (Mux/Mux1)^an + (Muy/Muy1)^an <= 1.0
   C. (Mux/Mux1) x (Muy/Muy1) <= 1.0
   D. Mux/Mux1 <= 1.0 only
   Answer: B. IS 456 cl. 39.6 uses the interaction equation with exponent an, where Mux1 and Muy1 are the uniaxial capacities at Pu. (Key point: Biaxial interaction check)

4. With Pu/Puz = 0.510, how is alpha_n obtained and what is its value?
   A. Fixed at 2.0 for all columns
   B. Linear between 1.0 at 0.2 and 2.0 at 0.8, giving about 1.52
   C. Linear between 1.0 at 0.8 and 2.0 at 0.2, giving about 1.48
   D. Equal to Pu/Puz, i.e. 0.51
   Answer: B. alpha_n = 1 + (0.510 - 0.2)/0.6 = about 1.52. (Key point: Exponent alpha_n)

5. Why were 12-20 and 8-25 rejected and 6-32 adopted?
   A. Their steel percentage was below 0.8%
   B. Their interaction sum was above 1.0 (1.418 and 1.303); 6-32 gave 0.996
   C. Their bars were too thick
   D. They gave a larger Puz than needed
   Answer: B. A trial is acceptable only if the interaction sum is <= 1.0; only 6-32 (p = 3.06%) satisfied it. (Key point: Trial selection)

6. For 32 mm main bars in a 350 mm wide column, the tie pitch is limited to
   A. 350 mm, the least lateral dimension
   B. 512 mm, i.e. 16 x 32
   C. 300 mm, the least of 350, 512 and 300
   D. 32 mm bar diameter
   Answer: C. Pitch <= min(least lateral dimension, 16 x smallest bar dia, 300 mm) = 300 mm. (Key point: Lateral ties)

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