---
title: "Slab design, 4x6 m, Fe500"
subject: "Design of RCC Structure"
chapter: 9
key: "semester_7/design_of_rcc_structure:slab-design-4x6-m-fe500"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/slab-design-4x6-m-fe500
site: BCT Engineering
---

# Slab design, 4x6 m, Fe500

Design of RCC Structure · Chapter 9: Design of slabs and staircase

- Asked in: 2065 Kartik (old course), 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/slab-design-4x6-m-fe500
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-9

## Question

Design a simply supported RC slab of 4 m x 6 m at limit state of collapse in flexure. Arrange the designed reinforcing bars and draw a neat sketch of slab showing arrangement of top and bottom reinforcements. Take, Live load = 4 kN/m$^2$, Floor finish = 1 kN/m$^2$, Grade of concrete = M20, Grade of steel reinforcement = Fe500.

## Short answer

Slab thickness 240 mm. Bottom reinforcement in both directions: 8 mm at 170 mm c/c. Top steel at discontinuous edges: 8 mm at 300 mm c/c extending 0.1 span. Designed as simply supported two-way slab with corners held down. Deflection satisfied.

## Answer

#### Data and assumptions
- Clear spans: $l_x = 4$ m (short), $l_y = 6$ m (long); support width 230 mm; $f_{ck} = 20$ MPa, $f_y = 500$ MPa.
- Live load = 4 kN/m², floor finish = 1 kN/m². Unit weight of concrete 25 kN/m³. Clear cover 20 mm (mild exposure, IS 456 Table 16); 10 mm bars assumed for working out $d$.
- 'Simply supported' slab with corners prevented from lifting (so top steel is provided at the corners); support width 230 mm assumed.
- Panel type: four edges discontinuous (corners held down); moment coefficients from IS 456 Annex D, Table 26 (case 9: four edges discontinuous).

#### Depth and effective span
Basic span/depth ratio = 20 (IS 456 cl. 23.2.1). Trial overall depth $D = 240$ mm, so $d = 240 - 20 - 5 = 215$ mm for the short-span bars and $d_y = 215 - 10 = 205$ mm for the long-span bars.
$$
l_x = \min(l_{clear} + d,\ c/c) = 4000 + 215 = 4215\ \text{mm},\quad l_y = 6000 + 215 = 6215\ \text{mm}
$$
$$
\frac{l_y}{l_x} = \frac{6215}{4215} = 1.474 < 2 \Rightarrow \text{two-way slab}
$$

#### Loads (per m²)
- Self-weight = 25 × 0.240 = 6.000 kN/m²
- Total working load $w$ = 6.000 + 1 + 4 = 11.000 kN/m²
- Factored load $w_u = 1.5 \times 11.000 = 16.500$ kN/m²

#### Bending moments
| Moment | Coefficient | Value (kN·m/m) |
|---|---|---|
| $M_x$ positive (mid-span, short) | 0.0880 | 25.79 |
| $M_y$ positive (mid-span, long) | 0.0560 | 16.42 |

$M = \alpha\, w_u\, l_x^2$ with $w_u l_x^2 = 16.500 \times 4.215^2 = 293.14$ kN·m/m. Coefficients are interpolated for $l_y/l_x = 1.47$.

#### Check of depth
$M_{u,lim} = 0.133\, f_{ck} b d^2 = 0.133 \times 20 \times 1000 \times 215^2 \times 10^{-6} = 122.96$ kN·m/m $>$ $M_{max} = 25.79$ kN·m/m. The section is under-reinforced; $d$ is adequate for flexure.

#### Reinforcement
$$
A_{st} = \frac{0.5 f_{ck}}{f_y}\left[1 - \sqrt{1 - \frac{4.6 M_u}{f_{ck} b d^2}}\right] b d
$$
Minimum steel = 0.12% of $bD$ = 288 mm²/m. Maximum spacing = min($3d$, 300 mm) = 300 mm (short), 300 mm (long).
| Location | $M_u$ (kN·m/m) | $A_{st}$ req (mm²/m) | Provide | $A_{st}$ prov |
|---|---|---|---|---|
| Short span, bottom (mid-span) | 25.79 | 288 | 8 mm @ 170 c/c | 296 |
| Long span, bottom (mid-span) | 16.42 | 288 | 8 mm @ 170 c/c | 296 |

Top steel at discontinuous edges (negative moment due to partial fixity, IS 456 Annex D) extends $0.1 l$ into the span: along the discontinuous long edges (short-span direction), 50% of mid-span steel = 148 mm²/m, provide 8 mm @ 300 c/c; along the discontinuous short edges (long-span direction), 50% of mid-span steel = 148 mm²/m, provide 8 mm @ 300 c/c.

#### Check for shear
$V_u = w_u l_x/2 = 16.500 \times 4.215/2 = 34.77$ kN/m
$\tau_v = \dfrac{V_u}{bd} = \dfrac{34774}{1000 \times 215} = 0.162$ MPa
Steel at support: $A_{st} = 148$ mm²/m, $p_t = 0.069\%$; $\tau_c = 0.28$ MPa (IS 456 Table 19, M20, $p_t \le 0.15\%$). For $D = 240$ mm, $k = 1.12$ (cl. 40.2.1.1), so $k\tau_c = 0.314$ MPa.
$\tau_v = 0.162 < k\tau_c = 0.314$ MPa, hence the slab is **safe in shear** and no shear reinforcement is needed.

#### Check for deflection
For the short-span bars: $p_t = \dfrac{100 \times 296}{1000 \times 215} = 0.138\%$, $f_s = 0.58 f_y \dfrac{A_{st,req}}{A_{st,prov}} = 279.9$ N/mm².
Modification factor from IS 456 Fig. 4 (for $p_t = 0.14\%$ and $f_s = 280$ N/mm²): $k_t \approx 1.7$
Permissible $l/d = 20 \times 1.7 = 34$; actual $l_x/d = 4215/215 = 19.6$. Since actual < permissible, **deflection is satisfied** (with a large margin, so a thinner slab could also be used).

#### Check for development length
$L_d = \dfrac{0.87 f_y \phi}{4\tau_{bd}}$ with $\tau_{bd} = 1.92$ MPa (M20, deformed bars, ×1.6): $L_d = \dfrac{0.87 \times 500 \times 8}{4 \times 1.92} = 453$ mm
Alternate bottom bars (148 mm²/m) run into the support: $M_1 = 13.59$ kN·m/m, $L_0 = \max(d, 12\phi) = 215$ mm.
$$
1.3\frac{M_1}{V} + L_0 = 1.3 \times \frac{13589014}{34774} + 215 = 723\ \text{mm}
$$
Since 723 mm $>$ $L_d = 453$ mm, **development length is satisfied**.

#### Torsion reinforcement at corners
Corner size $l_x/5 = 4215/5 = 843$ mm (say 850 mm) in both directions.
- All 4 corners have both edges discontinuous: provide top and bottom meshes, each layer 3/4 of the mid-span $A_{st}$ = 0.75 × 288 = 216 mm²/m. Provide 8 mm @ 170 c/c in both directions, top and bottom, over 850 mm × 850 mm.

#### Detailing
```text
PLAN  (long span horizontal, short span vertical)
  +------------------------------+
  |T                            T|
  | bot: 8mm@170 short span      |
  | bot: 8mm@170 long span       |
  |T                            T|
  +------------------------------+
  - / | : discontinuous edge (0.5 x mid-span top bars)
  T : torsion mesh top + bottom (850 x 850)

SECTION ALONG SHORT SPAN
  top bars at edges (0.5 x mid-span steel)
   ~~~                      ~~~
  |------ slab D = 240 mm ------|
   bottom 8mm @ 170 c/c, alt. bars bent up
  [support]            [support]
```

- Bottom bars: short-span bars in the lower layer, long-span bars above them; alternate bars may be bent up at $0.1 l$ from the support.
- All bars have 20 mm cover.
- Distribution/minimum steel $\ge$ 288 mm²/m is already satisfied by the bars provided.

**Answer:** $D = 240$ mm; short-span bottom 8 mm @ 170 c/c; long-span bottom 8 mm @ 170 c/c; top bars 8 mm @ 300 c/c at the edges; torsion mesh 8 mm @ 170 c/c at the corners. Shear, deflection and development length are satisfied.

## Check yourself

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

1. Which basic l/d ratio is used for this simply supported slab?
   A. 20
   B. 26
   C. 7
   D. 35
   Answer: A. IS 456 cl. 23.2.1 gives 20 for a simply supported span; 26 is for continuous spans. (Key point: Basic span/depth ratio)

2. Why is top steel provided at the corners of this simply supported slab?
   A. Corners are prevented from lifting, so torsion cracking must be controlled
   B. Because the slab is one-way
   C. To resist shear at midspan
   D. To replace distribution steel
   Answer: A. When corners are held down, the slab develops torsional moments at corners, so top and bottom meshes over lx/5 are provided. (Key point: Corner torsion)

3. Calculated Ast for the short span is below the minimum. What is provided?
   A. The smaller calculated value
   B. The minimum 0.12% of bD = 288 mm2/m
   C. Zero, as the moment is small
   D. 1% of bD
   Answer: B. Steel must not be less than 0.12% of bD (Fe500), so 288 mm2/m governs. (Key point: Governing steel)

4. Which Mu,lim coefficient applies for Fe500?
   A. 0.148 fck b d^2
   B. 0.138 fck b d^2
   C. 0.133 fck b d^2
   D. 0.87 fck b d^2
   Answer: C. Mu,lim = 0.133 fck b d^2 for Fe500. (Key point: Limit state constant for Fe500)

5. How many corners need the 3/4 Ast torsion mesh here?
   A. Four
   B. Two
   C. One
   D. None
   Answer: A. All four edges are discontinuous, so all four corners have both edges discontinuous. (Key point: Number of torsion corners)

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