Slab with two adjacent edges discontinuous, 4x5 m

Data and assumptions

• Clear spans: $l_x = 4$ m (short), $l_y = 5$ m (long); support width 275 mm; $f_{ck} = 20$ MPa, $f_y = 415$ 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 $d$.
• Panel type: two adjacent edges discontinuous; moment coefficients from IS 456 Annex D, Table 26 (case: two adjacent edges discontinuous).

Depth and effective span

Basic span/depth ratio = 26 (IS 456 cl. 23.2.1). Trial overall depth $D = 175$ mm, so $d = 175 - 20 - 5 = 150$ mm for the short-span bars and $d_y = 150 - 10 = 140$ mm for the long-span bars.

$$l_x = \min(l_{clear} + d,\ c/c) = 4000 + 150 = 4150\ \text{mm},\quad l_y = 5000 + 150 = 5150\ \text{mm}$$ $$\frac{l_y}{l_x} = \frac{5150}{4150} = 1.241 < 2 \Rightarrow \text{two-way slab}$$

Loads (per m²)

• Self-weight = 25 × 0.175 = 4.375 kN/m²
• Total working load $w$ = 4.375 + 1 + 4 = 9.375 kN/m²
• Factored load $w_u = 1.5 \times 9.375 = 14.062$ kN/m²

Bending moments

$M = \alpha\, w_u\, l_x^2$ with $w_u l_x^2 = 14.062 \times 4.150^2 = 242.19$ kN·m/m. Coefficients are interpolated for $l_y/l_x = 1.24$.

Check of depth

$M_{u,lim} = 0.138\, f_{ck} b d^2 = 0.138 \times 20 \times 1000 \times 150^2 \times 10^{-6} = 62.10$ kN·m/m $>$ $M_{max} = 15.03$ 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 $bD$ = 210 mm²/m. Maximum spacing = min($3d$, 300 mm) = 300 mm in both directions.

Top steel at the discontinuous edges (for the negative moment due to partial fixity, IS 456 Annex D) extends $0.1 l$ into the span. Along the discontinuous long edge (short-span direction): 50% of mid-span steel = 109 mm²/m, provide 8 mm @ 300 c/c; along the discontinuous short edge (long-span direction): 50% of mid-span steel = 107 mm²/m, provide 8 mm @ 300 c/c.

Check for shear

$V_u = w_u l_x/2 = 14.062 \times 4.150/2 = 29.18$ kN/m

$\tau_v = \dfrac{V_u}{bd} = \dfrac{29180}{1000 \times 150} = 0.195$ MPa

Steel at the support: $A_{st} = 296$ mm²/m, $p_t = 0.197\%$, so $\tau_c = 0.32$ MPa (IS 456 Table 19, M20). For $D = 175$ mm, $k = 1.25$ (cl. 40.2.1.1), so $k\tau_c = 0.40$ MPa.

$\tau_v = 0.195 < k\tau_c = 0.40$ MPa, hence the slab is safe in shear and no shear reinforcement is needed.

Check for deflection

For the short-span bars: $p_t = \dfrac{100 \times 219}{1000 \times 150} = 0.146\%$, $f_s = 0.58 f_y \dfrac{A_{st,req}}{A_{st,prov}} = 0.58 \times 415 \times \dfrac{215}{219} = 236$ N/mm².

Modification factor from IS 456 Fig. 4 (for $p_t = 0.15\%$ and $f_s = 236$ N/mm²): at such a low steel percentage the curve reaches its upper limit, so $k_t = 2.0$.

Permissible $l/d = 26 \times 2.0 = 52.0$; actual $l_x/d = 4150/150 = 27.7$. Since actual < permissible, deflection is satisfied.

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): $L_d = \dfrac{0.87 \times 415 \times 8}{4 \times 1.92} = 376$ mm

The check $L_d \le 1.3\dfrac{M_1}{V} + L_0$ (IS 456 cl. 26.2.3.3(c)) is made at the discontinuous edges. $L_0$ = straight length beyond the centre of the support $= 275/2 - 25$ (end cover) $= 112.5$ mm, plus the anchorage value of any bend.

• Short-span bars at the discontinuous long edge (alternate bars bent up, 109 mm²/m, $d = 150$ mm): $M_1 = 0.87 f_y A_{st} d\left(1 - \dfrac{A_{st} f_y}{b d f_{ck}}\right) = 5.84$ kN·m/m, $1.3\dfrac{M_1}{V} = 1.3 \times \dfrac{5.84 \times 10^6}{29180} = 260$ mm. Straight: $260 + 112.5 = 373$ mm, just below 376 mm; with a 90° bend ($8\phi = 64$ mm): $437$ mm $> 376$ mm.
• Long-span bars at the discontinuous short edge (107 mm²/m, $d = 140$ mm, same $V$ taken conservatively): $M_1 = 5.32$ kN·m/m, $1.3M_1/V = 237$ mm; with a 90° bend: $237 + 112.5 + 64 = 414$ mm $> 376$ mm.

With standard 90° bends at the discontinuous edges, the development length is satisfied.

Torsion reinforcement at corners

(Not required by the question; given for completeness.) Corner size $l_x/5 = 4150/5 = 830$ mm in both directions.

• Corner with both edges discontinuous (1 corner): top and bottom meshes, each layer 3/4 of the mid-span $A_{st}$ = 0.75 × 215 = 161 mm²/m. Provide 8 mm @ 235 c/c in both directions, top and bottom, over 830 mm × 830 mm.
• Corners with one edge discontinuous (2 corners): half of this = 0.375 × 215 = 81 mm²/m. Provide 8 mm @ 300 c/c in each direction, top and bottom.
• No torsion steel is needed at the corner where both edges are continuous (IS 456 D-1.10).

Detailing

PLAN  (long span horizontal, short span vertical)
  +==============================+
  #                             t|
  # bot: 8mm@230 short span      |
  # bot: 8mm@235 long span       |
  #t                            T|
  +------------------------------+
  = / # : continuous edge (top bars)
  - / | : discontinuous edge (0.5 x mid-span top bars)
  T : torsion mesh top+bottom;  t : half mesh

SECTION ALONG SHORT SPAN
  top bars over      top bars at
  continuous edge    discontinuous edge
  8mm@170            8mm@300
   ~~~~~~~            ~~~
  |------ slab D = 175 mm ------|
   bottom 8mm @ 230 c/c, alt. bars bent up,
   90 deg bend at the discontinuous edge
  [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.
• Top bars over continuous supports extend $0.3 l_x$ from the face of the support; all bars have 20 mm cover.
• Distribution/minimum steel $\ge$ 210 mm²/m is already satisfied by the bars provided.

Answer: $D = 175$ mm; short-span bottom 8 mm @ 230 c/c; long-span bottom 8 mm @ 235 c/c; top steel over the continuous long edge 8 mm @ 170 c/c (short direction) and over the continuous short edge 8 mm @ 215 c/c (long direction); 8 mm @ 300 c/c top bars at the discontinuous edges. The slab is safe in shear and deflection; development length is satisfied with 90° bends at the discontinuous edges.