Two-way slab with torsion, panel 1

Data and assumptions

• Clear spans: $l_x = 3$ m (short), $l_y = 5$ m (long); support width 250 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 1 is the corner panel: top and left edges are outer edges (discontinuous), right and bottom edges are continuous. Concrete M20 assumed.
• Panel type: two adjacent edges discontinuous; moment coefficients from IS 456 Annex D, Table 26.

Depth and effective span

Basic span/depth ratio = 26 (IS 456 cl. 23.2.1). Trial overall depth $D = 135$ mm, so $d = 135 - 20 - 5 = 110$ mm for the short-span bars and $d_y = 110 - 10 = 100$ mm for the long-span bars.

$$l_x = \min(l_{clear} + d,\ c/c) = 3000 + 110 = 3110\ \text{mm},\quad l_y = 5000 + 110 = 5110\ \text{mm}$$ $$\frac{l_y}{l_x} = \frac{5110}{3110} = 1.643 < 2 \Rightarrow \text{two-way slab}$$

Loads (per m²)

• Self-weight = 25 × 0.135 = 3.375 kN/m²
• Total working load $w$ = 3.375 + 1 + 4 = 8.375 kN/m²
• Factored load $w_u = 1.5 \times 8.375 = 12.562$ kN/m²

Bending moments

$M = \alpha\, w_u\, l_x^2$ with $w_u l_x^2 = 12.562 \times 3.110^2 = 121.51$ kN·m/m. Coefficients are interpolated for $l_y/l_x = 1.64$.

Check for depth

$M_{u,lim} = 0.138\, f_{ck} b d^2 = 0.138 \times 20 \times 1000 \times 110^2 \times 10^{-6} = 33.40$ kN·m/m $>$ $M_{max} = 9.74$ kN·m/m. 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 $bD$ = 162 mm²/m. Maximum spacing = min($3d$, 300 mm) = 300 mm (short), 300 mm (long).

Top steel at discontinuous edges (negative moment due to partial fixity, IS 456 D-1.6) extends $0.1 l$ into the span: along the discontinuous long edge(s) (short-span direction): 50% of mid-span steel = 97 mm²/m, provide 8 mm @ 300 c/c; along the discontinuous short edge(s) (long-span direction): 50% of mid-span steel = 84 mm²/m, provide 8 mm @ 300 c/c.

Check for shear

$V_u = w_u l_x/2 = 12.562 \times 3.110/2 = 19.53$ kN/m $\tau_v = \dfrac{V_u}{bd} = \dfrac{19535}{1000 \times 110} = 0.178$ MPa Steel at support: $A_{st} = 265$ mm²/m, $p_t = 0.241\%$; $\tau_c = 0.35$ MPa (IS 456 Table 19, M20). For $D = 135$ mm, $k = 1.30$ (cl. 40.2.1.1), so $k\tau_c = 0.455$ MPa. $\tau_v = 0.178 < k\tau_c = 0.455$ MPa, so the slab is safe in shear and no shear reinforcement is needed.

Check for deflection

For the short-span bars: $p_t = \dfrac{100 \times 193}{1000 \times 110} = 0.176\%$, $f_s = 0.58 f_y \dfrac{A_{st,req}}{A_{st,prov}} = 0.58 \times 415 \times \dfrac{191}{193} = 238$ N/mm². Modification factor from IS 456 Fig. 4 (for $p_t = 0.18\%$ and $f_s = 238$ N/mm²): $k_t \approx 1.9$. Permissible $l/d = 26 \times 1.9 = 49.4$; actual $l_x/d = 3110/110 = 28.3$. 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 Alternate bottom bars (97 mm²/m) run into the support: $M_1 = 3.77$ kN·m/m, $L_0 = \max(d,\ 12\phi) = 110$ mm.

$$1.3\frac{M_1}{V} + L_0 = 1.3 \times \frac{3769071}{19535} + 110 = 361\ \text{mm}$$

Since 361 mm $<$ $L_d = 376$ mm, the bars are given a standard 90° bend at the support (anchorage value $8\phi$ = 64 mm). The available length then becomes 361 + 64 = 425 mm $> L_d$, so development length is satisfied with bent ends.

Torsion reinforcement at corners

Corner size $l_x/5 = 3110/5 = 622$ mm (say 630 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 × 191 = 143 mm²/m. Provide 8 mm @ 300 c/c in both directions, top and bottom, over 630 mm × 630 mm.
• Corner with only one discontinuous edge (2 corners): half of 3/4 of the mid-span $A_{st}$ = 0.375 × 191 = 71 mm²/m. Provide 8 mm @ 300 c/c in each direction, top and bottom.
• No torsion steel is needed at corners where both edges are continuous (IS 456 D-1.10).

Detailing

PLAN  (long span horizontal, short span vertical)
  +------------------------------+
  |T                            t#
  | bot: 8mm@260 short span      #
  | bot: 8mm@300 long span       #
  |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 (0.5 x)
   ~~~~~~~                       ~~~
  |------ slab D = 135 mm ------|
   bottom 8mm @ 260 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.
• 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 $\ge$ 162 mm²/m is already satisfied by the bars provided.

Answer: $D = 135$ mm; short-span bottom 8 mm @ 260 c/c; long-span bottom 8 mm @ 300 c/c; top steel over continuous edge 8 mm @ 190 c/c (short direction); 8 mm @ 300 c/c (long direction). Shear, deflection and development length (with bent bar ends) are satisfied.