--- title: "Two-way slab with torsion, panel 1" subject: "Design of RCC Structure" chapter: 9 key: "semester_7/design_of_rcc_structure:two-way-slab-with-torsion-panel" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/two-way-slab-with-torsion-panel site: BCT Engineering --- # Two-way slab with torsion, panel 1 Design of RCC Structure · Chapter 9: Design of slabs and staircase - Asked in: 2066 Bhadra (old course), 12+4+4 marks - Video: [How to choose Two way slab as per IS:456-2000| Torsional reinforcement](https://www.youtube.com/watch?v=2vhOmYuv0pc) — cwill structures - 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/two-way-slab-with-torsion-panel - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-9 ## Question The floor slab system of a two-storeyed building is shown in figure. The slab system is supported on 250 mm wide beam as shown. Assuming a floor finish load of 1 kN/m$^2$ and a live load of 4 kN/m$^2$, design and detail the slab panel # 1 as indicated in the floor plan. Also check whether the section satisfies the deflection criteria. (Check for shear and development length not required). The torsional reinforcement should be designed. Use Fe415 steel. Assume mild exposure conditions. [Figure: floor plan with four panels; horizontal bays 5.0 m and 4.0 m, vertical bays 3.0 m and 4.0 m, beams 250 mm wide; Panel #1 is the top-left panel with clear size 3.0 m x 5.0 m] ## Short answer Slab thickness 135 mm. Short span bottom bars 8 mm at 260 mm c/c, top at continuous edge 8 mm at 190 mm c/c. Long span bottom bars 8 mm at 300 mm c/c, top at continuous edge 8 mm at 300 mm c/c; discontinuous edges 8 mm at 300 mm c/c. Torsional reinforcement provided at corners per IS 456. Deflection check satisfied. ## Answer #### 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 | Moment | Coefficient | Value (kN·m/m) | |---|---|---| | $M_x$ negative (continuous long edge) | 0.0802 | 9.74 | | $M_x$ positive (mid-span, short) | 0.0600 | 7.29 | | $M_y$ negative (continuous short edge) | 0.0470 | 5.71 | | $M_y$ positive (mid-span, long) | 0.0350 | 4.25 | $M = \alpha\, w_u\, l_x^2$ 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). | Location | $M_u$ (kN·m/m) | $A_{st}$ req (mm²/m) | Provide | $A_{st}$ prov | |---|---|---|---|---| | Short span, bottom (mid-span) | 7.29 | 191 | 8 mm @ 260 c/c | 193 | | Long span, bottom (mid-span) | 4.25 | 162 | 8 mm @ 300 c/c | 168 | | Short span, top over continuous long edge | 9.74 | 258 | 8 mm @ 190 c/c | 265 | | Long span, top over continuous short edge | 5.71 | 164 | 8 mm @ 300 c/c | 168 | 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 ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Panel 1 is the top-left corner panel. Which description of its edges is correct? A. Top and left edges discontinuous, right and bottom edges continuous B. All four edges continuous C. All four edges discontinuous D. Top and bottom edges continuous, left and right discontinuous Answer: A. Outer edges of the building are discontinuous, while the edges shared with neighbouring panels are continuous. (Key point: Corner panel edges) 2. With D = 135 mm, cover 20 mm and 10 mm bars, what is d for the short-span bars? A. 135 mm B. 115 mm C. 110 mm D. 100 mm Answer: C. d = D - cover - half bar diameter = 135 - 20 - 5 = 110 mm. The long-span bars sit above them, so d is 100 mm. (Key point: Trial depth and effective depth) 3. How is deflection checked in the answer? A. Actual lx/d is compared with permissible 26 x modification factor kt B. Deflection is computed from the bending moment directly C. The deflection check is skipped for two-way slabs D. Actual lx/d must exceed 26 x kt Answer: A. The span/depth method of IS 456 is used: permissible l/d = 26 x kt, from pt and fs. Here 28.6 > actual 28.3, so it is satisfied. (Key point: Deflection check) 4. For lx = 3110 mm, over what size is the corner torsion mesh provided? A. lx/2 = 1555 mm B. lx/10 = 311 mm C. lx/5 = about 630 mm square D. The whole panel Answer: C. Torsion reinforcement extends lx/5 from the corner in each direction, about 622, taken as 630 mm. (Key point: Torsion steel size) 5. What top steel is provided along the discontinuous edges, and why? A. None, because the moment there is zero B. Equal to full mid-span steel, as the edge is fixed C. About 50% of mid-span steel, for the negative moment caused by partial fixity D. Double the mid-span steel for shear Answer: C. Even a simply supported edge has some fixity, so top steel of 50% of mid-span steel extending 0.1 l into the span is given. 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