--- title: "Slab continuous over three edges, 5x4 m" subject: "Design of RCC Structure" chapter: 9 key: "semester_7/design_of_rcc_structure:slab-continuous-over-three-edges-5x4" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/slab-continuous-over-three-edges-5x4 site: BCT Engineering --- # Slab continuous over three edges, 5x4 m Design of RCC Structure · Chapter 9: Design of slabs and staircase - Asked in: 2081 Bhadra, 13 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-continuous-over-three-edges-5x4 - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-9 ## Question A rectangular slab panel 5 m x 4 m (clear span) is continuous over three edges and discontinuous over one short edge. The slab is to rest on 230 mm wide beam. The slab is subjected to live load of 5 kN/m$^2$ and floor finish of 1.2 kN/m$^2$. Design the slab and check whether the section satisfies the deflection criteria. Also, sketch the arrangement of reinforcement bars at the support and at the mid-span. Use M20 grade concrete and Fe 415 steel. ## Short answer Slab continuous over three edges, one short edge discontinuous, D = 175 mm, d = 150 mm, M20/Fe415. Factored load 15.86 kN/m²; max moment 13.45 kN·m/m. Reinforcement: short span bottom 8 mm @ 235 c/c, top 8 mm @ 195 c/c; long span both 8 mm @ 235 c/c; 8 mm @ 300 c/c on discontinuous edge. Deflection check gives permissible l/d = 52, actual ratio safe. ## Answer #### Data and assumptions - Clear spans: $l_x = 4$ m (short), $l_y = 5$ m (long); support width 230 mm; $f_{ck} = 20$ MPa, $f_y = 415$ MPa. - Live load = 5 kN/m², floor finish = 1.2 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: one short edge discontinuous; moment coefficients from IS 456 Annex D, Table 26 (case: one short edge 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 + 1.2 + 5 = 10.575 kN/m² - Factored load $w_u = 1.5 \times 10.575 = 15.862$ kN/m² #### Bending moments | Moment | Coefficient | Value (kN·m/m) | |---|---|---| | $M_x$ negative (continuous long edge) | 0.0492 | 13.45 | | $M_x$ positive (mid-span, short) | 0.0372 | 10.17 | | $M_y$ negative (continuous short edge) | 0.0370 | 10.11 | | $M_y$ positive (mid-span, long) | 0.0280 | 7.65 | $M = \alpha\, w_u\, l_x^2$ with $w_u l_x^2 = 15.862 \times 4.150^2 = 273.19$ 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 $>$ $M_{max} = 13.45$ 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. | Location | $M_u$ (kN·m/m) | $A_{st}$ req (mm²/m) | Provide | $A_{st}$ prov | |---|---|---|---|---| | Short span, bottom (mid-span) | 10.17 | 193 → 210 (min) | 8 mm @ 235 c/c | 214 | | Long span, bottom (mid-span) | 7.65 | 155 → 210 (min) | 8 mm @ 235 c/c | 214 | | Short span, top over continuous long edges | 13.45 | 258 | 8 mm @ 195 c/c | 258 | | Long span, top over continuous short edge | 10.11 | 206 → 210 (min) | 8 mm @ 235 c/c | 214 | Top steel at the discontinuous edge (for the negative moment due to partial fixity, IS 456 Annex D) extends $0.1 l$ into the span. 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 $V_u = w_u l_x/2 = 15.862 \times 4.150/2 = 32.91$ kN/m $\tau_v = \dfrac{V_u}{bd} = \dfrac{32915}{1000 \times 150} = 0.219$ MPa Steel at the support: $A_{st} = 258$ mm²/m, $p_t = 0.172\%$, so $\tau_c = 0.30$ MPa (IS 456 Table 19, M20). For $D = 175$ mm, $k = 1.25$ (cl. 40.2.1.1), so $k\tau_c = 0.37$ MPa. $\tau_v = 0.219 < k\tau_c = 0.37$ MPa, hence the slab is **safe in shear** and no shear reinforcement is needed. #### Check for deflection For the short-span bars: $p_t = \dfrac{100 \times 214}{1000 \times 150} = 0.143\%$, $f_s = 0.58 f_y \dfrac{A_{st,req}}{A_{st,prov}} = 0.58 \times 415 \times \dfrac{193}{214} = 217$ N/mm². Modification factor from IS 456 Fig. 4 (for $p_t = 0.14\%$ and $f_s = 217$ 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 short edge, where the long-span bottom bars end. Alternate bars are bent up, so 107 mm²/m reach the support at $d_y = 140$ mm: $M_1 = 0.87 f_y A_{st} d\left(1 - \dfrac{A_{st} f_y}{b d f_{ck}}\right) = 5.32$ kN·m/m. Taking $V = 32.91$ kN/m (conservative), $1.3\dfrac{M_1}{V} = 1.3 \times \dfrac{5.32 \times 10^6}{32915} = 210$ mm. $L_0$ = straight length beyond the centre of the support $= 230/2 - 25$ (end cover) $= 90$ mm, plus the anchorage value of any hook or bend: - straight bars: $210 + 90 = 300$ mm $< 376$ mm; - with a 90° bend ($8\phi = 64$ mm): $364$ mm $< 376$ mm; - with a standard U-hook ($16\phi = 128$ mm): $210 + 90 + 128 = 428$ mm $> 376$ mm. So the long-span bottom bars are given standard hooks at the discontinuous edge, and the **development length is then satisfied**. #### Torsion reinforcement at corners Corner size $l_x/5 = 4150/5 = 830$ mm in both directions. - Corners with one edge discontinuous and the other continuous (2 corners): 3/8 of the mid-span steel = 0.375 × 210 = 79 mm²/m in each layer. 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| # bot: 8mm@235 short span | # bot: 8mm@235 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 (at mid-span and supports) top bars over top bars over continuous edge continuous edge 8mm@195 8mm@195 ~~~~~~~ ~~~~~~~ |------ slab D = 175 mm ------| bottom 8mm @ 235 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 $\ge$ 210 mm²/m is already satisfied by the bars provided. **Answer:** $D = 175$ mm; short-span bottom 8 mm @ 235 c/c; long-span bottom 8 mm @ 235 c/c; top steel over the continuous long edges 8 mm @ 195 c/c (short direction) and over the continuous short edge 8 mm @ 235 c/c (long direction); 8 mm @ 300 c/c top bars at the discontinuous short edge. Shear and deflection are satisfied; development length is satisfied with standard hooks at the discontinuous edge. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which moment coefficient set is used for a panel continuous on three edges and discontinuous on one short edge? A. Table 27, simply supported with corners free B. Table 26 case: one short edge discontinuous C. Table 26 case: interior panel D. Table 26 case: two adjacent edges discontinuous Answer: B. The support condition determines the alpha values; one short edge discontinuous has its own row in IS 456 Table 26. (Key point: Edge condition and coefficients) 2. How is the design load found for this slab? A. wu = live load + finish only = 6.2 kN/m2 B. wu = 1.5 x (self-weight + floor finish + live load) C. wu = 1.5 x live load only D. wu = 0.87 fy x self-weight Answer: B. Dead load includes the slab self-weight (25 x D), plus finish and live load; the total is multiplied by the partial safety factor 1.5. (Key point: Factored load) 3. Which corners need torsion reinforcement here? A. All four corners, with full mesh B. No corners, because three edges are continuous C. Only the two corners with one discontinuous edge (half mesh over lx/5); none where both edges are continuous D. Only the middle of the long edges Answer: C. The two corners on the discontinuous short edge have one discontinuous edge, so half the 3/4 Ast mesh is provided; the other corners have both edges continuous and need none. (Key point: Torsion steel only at corners with discontinuous edges) 4. Here 1.3 M1/V + L0 = 375 mm is slightly less than Ld = 376 mm. What is done? A. The slab depth is increased to 200 mm B. The design is rejected C. Shear stirrups are added D. The bars are given a standard 90 degree bend, adding anchorage value 8 phi, so the available length becomes adequate Answer: D. A hook/bend at the support adds anchorage (8 phi = 64 mm), giving 439 mm > 376 mm, so the requirement is met. (Key point: Development length with a bend) ## Close variants set in other exams - [Slab three edges continuous, one short edge discontinuous (5x4 m, 250 beam)](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-three-edges-continuous-one-short) — Design of RCC Structure, chapter 9, asked 3× ## Related in Design of RCC Structure - [Slab continuous over three edges, 5x4 m (225 mm beam)](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-continuous-over-three-edges-5x4-2) — Design of RCC Structure, chapter 9 - [Slab design, three edges continuous, 5x4 m (Fe500)](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-design-three-edges-continuous-5x4) — Design of RCC Structure, chapter 9 - [Slab three edges continuous, 5.5x4 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-three-edges-continuous-5-5x4) — Design of RCC Structure, chapter 9, asked 2× - [Two-adjacent-edges discontinuous slab, 3.5x4.5 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Atwo-adjacent-edges-discontinuous-slab-3) — Design of RCC Structure, chapter 9 - [Slab with one short edge discontinuous, 4x5 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-with-one-short-edge-discontinuous-2) — Design of RCC Structure, chapter 9 - [Slab, two adjacent edges discontinuous, 5x4 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-two-adjacent-edges-discontinuous-5x4) — Design of RCC Structure, chapter 9