--- title: "Slab three edges continuous, one short edge discontinuous (5x4 m, 250 beam)" subject: "Design of RCC Structure" chapter: 9 key: "semester_7/design_of_rcc_structure:slab-three-edges-continuous-one-short" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/slab-three-edges-continuous-one-short site: BCT Engineering --- # Slab three edges continuous, one short edge discontinuous (5x4 m, 250 beam) Design of RCC Structure · Chapter 9: Design of slabs and staircase - Asked in: 2079 Bhadra, 10 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-three-edges-continuous-one-short - 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 250 mm wide beam. The slab is subjected to live load of 4 kN/m$^2$ and floor finish of 1.5 kN/m$^2$. Design the slab and check whether the provided section satisfies the deflection criteria. Also, sketch the arrangement of reinforcement bars at the support and at the midspan with torsional bars. ## Short answer Adopt total depth 170 mm and effective depth 145 mm; factored load is 14.63 kN/m2 giving maximum moment 12.37 kNm/m. Provide bottom 8 mm bars at 245 c/c both ways, top 8 mm at 205 c/c over continuous long edges, 245 c/c over continuous short edge, and 300 c/c at the discontinuous edge. Shear 0.209 MPa is below 0.38 MPa and deflection is within the permissible limit. ## Answer #### Data and assumptions - Clear spans: $l_x = 4$ m (short), $l_y = 5$ m (long); support width 250 mm; $f_{ck} = 20$ MPa, $f_y = 415$ MPa. - Grades are not stated in the question; M20 concrete and Fe415 steel are assumed. - Live load = 4 kN/m², floor finish = 1.5 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 $D = 170$ mm, so $d = 170 - 20 - 5 = 145$ mm for the short-span bars and $d_y = 145 - 10 = 135$ mm for the long-span bars. $$ l_x = \min(l_{clear} + d,\ c/c) = 4000 + 145 = 4145\ \text{mm},\quad l_y = 5000 + 145 = 5145\ \text{mm} $$ $$ \frac{l_y}{l_x} = \frac{5145}{4145} = 1.241 < 2 \Rightarrow \text{two-way slab} $$ #### Loads (per m²) - Self-weight = 25 × 0.170 = 4.250 kN/m² - Total working load $w$ = 4.250 + 1.5 + 4 = 9.750 kN/m² - Factored load $w_u = 1.5 \times 9.750 = 14.625$ kN/m² #### Bending moments | Moment | Coefficient | Value (kN·m/m) | |---|---|---| | $M_x$ negative (continuous long edge) | 0.0492 | 12.37 | | $M_x$ positive (mid-span, short) | 0.0372 | 9.36 | | $M_y$ negative (continuous short edge) | 0.0370 | 9.30 | | $M_y$ positive (mid-span, long) | 0.0280 | 7.04 | $M = \alpha\, w_u\, l_x^2$ with $w_u l_x^2 = 14.625 \times 4.145^2 = 251.27$ kN·m/m. Coefficients are interpolated for $l_y/l_x = 1.24$. #### Check of depth $M_{u,lim} = 0.138\, f_{ck} b d^2 = 0.138 \times 20 \times 1000 \times 145^2 \times 10^{-6} = 58.03$ kN·m/m $>$ $M_{max} = 12.37$ 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 $bD$ = 204 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) | 9.36 | 184 → 204 (min) | 8 mm @ 245 c/c | 205 | | Long span, bottom (mid-span) | 7.04 | 148 → 204 (min) | 8 mm @ 245 c/c | 205 | | Short span, top over continuous long edges | 12.37 | 245 | 8 mm @ 205 c/c | 245 | | Long span, top over continuous short edge | 9.30 | 197 → 204 (min) | 8 mm @ 245 c/c | 205 | 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 = 103 mm²/m; provide 8 mm @ 300 c/c. #### Check for shear $V_u = w_u l_x/2 = 14.625 \times 4.145/2 = 30.31$ kN/m $\tau_v = \dfrac{V_u}{bd} = \dfrac{30310}{1000 \times 145} = 0.209$ MPa Steel at the support: $A_{st} = 245$ mm²/m, $p_t = 0.169\%$, so $\tau_c = 0.30$ MPa (IS 456 Table 19, M20). For $D = 170$ mm, $k = 1.26$ (cl. 40.2.1.1), so $k\tau_c = 0.38$ MPa. $\tau_v = 0.209 < k\tau_c = 0.38$ MPa, hence the slab is **safe in shear** and no shear reinforcement is needed. #### Check for deflection For the short-span bars: $p_t = \dfrac{100 \times 205}{1000 \times 145} = 0.141\%$, $f_s = 0.58 f_y \dfrac{A_{st,req}}{A_{st,prov}} = 0.58 \times 415 \times \dfrac{184}{205} = 216$ N/mm². Modification factor from IS 456 Fig. 4 (for $p_t = 0.14\%$ and $f_s = 216$ 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 $l_x/d = 4145/145 = 28.6$. Since actual < permissible, the provided section **satisfies the deflection criterion**. #### 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 103 mm²/m reach the support at $d_y = 135$ mm: $M_1 = 0.87 f_y A_{st} d\left(1 - \dfrac{A_{st} f_y}{b d f_{ck}}\right) = 4.92$ kN·m/m. Taking $V = 30.31$ kN/m (conservative), $1.3\dfrac{M_1}{V} = 1.3 \times \dfrac{4.92 \times 10^6}{30310} = 211$ mm. $L_0$ = straight length beyond the centre of the support $= 250/2 - 25$ (end cover) $= 100$ mm, plus the anchorage value of any hook or bend: - straight bars: $211 + 100 = 311$ mm $< 376$ mm; - with a 90° bend ($8\phi = 64$ mm): $375$ mm, just below 376 mm; - with a standard U-hook ($16\phi = 128$ mm): $211 + 100 + 128 = 439$ 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 $l_x/5 = 4145/5 = 829$ mm (say 830 mm) in both directions. - Corners with one edge discontinuous and the other continuous (2 corners, at the ends of the discontinuous short edge): 3/8 of the mid-span steel = 0.375 × 204 = 76 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@245 short span | # bot: 8mm@245 long span | # t| +==============================+ = / # : continuous edge (top bars) - / | : discontinuous edge (0.5 x mid-span top bars) t : torsion mesh 8mm@300 top+bottom, 830 x 830 mm SECTION ALONG SHORT SPAN (supports and mid-span) top bars over top bars over continuous edge continuous edge 8mm@205 8mm@205 ~~~~~~~ ~~~~~~~ |------ slab D = 170 mm ------| bottom 8mm @ 245 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 $\ge$ 204 mm²/m is already satisfied by the bars provided. **Answer:** $D = 170$ mm; short-span bottom 8 mm @ 245 c/c; long-span bottom 8 mm @ 245 c/c; top steel over the continuous long edges 8 mm @ 205 c/c (short direction) and over the continuous short edge 8 mm @ 245 c/c (long direction); 8 mm @ 300 c/c top bars at the discontinuous edge; torsion mesh 8 mm @ 300 c/c at the two corners. 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. What torsion reinforcement is needed for this panel? A. 3/4 Ast mesh at all four corners B. Half mesh (0.375 Ast, top and bottom) over lx/5 at the two corners with one discontinuous edge; none at corners with both edges continuous C. No torsion steel ever D. Mesh over the whole slab Answer: B. IS 456 D-1.8 to D-1.10: corners with one discontinuous edge get half the 3/4 Ast mesh over lx/5; fully continuous corners need none. (Key point: Torsion mesh at corners) 2. What condition shows deflection is satisfied? A. Actual lx/d (28.6) must be below permissible 26 x kt (28.6 using the modification factor) B. lx/d must be above 26 C. Deflection is satisfied if Ast > minimum D. Deflection is not needed with M20 Answer: A. The deflection control compares the actual span/effective depth with the basic ratio multiplied by the tension-steel modification factor. (Key point: Deflection criterion) 3. Alternate bottom bars give 372 mm against Ld = 376 mm. What is done? A. Increase the concrete grade B. Add shear stirrups C. Reduce bar diameter to 4 mm D. Provide a 90 degree bend at the support, adding anchorage 8 phi Answer: D. A standard bend adds anchorage (8 phi = 64 mm), so the available length 436 mm exceeds Ld. (Key point: Anchoring bars when Ld not met) 4. How is the trial depth D = 170 mm confirmed adequate for flexure? A. Mu,lim = 0.138 fck b d^2 = 58.03 kN m/m exceeds the maximum moment 12.37 kN m/m, so section is under-reinforced B. Because D is above 150 mm C. Because span/depth is 26 D. Because 8 mm bars are used Answer: A. If the design moment is less than the limiting moment of resistance of the section, the section is under-reinforced and singly reinforced design is adequate. (Key point: Depth adequacy) ## Close variants set in other exams - [Slab continuous over three edges, 5x4 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-continuous-over-three-edges-5x4) — Design of RCC Structure, chapter 9, asked 2× - [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× ## 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 - [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 - [Slab with one short edge discontinuous, 6x4 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslab-with-one-short-edge-discontinuous) — Design of RCC Structure, chapter 9 ## Related in other subjects - [T-beam decking quantities (6 m clear span)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3At-beam-decking-all-quantities) — Estimating and Costing, chapter 6