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Design a roof slab on a room of clear dimension $3m \times 4m.$ The slab is simply supported on $230mm$ thick walls on all four sides.

Take live load as $3kN/m^2$ and floor finish load as $1kN/m^2$. Use $M20$ concrete and $Fe415$ steel. Adopt LSM.

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$$\dfrac {l_y}{lx}=\dfrac {4+023}{3+0.23}=\dfrac {4.23}{3.23}=1.3 < 2$$

$\text { It is two way slab} \\ d=\dfrac {\text {short effective span}} {\text {S⁄D ratio ×M.F}} =\dfrac {3230}{28×1.4}=82.39mm ≈100 \\ dr=100mm \\ \text {D=d+effective cover }\\ =100+25=125mm \\ \text {Load Calculation: } \\ D.L = D×25=0.125×25=3.125kN/m^2 \\ L.L =3kN/m^2 \\ F.F = 1 kN/m^2 \\ Total =7.125kN/m^2 \\ \text {Factored load }(w_d) =7.125×1.5=12=10.68kN/m \\ \dfrac {l_y}{lx}=4.23/3.23=1.3$

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$$M_{ux}=α_x wdlx^2 $$

$ =0.093×10.68×3.23^2 \\ =10.36 kNm \\ M_{uy}= α_y wdlx^2 \\ =0.138=0.055×10.68×3.23^2 \\ =6.128 kNm \\ M_{u\space max}=0.138f_{ck} bd^2 \\ =0.138×20×1000×100^2 \\ =27.6kNm \gt M_{ux}\space \& \space M_{uy} \\ Astx= \dfrac {0.5×20×1000×100}{415}×[1-\sqrt {1-\dfrac {4.6×10.36×10^6}{20×1000×100^2 }}]\\ =307 mm^2 \\ \text {Assume 8mm∅ bars } ∅_x=∅_y=8d_y=92mm \\ Asty= \dfrac {0.5×20×1000×92}{415}×[1-\sqrt{1-\dfrac {4.6×6.128×10^6}{20×1000×92^2 }}] \\ =193 mm^2 \\ Astmin=0.12/100 b×D=150 mm^2 \lt Astx \space \&\space Asty \\ \text { * Main steel in x-direction (8mm∅) } \\ spacing =\dfrac {b×c/s\space area}{Asty}=\dfrac {1000×\pi/4 ×8^2}{307}=163.7 ≈150mm \\ \text { * Main steel in y-direction (8mm∅) } \\ spacing =\dfrac {b×c/s \space area}{Asty}=\dfrac {1000×\pi/4 ×8^2}{193}=260.4 ≈250mm $

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