Question 5 of 6: Manure Tank Floor Slab — Typical Middle Span

Question 5: Manure Tank Floor Slab — Typical Middle Span (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. A trial slab thickness, the required flexural reinforcement (top and bottom), and a check of shear and deflection for a typical interior span of the 10-span continuous slab.

Approach. Trial a slab thickness, factor the total gravity load, use the standard CSA A23.3 approximate moment coefficients for an interior span of a multi-span continuous one-way slab ($wL_n^2/16$ positive at mid-span, $wL_n^2/11$ negative at the supports), then size reinforcement from the attached $K_r$–$\rho$ table and check shear and deflection.

1. Trial thickness and factored load. Try $t=200$ mm; self-weight $=24(0.2)=4.8$ kPa (Check: normal-density concrete unit weight assumed). Total dead load $=4.8+2.0=6.8$ kPa. $$w_f=1.25D+1.5L=1.25(6.8)+1.5(5.0)=8.5+7.5=\boxed{16.0\ \text{kN/m (per metre width)}}$$
2. Design moments (approximate coefficients, interior span). $$M_{f}^{+}=\frac{w_fL_n^2}{16}=\frac{16.0(3.4)^2}{16}=11.56\ \text{kN}\cdot\text{m/m (mid-span, bottom steel)}$$ $$M_{f}^{-}=\frac{w_fL_n^2}{11}=\frac{16.0(3.4)^2}{11}=\boxed{16.81\ \text{kN}\cdot\text{m/m (support, top steel, governs)}}$$
3. Reinforcement from $K_r$ table. With 40 mm cover and 15M bars ($d_b=16$ mm), $d=200-40-8=152$ mm (Check: $f'_c=30$ MPa assumed — a durable mix appropriate to the corrosive manure-adjacent exposure discussed in Question 6b). $$K_r=\frac{M_f\times10^6}{bd^2}=\frac{16.81\times10^6}{1000(152)^2}=0.73\ \Rightarrow\ \rho\approx0.22\%\ \text{(Table 2.1, interpolated)}$$ $$A_s=\rho\,b\,d = 0.0022(1000)(152)=334\ \text{mm}^2/\text{m}$$ Compare to the CSA A23.3 shrinkage/temperature minimum, $A_{s,min}=0.002\,b\,t=0.002(1000)(200)=400\ \text{mm}^2/\text{m}$ — the minimum governs (this lightly loaded slab is minimum-steel controlled, both faces).
4. Select bars and check shear/deflection. Use 15M @ 450 mm o.c. ($A_s=200/0.45=444\ \text{mm}^2/\text{m} > 400$, both top at supports and bottom at mid-span). Shear at the support face, $V_f=w_fL_n/2=16.0(3.4)/2=27.2$ kN/m, against $V_c=0.21\phi_c\sqrt{f'_c}\,b\,d_v=0.21(0.65)\sqrt{30}(1000)(144)\times10^{-3}=107.7$ kN/m $\gg27.2$  $\checkmark$ (no shear reinforcement needed). Deflection control (both ends continuous): $h_{min}=L_n/28=3400/28=121$ mm $< 200$ mm chosen $\checkmark$.