# Optimization Design of RC Elevated Water Tanks under Seismic Loads
## 2 Optimization Problem Definition
### Problem Definition
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Concrete Type** & **Water/Cement** & **Cement (kg)** & **Slump** & **Cost (6/m3)** \\ \hline C-35(1) & 0.65 & 250 & Flabby & 76.45 \\ C-35(2) & 0.60 & 275 & Flabby & 78.60 \\ C-35(3) & 0.60 & 300 & Flabby & 79.85 \\ C-35(4) & 0.60 & 325 & Flabby & 81.53 \\ C-35(5) & 0.60 & 350 & Flabby & 82.76 \\ C-35(6) & 0.55 & 300 & Flabby & 82.80 \\ C-35(7) & 0.50 & 300 & Flabby & 82.92 \\ C-35(8) & 0.50 & 325 & Flabby & 85.93 \\ C-35(9) & 0.45 & 350 & Flabby & 88.72 \\ C-35(10) & 0.65 & 250 & Plastic & 74.95 \\ C-35(11) & 0.60 & 275 & Plastic & 77.06 \\ C-35(12) & 0.60 & 300 & Plastic & 78.28 \\ C-35(13) & 0.60 & 325 & Plastic & 79.93 \\ C-35(14) & 0.60 & 350 & Plastic & 81.13 \\ C-35(15) & 0.55 & 300 & Plastic & 81.20 \\ C-35(16) & 0.50 & 300 & Plastic & 81.30 \\ C-35(17) & 0.50 & 325 & Plastic & 84.25 \\ C-35(18) & 0.45 & 350 & Plastic & 86.98 \\ C-40(1) & 0.50 & 300 & Flabby & 85.99 \\ C-40(2) & 0.50 & 325 & Flabby & 89.12 \\ C-40(3) & 0.45 & 350 & Flabby & 92.00 \\ C-40(4) & 0.50 & 300 & Plastic & 84.31 \\ C-40(5) & 0.50 & 325 & Plastic & 87.37 \\ C-40(6) & 0.45 & 350 & Plastic & 90.20 \\ C-45(1) & 0.50 & 300 & Flabby & 89.07 \\ C-45(2) & 0.50 & 325 & Flabby & 92.30 \\ C-45(3) & 0.45 & 350 & Flabby & 92.00 \\ C-45(4) & 0.50 & 300 & Plastic & 87.32 \\ C-45(5) & 0.50 & 325 & Plastic & 90.49 \\ C-45(6) & 0.45 & 350 & Plastic & 93.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Concrete unit costs for C-35, C-40 and C-45. foundation ground is a layer of sands. The internal friction angle of the sands is 35\({}^{\circ}\), and their cohesion is zero. The density of the sands is 2000 kg/m\({}^{3}\), and they have an SPT of 30. The water table is sufficiently deep so as not to affect the foundation. The permissible stress of the foundation is the minimum value obtained for the collapse load [36,37] and a settlement of 25.4 mm. The collapse load \(q_{h}\) in Equation (4) depends on the shape of the foundation, the eccentricity, and inclination of the load, as well as on the depth of the foundation:  
\[q_{h}=q\cdot N_{q}\cdot S_{q}\cdot i_{q}\cdot d_{q}+c\cdot N_{c}\cdot S_{c} \cdot i_{c}\cdot d_{c}+\frac{1}{2}\cdot B\cdot\gamma\cdot N_{\gamma}\cdot S_{ \gamma}\cdot i_{\gamma}\cdot d_{\gamma} \tag{4}\]  
where \(N_{q}\), \(N_{c}\) and \(N_{\gamma}\) are the capacity of load factors; \(S_{q}\), \(S_{c}\), and \(S_{\gamma}\) are the shape factors; \(i_{q}\), \(i_{c}\), and \(i_{\gamma}\) are the inclination factors; \(d_{q}\), \(d_{c}\) and \(d_{\gamma}\) are the depth factors; \(q\) is the stress on the foundation face prior to the execution of the footing; \(B\) is the width of the footing; and \(\gamma\) is the ground density. The settlement stress \(S\) is the stress that produces a maximum settlement of 25.4 mm. It is computed using Equation (5) [38] for granular grounds with a proportion in weight of particles coarser than 20 mm higher than 30%. \[S=f_{\mathrm{f}}\cdot f_{\mathrm{g}}\cdot q_{\mathrm{b}}\cdot B^{0.7}\cdot I_{c} \tag{5}\]  
where \(S\) is the settlement, \(f_{\mathrm{f}}\) is a correction factor that takes into account the existence of a rigid layer at a low depth under the footing, \(f_{\mathrm{g}}\) is a coefficient that depends on the shape of the footing, \(q_{\mathrm{b}}\) is the applied pressure at the foundation face required to cause the settlement \(S\), and \(I_{c}\) is the compressibility index based on the SPT value. Once the footing is checked for the two permissible stresses, the footing is checked against sliding and overturning. Sliding is checked considering a friction angle between the footing and the ground of 30\({}^{\circ}\) and a zero cohesion.