# Optimization Design of RC Elevated Water Tanks under Seismic Loads
## 2 Optimization Problem Definition
### Problem Definition  
The problem consists of the cost of the design optimization of an elevated RC water tank regarding its column and its foundation, the reservoir being constant during the optimization analysis. The study aims to minimize the cost of the objective function \(F\) in Equation (1) while satisfying the structural restrictions of the problem in Equation (2). Possible values of the \(x_{1}\), \(x_{2}\),..., \(x_{n}\) design variables are given in Equation (3). \[F(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1,r}p_{i}*m_{i}(x_{1},x_{2},\ldots,x_{n}) \tag{1}\]  
\[g_{j}(x_{1},x_{2},\ldots,x_{n})\leq 0 \tag{2}\]  
\[x_{i}\in(d_{i1},d_{i2},\ldots,d_{iq}) \tag{3}\]  
In addition, the structure has to comply with the structural constraints. All variables are discrete in the present model since one of the objectives of the study is that the resulting structure is fully defined and constructible. The rest of the necessary values to define the structure are the parameters. The main parameters are the height of the column of the water tank and the seismic action considered. The geometrical parameters are the shape and dimensions of the reservoir (533.08 m\({}^{3}\)), the height of the column, and the 5 m height of the climbing form. The height of the column varies from 20 to 40 m in steps of 5 m. The main seismic parameters are the reference peak ground accelerations considered, \(a_{gR}\), which are from 0.00\(g\) to 0.24\(g\) in steps of 0.04\(g\), where \(g\) is the gravity acceleration. Ground parameters are the density of the fill on top of the footing, the internal friction angle of the founding sands, the SPT of the founding sands, and the friction angle between the footing and the ground (Table 1). Steel reinforcement parameters are the \(f_{yk}\) = 500 MPa type of steel and the reinforcement of the lateral faces of the footing and plinth (\(\mathcal{O}\)12/20). Water tanks were analyzed in full compliance with the normative codes [31; 32; 33; 34]. Finally, there are parameters for the ambient exposure of the different elements: internal, external, and buried. The objective function in Equation (1) is the total cost of the water tank, where \(p_{i}\) are the unit prices and \(m_{i}\) are the measurements of the different units. The cost of each unit is obtained by multiplying the unit price by the measurement. Unit prices are summarized in Tables 2-4. Each concrete mix can be obtained with different water/cement ratios, kilograms of cement per cubic meter, and consistency measured by the Abrams cone, which can be plastic or flabby. All these concrete mix properties affect the verification of the durability constraints. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Height of column formworks & 5.00 m \\ Internal friction angle of the sands & 35\({}^{\circ}\) \\ Standard penetration test (SPT) sands & 30 \\ Ground– footing friction angle & 30\({}^{\circ}\) \\ Specific weight of the ground & 20.00 kN/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameters of geometry and foundation ground. \begin{table}
\begin{tabular}{c c c} \hline
**Unit** & **Cost (f/Unit)** \\ \hline kg steel reinforcement in columns (\(f_{yk}=500\))s & 1.99 \\ kg steel reinforcement in footing (\(f_{yk}=500\)) & 1.07 \\ m\({}^{2}\) formwork in footing & 18.19 \\ m\({}^{2}\) external formwork in columns & 48.19 \\ m\({}^{2}\) internal formwork in columns & 49.50 \\ m\({}^{3}\) concrete pumped placing in column & 26.03 \\ m\({}^{3}\) concrete not pumped placing in column & 27.34 \\ m\({}^{3}\) concrete not pumped placing in footing & 12.74 \\ m\({}^{3}\) excavation & 9.42 \\ m\({}^{3}\) earth fill & 4.81 \\ \hline \end{tabular}
\end{table} 
Table 2: Unit costs.