# Optimization Design of RC Elevated Water Tanks under Seismic Loads
## 4 Numerical Results  
This section presents and discusses the results for the 35 water tanks analyzed. As explained above, the five heights of the column vary from 20 to 40 m in steps of 5 m. Seven seismic exposures are considered characterized by \(a_{gR}/g\) from 0.00 to 0.24 in steps of 0.04. Results depend on the numerical nonlinearity of the seismic applied loads, which is the dominant condition in most cases studied. Table 5 shows the equivalent static force due to the seism with the tank full of water. This seismic hypothesis is the most relevant hypothesis for the check of the water tank using Eurocode 8 [33] and the approximate method by [PERSON] [5]. Figure 8 represents these forces and shows the high nonlinearity of the problem and the dependence of the force on the rigidity of the columns. Elevated water tanks have lower rigidity, and hence the seismic force is lower than those of the shorter water tanks. This implies that section and steel reinforcement results are not proportional to the height of the columns, as already observed in the study of seismic viaduct piers by [PERSON] et al. [46]. Regarding the value of \(a_{gR}/g\), larger values imply larger equivalent static forces. _Appl. Sci._**2022**, _12_, 5635 \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Column Height (m)** & \(a_{gR}\)/\(g\) & **Eurroc. 8 (kN)** & **Housner (kN)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 5: Horizontal seismic forces (kN). Figure 8: Horizontal seismic force versus height of the column. Figure 9 illustrates the variation of total concrete volume in the column for different heights and seismic exposures. It shows that concrete values increase with the height of the column and with the seismic degree of force. This measurement of concrete varies from a minimum of 170 m\({}^{3}\) to a maximum of 300 m\({}^{3}\). Figure 10 shows the measurement of steel reinforcement per meter of a column, which varies between 100 kg/m to a maximum of 700 kg/m. This graphic shows a linear increase in steel measurement with the column's height for water tanks not subjected to seismic loads. However, the measurement of steel remains quite the same as the height for water tanks subjected to the same seismic condition. Figure 10: Steel kg/m in the column versus height of the column. Figure 9: m\({}^{3}\) of concrete in the column versus height of the column. The width is one of the main geometrical variables of the optimization process. Figure 11 shows the column's width versus the column's height. The width of the column varies from a minimum of about 1.75 m to a maximum of about 4.75 m. Although the figure shows some peak values for the water tanks with large seismicity, the overall tendency is that the width of the column for the optimal tank is similar for the different heights studied given seismicity. Figure 12 shows the total area of the foundation as a function of column height and ground acceleration. This area varies from a minimum of about 20 m\({}^{2}\) for the lowest seismicity to a maximum of about 140 m\({}^{2}\) for the largest seismicity. Figure 13 shows the  
Figure 11: Width of the column versus height of the column. Figure 12: Area of the footing versus height of the column. total volume of concrete in m\({}^{3}\) in the foundation. The volume of concrete varies from a minimum of about 20 m\({}^{3}\) to a maximum of about 240 m\({}^{3}\). These two graphics can be divided into three result groups. The tendency in the group for \(a_{gR}/g\) from 0.00 and 0.08 is that the area of the foundation and the volume of concrete increase notably with the height of the water tank. The group for \(a_{gR}/g\) equal to 0.12 keeps sensibly constant the foundation area and the volume of concrete. Finally, the group for \(a_{gR}/g\) of 0.16, 0.20, and 0.24 shows a tendency to reduce the foundation area and the volume of concrete with the height of the water tank. It is worth noting that the results are highly nonlinear with the height of the column and the seismicity.