# Optimization Design of RC Elevated Water Tanks under Seismic Loads
## 2 Optimization Problem Definition  
### Elevated Water Tanks Object of Optimization  
The RC elevated water tanks consist of a high reservoir of conic trunk shape, a column with a hollow square cross-section, and a footing as a shallow foundation (Figures 1 and 2). The top reservoir keeps a constant geometry and steel reinforcement for the water tanks considered in the study. The top reservoir was analyzed by SAP2000 using two-dimensional shell finite elements of four nodes, whose parameters are the element thickness, together with the Young modulus, and the Poisson's ratio of the material. Alternative methods to the finite element method are the finite difference method [29] and the discrete element method [30]. Given the stress resultants, the reinforcement of the reservoir was computed by Eurocode 2 [31] and the Spaniard Code of Structures [32]. Figures 3-5 show the final reinforcement. The height of the conic trunk reservoir is 6 m, and the outer and inner diameters are 14 and 7 m, respectively. Man access of 1.20 m in diameter with walls of 0.20 m of thickness crosses the reservoir. This access goes from the column shaft to the top platform of the reservoir. The tank also has a handrail that surrounds the top slab. All the reservoir walls have a thickness of 0.30 m in concrete class C-30. The reservoir volume is 533.08 m\({}^{3}\), excluding the man's access. Figure 1: Typical plan view of an elevated water tank. Figure 2: Typical elevation of an elevated water tank. Figure 4: Reinforcement plan view of the top slab of the reservoir. Figure 3: Steel reinforcement elevation of the reservoir. An RC column supports the reservoir with square hollow sections founded on a footing (Figure 2). The column and the footing are optimized regarding their geometry and the steel reinforcement. The height and the cross-sections of the column play a key role in the structure's stiffness, which is crucial for the external forces and the stress resultants caused by earthquakes. Note that the present paper concentrates on the design interaction of the seismic action with the column's height and rigidity. This highly nonlinear problem is studied for a constant top reservoir. First and second-order stress resultants of the one-degree freedom oscillator are computed directly without any finite element simulation. The analysis considers five-column heights, from 20 to 40 m in steps of 5 m (Figure 2). Column stretches are defined by the variables that determine their cross-section, the concrete quality, and the steel reinforcement. The characteristics of each stretch are doubly symmetric since the seismic actions considered can act in any positive or negative direction (Figure 6). Index \(i\) in Figure 6 indicates a given column stretch since the wall thicknesses, reinforcement, and concrete quality can vary in each stretch. The external width of the column is a geometric variable that can vary between 1.80 and 4.90 m in steps of 0.10 m. This width is constant at the full height of the column. The limit of 4.90 is that of the size of the square inscribed in a circle of 7.00 m, which corresponds with the bottom dimension of the reservoir. The thickness of the column walls can vary in the different stretches, with a minimum of 0.25 m and a maximum of 0.75 m in steps of 2.5 cm. It is required that the thickness of the walls either keeps constant or reduces in height. The vertical reinforcement is placed in the outer and inner faces of the column. The number and diameter of the bars in each face define it. The vertical bars of the first stretch faces could be spaced between 0.100 and 0.300 m in steps of 2 mm. This gives 101 spacing possibilities. A given stretch face of the column can keep the number of bars in the stretch below or reduce it to half (half plus one bar if the number of bars in the stretch below is odd) unless the resulting spacing between bars would exceed 0.30 m. The possible diameters of the bars are 12, 16, 20, 25, and 32 mm.