# Optimization Design of RC Elevated Water Tanks under Seismic Loads

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimization Problem Definition

#### 2.1. Elevated Water Tanks Object of Optimization

^{3}, excluding the man’s access.

_{ck}in MPa, i.e., the compressive characteristic cylinder strength at 28 days. The concrete strength of a given stretch has to be equal to or smaller than that of the stretch below.

#### 2.2. Problem Definition

_{1}, x

_{2},…, x

_{n}design variables are given in Equation (3).

^{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 0.00g to 0.24g in steps of 0.04g, 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 (Ø12/20). 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.

_{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 Table 2, Table 3 and Table 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.

^{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}, N

_{c}and N

_{γ}are the capacity of load factors; S

_{q}, S

_{c}, and S

_{γ}are the shape factors; i

_{q}, i

_{c}, and i

_{γ}are the inclination factors; d

_{q}, d

_{c}and d

_{γ}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 γ is the ground density.

_{l}is a correction factor that takes into account the existence of a rigid layer at a low depth under the footing, f

_{s}is a coefficient that depends on the shape of the footing, q

_{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.

_{s}must be less than or equal to that resistance to slipping given by the product of the normal force N by the tangent of the friction angle plus the cohesion c multiplied by the width of the footing B.

_{v}is the overturning safety coefficient.

## 3. Optimization Method

_{ns}) is smaller than the cost of the previous solution (cost

_{ps}) and a specific parameter called threshold (U

_{f}). The process is updated, and the new solution becomes the previous solution.

_{f}is the new threshold, U

_{0}is the previous threshold, p is the number of solutions accepted in the searching process, and sol is the total number of feasible solutions considered, which in our case is taken as 1000. If the solution is accepted, then the threshold is reduced following Equation (10):

## 4. Numerical Results

_{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 Housner [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 Martínez-Martín et al. [46]. Regarding the value of a

_{gR}/g, larger values imply larger equivalent static forces.

^{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.

^{2}for the lowest seismicity to a maximum of about 140 m

^{2}for the largest seismicity. Figure 13 shows the 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. Consider that taller columns are more flexible, which reduces the column stress resultants due to ground acceleration. Figure 14 shows the total steel reinforcement in kg in the footing, which varies from a minimum of about 2000 kg to a maximum of about 14,000 kg. Again, the results can be divided into the same three groups with the same tendencies as Figure 12 and Figure 13.

^{3}/m. These values vary from a minimum of about 8 m

^{3}/m to a maximum of about 24 m

^{3}/m. Both figures show the same tendency. The amount of steel per unit column decreases with the height. This reduction is more accentuated for larger seismic degrees and shorter heights of the water tanks. Graphs tend to be horizontal when the seismic degree is low and the height of the tanks increases. All of this stresses the high nonlinearity of the problem since taller columns are more flexible and better accommodate the seismic response. Steel reinforcement directly depends on the equivalent seismic force of each of the 35 cases.

_{gR}/g, equal to 0.24, and a column height of 20 m. The minimum cost is 3749.51 €/m for a

_{gR}/g, equal to 0.00, and a column height of 40 m. Note that costs are per unit height of the water tanks.

## 5. Concluding Remarks

_{gR}/g varied from 0.00 to 0.24 in steps of 0.04. The foundation considered was a shallow RC footing with double symmetry. The foundation ground consisted of sands with an internal friction angle of 35° and no cohesion. The specific weight of the ground was 20 kN/m

^{3}, and the SPT was equal to 30. The water table considered was sufficiently deep not to alter the foundation conditions. Columns considered were RC square hollow sections with double geometry. The shape of the constant reservoir considered was a conic trunk. The height of the reservoir was 6 m, and the top and bottom diameters were 14 and 7 m, respectively. The optimization method considered is based on a hybrid OBA strategy with mutation operators named OBAMO. This method adjusts the search for new solutions as a function of the degree of success obtained with previous solutions, which resulted in the optimal outcome for the 35 water tanks analyzed. Regarding the analysis results, it is important to note the high nonlinearity of the problem due to the interaction between the seismic forces and the rigidity of the columns. It is worth noting that a full tank of water is determinant for seismic exposures a

_{gR}/g higher than 0.04. The amount of steel reinforcement and volume of concrete per unit height keeps relatively constant with the height for seismic zones of high degrees. Water tanks under no seismic conditions require more materials with height. Note also that the use of equivalent horizontal forces by the Eurocode 8 and the approximate method of Housner yield similar results. The study shows the algorithm’s applicability to the structure and provides engineers with guidelines for efficient design. The results provide information for a day-to-day design of RC elevated tanks.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Parameter | Value |
---|---|

Height of column formworks | 5.00 m |

Internal friction angle of the sands | 35° |

Standard penetration test (SPT) sands | 30 |

Ground– footing friction angle | 30° |

Specific weight of the ground | 20.00 kN/m^{3} |

Unit | Cost (€/Unit) |
---|---|

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 |

Concrete Type | Water/Cement | Cement (kg) | Slump | Cost (€/m^{3}) |
---|---|---|---|---|

C-25(1) | 0.65 | 250 | Flabby | 70.79 |

C-25(2) | 0.60 | 275 | Flabby | 72.78 |

C-25(3) | 0.60 | 300 | Flabby | 73.93 |

C-25(4) | 0.60 | 325 | Flabby | 75.49 |

C-25(5) | 0.60 | 350 | Flabby | 76.63 |

C-25(6) | 0.65 | 250 | Plastic | 69.40 |

C-25(7) | 0.60 | 275 | Plastic | 71.35 |

C-25(8) | 0.60 | 300 | Plastic | 72.48 |

C-25(9) | 0.60 | 325 | Plastic | 74.01 |

C-25(10) | 0.60 | 350 | Plastic | 75.12 |

C-30(1) | 0.65 | 250 | Flabby | 73.62 |

C-30(2) | 0.60 | 275 | Flabby | 75.69 |

C-30(3) | 0.60 | 300 | Flabby | 76.89 |

C-30(4) | 0.60 | 325 | Flabby | 78.51 |

C-30(5) | 0.60 | 350 | Flabby | 79.69 |

C-30(6) | 0.55 | 300 | Flabby | 79.66 |

C-30(7) | 0.50 | 300 | Flabby | 79.85 |

C-30(8) | 0.50 | 325 | Flabby | 82.75 |

C-30(9) | 0.65 | 250 | Plastic | 72.18 |

C-30(10) | 0.60 | 275 | Plastic | 74.20 |

C-30(11) | 0.60 | 300 | Plastic | 75.38 |

C-30(12) | 0.60 | 325 | Plastic | 76.97 |

C-30(13) | 0.60 | 350 | Plastic | 78.13 |

C-30(14) | 0.55 | 300 | Plastic | 78.10 |

C-30(15) | 0.50 | 300 | Plastic | 78.29 |

C-30(16) | 0.50 | 325 | Plastic | 81.13 |

Concrete Type | Water/Cement | Cement (kg) | Slump | Cost (€/m^{3}) |
---|---|---|---|---|

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 |

Column Height (m) | a_{gR}/g | Euroc. 8 (kN) | Housner (kN) |
---|---|---|---|

20 | 0.00 | 0.00 | 0.00 |

20 | 0.04 | 341.90 | 359.16 |

20 | 0.08 | 1232.35 | 1210.01 |

20 | 0.12 | 2144.68 | 2055.94 |

20 | 0.16 | 3479.77 | 3253.73 |

20 | 0.20 | 4198.74 | 3923.55 |

20 | 0.24 | 5702.18 | 5252.09 |

25 | 0.00 | 0.00 | 0.00 |

25 | 0.04 | 249.65 | 261.80 |

25 | 0.08 | 980.62 | 996.25 |

25 | 0.12 | 1344.01 | 1376.53 |

25 | 0.16 | 2245.38 | 2209.03 |

25 | 0.20 | 3468.79 | 3305.40 |

25 | 0.24 | 3477.36 | 3367.75 |

30 | 0.00 | 0.00 | 0.00 |

30 | 0.04 | 171.02 | 219.44 |

30 | 0.08 | 602.54 | 617.07 |

30 | 0.12 | 1568.58 | 1568.41 |

30 | 0.16 | 1694.09 | 1739.40 |

30 | 0.20 | 2049.89 | 2101.60 |

30 | 0.24 | 3082.29 | 3033.61 |

35 | 0.00 | 0.00 | 0.00 |

35 | 0.04 | 110.95 | 161.97 |

35 | 0.08 | 434.30 | 494.77 |

35 | 0.12 | 824.62 | 824.61 |

35 | 0.16 | 1356.35 | 1426.94 |

35 | 0.20 | 1661.61 | 1746.68 |

35 | 0.24 | 2063.15 | 2155.73 |

40 | 0.00 | 0.00 | 0.00 |

40 | 0.04 | 108.76 | 159.40 |

40 | 0.08 | 345.11 | 440.62 |

40 | 0.12 | 666.97 | 732.58 |

40 | 0.16 | 1153.28 | 1194.16 |

40 | 0.20 | 1390.90 | 1442.49 |

40 | 0.24 | 1707.13 | 1790.73 |

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**MDPI and ACS Style**

Martínez-Martín, F.J.; Yepes, V.; González-Vidosa, F.; Hospitaler, A.; Alcalá, J.
Optimization Design of RC Elevated Water Tanks under Seismic Loads. *Appl. Sci.* **2022**, *12*, 5635.
https://doi.org/10.3390/app12115635

**AMA Style**

Martínez-Martín FJ, Yepes V, González-Vidosa F, Hospitaler A, Alcalá J.
Optimization Design of RC Elevated Water Tanks under Seismic Loads. *Applied Sciences*. 2022; 12(11):5635.
https://doi.org/10.3390/app12115635

**Chicago/Turabian Style**

Martínez-Martín, Francisco J., Víctor Yepes, Fernando González-Vidosa, Antonio Hospitaler, and Julián Alcalá.
2022. "Optimization Design of RC Elevated Water Tanks under Seismic Loads" *Applied Sciences* 12, no. 11: 5635.
https://doi.org/10.3390/app12115635