## 4 Base-Isolated Liquid Storage Tank
### Surrogate Tank Model  
Numerical simulation of tanks is a difficult task due to the complexity of liquid-tank hydrodynamic behavior [29], since the explicit simulation of the liquid content results in computationally demanding models. In order to achieve an optimal balance among computational cost, model complexity and accuracy, valid surrogate models are frequently developed instead of complicated three-dimensional (3D) models. [PERSON] and [PERSON] [30] stated that simulating liquid storage tanks as mechanical analogs is suggested in contemporary seismic norms, e.g., Eurocode 8 [31], API 650 [32], NZSEE [33] and IITK-GSDMA [34]. Related studies have also shown that such simplified models can be effectively used in dynamic analyses of base-isolated tanks (e.g., [PERSON] and Whittaker [35]).  
Consequently, the examined tank is represented by modifying the \"Joystick model\", developed for fixed-base tanks by [PERSON] et al. [29]. Due to the cylindrical geometry of the tank, the isolators are arranged following a symmetrical radial layout, as shown in the surrogate model of Figure 3. As displayed in Figure 3, the surrogate model consists of a vertical beam that carries the impulsive mass and it is supported by an adequate number of rigid beam spokes (e.g., 12 spokes in this investigation), which are placed on top of the sliding bearings. The beam spokes of the Joystick model are used to simulate the rigid base of the tank. Additionally, the guidelines of Eurocode 8--Part 4 [31] have been used to calculate the impulsive mass and the properties of the vertical elastic beam using the equivalent stiffness, \(k_{i}\), (equal to \(7.8\times 109\) N/m) which corresponds to the impulsive period, \(T_{i}\), (equal to 0.16 s) and mass, \(m_{i}\) (equal to 5067 t). More details regarding the employed \"Joystick model\" can be found in [36]. This surrogate numerical model is quite reliable, as it accurately matches the impulsive fundamental period. Regarding the damping of the tank, it is set equal to 5%, as stated in Eurocode 8--Part 4 [31] recommendations for the ultimate limit state of liquid tanks. In the developed surrogate models, the lumped masses  
Figure 2: Performance levels for static conditions (left sketches) and various seismic scenarios (SLE = Service Level Earthquake, DBE = Design Basis Earthquake and MCE = Maximum Credible Earthquake) for: (**a**) SFPB and (**b**) TFPB (Reprinted from Computers and Structures, Volume 243, [PERSON] and [PERSON], Optimizing the seismic response of base-isolated liquid storage tanks using swarm intelligence algorithms, Pages No. 106407, Copyright (2022), with permission from Elsevier).  
represent the weight of the content, applied at the base beam spokes of the tank, i.e., the vertical loads are imposed on the bearings. It is noted that the weight of the tank walls can be neglected, as it is only a small percentage (\(\approx\)5%) of the total weight [37].  
As mentioned, the impulsive liquid component dominates the hydrodynamic response of the tanks. On the other hand, the convective liquid mass can be neglected, based on relevant studies (e.g., [PERSON] and [PERSON] [35]; [PERSON] et al. [38]), in which it is presented that although the impulsive pressure is reduced due to seismic isolation, the convective pressure remains practically constant. Furthermore, the impact of the convective part of liquid content can be estimated separately [29]. Figure 4 compares the predominant periods of the selected accelerograms with the periods of the impulsive and convective liquid components of the examined squat tank. It has to be stressed that the fundamental period of the examined tank isolated with different bearing configurations ranges from 2.29 s to 3.23 s. As shown in Figure 4, the convective period is close to 7 s; thus, it is evident that the dynamic response of the tank is mainly influenced by its impulsive period, while sloshing phenomena are not expected to occur.