Effects of soil-structure interaction in seismic analysis of buildings with multiple pressurized tuned liquid column dampers

In this paper soil-structure interaction (SSI) effects are investigated while an array of Pressurized Tuned Liquid Column Dampers (PTLCD) is employed for seismic vibration control of buildings. This device represents the most general case of a passive damper, with different reduction options to other control devices obtained by simplifying the involved parameters. Soil conditions considerably affect the control device functioning, because dynamic parameters such as natural frequency, damping factor, and natural modes depend on the soil properties. A simplified mathematical model is developed for the building with multiple degrees of freedom connected to a flexible base. For the time-domain analysis, a computational routine is developed for the linearization of the equilibrium equations of the PTLCD, as well as details for the reduction to the other types of passive dampers. Several numerical examples are selected for the analysis of the damper efficiency in reducing seismic vibration considering SSI. These simulations include Kobe earthquake data, which is applied to the model to evaluate the device performance under different scenarios. It is verified the influence of SSI in the natural frequency and structural response, which is related to the earthquake frequency components. Results confirm that the array of PTLCD’s can reduce the vibration amplitudes, being more effective for soils with higher stiffness values.


INTRODUCTION
Numerous technologies have been developed and studied to ensure structural safety and to prevent large vibration amplitudes in engineering structures, such as buildings, towers, and bridges. Control devices are designed with the purpose of damping, isolating and/or acting against vibrations induced by external forces in the structure in order to guarantee its use within project requirements. Passive devices act on the structure without the need for external power supply for its operation. This type of control device absorbs a fraction of the energy from the structure and dissipates it through its motion. In this sense, they are designed to oscillate at the same frequency of the structure, but with a lag, in order to produce an inertia force in the opposite direction to the structural vibration.
Passive control devices are generally defined within the following categories: Tuned Mass Damper (TMD); Tuned Liquid Damper (TLD); Tuned Liquid Column Damper (TLCD). Bigdeli and Kim (2016) conducted experimental studies to passive device connected to a structure subject to seismic action and under the influence of site conditions. Moreover, Jabary and Madabhushi (2018), through experimental analysis of multiple TMDs, evaluated their performance for the dynamic response. Although some papers on PTLCD are being extended for seismic analysis, few studies have been dedicated to the study of these effects concomitantly with SSI. Traditionally structures with vibration dampers have been analyzed using the fixed base model, which does not reflect the influence of the soil characteristics on the system response. As a result, the seismic response of a building with PTLCD for specific site conditions is of great interest to engineering practice.
This work presents an approach for the passive control of seismic induced vibrations in multi-floor buildings, where the effects of soil-structure interaction are applied to the definition of the type of control and the evaluation of the structural dynamic response. Initially, the mathematical model of PTLCD is developed, which is later coupled to a single degree of freedom model with soil-structure interaction. This model is extended to a multi-floor problem, which allows the inclusion of numerous passive devices at the top of the building, tuned according to the natural frequencies of the SSI problem. A generalized feature of the algorithm arises with the reduction of the PTLCD to any other type of passive attenuator (TLCD, TLD, TMD), obtained using a mathematical analogy with control parameters. Computational routines developed with MATLAB software are presented, enabling modal, harmonic, and transient analysis. Kobe earthquake data are applied to a 40-story building where the performance of the control devices is evaluated. The results in free and forced vibrations are in agreement with the simulations developed by Farshidianfar and Soheili (2013) for buildings with SSI in four different types of soil. Numerous complementary simulations are presented, with a study of: i) combining multiple dampers at different natural frequencies of the structure; ii) influence of soil stiffness and damping; iii) pressurizing as a flexibility factor in PTLCD tuning; (iv) linearization of the damping term of the PTLCD. Finally, the influence of soil-structure interaction on the natural frequency and dynamic response of the system is verified. The results confirm that the PTLCD is efficient in reducing displacement amplitudes, being more effective for soils of higher stiffness values. To the authors' knowledge, the proposed procedures, and the generalized mathematical model based on the PTLCD are original and allow an extensive approach for the passive control of seismic vibrations in buildings considering SSI effects.

THEORETICAL FORMULATION
The formulation for linear dynamic analysis of multiple degrees of freedom structures under seismic excitation is presented in this section, considering SSI at the base of the structure, represented by soil stiffness and damping, and a passive control device at the top of the structure, corresponding to an array of multiples PTLCD (MPTLCD).

Pressurized tuned liquid column damper analytical model
The TLCD, as well as the TMD, operate without the need for external power supply. This type of damper dissipates energy from the structure to which it is coupled by absorbing part of the energy produced in the structure during its vibration. The vibrating energy of the structure is dissipated by the relative movement of the liquid column in relation to the movement of the structure, as emphasized by Souza (2003). In contrast to TMD, which requires additional elements of stiffness and damping, the TLCD uses the gravity force and the energy dissipation of the liquid. These are generated by the movement of the fluid through the head loss positioned inside the attenuator as restoring and viscous forces, respectively.
The typical analytical model of the TLCD, consisting of a U-tube with liquid (usual water) of a constant section where z(t) represents the displacement of the liquid inside the pipe and x(t), the lateral displacement of the structure has the equation of motion expressed by: where ρ f is the density of water, A and D are the area and diameter of the cross-section of the TLCD, L and B are the total length and length of the horizontal section of the attenuator, A 0 is the area of the orifice and g is the acceleration of gravity. The term f is expressed as a function of the roughness of the tube ε and the Reynolds number Re. This coefficient is evaluated with the following expression: The liquid oscillation frequency ω D for the TLCD depends on the total length of the liquid column L: Thus, TLCD is distinguished as a long-term system. To solve this problem and increase its applicability, the vertical sections of the U-tube can be sealed and pressurized ( Figure 1). The presence of the pressurized chambers allows damper adjustment and tuning at the desired frequency to be set out with less effort, since it is not only dependent on the dimensions of the tube but also on the stiffnes arising in the pressurized cavity. By implementing compressed air at both ends of the vertical section the Pressurized Tuned Liquid Column Damper (PTLCD) is formed. The PTLCD, when subjected to dynamic loading, presents liquid displacement inside the sealed chambers that, consequently, causes a variation in the pressure of the gas column ( Figure 2). In the chamber, P represents the pressure, V its volume, dV the variation of this volume, dz the linear variation of the volume in the direction z, A the area of its cross section and l C the length of the pressurized section. Considering an isothermal process for the variation of the gas pressure and volume inside the PTLCD chamber: where η is a constant, and γ is the gas constant equal to 1.4. Thus, as demonstrated by Pedroso (1992) for the variation of the initial condition inside the cavity: The variation of the gas pressure of the chamber causes a variation in the force expressed by and considering the linear variation of volume and volume defined by: Results in a force variation dF given by: Supposing the stiffness of the pressurized cavity k P as a function of the force variation by displacement and with substitution of (10) in (11): The equation of motion of the PTLCD is given by: A comparison of Eqs. (1) and (13) shows that the PTLCD is a more general model when compared to the standard TLCD, with addition of gas cavity stiffness in the two vertical chambers of the device. In a more compact notation: with:

Soil structure interaction with PTLCD
During the event of an earthquake, the response of the structure is affected by the interaction of four connected systems: the PTLCD, the structure, the foundation, and the soil. The analysis with soil-structure interaction involves the joint response of these systems ( The free-field acceleration of the soil ẍ g (t) on the surface of the half-space results in interaction forces at the interface of the structure with its rigid foundation and produces shear force V f and moment M f at the base. These interacting forces unfold displacements x 0 (t) and rotations φ y (t) at the soil-foundation interface, together with the displacements of the structure x(t) and the movement of the liquid column z(t) caused by the earthquake.
For the simple case of a structure with a single degree of freedom (SDOF) of height h, supported by a foundation of mass m 0 and mass moment of inertia I 0 , attached to the elastic half-space, the absolute displacement of the structure is expressed by: x g , x 0 , φ y , and x denote, respectively, the free-field displacement of the soil, horizontal displacement, and rotation of the soil-foundation interface, and the structural relative displacement with respect to foundation in the time domain. From this decomposition, the dynamic equilibrium equations for the movement of the liquid inside the pressurized damper, the translation of the structural mass and the displacements/rotations at the base are established as proposed by Ghosh and Basu (2005). The equation of motion of the liquid column is given by and the equation of motion for the structure is where m, c, and k represent the mass, damping and stiffness of the structure, respectively. For the degrees of freedom of the foundation and elastic half-space interface, the dynamic equation for the translation of the foundation expressed by: and for the moment Mf caused by the interaction we have the following equation: The interaction forces V f and M f cause displacements and rotations at the soil-foundation interface. Such displacements and rotations are feasible due to the flexibility of the soil-foundation system, which contributes significantly to structural flexibility. The soil dissipates some of the vibration energy of the system, either by wave propagation or by the material behavior, considerably affecting the damping of the system. For the scenario of a foundation supported on the soil surface: The following linear system arises from the equilibrium of forces of a structure with one degree of freedom, foundation supported under the elastic half-space, and with PTLCD: the parameters m, c and k correspond, respectively, to the structural mass, damping and stiffness. The submatrices that form the mass matrix of the system are given by: The mass moment of inertia I of each building floor is given by: where b is the horizontal dimension of the floor (parallel to seismic excitation direction). The rigid foundation of radius r, mass m 0 , and mass moment of inertia I 0 is supported by homogeneous linear semi-elastic medium. For a circular shallow foundation, the following expression is valid for I 0 : The seismic force vector P applied to the structure (representing the excitation caused by ground movement) is defined as a function of the acceleration ẍ g (t) at the base, and expressed by: In this work the soil-foundation system is modeled by discrete models proposed by Dobry and Gazetas (1986), although in the computational routine other general parameters for the soil and the foundation can be used, and computed from other mathematical models, e.g.: Pais and Kausel (1988) and Mylonakis et al (2006). The stiffness and damping coefficients of the soil-foundation interaction are determined from simplified models for frequencyindependent impedance functions as presented in Table 1. The equations describe the stiffness and damping for translation (x) and for rotation (yy).
The soil shear modulus, G S , is determined from its modulus of elasticity, E S , and Poisson's coefficient, ν S , while the shear wave velocity V S is given as a function of the shear modulus and the specific mass of the soil:

Generalization for multiple degrees of freedom
For the scenario of a multi-floor building with multiple PTLCDs (MPTLCD) coupled to the last floor, as shown in Figure 4, the dynamic equilibrium matrix equation has its matrices formed of submatrices referring to each of the systems and to the interaction between the different systems as is expressed in Equation (29).
The mass matrix of the system comprises submatrices referring to individual systems and by submatrices that represent the coupling effect between these different systems. The submatrix m E defines the lumped mass matrix of the n floor structure, and also includes the contribution of the MPTLCDs masses present on the last floor: Latin American Journal of Solids and Structures, 2019, 16(8), e225 9/21 Considering a quantity q of PTLCDs the mass matrix m D and the submatrix of mass m ED are expressed by: The mass submatrices referring to the soil-structure interaction are given by: The stiffness and damping matrix do not present coupling between the different systems, and their formulations are unchanged, as presented in the matrix equation of a SDOF structure. The foundation stiffness sub-matrix k S and damping c S follow the same formulation presented in Eq. (21), while the stiffness matrix of the n-structure DOF and the matrix of dampers k D with q PTLCDs are respectively defined in expression Eq. (35).
The structural damping matrix c E is determined by the Rayleigh Damping: ω 1 and ω n are equivalent to the first and last natural undamped frequency of the building, and ξ is the damping ratio.
The MPTLCD damping submatrix c D is given by The vector force P for the case of the acceleration applied at the base of the system coupled with the load as a function of the acceleration at the base, ẍ g (t), is defined as while the dynamic response u of the system is given by:

COMPUTATIONAL ASPECTS
An algorithm was developed in MATLAB for the dynamic analysis of the SSI-MPTLCD systems, and it is available in GitHub. The code performs modal analysis, and transient dynamic analysis in both time and frequency domain (harmonic loading for a range of excitation frequencies).
It is possible to vary the quantity of PTLCDs and their features (dimensions), as well as the presence or absence of the pressurized chamber (simulating TLCD). Also, the submatrices containing the control device parameters can be easily adapted and extended to the TMD and TLD cases, where equivalent parameters can be considered for these control devices. An additional feature is the option of fixed or flexible base for the SSI. For the SSI model, it is possible to define different flexibility and damping coefficients for each type of soil, as well as the mass and the mass moment of inertia of the foundation.
The numerical integration of equilibrium equations is conceived with an adjustment in the method of constant average acceleration presented by Chopra (1995) apud Newmark (1959): a linearization procedure is applied to the damper equation at each time step.
To evaluate the increment in the displacement Δu i of the system, given by: it is necessary to determine ̂i and Δ � i for each time step Δt, since both assume different values throughout the integration, and the damping matrix of the system c i and the damping values of the PTLCD c Dj , in turn, depend on the liquid velocity inside the tube (Eq. 2b). Thus, at each time step, a new damping coefficient c Dj is defined and the damping matrix of system c i is updated and then ̂i and Δ � i are computed with: where ΔP = P (i + 1) + P i . Subsequently, the increase in velocity Δ̇i and the increase in acceleration Δü i are defined by: A general algorithm with each step of the computational routine described above is summarized in Table 2. Table 2 Algorithm for the evaluation of the dynamic response of the SSI system with MPTLCD.

Equivalent parameters for other types of passive control devices
The generalized aspect of the computational routine arises with the reduction of the PTLCD or any other type of passive damper (TLCD, TLD, and TMD) by setting equivalent parameters of the control device by means of a mathematical analogy.
Considering a SDOF structure, where m, c, and k define the structural mass, damping, and stiffness, respectively, with a coupled passive device, the dynamic equilibrium equation of the system is expressed as:

NUMERICAL EXAMPLES
This section presents the dynamic responses of MDOF buildings with pressurized dampers tuned to different frequencies of the structure on different soil types. The efficiency in reducing structural oscillations considering SSI under Kobe earthquake excitation, obtained from the data provided by the Center for Engineering Strong Motion Data (CESMD, 2018), is investigated through a parametric study in terms of mass ratio, damper frequency and soil properties

Effects of the soil-structure interaction on the coupled system natural frequency
In order to examine the effects of SSI on buildings, the model of shear building and soil adopted by Liu et al. (2008) and later by Farshidianfar and Soheili (2013) for the analysis of wind and earthquake vibrations, respectively, is employed. The structural properties are presented in Table 3. The flexible base of the model is determined by relating to three soil types with different mechanical parameters which, are defined as dense, medium and soft soil. Table 4 indicates the values of the stiffness and damping coefficients adopted for the springs and dashpots positioned at the base of the model for each soil type.  Table 4 Soil parameters and flexibility and damping coefficients. The values obtained by the algorithm implemented in MATLAB in comparison with the natural frequencies of the fixed base and flexible base are presented in Table 5. The results with asterisk represent the values obtained by Farshidianfar and Soheili (2013). Table 5 Undamped natural frequencies of the system. The maximum response of each system as a function of the excitation frequency is presented in Figures 5-7, where the peaks represent the natural frequency values of the structure and are in agreement with the values shown in Table 5. R 40 corresponds to the ratio between the relative displacement values for the 40 th floor, defined in Eq. (16), and the sinuidal acceleration amplitude at the base. By means of the graphical representation of the forced response of the system (Table 3 and 4), it can be noted that the fundamental frequency of the system, when compared to a fixed base value, is greatly reduced under a SSI hypothesis. These plots also provide a greater insight to the dynamic response, since

Effects of the soil-structure interaction on the system forced response
The system response under forced vibration is verified using the ground acceleration history recorded in Takatori station, Japan, during the 6.7 magnitude earthquake occurred on January 16, 1995, in Kobe. Figure 8 Figure 9 present the absolute displacement values for the 40 th floor produced by Kobe earthquake excitation, and compared to the results obtained by Farshidianfar and Soheili (2013). Table 6 indicates nearly identical results with relative differences inferior to 0.5%. Table 6 Maximum absolute displacements of the structure (40th floor) under Kobe earthquake excitation.

Model
Fixed base x 40 (m)

Soft soil x 40 (m)
Farshidianfar e Soheili (2013  The structural response for each type of soil is also examined in terms of relative displacements of the structure with respect to the foundation (Table 7). Initially SSI considering only soil flexibility influence (x 1 ), then the effect of adding soil damping (x 2 ). Analysis of the maximum and rms (roots mean square) relative displacements during a frame of 40 seconds of the structure 40 th floor, indicate that relative displacements are greatly reduced for the soft soil type, and that damping effects are almost negligible for the analyzed scenarios.

Analysis of multiples PTLCD for fixed base structure
The use of TLCD for control of structural vibrations of the building with parameters presented in Table 3 is possible only with the addition of the pressurized chamber to the device, i.e. using PTLCD. The TLCD, when tuned to the fundamental frequency of the structure under the fixed base scenario (1.65 rad/s), would require a length of only 7.2 m (Eq. 4). Thus, in order to achieve the required mass ratio μ (total mass of devices in relation to the mass of the structure) there would be two main problems: (i) dimensioning a pipe diameter in which there is a physically acceptable relationship between the length and diameter of the TLCD; and as a consequence, (ii) it would be necessary to install an excessive amount of dampers to achieve mass ratio values relevant for the control of structural vibration.
Before exploring the effectiveness of the MPTLCD coupled to SSI, a preliminary study of the vibration control applied only to the structure is carried out with the variation of device parameters. The efficiency of the vibration control of multiple PTLCD is analyzed for a given number of dampers, represented by the ratio μ and the influence of the ratio Ω (ratio of PTLCD frequency to structure) from changing the gas pressure of the attenuator. In this section, the base acceleration is represented by a sinusoidal function in resonance with the fundamental frequency of the structure and of amplitude (P 0 ) 0,2 N.
The PTLCD is tuned to the fundamental frequency of the structure with its dimensions and properties of the pressurized chamber shown in Table 8. From the analysis of the variation of the fluid coordinate z in the vertical section (Figure 10), it was ensured that this displacement was compatible with the length adopted for the pressurized chamber. Such damper dimensions may appear to be overstated; however, it should be noted that the device was designed for a 40-story high-rise building.  Using this PTLCD, vibration control versus the mass ratio μ is evaluated. For each simulation the peak displacement and mean rms of the displacement during a frame of 100 seconds at the top of the structure (40 th story) were observed. These values are compared with the displacement at the same point of the structure, with the absence of these devices. The values obtained from the analysis are presented in Fig. 11. It can be perceived that the increase in the mass ratio (or the number) of PTLCD causes a greater reduction in the displacements of the structure and is more notable for the rms values of displacement. The peak displacement values from μ equal to 5% and beyond show an asymptotic behavior in which the difference of values from μ to 5% and to 8% is only 6% for the displacement reduction. Simmilar behavior is shown by the rms plot. The dynamic response at the top of the structure for the values of μ equal to 1%, 2%, 4% and 8% are shown in Fig. 12. The reduction of structure displacements in the steady phase is highly sensitive to an increase of μ up to 4%, and there is a slight difference from 4% to 8%. Another parameter of great interest for design of the PTLCD is the tunning frequency of the device. In the case of seismic excitations, the condition with the highest vibration control capability of structures is that in which the damper is in resonance with the fundamental frequency of the structure (Ω = 1.0). To change the attenuator frequency, the pressure and height properties of the gas-sealed chamber are changed. The dynamic response results for μ equal to 2%, 3%, and 4% are shown in Fig. 13 for a variation in Ω.
As shown in Fig. 12, the mass ratio μ has a great influence on vibration control. The higher the value of μ, greater the efficiency in reducing the structural response. Through Fig. 13 it is also possible to state that the damper is less effective in the displacement control when shifting Ω from the fundamental frequency of the system. It is also worth noting that the greatest reduction of displacements occurs when the device is tunned with the fundamental frequency of the system, including the mass of the MPTLCD, and not necessarily for Ω = 1.0. This results from the fact that adding mass to the system (by adding dampers) and maintaining constant the stiffness of the structure leads to a reduction of the fundamental frequency of the structure-damper system. It can be observed that the minimum displacement values presented by the curve of μ equal to 3% and 4% present a shift to the left in these plots, positions where the device has a lower tuning frequency (Ω < 1.0), and are in resonance with their respective structure-damper frequencies.

Figure 13
Variation of rms of displacements at the top of the structure with control versus the ratio between the frequency of the MPTLCDs and the fundamental frequency of the structure.

Analysis of multiples PTLCD for flexible base structure
After investigating the behavior of multiple PTLCD coupled to the fixed base structure and subjected to an acceleration in the base represented by a sinusoidal function, the response of the same structure is evaluated with the same control devices, but now subjected to the accelerations generated during the Kobe earthquake, as shown in Figure 9. The structural parameters of the building and soil properties are shown in Tables 3 and 4, respectively. A series of results is presented for the vibration control of the structure versus μ for the different base properties and for MPTLCD tuned to the fundamental frequency of the structure ( Figure 14); the 1 st and 2 nd natural frequency of the structure ( Figure 15); and to the 1 st , 2 nd and 3 rd natural frequency of the structure (Figure 16). In order to tune the dampers to the 2 nd and 3 rd natural frequencies of the system, only the adjustment of the chamber pressure is done, that is, only the stiffness of the PTLCD is changed and the remaining features are kept constant (Table 8).
For the graphs in a) the dampers are tuned to the natural frequency of the fixed base structure, i.e., the tuning frequency of the control device is established without considering SSI. Although the design of the device disregards the interaction, in the forced vibration the soil-structure interaction is considered. Thus, these cases simulate the difference between theory and the application, using the theoretical scenario where vibration control neglects the effects of SSI (but in reality, the soil has an influence on the structural behavior). The graphs in b) present the results for the hypothesis that in the design of the PTLCD the natural frequencies of the structure were determined considering the SSI for each different type of soil. In summary, for tuning the device frequency, the natural frequencies of the structure considering SSI, which are shown in Table 5, were considered. The displacement values of each case, i.e., rigid base, dense soil, medium soil, and soft soil, are compared with their respective displacements shown in Table 6 for the second case, where the SSI effects are considered.   Concerning the displacements at the 40 th floor of the builing, it can be stated that the larger values of μ produce greater decreases in the displacements. Regarding the soil type, it is seen that the reduction of the maximum displacements of the structure by the PTLCD when considered the soft soil is inferior to the other models. In this case it is advisable to opt for another type of control device. The structure on the dense soil, which has the natural frequencies with the smallest difference to the theoretical fixed base structure, shows that SSI effects are almost negligible. Thus, it can be said that for high stiffness soils this hypothesis is legitimate. However, as described in item 4.2, when neglecting the soil-structure interaction, and specifically soil damping, an important source of energy dissipation of the system is disregarded. It can be observed that the control results for dense soil and medium soil were, respectively, slightly higher than the control for the fixed base case.
Also, under the consideration of MPTLCDs tuned to the different natural frequencies, and under the Kobe earthquake excitation, the greatest reductions of the mean displacements were obtained when the dampers were tuned solely for the 1 st natural frequency. Table 9 contrasts the cases investigated above for MPTLCD considering the largest mass ratio and when tuned to the natural frequency of the flexible base structure (numbered from 1 to 3 and following the same order as in Figures  14b -16b) with the results presented for the TMD to the same scenario (same structure, the same types of soil and Kobe earthquake). In general, using the vibration control with MPTLCD enables greater reductions in the response of the structure (case 1 and 2) or close values (case 3). Also, regardless of the damper model and case considered, the results obtained for the soft soil are entirely inferior to the other types of soil conditions.

CONCLUSION
In an overview, this paper presented a formulation for the seismic analysis of structures with multiple pressurized tunned liquid column damper (MPTLCD) and with the consideration of soil-structure interaction. The computational implementation was discussed for the analysis of vibration control under different types of soils of different mechanical properties.
From the study of the soil-structure interaction it is verified that the fundamental frequency of the SSI system is lower than the fundamental frequency of the fixed-base structure, and the reductions in natural frequencies are negligible for the structure under dense soil hypothesis, but for the structure under soft soil the SSI effects are considerable (Table 5).
For the analysis of vibration control using MPTLCD, it can be observed that an increase of the mass ratio μ presents greater efficiency in the reduction of structural displacements, but there is an asymptotic behavior in this displacement reduction, in which for the analyzed cases are mostly defined for μ=5%. On tuning the control devices, greater efficiency is achieved for vibration control when the mass of these devices is considered on the computation of the fundamental frequency.
For the scenario of MPTLCD tuned to the different natural frequencies, when the attenuators were set for the 1 st natural frequency, the largest reductions of the rms displacements were obtained under the Kobe earthquake excitation. The incorporation of pressurized chambers on the vertical sections of the "U" tube presents itself as a feature that enables excellent flexibility for the dimensioning of the control device.
The authors believe that the mathematical model developed for a flexible base structure provides a better understanding of the dynamic behavior of a building with multiple PTLCD during an earthquake. The conventional procedure of tuning the attenuator to the natural frequency of the fixed base structure, i.e., fixed base, demonstrates inaccuracy for the vibration control analysis of the structure under earthquake excitation, as shown in Figs. 14-16. It is important to emphasize that SSI results show that except for the dense soil type, results can be underestimated (medium soil case) or overestimated (soft soil case) if interaction effects are neglected.
The computational routines developed in this research are available for further studies and can be easily adapted for other types of passive dampers (https://github.com/MauricioVMendes?tab=repositories).