*Article* **Development of Vibration Control Structure on Suspended Ceiling Using Pulley Mechanism**

**Ryo Majima <sup>1</sup> , Shigeki Sakai <sup>2</sup> and Taiki Saito 1, \***


**Abstract:** A suspended ceiling system (SCS) is one of the most fragile and non-structural elements during earthquakes. However, effective seismic protection technologies for enhancing the suspended ceiling system have not been developed other than the steel bracing system. An innovative passive vibration control system is proposed in this paper, which equipped a damper-employed pulley amplification mechanism into the indirect suspended ceiling system, named the pulley–damper ceiling system (PDCS). Theoretical formulation and the detailed information on the system were presented first. In addition, a new rotational damper composition consisting of a non-linear viscous damper was developed to follow the large wire-cable stroke. Six types of the full-scale ceiling specimens of a 15.6-square meter area with different configurations were constructed for the preliminary experiments to evaluate the seismic performance and feasibility of PDCS under simulated earthquake motions. The comparative results of the shake table test demonstrated that the application of PDCS is capable of controlling both displacement and acceleration of the ceiling panels. This study also presents the nonlinear time history analyses by modeling a wire-cable as an equivalent truss element to transmit the relative displacement of the ceiling system to the damper. The analytical model accurately simulated the dynamic behavior of PDCS.

**Keywords:** suspended ceiling systems; passively controlled structure; pulley tackle mechanism; shake table test; simulation analysis

#### **1. Introduction**

The experience of countless earthquake events accelerated the development of seismic protection technologies for main structural frames, such as base isolation structures and vibration control devices. These systems are widely applied in structural design in Japan, especially after the 1995 Kobe earthquake, and their effectiveness in reducing the shaking was verified during the 2011 Great East Japan Earthquake [1]. Meanwhile, only a little attention has been paid to the seismic performance of nonstructural elements and/or secondary structures, including, for example, suspended ceilings, exterior walls, fire extinguishing equipment (fire door and sprinkler), piping installation, and lighting equipment [2,3].

During the consideration of nonstructural elements, a suspended ceiling system (hereinafter referred to as SCS), which is one of the most commonly used in large indoor public spaces, is fragile to the shaking during earthquake events [4]. The vulnerability of SCS in past earthquakes has been extensively documented. For instance, in Japan, the ceiling falls in indoor sports facilities, including a gymnasium in school, shopping mall, and airports terminal, were previously reported in the 2003 Tokachi-Oki earthquake [5] and the 2016 Kumamoto earthquake [6]. Moreover, the damage to the SCS has been reported in various countries, namely, in the 1994 Northbridge earthquake in the United States [7], the 2010 Chile earthquake [8], the 2016 Gyeongju earthquake, and the 2017 Pohang earthquake

**Citation:** Majima, R.; Sakai, S.; Saito, T. Development of Vibration Control Structure on Suspended Ceiling Using Pulley Mechanism. *Appl. Sci.* **2022**, *12*, 3069. https://doi.org/ 10.3390/app12063069

Academic Editor: Felix Weber

Received: 1 March 2022 Accepted: 15 March 2022 Published: 17 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

in Korea [9]. In addition, the failure and collapse of ceiling panels would cause, not only blocking evacuation paths and impending continuous operation of the building, but also human damage. During the 2011 Great East Japan Earthquake, there were more than 75 casualties caused by the collapsing of approximately 2000 SCSs [10,11].

The SCS is composed mainly of hanger elements, carrying ceiling runners, and ceiling panels. Based on the structure of the ceiling runner, the SCS can be classified into two types [12]: the direct-hung suspended ceiling system (direct-SCS) and indirect-hung suspended ceiling system (indirect-SCS). Figure 1 shows the schematic diagram of the SCS. The key difference between them is the connection of the cross runner and hanger elements. In the direct-SCS (Figure 1a), the primary runners are directly suspended from wires attached to the structural system above, and cross joints are applied to the secondary runners to insert between the span of the primary runners. Meanwhile, the indirect-SCS (Figure 1b,c) uses hanger bolts to suspend primary runners from the above floor, and primary runners and secondary runners are connected using clips.

**Figure 1.** Type of suspended ceiling systems: (**a**) Direct-hung suspended ceiling system; (**b**) indirecthung suspended ceiling system with lay-in panel; (**c**) indirect-hung suspended ceiling system with continuous panel.

 ଵ, ோଵ ଶ, ோଶ ൈ On the one hand, as the direct-SCS is widely used because of its convenience when changing the ceiling panels, relocating the attached air conditioner unit, and the lighting equipment in the practice, most of the previous studies focused on evaluating and understanding the seismic response of the direct-SCS by using a shake table for more than two decades. The main finding from those experiments was about the connection failure of the ceiling grid members due to the collision between ceiling panels and the surrounding structures, including walls. To overcome this issue and improve the seismic performance, several studies were performed on the hanger connection [12,13], the perimeter [14–16], retainer clip [17], and the cross joint [18,19]. In addition to these studies, few researchers proposed and investigated the feasibility of a pendulum-type ceiling system that does not use any rigid elements and is completely separated from the surrounding wall, with an aim to reduce ceiling damage in the earthquake [20,21]. Similarly, Fiorino [22] presented the application of flexible braces (lightweight steel) integrated into the SCS to absorb the input energy.

On the other hand, research on the indirect-SCS is recently becoming active. Based on their response behavior in the earthquake, the indirect-SCS can be divided into two types [23]: a continuous panel ceiling and a lay-in panel type (Figure 1). In the case of the lay-in panel SCS, the seismic performance decreases rapidly, owing to the lack of lateral stiffness when one panel falls, which is due to the unfastening of the ceiling runner. Therefore, the continuous panel SCS has been certified to perform low fragility because the larger area panels are fixed on the bottom surface of runners and the entire ceiling moves together [24]. The superiority of the continuous ceiling panel was also verified based on the seven types of direct and indirect SCSs using an array of two shaking tables [9].

In August 2013, a new technical bulletin of the suspended ceilings system "Determining a Safe Structural Method for Specific Ceilings and Specific Ceiling Structural Strengths" Notice No. 771 was established by the Ministry of Land, Infrastructure, Transport and Tourism (MLIT). According to this report, when the SCS is installed with more than 200 square meters with the end clearance and the mass per unit area is over 2 kg including the attached equipment, the required number of steel braces should be placed. The effect of damage reduction of the SCS with braces following the technical bulletin has been verified through a full-scale shaking table experiment of a school gym with approximately 550 square meters conducted at the E-Defense earthquake simulator in Japan [25]. However, Lee et al. [12] investigated the arrangement and the location of braces on the entire ceiling system through a shaking table test and indicated the necessity for consideration that the relative displacement occurs between the installed brace part and may cause the panel to fall out. Despite this, the separation distance of the braces for the installation is not explained in detail in the technical bulletin. Moreover, the installation of many braces greatly disturbs the facility design above the ceiling panel and decreases the advantage to adopt it. In addition, a rigid-brace-based ceiling design amplifies the acceleration from the above hanging structure and causes great influence on the interactions between SCS and other nonstructural elements [26–28]. Liangjie et al. [29] investigated the mechanism of the dynamic behavior between ceiling panels and suspended equipment, and revealed that the restriction by friction force between them is very limited and the air conditioner unit would collide even under a Level 1 earthquake. Furthermore, during a full-scale shaking table test of a five-story steel building using the E-Defense shake table, additional concern arose that the use of the bracing system increases the damage of SCS when subjected to the strong vertical excitation over about 1.0 G [30]. The rigid elements constrain the ceiling movement, so the acceleration difference between the fixed ceiling grid and the unfixed part makes the ceiling unstable and drop off.

In this study, an innovative passive control system employing pulley mechanisms into the SCS (hereinafter referred to as the pulley–damper ceiling system: PDCS) is proposed and the feasibility is evaluated through a full-scale shaking table test and simulation analysis. The proposed ceiling vibration control system is aimed to mitigate both the acceleration response and displacement response of the SCS and eliminate the possibilities of a ceiling system collapse even during a Level 2 earthquake. The concept of the pulley–damper system is to amplify the wire movement stretched between the SCS and surrounding structure as well as the force of the viscous damper device connected to the wire. Three types of full-scale ceiling system specimens: conventional-SC, braced-SCS, and damped-SCS, were designed for the shaking table tests.

#### **2. Pulley–Damper Ceiling System**

#### *2.1. Mechanism of the System*

Figure 2 explains the fundamental concept of PDCS, which consists of pulley tackle mechanisms, one energy-dissipating device (damper), and the SCS. A wire is reciprocally stretched *n* times with bilateral symmetry between Pulley *PL*1, *PR*<sup>1</sup> (attached on the SCS) and Pulley *PL*2, *PR*<sup>2</sup> (attached on the primary structure). Thus, according to the response displacement "*D*" of the SCS, the amount of damper deformation should be enlarged to *n* × *D* [31,32]. That is, the pulley tackle mechanism is employed in the proposed system to amplify the wire-cable movement and transmit it to significantly increase the energy absorption of the damper installed at the middle of the wire movement path.

**Figure 2.** Concept schematic for the proposed pulley–damper ceiling system (PDCS).

#### *2.2. Derivation of the Constitutive Equation*

 ଵ ଶ ோଵ ோଶ Constitutive equations were established for the proposed pulley–damper system considering wire-cable deformation under the static condition. Hence, the friction force, thermal changes, and the wire mass are not included in the equation. Figure 3 illustrates a schematic diagram for the PDCS. A continuous wire-cable is reciprocated *n* times between the movable pulley, *PL*1, and the fixed pulley, *PL*2, and similarly between the movable pulley, *PR*1, and the fixed pulley, *PR*2. When the SCS deformed displacement *D* in the horizontal direction and force, *f<sup>L</sup>* and *f<sup>R</sup>* act on the wire-cable each side (use plus for tension); the force–deformation relationship of the wire-cable in the diagonal part and horizontal part can be expressed both in the left-hand side of Equation (1) and the right-hand side of Equation (2), as follows:

$$f\_L = \frac{K\_{L1}}{n} \delta\_{L1} \cdots \text{Diagonal part} \tag{1a}$$

$$f\_{\mathbf{L}} = \mathbf{K}\_{\mathbf{L}2} \delta\_{\mathbf{L}2} \cdots \text{Horizontal part} \tag{1b}$$

$$f\_R = \frac{K\_{R1}}{n} \delta\_{R1} \cdots \text{Diagonal part} \tag{2a}$$

$$f\_R = \stackrel{\cdots}{K\_{R2}} \delta\_{R2} \cdot f^{\text{Horizonal part}}\_{\text{part}} \tag{2b}$$

 = ଶଶ ⋯ Horizontal part where *L*<sup>1</sup> and *δ*<sup>1</sup> are the wire-cable length and deformation in the diagonal part, respectively; *L*<sup>2</sup> and *δ*<sup>2</sup> are the wire-cable length and deformation in the horizontal part, respectively; *E* is the Young's module of wire-cable; *A* is the cross-sectional area of wire-cable; *K* is the axial stiffness of the wire-cable (= *EA*/*L*).

ோ = ோଵ ோଵ ⋯ Diagonal part In this derivation, the wire-cable is modeled as equivalent truss-elements. Thus, the force–deformation relationship can be converted into the axial force, *N*, and the deformation, *δ*, as follows for each side:

$$N\_{\rm L} = nf\_{\rm L} \tag{3a}$$

$$\delta\_L = \frac{(\delta\_{R1} + \delta\_{R2} \oint x)}{n} = \frac{(a\_L f\_L + x)}{n} \tag{3b}$$

$$N\_{\mathbb{R}} = nf\_{\mathbb{R}} \tag{4a}$$

$$
\delta\_R = \frac{(\delta\_{R1} + \delta\_{R2} - \mathfrak{x})}{n} = \frac{(\mathfrak{a}\_R f\_R - \mathfrak{x})}{n} \tag{4b}
$$

 = ⁄ where *α* = *n*/*K*<sup>1</sup> + 1/*K*<sup>2</sup>

<sup>ଵ</sup> <sup>ଵ</sup> <sup>ଶ</sup> <sup>ଶ</sup>

=

ଵ 

ோଵ 

= ⁄

As the PDCS consists of one continuous wire, if one side gets shorter, the other side must grow longer. Thus, the following relation should hold: ோ = ோଶோଶ ⋯ Horizontal part <sup>ଵ</sup> <sup>ଵ</sup> <sup>ଶ</sup> <sup>ଶ</sup>

$$(\delta\_{\rm L1} + \delta\_{\rm L2} + \mathbf{x}) + (\delta\_{\rm R1} + \delta\_{\rm R2} - \mathbf{x}) = \mathfrak{a}\_{\rm L} f\_{\mathbf{L}} + \mathfrak{a}\_{\rm R} f\_{\mathbf{R}} = \mathbf{0} \tag{5}$$

ଵ ଶ

ଵ ⋯ Diagonal part

ோଵ ⋯ Diagonal part

ோ

ோଵ ோଶ

In addition, the damper force can be obtained from the difference in the tensile force of both wire sides. Therefore, by substituting Equation (5) into Equation (6), the damper force and the displacement can be established as Equation (7):

$$Q(\mathbf{x}, \mathbf{\hat{x}}) = f\_{\mathbf{L}} - f\_{\mathbf{R}} \tag{6}$$

$$Q(\mathbf{x}, \mathbf{\dot{x}}) = \Omega \left( f - f' \right) = \left( \frac{\mathfrak{a}\_L + \mathfrak{a}\_R}{\mathfrak{a}\_R} \right) f\_L = -\left( \frac{\mathfrak{a}\_L + \mathfrak{a}\_R}{\mathfrak{a}\_L} \right) f\_R \tag{7}$$

Utilizing Equations (3a), (4a), and (7), the axial force of the truss elements can be finally written, as follows:

$$N\_{\rm L} = n \left( \frac{\alpha\_{\rm R}}{\alpha\_{\rm L} + \alpha\_{\rm R}} \right) Q(\mathbf{x}, \hat{\mathbf{x}}),\\ N\_{\rm R} = n \left( \frac{\alpha\_{\rm L}}{\alpha\_{\rm L} + \alpha\_{\rm R}} \right) Q(\mathbf{x}, \hat{\mathbf{x}}) \tag{8}$$

The deformation of the damper is given by:

$$\mathbf{x} = -n\delta\_L + \delta\_{L1} + \delta\_{L2} = -n\delta\_1 + \frac{\mathfrak{a}\_L N\_L}{n} \tag{9}$$

Therefore, the simulation analysis can be implemented by the following procedures:


#### **3. Component Details of the Pulley–Damper System**

#### *3.1. Friction Test of the Pulley Sheaves*

A friction test was conducted to evaluate the friction coefficient of pulley sheaves. A stainless wire-cable comprises 7 × 19 strands with 2.75 kN braking force, and 2.0 mm in nominal diameter was selected. The tensile force of the wire-cable during excitations was monitored through strain gauges placed on either side of a copper plate after the calibration in Figure 4. A triple pulley was used in this study as a deviator of wire-cable to smoothly transmit the ceiling displacement to the damper (Figure 5a). The pulley consists of three sets of a sheave of 30 mm in internal diameter, with a ball bearing inside.

**Figure 4.** (**a**) Installation of the tension measuring plate at the end of the wire in the specimen; (**b**) details of the tension measuring plate.

**Figure 5.** (**a**) Overview of the pulley; (**b**) details of the fixture between pulley and ceiling runners.

 , The purpose of this test is to discover the friction coefficient per sheave to predict friction force in the following shake table test and simulate it in the following simulation analysis. In this study, the friction coefficient *µ* per sheave was estimated by dividing the mean friction force, *F<sup>f</sup>* , by the number of touched sheaves, *N*, and pre-tension force, *T*, as follows:

$$\mu = \frac{F\_f}{N \times T} \tag{10}$$

$$\mu = \frac{F\_f}{N \times T} \tag{11}$$

$$\mu = \frac{F\_f}{N \times T} \tag{12}$$

 = ൈ ൈ Table 1 summarizes the three test cases for the friction test. The parameter in this friction test is the pre-tension force of the wire-cable and the number of wire-cable loops. Figure 6 shows the specimen view for the friction test. A steel test frame was prepared, and a movable pulley is hung from the top. The generated friction force when the movable pulley went up and down was calculated from the difference of the tensile force between the two strain gauges.

**Table 1.** Information on the friction test.


Figure 7a–c shows the force–displacement loop. The horizontal axis represents the movable pulley displacement, and the vertical axis represents the friction force, which comprises the overall activated sheaves. In all cases, if the number of loops stayed constant, the maximum friction force got larger as the initial tension increased. The friction coefficient was calculated from the displacement range as ±5 mm, using Equation (10). As Figure 7d shows the relation of the friction coefficient and the pre-tension force on the wire-cable,

the coefficient stays constant and the results of the average friction coefficient in this test were 0.014.

**Figure 6.** Test setup for characterizing friction behavior under sinusoidal wave excitation.

**Figure 7.** Experimental results of the pulley-friction test: (**a**–**c**) Force–displacement relationship under different number of the wire-cable loop and (**d**) friction coefficient with the average value.

#### *3.2. Dynamic Test of the Viscous Damper* 0.0 ൏ 1.0

Figure 8 explains an assembly of the rotational non-linear viscous damper using silicon oil, which was specially developed to enhance the potential damping performance of the pulley–damper system. The component consists of two rotary dampers of 60 mNm-rated torque, two ball bearings, and other fixtures. Wire-cable should be wrapped around the spool several times to allow for a large stroke of the amplified wire-cable movement. As the velocity-force relationship can be expressed by Equation (11), and to quantify the velocity dependence, the sinusoidal excitation test and the frequency range between 0.1 Hz and 1.0 Hz were conducted to understand the variable of the equation: 0.0 ൏ 1.0

$$F = \mathbb{C}V^{\mathbb{N}} \tag{11}$$

= <sup>ఈ</sup>

where *F* is the damping force, *C* is the coefficient of the fluid inside the damper, *V* is the velocity of the damper, and *<sup>α</sup>* is the exponent (0.0 <sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> 1.0). ∙ α

= <sup>ఈ</sup>

**Figure 8.** (**a**) Overview of the damper attached to the supporting beam in the specimen; (**b**) details of the damper components.

Figure 9 presents the test specimen for the sinusoidal wave excitation of the proposed rotational viscous damper. The damper is connected to actuator by the wire-cable to convert the horizontal movement of actuator to the rational motion in damper. Besides, the counterweight was set at the other side of wire end to provide initial wire tension to avoid the cable being slack during the excitations.

**Figure 9.** Test setup for characterizing the dynamic behavior of viscous damper under sinusoidal wave excitation.

The hysteresis loop of the force–displacement relationship and force–velocity relationship under the sinusoidal frequency test of the non-linear viscous damper is presented in Figure 10. The damping force gradually grows up as the input frequency increased. While the energy absorption of the viscous damper is relatively limited at approximately 25 N, the velocity power coefficient of *α* = 0.9 and the damping coefficient *C* = 0.06 N·(s/mm)α was decided based on the SRSS (square root of the sum of squares) of all the test results and modeled in the following simulation analysis.

**Figure 10.** Hysteresis loop of viscous damper measured under sin wave excitation test at different frequencies: (**a**) Force–displacement loops; (**b**) force–velocity loops.

#### **4. Full-Scale Shake Table Test**

*4.1. Shake Table and Test Specimens*

The unidirectional dynamic tests were performed using the earthquake simulator at the structural engineering laboratory of the Hazama Ando Technical Research Institute to evaluate and qualify the ceiling vibration control system in January 2020 and 2021. The performance envelops of the simulator are a ±300 mm displacement and 3.0 g acceleration at a payload of 35 tf in the horizontal direction.

An overall 5800 mm (length) × 3500 mm (width) × 3200 mm (height) rigid steel test frame was properly designed and built to hang the SCS specimens test (Figure 11) At the ceiling level, 1500 mm from the top, steel beams were fixed to the columns on two sides in a Y-direction to provide support for the pulley installation.

The ceiling specimen assemblies consisted of ceiling panels, hanger bolts, carrying channels, and seismic clips. The ceiling system, 4020 mm × 3200 mm in size, was suspended 1500 mm by the hanger bolt (9 mm diameter) under the beams attached to the structural frame (Figure 12). In the specimens, the primary runners were arranged along the Y-direction with a spacing of 900 or 600 mm. The secondary runners were placed under primary runners along the X-direction with a spacing of 364 mm. The interaction points of the ceiling runner were tightly connected by seismic clips. Then, plasterboard (1820 mm × 910 mm) was fixed on the bottom surface of secondary runners. Here, in order to avoid the torsion of the entire system, V-braces were placed in the Y-direction for all the test cases.

#### *4.2. Test Cases for the Shaking Table Tests*

In this study, four types of indirect-SCSs with continuous panels were designed to understand the dynamic behavior of the proposed system: conventional SCS (Case C), braced SCS (Case CB), wired SCS (Case CW), and damped SCS (Case CWD), which places the proposed damper in Section 3.1. Moreover, to investigate the influence of

the wire stretching angle on seismic performance, two types of system arrangement, the horizontally stretched case (0-degree angle) and diagonally stretched case (60-degree angle), were prepared. The list of specimens is summarized in Table 2. The wire was symmetrically reciprocated 2.5 times between pulleys on the left and right sides, with 50 N of initial tensile force.

**Figure 11.** Overall view of the SCS specimen on the shake table.

**Figure 12.** Schematic view of overall specimen: (**a**) Plan view; (**b**) cross-section view in X-direction; (**c**) cross-section view in Y-direction.

#### *4.3. Measurement Plan and Input Excitations*

A total of 27 accelerometers, in a triaxial direction and with 9 locations (four of each on the specimen top and ceiling panel, and the shake table surface), and 4 laser displacement sensors (two of each in the X-direction and Y-direction) were used for monitoring the outputs in the tests with 500 Hz sampling frequency. During the excitation with the damper system, one draw-wire displacement sensor, and four tension measuring plates with strain gauges were installed for each pulley–damper system. The location and details of these instruments are shown in Figure 13.

**Table 2.** Information on the SCS specimen.

**Figure 13.** Measurement instrument location for the full-scale suspended ceiling system.

White noise with random phase tests (frequency range: 0.1 Hz to 49 Hz) were conducted for the initial tests to analyze the dynamic properties of the SCS and the test steel frame. Then, earthquake excitation tests using a simulated earthquake ground motion with random phases, based on the design spectrum specified in the Building Standards Law of Japan, were carried out. The amplitude was gradually scaled according to the condition of the specimens. Figures 14 and 15 show the acceleration time histories and spectrum of the input motions given to the shake table.

#### *4.4. Full-Scale Shake Table Test Results*

Figure 16 displays the amplitude of acceleration transfer functions from the shake table surface to the ceiling panel surface under random waves for a better understanding of the dynamic response of six specimens and to estimate the natural periods. The first natural periods of Case C and Case CB were 1.37 and 0.14 s, respectively. In other cases, for example, Case CW and Case CWD, a distinct peak point was not confirmed. That is, the implementation of PDCS into the SCS provides a great effect to suppress vibration and prevent resonance damage to the ceiling.

**Figure 14.** White noise wave used in shake table test: (**a**) Acceleration time histories; (**b**) Fourier amplitude spectrum.

**Figure 15.** Simulated earthquake wave used in shake table test: (**a**) Acceleration time histories; (**b**) spectral acceleration.

**Figure 16.** Amplitude of acceleration transfer function under white noise wave.

Figure 17a depicts the relationship between the maximum displacement response of the ceiling panel and the peak ground acceleration (PGA) at the shake table surface for simulated earthquake waves up to PGA 5 m/s/s. Table 3 summarizes the ratio of the maximum displacement response for each test case to Case C under major PGAs.

faster than Case C.

**Figure 17.** Displacement response of the SCS surface under simulated wave: (**a**) Maximum displacement for different PGAs; (**b**) displacement time histories under PGA 2 m/s/s.


**Table 3.** The response ratio of maximum displacement to the Case C under major PGAs.

As shown in Figure 17a and Table 3, Case CB, which was rigidly fixed to the floor above by the steel brace, exhibited the largest response reduction, of an average of 3% for the unreinforced Case C. Similarly, the specimens applying the proposed system, Case CWD0 and Case CWD60, effectively reduced the displacement response to 6% in Case CWD0 and 27% in Case CWD60, respectively, compared to Case C. Case CW0 mitigated the displacement response more than Case CW60, because the horizontal stiffness given by the stretched wire of Case CW0 is larger than that of Case CW60.

Figure 17b compares the displacement time history of the ceiling panel for each case under the simulated earthquake motion at PGA 2 m/s/s. As the displacement time history demonstrates, the displacements in Case CWD0 and CWD60 were much smaller than Case C, between 20 to 40 s, and they suppressed the shaking faster than Case C.

Figure 18a displays the relationship between the maximum acceleration response at the ceiling panel and the peak ground acceleration (PGA) at the shake table surface for simulated earthquake waves up to PGA 5 m/s/s. The ratio of the maximum acceleration response for each test case to Case CB under major PGAs is presented in Table 4.

As shown in Figure 18a and Table 4, the acceleration response of Case CB was amplified twice as compared with Case C; meanwhile, the Case CB provided great displacement reduction effect. Moreover, in the Case CB, the buckling of the hanger bolts after the excitation of the simulated earthquake motion at PGA 6 m/s/s was observed due to the large acceleration response. The accelerations in other cases were reduced to almost 20% of Case CB, and there were no significant differences at the maximum acceleration response between Case CWD0 and Case CWD60.

Figure 19 displays the amplification factor of PDCS under the simulated ground motion at different PGAs. This amplification factor is the ratio of the maximum damper displacement to the maximum ceiling panel displacement. The theoretical amplification factor for Case CW0/CWD0 and Case CW60/CWD60 are 5.0 and 2.5, respectively. While little fluctuation of the factor can be observed, especially in Case CWD0 when PGAs are small, the mean values of the experimental results are close to the theoretical values.

**Figure 18.** Acceleration response of the SCS surface under simulated wave: (**a**) Maximum acceleration for different PGAs; (**b**) acceleration time histories under PGA 2 m/s/s. ―



**Figure 19.** Amplification factor for different PGAs.

#### **5. Simulation Analysis**

*5.1. Simulation Model*

With the aim of observing the verification of the structural modeling using equivalent truss elements for the wire-cable, a simulation analysis was carried out and the modeling results were compared with the experimental results. The 3D-frame simulation model of the experimental test specimen was constructed by using the STructural Earthquake Response Analysis 3D (STERA 3D Version 10.8) software, which is a finite element-based program developed by one of the authors considering the material and geometric nonlinearities [33]. In the model, as illustrated in Figure 20, the steel beam element is presented by a line element and two nonlinear flexural springs at both ends. The steel column element is presented by a line element with nonlinear vertical springs in both end sections to consider nonlinear axial-moment interaction.

**Figure 20.** Moment-rotation relationship for bending spring. ሶ

Figure 21 shows the simulation model in the STERA 3D. The ceiling panel is suspended 1500 mm from the supporting beams, and the model was analyzed applying the fixedended connection to all the joints. In addition, recorded acceleration data at the shake table surface during the excitation were used as the input seismic motions.

ௗ

<sup>ଵ</sup>

ௗ

௬

Figure 22 shows the mechanical model of the bilinear and non-linear viscous damper. The friction force, *F<sup>f</sup>* , generated on pulley-wire contacts, was modeled as one bilinear model, and the yield strength, *Ff y*, was estimated by multiplying the number of pulleys and the friction coefficient, using Equation (10). The initial and second stiffnesses of the bilinear hysteresis are *K*<sup>0</sup> and *K*1, respectively. The hysteresis behavior of the fluid viscous damper is modeled based on the element test results in Figure 10. The damping force *F<sup>d</sup>* is .

quantified in proportion to an exponential *α* of the velocity *d* (Equation (11)).

The technical parameters for the one set of the pulley–damper system in the simulation model are summarized in Table 5. The damping factor in the SCS was predicted using a free-vibration range during shake table excitation, and 1.5% was selected for the inherent damping of the SCS by the logarithmic attenuation method.

#### *5.2. Simulation Results*

In Figure 23, the experimental results of maximum response values and time history data in displacement and acceleration at the SCS panel are compared with the analytical results for all the experimental cases under simulated earthquake waves. During the comparison of the maximum response value, while the experimental results are relatively larger than the analytical results, the experimental and numerical models' responses are in good agreement.

**Figure 22.** Restoring force characteristics of damper in the simulation model.

**Table 5.** Damper parameter in the analytical model.


ௗ α = \* the unit of the damping coefficient *C<sup>d</sup>* changes with velocity exponent α (i.e., when *α* = 1 is N·s/mm, and when *<sup>α</sup>* <sup>=</sup> 0.9 is N·(s/mm)0.9).

1 ∙ = 0.9 ∙ It is considered that the repeated earthquake excitation to the specimen decreased the pre-tension force of the stay cable, and the joint rigidness caused the differences between the experimental and analytical results. The shake table test of Case CW60 was carried out after the Case CW0 in 2020. Therefore, due to the cumulative of the previous test case damage, the error in Case CW60 was made to become larger than Case CW0.

The coefficient of determination, *R 2* , is selected to quantitatively measure the accuracy of the simulation model in this study, and it is defined as Equation (12). In addition, Table 6 shows the *R 2* for the validation of each simulation model. The *R <sup>2</sup>* value for all cases of the maximum acceleration and displacement response are 0.83 and 0.91, respectively. In particular, the minimum *R <sup>2</sup>* value was 0.69 in the Case CW60. Besides, the simulation model accurately reproduced the time histories responses of the accelerations and displacements, including the frequency and the location of the wave crest.

$$R^2 = 1 - \frac{\sum\_{i=1}^{n} (Exp\_i - Ann\_i)^2}{\sum\_{i=1}^{n} \left(Exp\_i - \overline{Exp} \right)^2} \tag{12}$$

where *Exp<sup>i</sup>* is the output by the shake table test, *Ana<sup>i</sup>* is the output by the simulation analysis, and *Exp* is the mean of *Exp<sup>i</sup>* .

The results of the force–velocity relationship of the damper device in the shake table tests and simulation analyses are contrasted in Figure 24. The gaps in the hysteresis loop shape were confirmed; however, the error between the experimental and analytical results of the time histories are reproduced well. Thereby, the effect of displacement amplification was successfully simulated in the analytical model.

**Figure 23.** Results comparison of the SCS response between shake table test and simulation analysis. **Figure 23.** Results comparison of the SCS response between shake table test and simulation analysis. **Table 6.** Coefficient of correlation for the validation of simulation model.


**Figure 24.** Results comparison of the damper response between shake table test and simulation anal-**Figure 24.** Results comparison of the damper response between shake table test and simulation analysis.

#### **6. Conclusions**

This paper proposed an innovative passive vibration control structure for an indirect suspended ceiling system using pulley tackle mechanisms, named the pulley–damper ceiling system (PDCS), and provided detailed information on the system configuration. A preliminary experimental investigation of the dynamic characteristics of the proposed system was conducted. A main goal was to understand the behavior of the entire system under seismic motion and evaluate the feasibility. The results obtained in this study are summarized below:


**Author Contributions:** Conceptualization, T.S.; methodology, S.S.; software, T.S.; validation, R.M. and S.S.; formal analysis, R.M.; investigation, R.M.; data curation, R.M.; writing—original draft preparation, R.M.; writing—review and editing, T.S.; visualization, R.M.; supervision, T.S.; project administration, S.S.; funding acquisition, T.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The authors gratefully thank the practice committee of the development of the ceiling vibration control system: M. Uchikawa (Sato Kogyo Co., Ltd.), E. Nishimura (Toda Corp.), Y. Yamasaki (Nishimatsu Construction Co., Ltd.), H. Ryujin (Maeda Co., Ltd.), and R. Doi (Kumagai Gumi Co., Ltd.).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Mostafa Farajian 1,2 , Mohammad Iman Khodakarami 1, \* and Pejman Sharafi 2**


**\*** Correspondence: khodakarami@semnan.ac.ir

**Abstract:** Cylindrical liquid storage tanks are vital lifeline structures, playing a critical role in industry and human life. Damages to these structures during previous earthquakes indicate their vulnerability against seismic events. A novel strategy to reduce the seismic demands in the structures is the use of metamaterials, being periodically placed in the foundation, called MetaFoundation (MF). The periodic configuration of metamaterials can create a stop band, leading to a decrease in wave propagation in the foundation. The aim of this paper is to study the effect of MF on the dynamic behaviour of liquid storage tanks. To that end, the governing equations of motion of the liquid storage tank equipped with MF are derived and solved in the time domain to obtain the time history of the responses under a set of ground motions. Then, the peak responses of tanks, mounted on MF, are compared with the corresponding responses in the fixed base condition. Besides, a parametric study is performed to assess the effect of the predominant frequency of earthquakes, the number of layers of metamaterials, the thickness of soft material, and the damping ratios of soft material on the performance of the MF. The obtained results indicate that the MF improves the dynamic behaviour of the squat tank, in which the mean ratio of responses using MF to the ones in the fixed base conditions equals 0.551 for impulsive displacement, overturning moment, and base shear.

**Citation:** Farajian, M.; Khodakarami, M.I.; Sharafi, P. Effect of MetaFoundation on the Seismic Responses of Liquid Storage Tanks. *Appl. Sci.* **2022**, *12*, 2514. https:// doi.org/10.3390/app12052514

Academic Editor: Felix Weber

Received: 28 January 2022 Accepted: 22 February 2022 Published: 28 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Keywords:** liquid storage tank; MetaFoundation; time domain analysis; earthquake; passive control; stop band

#### **1. Introduction**

Cylindrical liquid storage tanks made of steel or concrete materials are strategic structures, having been employed to store water for drinking or firefighting, oil and chemical products in urban areas, and industrial plants. Damages to these structures may have catastrophic consequences such as economic losses, fire due to flammable materials, environmental pollution, and disruption of human lives. The reported failure modes of these structures during past earthquakes indicate their inappropriate seismic performance [1,2] and the need for a safeguard against seismic events. The seismic behaviour of liquid storage tanks has been studied extensively (e.g., [3–6]). Generally, two approaches can be followed to improve the performance of liquid containers against seismic events. The first approach is the strengthening of tanks by increasing their wall thickness. However, increasing the thickness of the wall will increase the input seismic energy. The second approach, aiming to decrease and dissipate the input seismic energy, is exploiting the advantages of control devices, such as base isolation systems, dampers, or other innovative control devices [7].

For more than four decades, a wide variety of studies have been conducted to develop various base isolation devices and to study their effects on the improvement of the seismic performance of steel tanks [8–14]. However, several factors influence the seismic responses of structures [15,16], and in the case of base-isolated liquid storage tanks, they may have adverse effects leading to an increase in seismic demands compared to the fixed base condition [17–19]. For example, Bagheri and Farajian [17] showed that the base isolation may experience an excessive displacement when the system is subjected to near-fault excitations with severe pulses, resulting in a dramatic increase in the displacement of the impulsive mass and, consequently, base shear and overturning moment. The same results were observed through experimental tests conducted on the seismic response of scaled structures isolated by highly efficient, low-cost PVC-Rollers Sandwich seismic isolation [20,21]. Besides, the traditional base isolation devices are less capable of improving the seismic performance of structures built on soft soil and cannot isolate structures from the vertical component of ground motions. Therefore, it would be beneficial to mitigate the seismic demands through other control systems without the shortages mentioned above.

In recent years, the progress in the area of solid-state physics showed that the structures being placed periodically display distinctive properties called frequency stop band, which can be explained as a wave is blocked to propagate through a continuum model if its frequency falls in the stop band frequency and can propagate to the model for the frequencies other than the frequencies of stop band [22–24]. This fascinating feature attracted considerable attention, stimulating researchers to construct materials with periodic structures to block the propagation of waves. Locally resonant metamaterials (LRMs) and phononic crystals (PCs) have been known as two types of periodic structures which are employed to avoid the propagation of waves [25]. Compared to PCs, LRMs have a more suitable performance to diminish vibrations with low frequency because the local resonances have the ability to create the stop band. LRMs, made from heavy materials (metamaterials) coated with a soft layer in a stiff matrix [26] (usually concrete), have been employed as foundations and barriers to decrease the propagation of seismic waves into structures and to protect them against the harmful consequences of earthquakes. The benefits of such a foundation would be magnified in the case of different structures, such as liquid storage tanks as well as modular structures, in which their performance under seismic actions is still a big question [27,28]. Jia and Shi [29] investigated the effect of physical as well as geometrical properties of periodic foundation on the stop band. They concluded that a lower band gap can be achieved by considering a higher mass density for the core. Bao et al. [24] compared the seismic behaviour of a seven-story building mounted on a periodic foundation with the traditional foundation and base isolation system. They observed that the periodic foundation has a better performance compared to base isolation systems in attenuating the seismic responses when the predominant frequency of the wave incident lies in the stop band. Mitchell et al. [30] proposed metaconcrete in which spherical metal cores coated with soft material are used instead of standard concrete. The coated metal core behaved as a resonator, activating when a dynamic blast load is applied with the frequency at or near the resonator frequency. Hence, the overall system exhibits negative effective mass, leading to the reduction of the amplitude of the applied blast wave. Dertimanis et al. [31] used mass-in-mass barriers being placed periodically to investigate the effect of locally resonant metamaterials. They connected internal mass to the outer mass by tendons and found that the input energy is filtered if the frequency falls in the stop band. Maleki and Khodakarami [32] conducted numerical analyses to evaluate the effect of MetaSoil on the amplification of in-plane waves due to topography irregularities. More recently, Basone et al. [33] employed the concept of LRMs to alleviate the demand in liquid tanks due to seismic actions. The foundation was made of steel columns, and the concrete-type resonators were connected to these columns. They conducted an optimization procedure to obtain the optimized damping and frequency corresponding to the resonators. The obtained results demonstrated that the proposed system could decrease the base shear in a slender tank up to 30%. Aguzzi et al. [34] investigated the propagation of flexural waves in a thin reticulated plate augmented with two classes of metastructures for wave mitigation. Despite all the research on metamaterials, no study has been conducted on the performance of MetaFoundation, composed of concrete matrix and steel core, in liquid storage tanks. Moreover, it is vital to investigate the effect of various parameters influencing the performance of MF on the seismic behaviour of the superstructure. Particularly, due to

the fact that the MF works in a range of frequencies, it is essential to investigate the effect of earthquake frequency on the seismic behaviour of MF.

This paper aims to investigate the efficiency of a foundation made by concrete and locally resonant materials coated with a soft layer called MetaFoundation (MF) on the dynamic behaviour of cylindrical tanks made from steel materials. To that end, the theory and background of the MF based on Bloch's theorem are studied, and a simplified mass– spring model is suggested for the dynamic analysis of the coupled MF–tank system. Then, the governing equations of motion of the system are derived and solved in the time domain. For the numerical study, two types of tanks, namely squat and slender, are subjected to a set of ground motions with far-field characteristics to compare the seismic responses of the tanks mounted on MF with corresponding responses in the fixed base condition. Besides, a parametric study is performed to investigate the influence of different parameters on seismic responses of considered tanks with MF. These parameters are the predominant frequency of ground motions, the number of layers of metamaterials, the thickness of the soft material, and the damping of the soft material.

#### **2. Simplified Model of MF**

The 3D view of an MF is illustrated in Figure 1. Compared to the traditional foundation, which is made of concrete and longitudinal and transversal rebars, the MF contains heavy cores with cubic shapes that are coated with soft material in addition to the concrete and rebars. It is assumed that the MF is subjected to dynamic excitations in *X* and *Y* directions. Therefore, it can be divided into *N* unit cells in *X*, *Y,* and *N*-layers in *Z* directions.

**Figure 1.** Three-dimensional view of the proposed MetaFoundation.

Assuming that the model is continuous, isotropic, made by materials with perfectly plastic behaviour, small deformation and insignificant damping, the governing equation of an inhomogeneous solid is written as [35]:

$$\rho(\mathbf{r})\frac{\partial^2 \mathbf{u}}{\partial t^2} = \nabla \left\{ [\lambda(\mathbf{r}) + 2\mu(\mathbf{r})](\nabla.\mathbf{u}) \right\} - \nabla \times \left[ \mu(\mathbf{r})\nabla \times \mathbf{u} \right] \tag{1}$$

where *ρ* is the mass density, **u** = (*ux*, *uy*, *uz*) is the displacement vector, *t* is the time parameter, **r** = (*x*, *y*, *z*) the coordinate vector, *λ* and *µ* are Lame's constants and ∇ is the Laplace operator. Decoupling the out-of-plane modes from those of in-plane ones, Equation (1) is rewritten as [35]:

$$\rho(\mathbf{r})\frac{\partial^2 u\_j}{\partial t^2} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \lambda(\mathbf{r}) \frac{\partial u\_l}{\partial \mathbf{x}\_j} \right] + \frac{\partial}{\partial \mathbf{x}\_l} \left[ \mu(r) \left( \frac{\partial u\_l}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_l} \right) \right] \tag{2}$$

where *u<sup>j</sup>* = *u<sup>j</sup>* (*x*, *y*) is the in-plane displacement vector, *j*, *l* = 1, 2 and **r** = (*x*, *y*). Using the Floquet–Bloch theorem for the periodic configuration of unit cells in uniaxial direction, the solution of Equation (2) is given as:

$$u(\mathbf{x},t) = e^{i(q\mathbf{x}-\omega t)}u \tag{3}$$

where *q* is the wave vector in reciprocal space, *ω* is the circular frequency. The wave amplitude is shown by *u,* which is a periodic function and is expressed as:

$$
\mu(\mathbf{x}) = \mu(\mathbf{x} + A) \tag{4}
$$

where *A* is the unit cell size (shown in Figure 1). The stop band can be calculated through the dispersion analysis of the periodic structure. Two different approaches can be employed to conduct the dispersion analysis and obtain the stop band. The first approach is the 3D finite element modelling of the MF, which requires potential workload and high computational effort due to the complexity in modelling. The second approach is the use of the simple model of MF, which is represented by a set of masses and springs. Since the unit cells of each layer operate in parallel under seismic excitation, they can be represented by an equivalent unit cell through dynamic condensation. Figure 2 shows the simplified mass–spring model of an MF with *N*-layers in the *Z* direction.

**Figure 2.** Simplified mass–spring model of an MF.

The total mass of the concrete in each layer is shown by the external mass (*m*1), and the total mass of the metamaterials in each layer is shown by the internal mass (*m*2). The stiffness corresponding to each layer of foundation and soft material is represented by springs whose stiffnesses are shown by *k*<sup>1</sup> and *k*2, respectively. The stiffness corresponding to the soft material for each unit cell is obtained by:

$$k\_2 = \frac{E\_{SM}A\_{SM}}{t\_{SM} - (|\boldsymbol{\mu}\_2 - \boldsymbol{\mu}\_1|)}\tag{5}$$

where *ESM* and *tSM* are the modulus of elasticity and thickness of the soft material, respectively. The area of the soft material perpendicular to the applied wave direction is represented by *Asm*; *u*<sup>1</sup> and *u*<sup>2</sup> are displacements of external and internal masses. Note that, in this study, the stiffness corresponding to the soft material will be updated during the analysis based on the relative displacement of the internal and external masses. Therefore, more accurate seismic responses will be obtained. The damping associated with the external and internal masses are shown by *c*<sup>1</sup> and *c*<sup>2</sup> and expressed by

$$\mathbf{c}\_1 = 2\zeta\_{con}\sqrt{k\_1(m\_1+m\_2)}\tag{6}$$

$$c\_2 = 2\zeta\_{SM}\sqrt{k\_2 m\_2} \tag{7}$$

where *ζcon* and *ζSM* are the damping ratios of concrete and soft material, respectively. Due to the periodicity, it is possible to reduce the dispersion analysis of the infinite structure to the dispersion analysis of a single unit cell with the same boundary condition. The response of boundary conditions is calculated by substituting Equation (4) into Equation (3) as follows:

$$u(\mathbf{x} + A, t) = e^{i q A} u(\mathbf{x}, t) \tag{8}$$

The stop band is estimated by conducting the Eigen-frequency analysis of an undamped unit cell for a considered Bloch wave vector (*q*). The governing equations of motion of *j*-th unit cell are written as:

$$m\_1^j \ddot{u}\_1^j + k\_1(2u\_1^j - u\_1^{j-1} - u\_1^{j+1}) + k\_2(u\_1^j - u\_2^j) = 0\tag{9}$$

$$k m\_1^j \ddot{u}\_1^j + k\_2 (u\_2^j - u\_1^j) = 0 \tag{10}$$

Equations (9) and (10) are rewritten by substituting Equation (8) into Equation (9) and under the consideration of the trigonometric relationship of *e iqA* = cos (*qA*) + *i* sin (*qA*), as follows:

$$2m\_1^j \ddot{u}\_1^j + (2k\_1 - 2k\_1 \cos(qA) + k\_2)u\_1^j - k\_2 u\_2^j = 0 \tag{11}$$

$$k\nu\_2^j \ddot{u}\_2^j - k\_2 u\_1^j + k\_2 u\_1^j = 0 \tag{12}$$

Equations (11) and (12) are formulated by Equation (13) to obtain the dispersion relation of the considered MetaFoundation.

$$[\mathbf{K}(q) - \omega^2 \mathbf{M}]\mathbf{u} = \mathbf{0} \tag{13}$$

where *K*(*q*) and *M* are the stiffness and mass matrices of the unit cell, respectively. Equation (13) shows that the stiffness of the unit cell is dependent on the Bloch wave vector (*q*). The nontrivial solution of the eigenvalue problem is

$$m\_1 m\_2 \omega^4 - [(m\_1 + m\_2)k\_2 + 2m\_2k\_1(1 - \cos(qA))]\omega^2 + 2k\_1k\_2(1 - \cos(qA)) = 0\tag{14}$$

Equation (14) has two responses corresponding to two dispersion curves, known as acoustic and optical. The lower response is related to the acoustic branch, and the higher response is associated with the optical branch. The stop band gap falls between these two dispersion curves.

#### **3. Structural Model of the Liquid Storage Tank**

The three-dimensional finite element modelling and analysis of a cylindrical liquid tank is a complicated process and requires computational effort mainly due to the interaction of tank and fluid. Besides, the MF, which is made of different materials, increases its modelling and analysing complexity. Therefore, the use of a simple and accurate model with the capability to determine the response of a liquid storage tank under seismic ground motions is of interest. As an alternative approach of finite element modelling, the simplified mass–spring model, which is accepted by standard codes such as API 650 [36], ASCE 7- 16 [37], and Eurocode 8 [38] can be employed to model and analyse of tank containers. This simplified mass–spring model is generally based on the work conducted by Housner [39]. He evaluated the effect of hydrodynamic actions in liquid storage tanks, assuming that the tank wall has a rigid behaviour. According to Housner's model, the hydrodynamic response of a fluid tank system is determined by the superposition of two components, namely convective and impulsive components. The convective component is generated by convective mass (*mc*), and the impulsive component is produced by impulsive mass (*m<sup>i</sup>* ). The convective mass refers to the portion of the liquid in the upper part of the tank near the free surface, which experiences a long period of sloshing motion during dynamic loading. Conversely, the portion of the filling liquid near the base of the liquid storage tank accelerating in unison with the tank wall is represented by impulsive mass. Haroun and Housner [40] modified the simplified mass–spring model of Housner to consider the tank wall flexibility. Malhotra et al. [41] offered a simple and accurate procedure to assess the seismic responses of liquid storage tanks. They combined the first impulsive modal mass with the higher impulsive modal mass and the first convective modal mass with the higher convective modal mass. As a result of such a combination, the tank liquid system was represented by two modes only. Figure 3 demonstrates a cylindrical liquid tank mounted

on an *N*-layer MF. The geometrical properties of the tank are the height of the liquid (*H*), the thickness of the wall of the tank (*tw*) and the radius of the tank (*R*).

**Figure 3.** A liquid storage tank mounted on an MF.

In this paper, the mass–spring model suggested by Malhotra et al. is used for the dynamic time history analysis of the liquid storage tank (Figure 4).

**Figure 4.** Simplified model of the liquid storage tank.

The impulsive and convective masses (*m<sup>i</sup>* and *mc*) are connected to the wall of the tank by linear springs whose stiffnesses are *k<sup>i</sup>* and *kc*, respectively. The stiffness of springs is influenced by the properties of the filling fluid, the tank material, and the geometric properties of the tank. The damping coefficients corresponding to the impulsive and convective masses are shown by *c<sup>i</sup>* and *cc*, respectively.

The damping coefficient and stiffness corresponding to impulsive and convective masses are expressed as follows:

$$c\_i = 2\zeta\_i m\_i \times \frac{2\pi}{T\_i} \tag{15}$$

$$\mathfrak{c}\_{\mathfrak{c}} = 2\mathfrak{zeta}\_{\mathfrak{c}}\mathfrak{m}\_{\mathfrak{c}} \times \frac{2\pi}{T\_{\mathfrak{c}}} \tag{16}$$

$$k\_{\dot{l}} = m\_{\dot{l}} \times \frac{4\pi^2}{T\_{\dot{l}}^2} \tag{17}$$

$$k\_c = m\_c \times \frac{4\pi^2}{T\_c^{\circ}}\tag{18}$$

where *ζ<sup>i</sup>* and *ζ<sup>c</sup>* are the damping ratios of the impulsive and convective masses. *T<sup>i</sup>* and *T<sup>c</sup>* are natural periods corresponding to impulsive and convective responses, given by [41]:

$$T\_i = \mathcal{C}\_i \frac{H\sqrt{R}}{\sqrt{t\_w/R} \times \sqrt{E\_s}} \tag{19}$$

$$T\_{\mathfrak{C}} = \mathbb{C}\_{\mathfrak{C}} \sqrt{\mathbb{R}} \tag{20}$$

where *E<sup>s</sup>* is the modulus of elasticity of the tank material and *ρ<sup>w</sup>* is the mass density of filling fluid, respectively. The ratio of impulsive and convective masses to the total mass (*mi*/*m*) and (*mc*/*m*), relative heights of impulsive and convective masses (*h<sup>i</sup> /H*) and (*hc/H*), and the coefficients *C<sup>i</sup>* and *C<sup>c</sup>* are suggested by Malhotra et al. [41]. The filling fluid mass (*m*) equals to *πR* <sup>2</sup>*Hρw*.

#### **4. Governing Equations of Motion MF-Tank System**

The simplified mass–spring model of an *N*-layer MF-tank system is depicted in Figure 5.

**Figure 5.** Simplified mass–spring model of the liquid storage tank mounted on an N-layer MF.

The whole system comprises 2*N* degrees of freedom representing the MetaFoundation and two degrees of freedom demonstrating the convective and impulsive masses, respectively.

The governing equations of motion of the MF tank system can be written in matrix form:

$$\mathbf{M}\_T \ddot{\boldsymbol{\mu}} + \mathbf{C}\_T \dot{\boldsymbol{\mu}} + \mathbf{K}\_T \boldsymbol{\mu} = -\mathbf{M}\_T \mathbf{r} \ddot{\boldsymbol{\mu}}\_{\mathcal{S}} \tag{21}$$

where *MT*, *KT*, and *C<sup>T</sup>* are the mass, stiffness, and damping matrices of the coupled tank system mounted on the MetaFoundation, respectively expressed by

$$\mathbf{M}\_{T} = \begin{bmatrix} \mathbf{M}\_{\rm MF} & \mathbf{0} \\ \mathbf{0} & \mathbf{M}\_{\rm Tank} \end{bmatrix} \tag{22}$$

$$\mathbf{C}\_{T} = \begin{bmatrix} \mathbf{C}\_{MF} + \mathbf{C}\_{Tank} & -\mathbf{C}\_{Tank} \\ -\mathbf{C}\_{Tank} & \mathbf{C}\_{Tank} \end{bmatrix} \tag{23}$$

$$\mathbf{K}\_T = \begin{bmatrix} \mathbf{K}\_{MF} + \mathbf{K}\_{Tank} & -\mathbf{K}\_{Tank} \\ -\mathbf{K}\_{Tank} & \mathbf{K}\_{Tank} \end{bmatrix} \tag{24}$$

*MMF* = diagonal (*m*1, *m*<sup>2</sup> . . . *m*1, *m*2)2*N*×2*N*; *MTank* = diagonal (*m<sup>i</sup>* , *mc*)2×2; (25)

$$\mathbf{C}\_{\rm MF} = \begin{bmatrix} 2c\_1 + c\_2 & -c\_2 & -c\_1 & \cdots & 0 \\ -c\_2 & c\_2 & 0 & \cdots & 0 \\ & -c\_1 & 0 & \ddots & \cdots & \vdots \\ & \vdots & \vdots & \vdots & c\_1 + c\_2 & -c\_2 \\ & 0 & 0 & \cdots & -c\_2 & c\_2 \end{bmatrix} \tag{26}$$

$$\mathbf{C}\_{\text{Tank}} = \text{diagonal } (c\_{i\prime}c\_{c})\_{2 \times 2} \tag{27}$$

$$\mathbf{K}\_{MF} = \begin{bmatrix} 2k\_1 + k\_2 & -k\_2 & -k\_1 & \cdots & 0 \\ -k\_2 & k\_2 & 0 & \cdots & 0 \\ & -k\_1 & 0 & \ddots & \cdots & \vdots \\ & \vdots & \vdots & \vdots & k\_1 + k\_2 & -k\_2 \\ & & & & \ddots & \vdots \end{bmatrix} \tag{28}$$

0 0 · · · −*k*<sup>2</sup> *k*<sup>2</sup>

$$\mathbf{K}\_{\text{Tank}} = \text{diagonal} \ (k\_{i\prime} k\_c)\_{2 \times 2} \tag{29}$$

$$\boldsymbol{\mu} = \begin{bmatrix} \boldsymbol{\mu}\_1 \ \boldsymbol{\mu}\_2 \ \dots \ \boldsymbol{\mu}\_{1\prime} \ \boldsymbol{\mu}\_{2\prime} \ \boldsymbol{\mu}\_{i\prime} \ \boldsymbol{\mu}\_c \end{bmatrix}^T \tag{30}$$

The superscript *MF* and *Tank* stand for MetaFoundation and tank system, respectively; *u* is the displacement of the system relative to the ground; *x*<sup>2</sup> = (*u*<sup>2</sup> − *u*1), *x<sup>i</sup>* = (*u<sup>i</sup>* − *u*1), *x<sup>c</sup>* = (*u<sup>c</sup>* − *u*1) are displacement of internal, impulsive, and convective masses relative to the external mass, respectively. In Equation (21), the earthquake acceleration is shown by *ü*g, and *r* is a column vector of one.

To obtain the time history of responses of the liquid storage tanks with MF, the governing equations of motion are derived and transferred to the first-order differential equations. Then, these equations are solved in each time step using the state-space representation and ODE15s in MATLAB programming language. The response quantities of interest are vertical displacement of the surface of the fluid (*dv*), impulsive displacement (*x<sup>i</sup>* ), overturning moment (*M*) at the top of the MF, and structural base shear of the tank (*Fs*). While the impulsive displacement can be obtained directly through the solving of equations of motion, the vertical displacement of the free surface, the overturning moment at the top of the foundation, and structural base shear are calculated according to Equations (31)–(33), respectively [8]. The overturning moment determines the generated hydrodynamic forces in the tank wall, which is proportional to the axial compressive force. The axial compressive force is the main reason of buckling of the tank walls, either in the form of elephant foot buckling or diamond shape buckling. Conversely, the vertical displacement of the free surface caused by the convective component controls the required freeboard. In the case of lack of sufficient freeboard, the tank may experience a leak of fluid, tear of the shell or break of the shell–roof connection.

$$d\_{\upsilon} = 0.837 \text{R} \frac{\omega\_c^2 \mathbf{x}\_c}{\mathcal{g}} \tag{31}$$

$$M = -\left\{ m\_c h\_c (\ddot{\boldsymbol{u}}\_c + \ddot{\boldsymbol{u}}\_\mathcal{g}) + m\_i h\_i (\ddot{\boldsymbol{u}}\_i + \ddot{\boldsymbol{u}}\_\mathcal{g}) \right\} \tag{32}$$

$$F\_{\mathbb{S}} = -\left\{ m\_{\mathbb{c}} (\ddot{\boldsymbol{u}}\_{\mathbb{C}} + \ddot{\boldsymbol{u}}\_{\mathbb{S}}) + m\_{\mathbb{I}} (\ddot{\boldsymbol{u}}\_{\mathbb{I}} + \ddot{\boldsymbol{u}}\_{\mathbb{S}}) \right\} \tag{33}$$

where *ω<sup>c</sup>* is the frequency of the convective mass (*ω<sup>c</sup>* = 2π/*Tc*). For simplicity and better comparison of the results, the performance index of vertical displacement of the surface of the liquid (PI *dv*), the displacement of impulsive mass (PI *x<sup>i</sup>* ), overturning moment on the foundation (PI *M*), and structural base shear (PI *Fs*) are defined as the ratio of seismic responses of the liquid storage tank mounted on the MF (*d<sup>v</sup> MF* , *x<sup>i</sup> MF* , *MMF* and *F<sup>s</sup> MF*) to the corresponding responses of the tank in fixed base condition (*d<sup>v</sup> F* , *x<sup>i</sup> F* , *M<sup>F</sup>* and *F<sup>s</sup> F* ), according to Equations (34)–(37). Therefore, a PI of less than one indicates that the MF is an efficient solution to improve the dynamic behaviour of liquid storage tanks. However, the performance index of more than one shows that the response is amplified due to the implementation of MF. Therefore, the MF has an adverse effect.

$$\text{PI} \, d\_{\upsilon} = \frac{d\_{\upsilon}^{\text{MF}}}{d\_{\upsilon}^{F}} \tag{34}$$

$$\text{PI } \mathfrak{x}\_{i} = \frac{\mathfrak{x}\_{i}^{\text{MF}}}{\mathfrak{x}\_{i}^{\text{F}}} \tag{35}$$

$$\text{PI } M = \frac{M^{MF}}{M^F} \tag{36}$$

$$\text{PI } F\_s = \frac{F\_s^{MF}}{F\_s^F} \tag{37}$$

In addition to the time history of responses, it is interesting to evaluate the efficiency of the proposed MetaFoundation in the frequency domain through the transmission ratio (*TR*) of displacement above the MF against different frequencies (*f*). The transmission ratio is expressed as

$$TR(f) = 20\log\_{10}(\frac{u\_t(f)}{u\_0(f)})\tag{38}$$

In Equation (38), the amplitude of the input displacement at the bottom of the MF is shown by *u*<sup>0</sup> (*f*), and the amplitude of the displacement measured at the top of the MF is shown by *u<sup>t</sup>* (*f*), respectively. In detail, in order to obtain the displacement transmission ratio, a harmonic displacement at various frequencies is imposed at the bottom of the foundation with amplitude *u*0, and the displacement response *u<sup>t</sup>* on the top of the foundation is observed.

#### **5. Numerical Study**

For practical applications, the periodic MetaFoundation must be constructed by available materials. In this paper, it is assumed that the foundation is made of concrete, and the heavy cores are made of steel coated by rubber. The material properties of different parts of unit cells, including modulus of elasticity and density, are listed in Table 1.

**Table 1.** Material properties of a unit cell.


For the preliminary time history analysis, it is assumed that the MetaFoundation comprises unit cells with dimensions of 0.305 × 0.305 × 2.0 m corresponding to the length, width, and height, respectively. In addition, the cores of unit cells have a dimension of 0.10 m. The thickness of the rubber is assumed to be 0.05 m. The damping ratios corresponding to concrete and rubber are assumed to be 5% and 30%, respectively. A dispersion analysis has been conducted to calculate the dispersion relation and stop band corresponding to an infinite unit cell. The optical and acoustic branches obtained from the dispersion analysis are illustrated in Figure 6.

**Figure 6.** Dispersion analysis of considered unit cell.

The results show that the stop band forms in the frequency range of 18.733 to 18.984 Hz, where the elastic waves cannot propagate through the MF according to the Floquet–Bloch theory. However, the obtained stop band is valid for an infinite number of unit cells. For the case of the foundation comprising finite unit cells, additional modal analysis is required to calculate the stop band.

Two different types of cylindrical tanks have been considered as case studies from [8]. The considered tanks have different aspect ratios (height to radius). The ratio of height to the radius of the assumed tanks (*S* = *H*/*R*) is 0.6 and 1.85, corresponding to squat and slender tanks, respectively. The filling fluid is assumed to be water, with a mass density of *ρ<sup>w</sup>* = 1000 kg/m<sup>3</sup> . Table 2 shows the geometrical properties of considered tanks. A foundation that has a square shape is considered for both squat and slender tanks. The dimension of the MF for the squat tank is assumed to be 50.0 × 50.0 m, corresponding to its width and length, respectively; for the slender tank, a MetaFoundation with 15.0 m length and width is considered. As mentioned above, the height of each layer of MF is assumed to be 2.0 m.


**Table 2.** Geometrical properties of the selected tanks used as case studies.

The natural period, relative masses, and relative heights of the simplified mass–spring model depend on the geometrical properties of the cylindrical tanks. The parameters of the equivalent mechanical model of the tank have been calculated and tabulated in Table 3. The damping ratios of the convective (*ζc*) and impulsive masses (*ζ<sup>i</sup>* ) are assumed to be 0.5% and 2%, respectively, as suggested by [8].

**Table 3.** Resultant parameters of the equivalent mechanical model for the selected squat and slender tanks.


An adequate number of earthquakes should be considered for the required time history analysis to assess the efficacy of the proposed MF on the seismic responses of liquid storage tanks. Federal Emergency Management Agency (FEMA) [42] suggested three sets of ground motions for quantifying seismic performance factors of buildings. The three sets

include far-field (FF), near-fault without pulse (NF-WO Pulse), and near-fault with pulse (NF-W Pulse) ground motions. In this paper, the FF set, which comprises twenty-two pairs of records with an average moment magnitude of *M<sup>w</sup>* = 7.0, has been used. Each record has two horizontal components; therefore, the system is subjected to forty-four ground motions. All of the considered earthquakes with a distance from the fault rupture of more than 10 km were recorded on site classes C or D based on the NEHRP classification, and they do not reveal any pulse in their velocity time history. Besides, the selected excitations cover different intensities in terms of their PGA. The ground motions were downloaded from the strong ground motion database of the PEER NGA-West2 (pacific earthquake engineering research) centre. The properties of the considered earthquakes, including the station name, *Mw*, PGA, and PGV, have been listed in detail in Table 4. Figure 7 a,b depicts the pseudoacceleration and spectral displacement of the selected earthquake ground motions for 2% and 0.5% damping ratios corresponding to the impulsive and convective masses, along with median as well as 2.5% and 97.5% percentile.

**Table 4.** Selected earthquakes for time history analysis.


**Figure 7.** (**a**) Pseudo-acceleration and (**b**) response displacement spectra of considered earthquake ground motions.

#### **6. Verification**

In order to verify the obtained governing equations of motion, the vertical displacement of free surface (*dv*) and impulsive mass displacement (*x<sup>i</sup>* ) of both squat and slender tanks mounted on an MF with a small dimension of the core are compared with corresponding responses in fixed base conditions. To that end, it is assumed that the core has a dimension of 0.001, 0.001*,* and 0.001 m corresponding to width, length, and height, respectively, and the thickness of the soft layer is 0.001 m. These small dimensions have been assigned to the core and the soft layer to avoid difficulties during solving of the governing equations of motion. It is expected that this MetaFoundation behaves as a traditional foundation made by concrete, and the resonators have no significant influence on the seismic responses of tanks.

Figure 8 compares the time history of the vertical displacement of the free surface as well as the impulsive displacement of both squat and slender tanks subjected to the Friuli, A-TMZ270 earthquake. The obtained responses verify the governing equations of motion and the accuracy of responses of liquid storage tanks with MF. The maximum difference between the responses is found to be less than 1%.

**Figure 8.** Verification of the obtained responses subjected to the Friuli, A-TMZ270 earthquake.

#### **7. Results and Discussions**

#### *7.1. Effect of MetaFoundation*

To evaluate the influence of the proposed MetaFoundation in the time domain, the time transient analysis is performed under selected ground motions. The seismic responses of liquid storage tanks mounted on MF are compared with the corresponding responses in fixed base condition. The parameters of the MF are described in the previous section.

Figure 9 shows the time history of seismic responses of considered tanks under ABBAR—L, Manjil earthquake.

**Figure 9.** Time history of responses of squat and slender tank subjected to ABBAR—L, Manjil ground motion for w/o and w MF.

The seismic responses are depicted for both squat and slender tanks in fixed base condition and mounted on MF, respectively. The performance index of seismic responses of the selected liquid storage tank tanks subjected to various ground motions is depicted in Figure 10. According to the obtained responses, two different trends were observed for the performance of the MF on the seismic responses of squat and slender tanks.

First, in the squat tanks, it is observed that the displacement of the impulsive mass, overturning moment on top of the foundation, and base shear have been reduced in most cases due to the implementation of the MF. The mean performance index of the displacement of impulsive mass, overturning moment on the foundation, and base shear are 0.551, 0.551, and 0.551, illustrating that the MF reduces the input excitation. Therefore, the use of the MF in the squat tank is an effective solution to improve its dynamic behaviour. The best performance of the MF was observed when the system was subjected to G03000, Loma Prieta ground motion in which the PI of the displacement of the impulsive mass, overturning moment on the MF, and base shear equalled 0.091, 0.09, and 0.091, respectively. However, a performance index of more than one is seen in some cases. For example, a PI equal to 1.491, 1.480, and 1.476 for the displacement of impulsive mass, the overturning moment on the foundation and base shear was observed when the system was excited by HEC090, showing that the MF increases the responses compared to fixed base condition. This is attributed to the fact that the performance of MF highly depends on the frequency content of input earthquake ground motion as well as the stop band. According to Figure 11, it is seen that in some frequency ranges, the TR is magnified. This illustrates that the amplitude of the input wave will be intensified, and therefore, the responses of liquid storage tanks are increased compared to the fixed base condition. The reduction of the

overturning moment leads to a decrease in the axial demand in the tank wall, which is the main reason for the buckling of the tank wall. Hence, the more economical and reliable design of the tank is achievable by the use of MF.

**Figure 10.** Performance index (PI) of various seismic responses of squat and slender tanks.

**Figure 11.** TR of the displacement above the MF.

Conversely, it is seen that the MF increases the vertical displacement of the free surface of the liquid. The mean performance index of the vertical free surface displacement is 1.5. Therefore, it is vital to consider a more freeboard to avoid the disrupting consequences corresponding to vertical displacement of the free surface.

Second, with reference to Figure 10, one can conclude that the displacement of impulsive mass, overturning moment on the foundation, and base shear in the slender tank are magnified in most cases. The mean PI of the impulsive displacement, overturning moment, and base shear in the slender tank are 1.315, 1.315, and 1.35, respectively. Therefore, it can be concluded that the MF has an adverse effect on the seismic responses of the slender liquid storage tank. The best performance of the MF is observed under the ABBAR—L, Manjil earthquake in which the PI corresponding to the displacement of impulsive mass, overturning moment on top of foundation, and base shear are 0.742, 0.742, and 0.740, respectively. In terms of the vertical displacement of the free surface, it is seen that the MF has no significant effect. As is seen in Figure 10, the PI of the vertical displacement of the free surface approximately equals one for all of the earthquake ground motions.

To demonstrate the filtering effect of the MF, the TR above the MF in the absence of the liquid storage tank is demonstrated in Figure 11.

A TF equal to zero implies that the displacement output on the top of the foundation equals the input displacement induced at the bottom of the foundation. After that, the TR corresponding to the regions of less than zero indicates that the amplitude of the output displacement is less than the amplitude of the input displacement. This is due to the fact that the incident wave is blocked and cannot pass through the MF. Conversely, when the amplitude of the input displacement increases, the value of TR becomes more than zero; hence, the MF has an adverse effect. With reference to Figure 11, an amplification area is seen for the frequency region below 18.733 Hz, followed by an attenuation region from 18.733 to 18.984 Hz where the wave cannot propagate throughout the MF. The second attenuation zone is seen for frequencies over 22.4 Hz. Based on the obtained responses, the MF reduces the amplitude of waves with frequencies falling in the stop band. Since an earthquake is a random phenomenon with a wide range of frequencies, and its frequency content cannot be predetermined, the effectiveness of MF is limited to those ranges of frequencies which it is designed for.

#### *7.2. Effect of the Frequency Content of the Excitation*

To evaluate the influence of the characteristics of the input excitation, the PI of the seismic responses of both squat and slender tanks are depicted in Figure 12 against the predominant frequency of the considered earthquake. The predominant frequency is determined through the signal processing of considered earthquake ground motions and by selecting the maximum spectral acceleration occurring in their acceleration response spectra.

Besides, a linear regression line is plotted to observe the overall trend of the responses over the frequencies of the considered earthquake ground motions. Generally, it is seen that the PI of the displacement of the impulsive mass, overturning moment, and base shear of the squat tank is less than one for all of the ground motions with the predominant frequency higher than the frequency corresponding to impulsive mass. Conversely, both a PI of less and more than one were observed for the aforementioned responses when the system was subjected to ground motions with the predominant frequency lower than the frequency of the impulsive mass. In the case of the slender tank, the PI of more than one was observed for the impulsive displacement, overturning moment, and base shear when the system was subjected to excitations with the predominate frequency lower than the impulsive frequency. Besides, for those earthquakes with a predominant frequency higher than the impulsive mass frequency, a PI of less and more than one can be observed.

**Figure 12.** Effect of the frequency content of input earthquake on performance index of seismic responses of squat and slender tanks.

#### *7.3. Effect of Number of Layers*

In this section, the effect of the number of layers of MetaFoundation on the seismic responses of squat and slender liquid storage tanks is investigated. The height of each layer is kept constant at 2.0 m, and the properties of unit cells are the same as those mentioned in the previous sections. For the required analysis, five types of MF with one, two, three, four, and five layers are considered. For a better comparison of the obtained results, the numerical results provided in terms of performance index under various considered ground motions are shown in Figure 13.

In the case of the squat tank, one can see that the performance index of the displacement of impulsive mass, overturning moment on the foundation, and structural base shear decreases as the number of layers. Therefore, the more layers there are, the more reduction in the responses. Conversely, for the slender tank, it is seen that as the number of layers of MetaFoundation increases, the displacement of the impulsive mass, overturning moment on the MF, and base shear significantly increase. According to the TR shown in Figure 14, the boundary of the stop band becomes steeper when the number of layers of MF increases, leading to more attenuation of the input energy in this region. Conversely, for other regions other than the stop band, the TR increases, causing the intensify of the input energy. Therefore, the seismic responses of the squat tank are decreased, and those of the slender tank are magnified.

**Figure 13.** Effect of the number of layers on various seismic responses of squat and slender liquid storage tanks.

**Figure 14.** Effect of the number of layers on the TR on top of the foundation.

In terms of the vertical displacement of the free surface, it is observed that the mean PI of vertical displacement of the free surface remains constant for the different number of layers in the squat tank. However, a slight increase is observed in the slender tank.

#### *7.4. Effect of Rubber Thickness (tr)*

The effect of rubber thickness on the seismic responses of liquid storage tanks is investigated here. Five different rubber thicknesses, *t<sup>r</sup>* = 0.01, 0.02, 0.03, 0.04, 0.05 m were considered for the models. The dimension of the unit cell and the heavy core is the same with other sections. The obtained responses of squat and slender tanks subjected to various ground motions are illustrated in Figure 15.

**Figure 15.** Effect of thickness of rubber (*tr*) on various seismic responses of squat and slender liquid storage tanks.

From the obtained response, one can observe that the free vertical surface displacement (*dv*) is not sensitive to rubber thickness (*tr*), and these responses are almost constant for various values of rubber thickness. Conversely, other seismic responses are changed with the change in rubber thickness. The obtained results show that as the thickness of rubber increases, the displacement of the impulsive mass (*x<sup>i</sup>* ), overturning moment on the MF, and base shear (*M* and *Fs*) decreases in the squat tank, leading to the improvement in the performance of the liquid storage tank. Conversely, a slight increase is observed in the impulsive mass displacement, overturning moment, and base shear of the slender tank. However, a fluctuation trend is seen for some earthquakes in both squat and slender tanks. This is attributed to the frequency content of earthquake ground motion, which affects the performance of MF.

#### *7.5. Effect of Damping of Soft Material*

In this section, the effect of damping of soft material on the seismic response of considered liquid storage tanks is investigated. In order to study the effect of damping, five different damping ratios of *ζSM* are considered as 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3, respectively. The dimension of unit cell, heavy core, and thickness of soft material are assumed to be the same with the previous sections. Figure 16 illustrates the effect of damping ratios on the maximum responses of both liquid storage tanks.

**Figure 16.** Effect of soft material damping ratio on the seismic response of squat and slender liquid storage tanks.

The results obtained from time history analysis indicate that the damping ratios of soft material affects the maximum responses of structures. The more damping ratios there are, the more the impulsive mass displacement (*x<sup>i</sup>* ) becomes reduced. As a result of such reduction, the overturning moment (*M*) recorded on the MF and the structural base shear (*Fs*) are decreased. However, it is seen that the damping ratio of soft material has no significant effect on the vertical displacement of free surface (*dv*).

#### **8. Conclusions**

This paper evaluates the effectiveness of a novel foundation called MetaFoundation (MF) on the seismic response of liquid storage tanks. The MF comprises concrete and cubic shape steel coated with rubber as a soft layer. The concept of the MF is based on the finite

locally resonant metamaterials, which produce a stop band to avoid the propagation of waves. The obtained results indicate that the MF has the capability to reduce the seismic response of squat liquid storage tanks, such as displacement of the impulsive mass, overturning moment, and base shear. Conversely, the obtained performance indexes indicate that the use of MF will lead to an increase in the seismic responses of the slender tank.

The effects predominant frequency of earthquakes, the number of layers of metamaterial, rubber thickness, and damping ratios of soft material were studied. The results confirm that as the number of metamaterials increases, more reductions can be observed in the TR of the stop band; therefore, the displacement of the impulsive mass, overturning moment on the top of the foundation, and base shear of squat tank decreases. However, the displacement of free vertical surface displacement was not significantly affected. Besides, the obtained responses show that the rubber thickness has an insignificant effect on the vertical displacement of the free surface of the squat tank, while other responses such as impulsive mass displacement, overturning moment, and structural base shear reduce when the rubber thickness increases. Furthermore, it was observed that the use of MF amplified the seismic responses of the slender tank. The results indicated that as the number of layers of metamaterial increases, the results become more reduced.

**Author Contributions:** Conceptualization, M.F. and M.I.K.; methodology, M.F. and M.I.K.; software, M.F.; validation, M.F., M.I.K. and P.S.; formal analysis, M.F.; investigation, M.F.; resources, P.S.; data curation, M.F.; writing—original draft preparation, M.F.; writing—review and editing, P.S.; visualization, M.I.K.; supervision, M.I.K.; project administration, M.I.K.; funding acquisition, P.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** No specific funding was received for this article.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

