Adaptation of metamaterials at micro- to nanometer scales to metastructures at much larger scales offers a new alternative for seismic isolation systems. These new isolation systems, known as periodic foundations, function both as a structural foundation to support gravitational weight of the superstructure and also as a seismic isolator to isolate the superstructure from incoming seismic waves. Here we describe the application of periodic foundations for the seismic protection of nuclear power plants, in particular small modular reactors (SMR). For this purpose, a large-scale shake table test on a one-dimensional (1D) periodic foundation supporting an SMR building model was conducted. The 1D periodic foundation was designed and fabricated using reinforced concrete and synthetic rubber (polyurethane) materials. The 1D periodic foundation structural system was tested under various input waves, which include white noise, stepped sine and seismic waves in the horizontal and vertical directions as well as in the torsional mode. The shake table test results show that the 1D periodic foundation can reduce the acceleration response (transmissibility) of the SMR building up to 90%. In addition, the periodic foundation-isolated structure also exhibited smaller displacement than the non-isolated SMR building. This study indicates that the challenge faced in developing metastructures can be overcome and the periodic foundations can be applied to isolating vibration response of engineering structures.

Important structures such as hospital buildings, fire, rescue, and other emergency response facilities, in general, are designed to withstand large earthquakes with minor damage or an immediate occupancy performance level as regulated by FEMA 273.1 Furthermore, structures housing sensitive equipment such as nuclear power plants should experience very little vibrations and therefore maintain the safety of the nuclear facilities. However, such high structural performance is often costly and difficult to achieve, especially in the high seismic regions.

The current effective solution used to reduce the seismic demand on the structures is to equip them with seismic isolation systems,2 such as rubber bearings and friction pendulum systems. The conventional isolation systems work by introducing low lateral stiffness devices at the base of the structure, hence lengthening the natural period of the structural system and reducing the input acceleration accordingly. However, these conventional isolation systems are not without shortcoming. The conventional isolation systems are incapable of isolating vertical earthquakes. Therefore, development of new seismic isolation systems that can overcome the aforementioned drawback has become an attractive area of research.

The research of metamaterials in the field of solid state physics has discovered periodic materials or phononic crystals. These materials can manipulate incoming waves by utilizing their frequency band gap property. An infinite-layer of the phononic crystal lattice, in theory, can prevent the propagation of elastic waves having frequencies within the frequency band gaps to propagate through the crystal’s medium.3,4 According to the number of directions where the unit cell is repeated, phononic crystals can be classified as one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) phononic crystals,5 as shown in Figure 1.

FIG. 1.

Classification of phononic crystal: (a) 1D; (b) 2D; (c) 3D.

FIG. 1.

Classification of phononic crystal: (a) 1D; (b) 2D; (c) 3D.

Close modal

At centimeter or smaller scales, the phononic crystals are mostly utilized to isolate acoustic waves.4–6 The unique property of the frequency band gap in phononic crystals can be replicated at larger scales as metastructures for structural engineering applications, particularly for vibration isolation. A few examples of studies on metastructures are: periodic rods on offshore platforms to isolate sea waves,7 periodic materials on pipes conveying fluid to isolate flexural waves,8 and a mesh of vertical empty inclusions bored in soil to isolate seismic surface waves.9 

The recent advance in metastructures has developed a new type of seismic base isolation systems known as periodic foundations.10–12 This type of foundation can support any superstructure and isolate the superstructure from the incoming seismic waves since it possesses frequency band gaps. Moreover, since this blocking wave mechanism is effective for both the vertically and horizontally propagating waves, periodic foundations can isolate the superstructure from seismic excitations in both the horizontal and vertical directions. These advantages can easily solve the shortcoming of conventional seismic isolation systems.

A feasibility study of periodic foundations has been conducted to investigate the existence of frequency band gaps and their functionality in seismic isolation. At this level, researchers have conducted experimental tests of 1D,10 2D,11 and 3D12 periodic foundations. In each of the tests, a simple structure with a periodic foundation was tested simultaneously with a non-isolated counterpart. The test results show that the structures isolated with periodic foundations have a much less acceleration response compared to the non-isolated structures.

Guided by results in the feasibility study, we proceed with the application into the real engineering structures. In this study, a 1D periodic foundation is employed to isolate a small modular reactor (SMR) building model. The study reported in this paper begins with the theoretical derivation of the dispersion relation in the 1D periodic materials to design the frequency band gaps on the test specimens. The subsequent section then describes the design and fabrication of the 1D periodic foundation. Finally, the shake table test results of the 1D periodic foundation structural system are presented and discussed.

The frequency band gaps of periodic materials can be obtained by analyzing a single unit cell model with periodic boundary conditions. This section briefly shows the derivation of frequency band gaps for 1D periodic materials using the transfer matrix method. Consider a 1D periodic material with a crystal lattice repeated in the z direction, as shown in Figure 2. Due to symmetry, a periodic medium can be reduced to a single unit cell model with a periodic boundary condition. Suppose that the unit cell of the periodic foundation consists of N layers. Hence, the elastic wave equation in each layer n of the unit cell is shown in Eq. (1). The Cn constant is expressed in Eqs. (2) and (3) for input waves of the transverse wave (S-Wave) and the longitudinal wave (P-Wave), respectively.

2unt2=Cn22unzn2
(1)
Cn=μn/ρn
(2)
Cn=(λn+2μn)/ρn
(3)
unzn,t=eiωtunzn
(4)
Cn22un(zn)zn2+ω2un(zn)=0
(5)
un(zn)=Ansin(ωzn/Cn)+Bncos(ωzn/Cn)
(6)
FIG. 2.

A unit cell with N layers in one dimensional periodic material.

FIG. 2.

A unit cell with N layers in one dimensional periodic material.

Close modal

In which ρn is the material density and λn and μn are the first and second Lamé parameter constants,13 respectively. Consider a steady state oscillatory wave of angular frequency, as shown in Eq. (4). Substitution of Eq. (4) into Eq. (1) yields Eq. (5). The general displacement solution of Eq. (5) is shown in Eq. (6).

The terms An and Bn in Eq. (6) are the amplitudes of the general displacement solution on layer n. In the elastic body of each of the layers, the constitutive equations for normal and shear stresses are shown in Eqs. (7) and (8), respectively. To obtain the dispersion curve of the S-Wave, Eqs. (6) and (8) are arranged into matrix form, as shown in Eq. (10). For the S-Wave propagation, the Cn constant for the respective wave is used.

σn(zn)=(λn+2μn)un/zn=(λn+2μn)ω[Ancos(ωzn/Cn)Bnsin(ωzn/Cn)]/Cn
(7)
τn(zn)=μnun/zn=μnω[Ancos(ωzn/Cn)Bnsin(ωzn/Cn)]/Cn
(8)
un(zn)τn(zn)=sin(ωzn/Cn)cos(ωzn/Cn)μnωCncos(ωzn/Cn)μnωCnsin(ωzn/Cn)AnBn
(9)
or wn(zn)=Hn(zn)ψn
(10)

The left-hand side vector of Eq. (10) at the bottom of layer n is defined as wnb [Eq. (11)], which gives information regarding the displacement and the stress at the bottom of layer n. As for the top of layer n, the left-hand side vector is defined as wnt [Eq. (12)], which gives the information of displacement and the stress at the top of layer n. The height of layer n is denoted by hn. Eq. (11) can be related to Eq. (12) through a transfer matrix Tn. as shown in Eq. (13). Hence, the transfer matrix Tn for a single layer n is denoted in Eq. (14)

wnbwn(0)=Hn(0)ψn
(11)
wntwn(hn)=Hn(hn)ψn
(12)
wnt=Tnwnb
(13)
Tn=Hn(hn)Hn(0)1
(14)

Each layer’s interface of the unit cell is assumed to be perfectly bonded; hence the displacement and shear stress need to satisfy continuity. Therefore, the displacement and shear stress of the top of layer n are equal to that of the bottom of layer n+1, as indicated in Eq. (15). Subsequently, the relationship of displacement and shear stress of the bottom and top surfaces of the unit cell containing the N layers can be expressed as in Eq. (16). By changing the displacement and shear stress vector of the top and bottom surface of the unit cell as wt=wNt and wb=w1b, respectively, Eq. (16) can be shortened into Eq. (17).

wn+1b=wnt
(15)
wNt=TNwNb=TNwN1t=TNTN1wN1b=...=TNTN1...T1w1b
(16)
wt=T(ω)wb
(17)

The transfer matrix for a unit cell of the 1D periodic foundation is T(ω) = TNTN-1T1. Based on the Bloch-Floquent theorem, the periodic boundary conditions can be expressed as in Eq. (18), in which h=1Nhn is the unit cell thickness, k is the wavenumber in reciprocal lattice space and i is the imaginary number. Subtraction of Eq. (18) from Eq. (17) yields Eq. (19) and the nontrivial solution can be achieved when the determinant is equal to zero, as shown in Eq. (20).

wt=eikhwb
(18)
T(ω)eikhIwb=0
(19)
T(ω)eikhI=0
(20)

Eq. (20) is the so called Eigenvalue problem, with eikh equal to the Eigenvalue of the transformation matrix T(ω). Thus the relationship between wavenumber k and frequency ω can be obtained by solving the corresponding Eigenvalue problem. The relationship between the wavenumber and frequency forms the S-Wave dispersion curve. The curves are related to real wavenumbers and the frequency band gaps are related to complex wavenumbers. Although the wavenumber k is unrestricted, it is only necessary to consider k limited to the first Brillouin zone,14 i.e. kπ/h,π/h, to obtain the frequency band gaps. Likewise, the P-Wave dispersion curve can be obtained through a similar approach by arranging Eqs. (6) and (7) into the matrix form of Eq. (9) and using the Cn constant for the P-Wave.

In this study, the prototype used as the superstructure is a representative of an SMR building by NuScale Power.15 As shown by the symmetric cut of the SMR building in Figure 3(a), the SMR building can host up to twelve SMRs inside the water pool. The dimensions and masses of the building structure were estimated from the sketches and the rendering pictures provided by NuScale Power. The length, width, and height of the SMR building were selected as 100 m, 40 m, and 40 m, respectively. It is assumed that the building structure is made of reinforced concrete (RC) material. Based on the information, a finite element (FE) model was built to study the dynamic characteristic of the prototype building [see Figure 3(b)]. The model was constructed from 13197 tetrahedron and 1100 brick elements using Abaqus software. A total of 29.68 million kg of nonstructural masses was added into the structure. The nonstructural masses represent nonstructural elements attached to the building which include water in the reactor pool, all twelve SMRs, crane, and utilities. Modal analysis results show that the first expected mode of the superstructure building is a translational mode with a natural frequency of 6.77 Hz.

FIG. 3.

Design process of superstructure model: (a) NuScale SMR Building; (b) Finite element model of prototype building; (c) Modal analysis result of prototype building; (d) Modal analysis result of scaled model.

FIG. 3.

Design process of superstructure model: (a) NuScale SMR Building; (b) Finite element model of prototype building; (c) Modal analysis result of prototype building; (d) Modal analysis result of scaled model.

Close modal

Due to the limitations of the shake table facility, the experimental test cannot be performed at a full-scale size. Therefore, a scaled-down model was designed for this study. The scaling parameters follow the similitude requirements for true ultimate strength16 with the selected length scale (lr) of 1/22. The goal in this scaling is to obtain a superstructure model that allows its natural frequency to satisfy the frequency scale requirement (ωr=lr1/2=22=4.69). Upon satisfying the frequency scale requirement, the displacement demand on the scaled model will reflect that on the prototype structure by the length scale. Note that in this study, the structural systems are assumed to be elastic. For the ease of construction and sensor installation, a steel frame structure was chosen as the SMR building model with the details shown in Figure 4. Wide flange sections of 150X150X7X10 and 200X200X8X12 and angle sections of 65X65X6 and 50X50X5 were selected for beams, columns, and longitudinal and transverse braces, respectively. Additional masses of 1750 kg and 8500 kg were provided on the roof and on the floor of the steel frame, respectively. Modal analysis on the FE model shows that the expected natural frequency of the steel frame is 31.1 Hz, which satisfies the frequency scale requirement.

FIG. 4.

Designed superstructure model (units in mm).

FIG. 4.

Designed superstructure model (units in mm).

Close modal

A four-layer unit cell was designed as the seismic wave isolator to protect the superstructure model. The unit cell consists of two RC layers and two polyurethane layers arranged alternately, as shown in Figure 5. The material properties for each of the base materials are shown in Table I. A four-layer unit cell can be very beneficial because it possesses low and wide frequency band gaps with very thin pass bands, in which the response amplification can be easily damped out by material damping.17 The dispersion curves of the designed unit cell for both the S-Wave and P-Wave are shown in Figures 5(b) and 5(c), with frequency band gaps marked with yellow shaded areas. The first two frequency band gaps for the S-Wave are observed to be in 18.41-20.76 Hz and 29.43-74.06 Hz, while those for the P-Wave are observed to be in 70.15-79.07 Hz and 112.1-282.2 Hz.

FIG. 5.

Designed 1D periodic foundation: (a) Unit cell; (b) Dispersion curve for S-Wave; (c) Dispersion curve for P-Wave.

FIG. 5.

Designed 1D periodic foundation: (a) Unit cell; (b) Dispersion curve for S-Wave; (c) Dispersion curve for P-Wave.

Close modal
TABLE I.

Unit cell’s material properties.

Material properties
MaterialYoung’s modulus (MPa)Density (kg/m3)Poisson’s ratio
Reinforced concrete 31,400 2300 0.2 
Polyurethane 0.1586 1100 0.463 
Equivalent superstructure 200,000 23029.2 0.3 
Material properties
MaterialYoung’s modulus (MPa)Density (kg/m3)Poisson’s ratio
Reinforced concrete 31,400 2300 0.2 
Polyurethane 0.1586 1100 0.463 
Equivalent superstructure 200,000 23029.2 0.3 

Since a periodic foundation is also designed to support a superstructure, the presence of the superstructure may affect the frequency band gaps. The shift in frequency band gaps can be approximated by converting the superstructure into an additional equivalent structure layer on the unit cell [see Figure 6(a)] and subsequently by finding the dispersion curves of the corresponding unit cell.17 To calculate the equivalent superstructure layer, the length and width of the periodic foundation need to be determined beforehand. In this study, the 1D periodic foundation is designed to be 4.6 m long and 2.06 m wide. The thickness for the equivalent layer can be assumed to be the same as the thickness of the uppermost layer, i.e. RC layer 2. The total mass of the superstructure is then divided with the equivalent layer volume to obtain the density. The Young’s modulus and the Poisson’s ratio simply follow the base material of the superstructure. Figures 6(b) and 6(c) show the dispersion curves of the unit cell with an equivalent structure layer. The frequency band gaps are obviously located in the frequency region lower than that without the structure layer. The obtained frequency band gaps for the S-Wave are 6.12-20.34 Hz and 21.44-50 Hz, while that for the P-Wave is 18.97-50 Hz. The analysis was ceased at 50 Hz since the frequency range of interest in this study is below 50 Hz.

FIG. 6.

Designed 1D periodic foundation considering superstructure weight: (a) Unit cell with an equivalent structure layer; (b) Dispersion curve for S-Wave; (c) Dispersion curve for P-Wave.

FIG. 6.

Designed 1D periodic foundation considering superstructure weight: (a) Unit cell with an equivalent structure layer; (b) Dispersion curve for S-Wave; (c) Dispersion curve for P-Wave.

Close modal

Figure 7 shows the designed 1D periodic foundation supporting the SMR building model. The designed 1D periodic foundation consists of a single four-layer unit cell. A single unit cell has proven to possess the same frequency band gaps as those with an infinite number of unit cells17 and has sufficient response reduction inside the frequency band gaps.

FIG. 7.

Designed 1D periodic foundation structural system (units in mm).

FIG. 7.

Designed 1D periodic foundation structural system (units in mm).

Close modal

Figure 8 shows the fabrication process of the 1D periodic foundation. The 1D periodic foundation was assembled layer by layer. The interfaces of the RC layers (gray) and the polyurethane layers (black) were glued using a polyurethane based glue with nominal tensile and tear strengths equal to 1.5 MPa and 8 MPa, respectively. To compare the 1D periodic foundation-isolated SMR building with a non-isolated one, an RC foundation was also designed to support the SMR building model. A bolt type connection was chosen to connect the SMR building model to either the periodic foundation or the RC foundation.

FIG. 8.

Construction of 1D periodic foundation.

FIG. 8.

Construction of 1D periodic foundation.

Close modal

Four test cases (see Figures 9–12) were designed for the experimental program. Each of the test cases serves different purposes in understanding the behavior of the 1D periodic foundation and its counterpart. In Case 1, only the RC foundation is tested. Through this case, the wave propagation in the homogeneous material is examined. In Case 2, the SMR building model is placed on the top of the RC foundation. This case shows the behavior of the non-isolated SMR building. In Case 3, only the 1D periodic foundation is tested. This test would confirm the existence of frequency band gaps on the 1D periodic foundation. In Case 4, the SMR building model is placed on the top of the 1D periodic foundation. The purpose of this test is to investigate the interaction between the 1D periodic foundation and the superstructure as well as to study the behavior of the structural system subjected to various harmonic and seismic waves in different directions.

FIG. 9.

Test setup of Case 1 (RC foundation).

FIG. 9.

Test setup of Case 1 (RC foundation).

Close modal
FIG. 10.

Test setup of Case 2 (RC foundation with SMR building structure).

FIG. 10.

Test setup of Case 2 (RC foundation with SMR building structure).

Close modal
FIG. 11.

Test setup of Case 3 (periodic foundation).

FIG. 11.

Test setup of Case 3 (periodic foundation).

Close modal
FIG. 12.

Test setup of Case 4 (periodic foundation with SMR building structure).

FIG. 12.

Test setup of Case 4 (periodic foundation with SMR building structure).

Close modal

In each case, a concrete base with a size of 5.12 m long by 3 m wide by 0.2 m thick was placed underneath the specimen to connect the specimen with the shake table. The shake table test facility belongs to the National Center for Research on Earthquake Engineering (NCREE). The five by five meter shake table can simulate motions in six degrees of freedom with a minimum frequency of 1 Hz and a maximum frequency of 50 Hz.

Three types of instrumentation are used to record the response of the structure, namely an accelerometer (Model 141 by Setra System, Inc.18 with a sensitivity of 0.012 g/g), a traditional displacement transducer (Temposonic RH Rod-style model by MTS Sensors19 with an accuracy of 2 micrometer) and a vision-aided measurement system (NDI optotrak optical measurement20 with an accuracy of up to 0.1 mm and resolution of 0.01 mm). The accelerometer was used to measure acceleration response, while both the temposonic and NDI systems were used to measure displacement response. Figure 13 shows the sensors arrangement in Case 4. Each of the letters shown in Figure 13 refers to a single sensor. Letter A refers to accelerometer, letter T refers to temposonic, and letter N refers to the NDI marker. The sensors were placed on each corner of the shake table, middle RC layer, the top of the 1D periodic foundation, and roof of the superstructure. In addition, NDI markers were also pasted on each corner of each of the polyurethane layers. In Case 2, the markers were attached to each corner of the shake table, the top of the RC foundation, and roof of the superstructure. The sensors arrangement in Cases 1 and 3 are similar to those in Cases 2 and 4 except without sensors on the superstructure.

FIG. 13.

Sensors arrangement in Case 4: (a) Elevation view; (b) Plan view.

FIG. 13.

Sensors arrangement in Case 4: (a) Elevation view; (b) Plan view.

Close modal

Both of Cases 1 and 3 specimens were subjected to frequency sweeping tests. The tests were conducted to verify the existence of frequency band gaps on the 1D periodic foundation and the wave propagation on the RC foundation. In the test, the input wave was a stepped sine wave starting from 1 Hz to 50 Hz with an increment of 0.5 Hz. From the frequency sweeping tests, the accelerations recorded at the top of the foundations can provide information as to which frequency regions the responses are attenuated. Those regions correspond to the so-called attenuation zones or frequency band gaps. To obtain the attenuation zones, the acceleration records in the time domain were transformed into the frequency domain using the Discrete Fourier Transform (Welch method21). The frequency response function (FRF) curves were then generated from the Fourier spectra. The FRF for each frequency on each response-measured location can be calculated using FRF = 20log(ao/ai) where ao is the acceleration amplitude at the location of interest and ai is the acceleration amplitude at the shake table.

Figure 14 shows the FRF curves for both the RC (black curves) and 1D periodic foundation (red curves) from the frequency sweeping tests in the horizontal and vertical directions and in the torsional mode. For comparison, the theoretical frequency band gaps obtained in Section III are also plotted in the figure. The negative FRF values indicate response reduction at particularfrequencies while the positive values show response amplification. The attenuation zone, the zone with negative FRF values, starts from 12.75 Hz to 50 Hz for shaking in the horizontal direction. This attenuation zone overlaps the theoretical frequency band gaps of the S-Wave in the range of 0 to 50 Hz. For the shaking in the vertical direction, no attenuation zone can be found in the range of 0 to 50 Hz. This result also agrees with the theoretical frequency band gaps of the P-Wave. The attenuation zone in the torsional mode is found starting from 12.65 Hz to 50 Hz, which is very similar to the result in the horizontal direction. This is reasonable since torsional movement is also a form of shear. Therefore, the test result is comparable and is observed to be overlapping with the theoretical frequency band gaps of the S-Wave. The test results on the RC foundation show that the linear lines at FRF are equal to zero, which means the response of the RC foundation is the same as the input, or the input waves propagate from the shake table through the RC foundation.

FIG. 14.

Frequency sweeping test results on foundations: (a) In horizontal direction; (b) In vertical direction (c) In torsional mode.

FIG. 14.

Frequency sweeping test results on foundations: (a) In horizontal direction; (b) In vertical direction (c) In torsional mode.

Close modal

The structural systems of Cases 2 (SMR building model on RC foundation) and 4 (SMR building model on 1D periodic foundation) were subjected to four types of tests, i.e. White noise, frequency sweeping, seismic, and harmonic tests. Each test serves a different purpose which will be elaborated in the next subsections.

1. White noise tests

In this test, Cases 2 and 4 were subjected to a white noise wave in the horizontal and vertical directions and the torsional mode to obtain the natural frequencies of the structural systems. White noise we have used consists of random waves with a uniform main frequency content ranging from 1 Hz to 50 Hz. When a structural system is subjected to white noise in a particular direction, the response will show amplification at a specific frequency. This specific frequency is the natural frequency of the structural system in that particular direction. Figure 15 shows the normalized Fourier spectra ratio of the white noise test results. The term “input” in Figure 15 corresponds to the Fourier spectra of the accelerations recorded at the shake table while the term “output” represents the Fourier spectra of the accelerations recorded at the roof of the superstructure. The peak ratio of output by input indicates that there exists response amplification at the particular frequency. Each of the output by input ratios was normalized with its peak value so that the results from Cases 2 and 4 can be shown in the same figure. The natural frequency of the Case 2 structural system in each of the horizontal direction, vertical direction and torsional mode was found to be 16.41 Hz, 26.86 Hz and 46.88 Hz, respectively. While that of the Case 4 structural system in each of the horizontal and vertical directions and torsional mode was found to be 2.49 Hz, 19.63 Hz, and 3.42 Hz, respectively.

FIG. 15.

Transfer function of white noise test results on structural systems: (a) In horizontal direction; (b) In vertical direction; (c) In torsional mode.

FIG. 15.

Transfer function of white noise test results on structural systems: (a) In horizontal direction; (b) In vertical direction; (c) In torsional mode.

Close modal

The natural frequency of the non-isolated SMR building in the horizontal direction was designed to be 31.1 Hz. Instead, the test results show that the natural frequency is 16.41 Hz. The difference may come from different boundary conditions in the FE analysis and in the test. In the analysis, the FE model was fixed on each node of the steel floor making the superstructure much stiffer. In the test, however, the SMR building model was connected to the RC foundation using 34 bolts along the perimeter (lines 1, 2, A, and C in Figure 4) which makes the superstructure relatively less constrained. Moreover, the actual mass blocks placed on the roof of the superstructure and the floor of the superstructure were measured to be 1830 kg and 8368 kg, respectively.

2. Frequency sweeping tests

Cases 2 and 4 were also subjected to frequency sweeping tests in the horizontal and vertical directions and the torsional mode. The same stepped sine wave used in the tests of Cases 1 and 3 were also used as the input in these tests. Figure 16 shows the FRF curves processed from the frequency sweeping test results. In these tests, the FRF curves were obtained for the results measured at both the top of the foundations and the roof of the superstructures. The FRF curves are also compared with the theoretical frequency band gaps shown in Section III.

FIG. 16.

Frequency sweeping test results on structural systems: (a) In horizontal direction; (b) In vertical direction; (c) In torsional mode.

FIG. 16.

Frequency sweeping test results on structural systems: (a) In horizontal direction; (b) In vertical direction; (c) In torsional mode.

Close modal

In the horizontal direction, the attenuation zone measured at the top of the 1D periodic foundation is found to be at 3.7–50 Hz [red curve in Figure 16(a)] while that at the roof of the superstructure is found to be at 3.8–50 Hz [brown curve in Figure 16(a)]. The results show that the input waves have been completely obstructed by the 1D periodic foundation before reaching the superstructure. In the vertical direction, the attenuation zones measured at the top of the 1D periodic foundation are found to be at 20.75–24.1 Hz and 32.5–36 Hz, as shown by the red curve in Figure 16(b), while those at the roof of the superstructure are seen at 22.8–27.83 Hz and 32.4–40.7 Hz [brown curve in Figure 16(b)]. In the torsional mode, the attenuation zone measured at the top of the 1D periodic foundation is located at 5.1–50 Hz [red curve in Figure 16(c)], while that measured at the roof of the superstructure is located at 5.2 –50 Hz [brown curve in Figure 16(c)].

From the test results, one can see that the attenuation zones in the horizontal direction and torsional mode overlap with the theoretical frequency band gaps of the S-Wave. However, the attenuation zones in the vertical direction are somewhat different than the theoretical frequency band gap of the P-Wave. The discrepancy in the vertical direction may come from the construction errors such as an imperfect flat surface of RC layers and non-uniform glue distribution. Therefore, in some parts of the 1D periodic foundation the polyurethane surface is not perfectly connected to the RC surface. The imperfect interface connection between each layer on the 1D periodic foundation is affected the most when the structure is subjected to the wave in the vertical direction. The imperfect interface connection causes non-uniform axial stresses and load transfer between each layer making the wave propagation different from the theory. In addition, the bolt connection between the steel floor of the superstructure and the top of the 1D periodic foundation is also not a perfect connection. The threaded bolts that tighten the connection could not provide a perfect rigid interface when the bolts are in tension. The imperfection of the interface connection, however, has little influence when the structure is subjected to incoming waves in the horizontal direction and torsional mode. Large superstructure weight allows sufficient friction between each layer on the 1D periodic foundation as well as between the 1D periodic foundation with the superstructure to make the shear stress uniform. Therefore, the attenuation zones in the horizontal direction and torsional mode are very close to the theoretical frequency band gaps. It is also observed that the attenuation zones of the 1D periodic foundation structural system are lower than that of the 1D periodic foundation only. Therefore, the tests confirmed the shift in attenuation zones or frequency band gaps due to the presence of the superstructure. On the other hand, there is no attenuation zone in the RC foundation as is clearly shown by close to zero FRF values in Figures 16(a)–16(c). The responses at the roof of the structure with the RC foundation are mostly amplified in all three directions.

3. Seismic tests

Based on the frequency sweeping test results, the attenuation zones of Case 4 in the vertical direction are insufficient to cover the majority of the main frequency contents of the vertical earthquakes. Therefore, the seismic tests were conducted only in the horizontal direction and torsional mode. For these tests, four earthquake records were selected as input motions. The earthquake records were obtained from the Pacific Earthquake Engineering Research Center (PEER) Ground Motion Database.22 All of the earthquake records used as the input waves for horizontal shaking had their peak ground accelerations (PGAs) scaled to 0.4 g. One of the earthquake records was modified into a torsional acceleration with the PGA scaled to 25 deg/s2. The time of the earthquake records were compressed by time scale tr=lr=1/22 to comply with the similitude requirement.

a. Acceleration responses.

Figure 17 shows the test results of both structural systems in the horizontal direction. Significant acceleration response reductions were observed in the 1D periodic foundation structural system. It is observed that after passing through the 1D periodic foundation, the maximum acceleration recorded on top of the 1D periodic foundation subjected to each of the Anza, Bishop, Gilroy, and Oroville Earthquakes is reduced by 88.12 %, 91.55 %, 90.48 %, and 91.23 %, respectively, in comparison to that recorded on the shake table. The term “maximum acceleration” used in this paper refers to the absolute peak acceleration throughout the time series. From the top of the periodic foundation to the roof of the superstructure, the maximum accelerations barely change. Compared to the PGA recorded on the shake table, the maximum accelerations recorded on the roof of the superstructure are reduced by 88.48 %, 90.65 %, 83.38 %, and 91.74 %, respectively, for each of the Anza, Bishop, Gilroy, and Oroville Earthquakes. On the other hand, the acceleration responses of the RC foundation structural system subjected to all four earthquakes are observed to be much larger than the input accelerations. The results clearly indicate the capability of the 1D periodic foundation to isolate seismic waves.

FIG. 17.

Seismic test results in the horizontal direction: (a) Anza Earthquake; (b) Bishop Earthquake; (c) Gilroy Earthquake; (d) Oroville Earthquake.

FIG. 17.

Seismic test results in the horizontal direction: (a) Anza Earthquake; (b) Bishop Earthquake; (c) Gilroy Earthquake; (d) Oroville Earthquake.

Close modal

The isolation mechanism of the 1D periodic foundation can be observed from the Fourier spectra of the recorded time series. Figure 18 shows the Fourier spectra of the test results of Case 4. The spectra show that the majority of the main frequency contents of all four earthquakes are located within 3.7-50 Hz, which is overlapping with the tested attenuation zone and the theoretical frequency band gaps. As one can clearly see, the frequency content of the input waves in this region is effectively filtered out.

FIG. 18.

Fourier spectra of seismic test results of Case 4 in the horizontal direction: (a) Anza Earthquake; (b) Bishop Earthquake; (c) Gilroy Earthquake; (d) Oroville Earthquake.

FIG. 18.

Fourier spectra of seismic test results of Case 4 in the horizontal direction: (a) Anza Earthquake; (b) Bishop Earthquake; (c) Gilroy Earthquake; (d) Oroville Earthquake.

Close modal

Seismic tests were also conducted on the structural systems in the torsional mode. The Bishop Earthquake record was modified into a torsional acceleration time series with a PGA of 25 deg/s2. Figure 19 shows the seismic test results in the torsional mode where a large response reduction is observed on the Case 4 structural system. The acceleration response at the top of the 1D periodic foundation is reduced by 95.4% while that on the roof of the superstructure is reduced by 93.6%. Contrary to the response reductions in Case 4, large amplification is observed in the structural response of Case 2.

FIG. 19.

Seismic test results in the torsional mode.

FIG. 19.

Seismic test results in the torsional mode.

Close modal

The isolation mechanism of the 1D periodic foundation in the torsional mode is similar to that in the horizontal direction. The Fourier spectra of the input acceleration and the acceleration response of Case 4 show a filtering effect in the range of 5.1-50 Hz (see Figure 20) which is similar to the tested attenuation zone and is overlapping with the theoretical frequency band gaps of the S-Wave.

FIG. 20.

Transfer function of seismic test results of Case 4 in torsional mode.

FIG. 20.

Transfer function of seismic test results of Case 4 in torsional mode.

Close modal
b. Displacement responses.

The seismic test results shown above have proven that the 1D periodic foundation is capable of filtering incoming seismic waves when the incoming seismic waves have their main frequency contents falling inside the attenuation zones. Due to the filtering mechanism, it is expected that 1D periodic foundation (also periodic foundations in general) does not produce large relative displacement during seismic excitations. To prove the hypothesis, the roof-to-shake table relative displacements of both Cases 2 and 4 in the horizontal direction are compared. As shown in Figure 21, the relative displacements of the structure with the 1D periodic foundation (Case 4) are smaller than that with the RC foundation (Case 2) when subjected to the Bishop and Oroville Earthquakes. The main frequency contents of the Bishop and Oroville Earthquakes are completely inside the frequency band gap, as shown in Figures 18(b) and 18(d).

FIG. 21.

Roof of superstructure to shake table relative displacement: (a) Bishop Earthquake; (b) Oroville Earthquake.

FIG. 21.

Roof of superstructure to shake table relative displacement: (a) Bishop Earthquake; (b) Oroville Earthquake.

Close modal

The application of periodic foundations as seismic metastructures to isolate an SMR building has been experimentally validated. The designed 1D periodic foundation can isolate the incoming waves in the horizontal and vertical directions as well as the torsional mode using its frequency band gap property. Incoming waves having frequencies falling inside the frequency band gaps or attenuation zones were filtered out by the 1D periodic foundation. Moreover, the 1D periodic foundation can isolate the superstructure without introducing large relative displacement when the frequencies of the incoming waves are located inside the frequency band gaps. In comparison to the non-isolated structure, the structure isolated with the 1D periodic foundation had performed extremely well with the peak acceleration response in time domain reduced by 90% compared to the input peak acceleration.

The research described in this paper is financially supported by the U.S. Department of Energy NEUP program (Project No. CFA-14-6446). The opinions expressed in this study are those of the authors and do not necessarily reflect the views of the sponsor.

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