Effect of geofoam as cover material in cut and cover tunnels on the seismic resp

Sadri Shadabi, Masoud Rabeti Moghadam and Mansour Parvizi

Department of Civil Engineering, Faculty of Engineering, Yasouj University, Yasouj 0098, Iran

Abstract: In recent years, special attention has been given to the effects of underground spaces and structures on the seismic response of adjacent ground. Nevertheless, to the best of the authors′ knowledge, no method has yet been considered to reduce these effects in technical literature. The present study aims to investigate the role of geofoam as the cover material in cut and cover tunnels on the seismic response of the ground surface. For this purpose, a numerical study was performed using FLAC 2D, a finite difference software, and verified against previous studies. The effects of parameters such as the geofoam type, thickness of the geofoam blocks and interfaces between the blocks, on the ground surface amplification pattern were investigated. Parametric studies demonstrate that the geofoam has a strong potential to attenuate the seismic horizontal movements of the ground surface. As the thickness of the geofoam blocks is decreased, its attenuation ability is increased.It was also determined that the interface between the geofoam blocks plays a key role in the attenuation of the aboveground seismic responses. Based on the results, it was concluded that geofoam is a proper material to attenuate seismic amplifications at the ground surface, induced by underground tunnels.

Keywords: cut and cover tunnel; ground surface; seismic response; amplification ratio; geofoam; attenuation

1 Introduction

Local site effects on the c haracteristics of earthquakes are well known. The local conditions of the site provide significant effects on the important characteristics of strong ground motions including amplitude, frequency content and duration. Some cases such as site geometry,material properties of the subsurface layers, site topography, and incident motion characteristics are known factors for site effects, and have gradually been considered by building regulations. Among the issues that have attracted the attention of many researchers in recent years is the impact of underground spaces and structures on the characteristics of incident waves, especially on the ground surface. According to findings in the technical literature, when an underground structure is constructed at shallow depth, its effects on the ground motions at the ground surface become prominent. Because the cut and cover tunnels are typically constructed at shallow depths with respect to deep tunnels, the effect of such structures is more prominent on the seismic response of the adjacent ground surface.

Based on studies conducted by many researchers(Lee and Trifunac, 1979; Lee and Karl, 1993; Lianget al., 2003, 2004; Rodrıguez-Castellanoset al., 2006;Yiouta-Mitraet al., 2007; Smerziniet al., 2009; Liang and Liu, 2009; Yu and Dravinski, 2010; Cilingir and Madabhushi, 2011; Sgarlatoet al., 2011; Lanzanoet al.,2012; Sicaet al., 2014; Baziaret al., 2014; Alielahiet al., 2015; Ulgenet al., 2015; Alielahi and Ramezani,2016; Rabeti Moghadam and Baziar, 2016; Baziaret al., 2016; Amornwongpaibunet al., 2016; Abate and Massimino, 2017; Xuet al., 2017; Alielahi and Adampira, 2018; Sunaet al., 2019; Huanget al., 2019;Panji and Habibivand, 2020; Rahnemaet al., 2021),various parameters affect the seismic response of the ground surface near an underground structure. Yu and Dravinski (2010) found that ground surface responses due to an underground cavity can be classified into two distant areas: structure field (|X/a|>5) and free field(|X/a|<5), whereX/ais the dimensionless distance.Therefore, in the free field distance, the ground surface response is similar to the free field response, while in the structure field, the ground surface response is affected by the scattering of waves due to collision with the cavity.Yiouta-Mitraet al. (2007), by using a numerical study,presented the amplification of the horizontal component of motion from 20% to 85%, due to the presence of a circular cavity, up to a distance of 11 times the radius of the tunnel from the main axis of the tunnel (-11≤X/a≤11).Smerziniet al. (2009) concluded that the existence of underground structures in low periods of the acceleration response spectrum (in particular from 0.05 to 0.2 s)has a significant effect (with an amplification of up to 1.5) on the seismic response of the ground. Alielahiet al. (2016) concluded that the seismic response of the ground surface above the tunnel in the dimensionless distance of -10≤X/a≤10 is significantly different when compared to the seismic response at the free field. Rabeti Moghadam and Baziar (2016), based on the results of an experimental and numerical study, concluded that the tunnel amplifies the motions with dimensionless period(λ/D) in the range of 3 to 10 (3<λ/D<10).

Although the effects of underground structures on the seismic response at the ground surface have been confirmed, to the best of the authors′ knowledge, no method has yet been considered to reduce these effects.The appearance of new materials, such as geofoams, have been widely used for solving challenges in geotechnical engineering. According to the experiments and modeling reported in the literature (Wang, 2011; Kiryuet al., 2012), it can be seen that geofoam performs as a relatively strong seismic isolation for the tunnels. This lightweight material, due to its ability to attenuate the seismic waves, is widely used for retrofitting retaining systems in high seismicity areas (Trandafir and Ertugrul,2011; Zekkoset al., 2012; Bathurst and Zarnani, 2013;Lalet al., 2014; Azzam and Abdel Salam, 2015; Abdel Salam and Azzam, 2016; Niet al., 2016; Ertugrulet al., 2017; Hasanpouri Notash and Dabiri, 2018; Dabiri and Hasanpouri Notash, 2019). The capabilities of the expanded polystyrene (EPS) geofoam in reducing the seismic demands imposed on earth retaining structures have been highlighted by others (Abdel Salam and Azzam, 2016; Zarnani and Bathurst, 2008). Years of practical experience with polymeric geofoam have proved its ability to withstand vertical and lateral stresses when used in the construction of earthworks. Stability,durability and resistance to moisture and deterioration are known as superior characteristics of the geofoam(Ertugrul and Trandafir, 2017). Reported applications of geofoam in geotechnical works include reduction of static lateral earth pressures on rigid and non-yielding retaining structures (Ertugrul and Trandafir, 2011)and attenuation of seismic forces on retaining walls(Trandafir and Ertugrul, 2011; Ertugrul and Trandafir,2012).

Some researchers have used geofoam as a protection against blast loading on underground structures. Davies(1994) used trenches filled with polystyrene foam as a barrier against blast loading. De and Zimmie (2007)and Deet al. (2016) used polyurethane geofoam as a barrier against blast loading on an underground tunnel.The surrounding area of the tunnel was covered with polyurethane geofoam. These studies demonstrate the benefits of polyurethane geofoam as a barrier against surface explosion to reduce the effects of explosions on underground tunnels. Baziaret al. (2018) investigated the effect of dynamic loading on an underground structure and the performance of geofoam as a barrier by using centrifuge tests. The results of the tests confirmed the effectiveness of the geofoam barrier against dynamic loading effects. Yanget al. (2020) studied the isolation performance of trenches on train-induced ground-bore vibration from underground tunnels. A series of model tests was conducted to perform the study. The results showed that the open trench and the EPS-filled trench increased the dynamic response of the surface soil in front of the trench, but the Duxseal-filled trench had little effect on the dynamic response of the soil on the surface in this area.

The diversity of geofoam properties has increased the scope of its use in such projects. Due to its lightweight, ease of implementation and transportation,these materials are of interest to engineers and designers.In this regard, the present study aims to investigate the effect of geofoam, as cover material, in cut and cover tunnels, on the seismic response of ground surface through the use of numerical modeling.

2 Definition of the problem and numerical modeling

2.1 Definition of the problem

In this study, by using numerical analysis, the effect of geofoam, as cover material, in cut and cover tunnels, on the seismic response of the ground surface is investigated. The effects of parameters such as geofoam blocks thickness (t), geofoam type, interface between geofoam blocks and frequency of incident wave (f) for both harmonic and real motions, on amplification pattern at the ground surface are assessed.

Figure 1 schematically shows the geometry of the problem. A square tunnel with dimensionD(D=2a),is located at depthdfrom the ground surface and subjected to an in-plane shear wave (SV) motion at the base of the model. The influence of EPS geofoam on the seismic response at the ground surface is reflected by obtaining the amplification ratio at the ground surface at different dimensionless distances (X/a).

Fig. 1 Schematic geometry of the problem

2.2 Numerical modeling

The numerical modeling was performed by a twodimensional finite difference software, FLAC 2D v7.0.The two-dimensional model has a height of 50 m and a width of 180 m, with a square tunnel, withD=8 m,in the middle of the model at a specific depth (d=8 m).Due to the fact that cut and cover tunnels are shallow structures, the dimensionless depth (d/a) was selected to be 2, which is equivalent to 8 m surcharge above the tunnel. In the numerical model, soil was considered as viscoelastic material. Shear wave velocity, density and Poisson ratio of a soil material were considered to be 400 m/s, 2000 kg/m3and 0.33, respectively. Soil damping is also considered as Rayleigh damping with a critical damping ratio of 2%. Plane strain rectangular elements were used to mesh the model. Mesh sizes were selected as small enough to properly simulate the wave propagation.For this purpose, according to the recommendations of Kuhlemeyer and Lysmer (1973), the dimensions of the elements are selected to be less than one-tenth of the wavelengths of shear waves propagated in the soil.Therefore, 9000 square elements with 1 m length were considered. In the static analyses, the bottom boundary of the model was fixed in both the horizontal and vertical direction, while the side boundaries were fixed in horizontal direction. In the dynamic analyses, a quiet boundary was used at the bottom of the model and input motion was applied as shear stress to the base of the model. The quiet boundary simulates radiation damping in the model. For lateral boundaries of the model, a freefield boundary is used to prevent reflection of outward waves into the model. The free-field boundary actually comprises a single-column of soil that simulates the behavior of the infinite peripheral environment outside the model.

As cut and cover tunnels have been constructed by the stage construction method, numerical modeling of the tunnel was carried out in several steps to simulate the construction process. In the first stage, initial stress was developed in the soil model. Then, the soil was excavated to the desired depth. To stabilize the excavation, a boundary condition which simulates the retaining system was considered. In this regard, a boundary condition was introduced in the numerical model to simulate the reciprocal support retaining system. Because of the symmetry, at two reciprocals facing the excavation, all nodes with the same height were forced to have the same horizontal displacement at the same direction using the SLAVE command in FLAC. The interaction between the tunnel and the surrounding soil was modeled using the interface elements. The tunnel material was considered to be concrete with a unit weight of 2400 kg/m3and modulus of elasticity of 30 GPa. The thickness of the tunnel lining was obtained to be 0.88 m, considering that the flexibility ratio (J) is equal to 8.

After analyzing the excavation stage, the tunnel was placed and analysis was performed. Figure 2 shows the cut and cover tunnel placement, after the excavation stage, in the numerical model.

Then, to cover the tunnels, backfill soil material and geofoam layers were placed surrounding the tunnel.Figure 3 indicates the tunnel covered with the geofoam blocks as the cover material. Geofoam blocks with 1-m thickness were used. EPS19 was considered in the parametric studies with the modulus of elasticity, density and Poisson ratio equal to 4000 kPa, 18.4 kg/m3and 0.1,respectively. In the models with and without geofoam,the backfill soil properties were assumed to be same as the surrounding soil properties in order to clearly examine the effect of the geofoam.

The slip potential between geofoam blocks is one of the effective factors for controlling movements in highintensity earthquakes (Bartlett and Lawton, 2008). The evaluation of slips between the geofoam blocks is one of the factors influencing the resistance of high-intensity movements during an earthquake or any horizontal forces (Amini, 2013).

The traditional geofoam block placement for embankment construction creates continuous horizontal planes. The interface shear resistance may be insufficient to resist the horizontal driving forces during significant seismic events. If shear resistance in between layers of geofoam blocks against horizontal forces is insufficient,additional interface friction resistance can be provided by mechanical connectors along the horizontal interfaces of the geofoam block (Ozer and Akay, 2016). Some methods are introduced and evaluated in the literature in order to increase shear resistance of the interface between the geofoam blocks:

Fig. 2 Numerical modeling of the cut and cover tunnel

Fig. 3 Numerical modeling of the filling stage in the cut and cover tunnel

(1) The design guideline prepared by Arellanoet al.(2011), which was funded by the National Cooperative Highway Research Program (NCHRP), recommends using mechanical connectors in all horizontal surfaces between geofoam blocks as a conservative approach for geofoam slope systems designed under seismic loading. GeoGripper plate, which is a galvanized steel multibarbed connector, is the most commonly used connector in North America.

(2) Bartlett and Lawton (2008) proposed the shear key concept for geofoam embankments designed for seismic loading.

(3) In order to increase the EPS/EPS interface shear properties, polyurethane adhesives can be used as a binding agent (Barrett and Valsangkar, 2009).

(4) Kurose and Tanaka (1996) introduced an interlocking concept by using “C” and “H” shaped geofoam blocks. In this technique, “C” shaped blocks with the opening upward are placed first and then “H”shape blocks are placed in the opening of the “C” shaped blocks to create interlocked surfaces.

Bartlett and Lawton (2008) evaluated the seismic stability and performance of a freestanding geofoam embankment in Salt Lake City, Utah. They modeled the geofoam embankment using FLAC 2D. FLAC requires Mohr-Coulomb properties and normal and shear stiffness at all interfaces. The required properties include: friction,cohesion, tensile strength, normal stiffness, and shear stiffness. However, they applied no cohesion, dilation,or tensile bond strengths at these interfaces. In addition,the effects of gripper plates, which are commonly placed between geofoam layers during construction, were not considered a significant source of sliding resistance due to their relatively small size and were neglected in the analyses.

In the current study, in order to allow blocks to slip against each other, an interface was used between the geofoam blocks, and the parameters of the interface were derived from the results of shear tests conducted by Bartlett and Lawton (Bartlett and Lawton, 2008).For interaction modeling between the geofoam blocks,the unglued type interface was used with parameters ofKS=2.6 MPa,Kn=7.2 MPa and friction angle of 41 degrees (Amini, 2013).

Before excavation, at-rest earth pressure was developed as the initial stress condition in the model.The contour of the geostatic stress regime in the model is presented in Fig. 4(a). In subsequent stages, the stress distribution in the model was developed by performing the stage construction and analyzing the model under the specified loading conditions (Fig. 4(b)). With the presence of the tunnel in the soil, the amount of stress around the tunnel is changed. Due to the low density of the geofoam material, vertical stress above the tunnel is near zero (Fig. 4(c)).

For dynamic analysis of the model, harmonic inplane shear waves (SV) with different frequencies and real earthquake motions were used. The maximum amplitude of the applied input motions was 0.2 g. In this study, viscoelastic behavior is assumed for the soil material, as considered by the studies used for verification. In this regard, a low level of PGA was considered in the analyses and nonlinear behavior of the soil was not modeled in this study. The input wave was applied to the base of the model in the form of shear stress (σS) time history as presented in Eq. (1). In this equation,ρis the density of the soil,VSis shear wave velocity of the soil, andVSis velocity time history of the incident wave.

In the next section, results of the numerical model are compared with previous studies.

3 Verification of the numerical model

The results of the numerical model were compared with the results from previous research, specifically with Yiouta-Mitraet al. (2007) and Alielahi and Ramezani(2016).

Fig. 4 Vertical stress distribution in different construction stages of the model, a) geostatic stress, b) after excavation and c) placement of tunnel with geofoam cover

3.1 Verification of the numerical model with Yiouta-

Mitraet al. (2007)

To investigate the effect of underground structures on the seismic response of the ground surface, Yiouta-Mitraet al. (2007) conducted a series of plane strain dynamic numerical analyses. A circular tunnel was subjected to harmonic shear waves (SV) in a viscoelastic half-space environment. The numerical model of the problem,modeled in FLAC 2D, is presented in Fig. 5.

The stiffness of a tunnel relative to the surrounding ground is quantified by the flexibility ratio (J), which measures the flexural stiffness (resistance to racking) of the lining relative to the surrounding medium. In circular tunnels, the flexibility ratio can be expressed as follows(St. John and Zahrah, 1987):

in whichEsoilandυsoilare Young′s modulus and Poisson′s ratio of the medium andElining,υlining,Randtare,respectively, Young′s modulus, Poisson′s ratio, radius,and thickness of the tunnel lining.

The dimensionless frequency (η), is ratio of excitation wavelength (λ) to the underground structure diameter (D), which is defined as:

In Eq. (3),VSis the shear wave velocity of the soil,Tis period of harmonic wave andais half of the tunnel width/diameter.

The results of the numerical model are compared with the results of Yiouta-Mitraet al. (2007) in Figs. 6 and 7. The amplification patterns at the ground surface are plotted against dimensionless distance of -18≤X/a≤18 for two values of the flexibility ratio (J=5 andJ=150).Here, the amplification ratio is obtained by dividing the maximum acceleration values in the model with the tunnel to the maximum acceleration at the free field model for eachX/aat the ground surface.

As shown, the developed numerical model captures the trend and values of the amplification ratios obtained in Yiouta-Mitraet al. (2007) for two values of the flexibility ratio.

Considering that the aim of this study is to investigate cut and cover tunnels and these tunnels are mostly box shaped, the model of Alielahi and Ramezani (2016) is also used for verifications of the numerical model.

3.2 Verification of the numerical model with Alielahi and Ramezani (2016)

Alielahi and Ramezani (2016) investigated the effect of a box-shaped underground tunnel on the ground surface seismic response. For numerical modeling,two-dimensional finite-difference software, FLAC 2D,was used. Using parametric analysis, they studied the effects of parameters such as tunnel depth (d/a), tunnel lining flexibility ratio (J), and frequency content of input motion (η) on the amplification at the ground surface.For rectangular tunnels, the following equation has been proposed by Wang (1993) to calculate the flexibility ratio:

In Eq. (4),WandHare the width and height of the rectangular tunnel section, respectively.Gsoilis the shear modulus of the surrounding soil. For simple rectangular tunnels,IRis the moment of inertia for the roof andIWis the moment of inertia of the side walls. The numerical model of the box-shaped tunnel, with 8 m height and 8 m width, based on Alielahi and Ramezani (2016), is shown in Fig. 8. Here, the amplification ratio is obtained by dividing the maximum acceleration values in the model with the tunnel to the maximum acceleration at the free field model for eachX/aat the ground surface.

Fig. 5 Numerical model of Yiouta-Mitra et al. (2007) study,modeled in FLAC 2D

Fig. 6 Comparison of the numerical model results with Yiouta-Mitra et al. (2007) for η=0.2, d/a=2 and J=5

Fig. 7 Comparison of the numerical model results with Yiouta-Mitra et al. (2007) for η=0.2, d/a=2 and J=150

In Fig. 9, the results of the numerical model are compared with the results of Alielahi and Ramezani(2016) forJ=150,d/a=2 and different dimensionless frequencies (η). As can be seen, for all dimensionless frequencies, the results are in close agreement with the study of Alielahi and Ramezani (2016).

Comparison of the numerical model results with the results of Alielahi and Ramezani (2016) ford/a=0.5,J=150 andη=0.1, 0.2 and 0.3 are shown in Fig. 10.Comparative diagrams show an acceptable agreement at-10 ≤X/a≤10. As can be seen, for all three dimensionless frequencies, the numerical modeling results are in close agreement with the study of Alielahi and Ramezani(2016).

The verification results prove that the proposed numerical model can predict the amplification patterns and amplitudes at different distances from the underground structure at the ground surface, for both circular and rectangular tunnels.

4 Parametric studies

To study the effect of geofoam as cover material on the accelerations at the ground surface, two models were used in the parametric studies, as shown in Fig. 11. In the first model, the soil was used as the cover material and in the second model, the geofoam was used as the cover material. The properties of the cover soil in the first model were the same as the surrounding soil material. However,the characteristics of the backfill soil may be different from the in-situ soil, but for the sake of simplicity, it has been assumed that the backfill has the same properties as the in-situ soil. As shown in Fig. 11, the cover material has a thickness of 8 m. Geofoam blocks have a length of 18 m and a thickness of 7 m and cover the top of the tunnel. A thin layer of soil, with 1 m thickness,was used at the top of the geofoam layers in order to simulate a pavement layer at the ground surface.

In the parametric studies, the effect of geofoam on the amplifications has been studied for different values of the dimensionless period. The dimensionless period, , is the ratio of excitation wavelength (λ) to the underground structure width (D), which is defined as:

In this relation,VSis the shear wave velocity of the soil,Tis the period of the harmonic wave andais half of the tunnel width. The inverse of the dimensionless period is known as the dimensionless frequency (η) defined in Eq. (4). Theλ/Dvalues of 3, 6, 8, 10 and 12.5 were considered in the analyses based on previous studies.

Two types of geofoam are considered in the dynamic analysis, to investigate the effect of the type of geofoam on the results. The properties of the geofoams are given in Table 1. The thickness of the geofoam blocks was also considered as one of the parameters that may affect the ground surface response. Geofoam blocks with thicknesses of 1, 2 and 7 m were used in the parametric study. Moreover, the effect of the presence or absence of interfaces between the geofoam blocks was also considered in the parametric study.

The amplification ratio was defined to demonstrate the effect of geofoam blocks on the seismic response at the ground surface. The amplification ratio was obtained by dividing the maximum recorded acceleration at the ground surface in the geofoam model to the maximum acceleration recorded in the soil model at the same point. Amplifications ratios greater than one imply that geofoam amplifies the responses with respect to the soil.Amplifications ratios lower than one means that the geofoam blocks can attenuate the ground response with respect to the soil model. In each model, the amplification pattern was drawn from the range of -20≤X/a≤20 at the ground surface. Note that the fundamental frequency of the models was 2.0 Hz.

Fig. 8 Numerical model of the box-shaped tunnel, studied by Alielahi and Ramezani (2016)

Fig. 9 Comparison of the numerical model results with Alielahi and Ramezani (2016) for d/a=2 and J=150(a) η=0.1, (b) η=0.2 and (c) η=0.3

Fig. 10 Comparison of the results of the numerical model with the study of Alielahi and Ramezani (2016) for d/a=0.5,J=150 (a) η=0.1, (b) η=0.2 and (c) η=0.3

4.1 Effect of geofoam on the amplification ratio

4.1.1 Harmonic waves

In this section, the effect of geofoam blocks, as cover material, on the amplifications at ground surface are investigated. EPS19 was considered as the cover material in the parametric studies of this section. The geofoam blocks have one-meter thickness and are extended in the range of -2.25≤X/a≤2.25 above the tunnel. In Fig. 12,the amplification ratio is plotted againstX/afor different values ofλ/D.

Forλ/D=12.5, the amplification ratio fluctuates around 1 with ±10% tolerance for the range of-20≤X/a≤20, which means that the geofoam blocks do not have a notable effect on the ground response.λ/D=12.5 is related to the motions with longer periods or lower frequencies. Forλ/D=10, the amplification ratio is around 0.6 above the tunnel at the ground surface (-2.5≤X/a≤2.5). This means that geofoam blocks attenuate the ground response about 40% with respect to the soil material. Out of this range ofX/a,the amplification ratio is larger than 1, with a maximum value of 1.1. At larger distances from the tunnel and geofoam, the amplification ratio tends to 1.

Fig. 11 Two models used for the parametric studies (a) soil is used as the cover material, (b) geofoam is used as the cover material

Table 1 Properties of geofoams considered in dynamic analysis

Forλ/D=8, the amplification ratio is less than 1.0 for -3.0≤X/a≤3.0. The amplification ratio is about 0.3 in this distance. This means that geofoam blocks attenuate the ground response about 70% with respect to the soil material. This implies the effectiveness of geofoam blocks to attenuate the ground surface response. Out of this range ofX/a, the amplification ratio becomes larger and reaches 1 atX/a=±5. The maximum value of the amplification ratio occurs at 1.2 atX/a=±15.

Forλ/D=6, the amplification ratio is about 0.2 above the tunnel at the ground surface (-3.0≤X/a≤3.0).This means that geofoam blocks attenuate the ground response about 80% with respect to the soil material. Out of this range ofX/a, the amplification ratio is larger than 1, with a maximum value of 1.36. At larger distances from the tunnel and geofoam, the ground response is attenuated in the presence of the geofoam.

Forλ/D=3, the amplification ratio is about 0.1 above the tunnel at the ground surface (-3.0≤X/a≤3.0). This means that the geofoam blocks attenuate the ground response about 90% with respect to the soil material for the high frequency motions. Out of this range ofX/a, the amplification ratio is less than 1. The geofoam attenuates the seismic response at the ground surface in all dimensionless distances of -20≤X/a≤20 forλ/D=3.

It can be concluded that in all dimensionless periods,the presence of geofoam above the tunnel reduces the horizontal acceleration of the ground surface in the range of -2.5≤X/a≤2.5. This distance is the range which the geofoam is extended above the tunnel. As shown in the diagrams, attenuation increases as the dimensionless period decreases. The maximum attenuation ratio was equal to 0.10 atλ/D=3. This parametric study reveals the potential of the geofoam material to decrease the seismic response at the ground surface due to an underground tunnel.

4.1.2 Earthquake motions

Four different earthquake motions were selected and used in the analyses to investigate the effect of the geofoam on the ground surface acceleration, for using actual motions. The motions with predominant frequency in the frequency range of the harmonic waves were selected, because the tunnel effects on the ground surface are predominant in this frequency range (3≤λ/D≤12.5). The specifications of the selected motions are tabulated in Table 2. The PGA of the motions was scaled to 0.2 g in the numerical analyses.

Figure 13 illustrates the amplification pattern for four different earthquake motions. As seen from the figure,the patterns are similar to the patterns obtained for the harmonic waves. However, the maximum and minimum values of the amplification ratios differ from motion to motion. For the Niigata earthquake motion, geofoam has better performance in attenuating the ground surface accelerations, when compared to the other motions.The Umbria and Big Bear motions have almost the same amplification pattern. For the Izmir motion, the geofoam has little effect on the amplifications. As seen,the geofoam attenuates the real motions above the tunnel with the maximum attenuation ratio of about 0.2. The patterns of the amplification and amplification ratios strongly depend on the frequency content of the motions.According to the earthquake characteristics in Table 1, for earthquakes with higher predominant frequency(lowerλ/D), such as the Niigata motion, the performance of the geofoam is more pronounced in attenuation. As seen for harmonic input motions, as the frequency of the motion is increased, the geofoam presents a greater attenuation ratio at the ground surface. This finding is observed for real earthquake motions.

4.1.3 Discussion on the geofoam attenuation potential There are several issues involved in discussing

Fig. 12 Amplification ratio vs X/a for different values of λ/D

Fig. 13 Amplification ratio vs X/a for the real motion

Table 2 Characteristics of the selected earthquake motions

the effect of geofoam on the attenuation of the ground response, which are: impedance ratio, damping of the geofoam material and the interface between layers of geofoam.

The impedance ratio (I) is given in Eq. (6), whereρ1,V1,ρ2andV2are density and shear wave velocity of layers 1 and 2, respectively.

When the impedance ratio is less than one, it means that the wave enters a softer environment. In this case,the amplitude of the transmitted waves will reduce.In other words, as the impedance ratio decreases, the amplitude of the transmitted wave decreases. In this study, the impedance ratio is equal to 0.007 for EPS19(layer 2) with respect to the soil (layer 1). Therefore,the input wave experiences attenuation of the amplitude when it propagates from the soil to the geofoam layer.

Based on an experimental study conducted by Athanasopouloset al. (1999) on dynamic properties of EPS geofoam, it was concluded that the damping ratio values of the EPS geofoam are very low (< 1.5%) for strains less than 1%; however, the energy absorbing characteristics become pronounced for greater strains and reach a value of 10% for strains on the order of 10%. This finding suggests that the efficiency of EPS geofoam for dissipating low-strain vibrations is due to its low stiffness values rather than its capability to absorb vibration energy. Therefore, the attenuation potential of the geofoam is primarily due to the low values of its impedance ratio.

Regarding the interface between the geofoam layers, Amini (2013) studied the internal sliding evaluations to evaluate the internal stability of an EPS embankment for seismic and other horizontal loadings.The study showed that the large difference between the maximum acceleration values calculated at the top of the EPS geofoam embankment is caused by the relative movement of the EPS at the interfaces, which appears to be a very efficient energy dissipating mechanism.

In the subsequent sections, the effect of the main parameters on the geofoam attenuation potential is investigated.

4.2 Effect of geofoam blocks thickness on the amplification ratio

As another parameter of the geofoam, thickness of each geofoam block (t) on the amplification ratio at ground surface was investigated. Three different values were selected for the thickness of the blocks, equal to 1, 2 and 7 m. The models corresponding to 1 m, 2 m and 7 m thickness of the geofoam blocks are presented in Figs. 14(a), 14(b) and 14(c), respectively. EPS19 was considered as the cover material in the parametric studies. The amplification ratio for these models with different thicknesses of the geofoam blocks was obtained forλ/D=3, 6, 8, and 10, and plotted versusX/a.

Figure 15(a) shows the amplification ratio for three models with different thickness of geofoam block (t) forλ/D=3. In a model witht=1 m, the amplification ratio is close to 0.1 in the center of the tunnel at -3≤X/a≤3.However, the amplification ratio is about 2.0 for the models witht=2 m andt=7 m. Two different behaviors can be seen because of the difference in the thickness of the geofoam atλ/D=3.

Figure 15(b) shows the amplification ratio for three models with different thicknesses of the geofoam block(t) forλ/D=6. In a model witht=1 m, the amplification ratio is close to 0.2 in the center of the tunnel. The amplification ratio is about 1.4 and 0.6 for the models witht=2 m andt=7 m, respectively. Therefore, as the thickness of geofoam blocks is decreased, its potential to attenuate the seismic response at the ground surface is increased. The amplification ratio is less than 1 for-3≤X/a≤3. ForX/afrom 3 to 15, the amplification ratio is increased with the maximum value of 1.5. For larger distances, the amplification ratio is decreased and reaches to 1. However, the effect of geofoam thickness on the amplification ratio is prominent at -3≤X/a≤3 and its effect becomes less important at larger distances.

Figure 15(c) shows the amplification ratio for three models with different thicknesses of the geofoam block(t) forλ/D=8. In a model witht=1 m, the amplification ratio is close to 0.3 above the tunnel. The amplification ratio is about 0.5 and 0.7 for the models witht=2 m andt=7 m, respectively. Therefore, as the thickness of the geofoam blocks is decreased, its potential to attenuate the seismic response at the ground surface is increased.The amplification ratio is less than 1 for -3≤X/a≤3.ForX/alarger than 5 to 15, the amplification ratio is increased with the maximum value of 1.1. For larger distances, the amplification ratio is decreased. However,the effect of geofoam thickness on the amplification ratio is prominent at -3≤X/a≤3 and its effect becomes less important at larger distances.

Figure 15(d) shows the amplification ratio for three models with different thicknesses of geofoam block (t)forλ/D=10. In a model witht=1 m, the amplification ratio is close to 0.4 in the center of the tunnel. The amplification ratio is approximately 0.6 and 0.8 for the models witht=2 m andt=7 m, respectively. Therefore,as the thickness of geofoam blocks is decreased, its potential to decrease the seismic response at the ground surface is increased. The amplification ratio is less than 1 for -3≤X/a≤3. ForX/alarger than 3, the amplification ratio fluctuates around 1.0 with maximum 10%tolerances. However, the effect of geofoam thickness on the amplification ratio is prominent at -3≤X/a≤3 and its effect becomes less important at larger distances.

4.3 Effect of geofoam type on the amplification ratio

To study the effect of geofoam type on the amplification ratio at the ground surface, two types of geofoam (EPS19 and EPS12) have been used in the parametric study with the specifications given in Table 3.

The amplification ratio at the ground surface for two models with EPS19 and EPS12 as cover material is depicted in Fig. 16 for different values ofλ/D. As can be seen, EPS12 has more attenuation potential at ground responses with respect to the EPS19 for -2.5≤X/a≤2.5.ForX/alarger than 2.5, the effect of geofoam type on the amplification ratio is eliminated and the responses are the same for the two types of geofoam. Additionally, with reference to Fig. 16, as theλ/Dis increased (long period motions), the type of geofoam becomes more prominent.

According to Table 3, impedance ratios are equal to 0.007 and 0.003 for EPS19 and EPS12 with respect to the soil, respectively. As the impedance ratio decreases,the amplitude of the transmitted wave stress decreases.According to the numerical modeling results, EPS12 with a lower impedance ratio shows better performance in decreasing the responses with respect to EPS19. Zarnani and Bathurst (2009) investigated the performance of EPS geofoam on earthquake-induced loads acting on rigid retaining wall structures. They demonstrated that lower densities of EPS will produce higher isolation efficiencies because they are less stiff and hence produce a more complete compressible inclusion.

4.4 Effect of the interface between geofoam blocks on the amplification ratio

Fig. 14 Numerical models related to the geofoam blocks with different thicknesses (a) t=1 m (b) t=2 m (c) t=7 m

Fig. 15 Amplification ratio versus X/a, considering different values of geofoam thickness (t), (a) λ/D=3, (b) λ/D=6, (c) λ/D=8 and(d) λ/D=10

Table 3 Specifications of the geofoams

According to the study performed by Amini (2013),the presence and absence of an interface between the geofoam blocks can affect the seismic responses at the ground surface. Therefore, two different models were considered to investigate the effect of the interface between the blocks on the amplification ratio: a model with the interface between the blocks (Fig. 17(a)) and the other without the interface between the blocks (Fig. 17(b)).EPS19 was considered as the cover material in the parametric studies.

Fig. 16 Comparison of the amplification ratio for EPS12 and EPS19 (a) λ/D=3 (b) λ/D=6 (c) λ/D=8 (d) λ/D=10

Fig. 17 Geometry of the models (a) with the interface between the blocks, (b) without the interface between the blocks

Fig. 18 Comparison of the amplification ratio for models with/without interface (a) λ/D=3, (b) λ/D=6, (c) λ/D=8, (d) λ/D=10

Figure 18 presents a comparison of the amplification ratio for models with and without interfaces between the geofoam blocks. The comparison is performed forλ/D=3, 6, 8 and 10. As can be seen, the amplification ratio is mainly affected by the presence of the interface.In models with an interface, the responses are attenuated for -2.5≤X/a≤2.5 up to 90%; while for models without an interface, the maximum attenuation is about 20%.ForX/alarger than 5, the effect of the interface on the amplification ratio is eliminated and the responses are the same as the models with and without interface. The same trend can be found for the otherλ/Dconsidered herein.

Amini (2013) found that the large difference between the maximum acceleration values calculated at the top of the EPS geofoam embankment is caused by the relative movement of the EPS at the interfaces, which appears to be a very efficient energy dissipating mechanism. In the current study, it is demonstrated that the interface between the geofoam layers plays a significant role in the attenuation potential of the geofoam in seismic events.

5 Conclusion

This study investigates the effect of geofoam as cover material in cut and cover tunnels on the seismic response of the ground surface through the numerical approach. After verification of the numerical model with the results from previous research, a series of parametric studies were carried out to explore the effect of the main parameters of geofoam on the amplification ratio.

It was concluded that geofoam has strong potential to reduce the seismic horizontal movements of the ground surface. The effect of geofoam on the amplification ratio is more prominent at the distances right above the geofoam layers and its effect becomes less important at farther distances. The type of geofoam, thickness of geofoam blocks, interface between the blocks and dimensionless period (λ/D) are the main parameters that dictate the geofoam effect on the ground surface response. Among the parameters investigated in this study, the effect of the interface between the layers and thickness of the blocks are more prominent than the type of geofoam on the accelerations at the ground surface. As the thickness of the geofoam blocks is decreased, higher attenuation ratio of the response at the ground surface is achieved. It was also found that the interface between the geofoam blocks plays a key role in decreasing the seismic responses above ground. In the current study,an attenuation ratio of up to 90% was achieved by the geofoam blocks.

Based on the results of this study, it is concluded that geofoam is a proper material to attenuate the seismic amplifications at the ground surface, induced by an underground tunnel in seismic loading. However, note that although the parametric studies were performed for cut and cover tunnels, this mitigation technique is applicable to other types of underground structures and utilities.