Evaluating the Role of Geofoam Properties in Reducing Lateral Loads on Retaining Walls: A Numerical Study

: Expanded polystyrene (EPS) geofoam is a lightweight compressible material that has been widely used in various civil engineering projects. One interesting application of EPS in geotechnical engineering is to reduce the lateral earth pressure on rigid non-yielding retaining walls. The compressible nature of the EPS geofoam allows for the shear strength of the backﬁll soil to be mobilized, which leads to a reduction in lateral earth pressure acting on the wall. In this study, a ﬁnite element model is developed and used to investigate the role of geofoam inclusion between a rigid retaining wall and the backﬁll material on the earth pressure transferred to the wall structure. The developed model was ﬁrst calibrated using experimental data. Then, a parametric study was conducted to investigate the effect of EPS geofoam density, relative thickness with respect to the wall height, and the frictional angle of backﬁll soil on the effectiveness of this technique in reducing lateral earth pressure. Results showed that low-density EPS geofoam inclusion provides the best performance, particularly when coupled with backﬁll of low friction angle. The proposed modeling approach has shown to be efﬁcient in solving this class of problems and can be used to model similar soil-geofoam-structure interaction problems.

Retaining structures are integral components of many important structures including bridges, ports, highways, railway, and underground structures. These retaining structures are typically designed to resist lateral earth pressure exerted by the surrounding soil mass. The design of a retaining wall is strongly related to the magnitude and distribution of earth pressure acting on the wall. The higher the earth pressure, the more the cost of the wall structure. Various methods have been suggested to reduce the magnitude of lateral earth pressure on retaining walls. Pressure reduction can be achieved by allowing controlled yielding of the backfill soil by introducing compressible material between the wall and backfill soil. Partos and Kazaniwsky [30] introduced EPS geofoam of low stiffness between a relatively higher stiffness retaining structure and the retained soil mass (see Figure 1). The least stiff material (EPS geofoam) generally compresses more than the other two materials, which causes reduction in the lateral earth pressure due to the mobilization of the shear strength within the adjacent soil [27]. Depending on the response of the earth retaining structures under the applied lateral loading, retaining walls can be broadly classified as follows: (i) Non-yielding retaining walls, when the structure is not allowed to displace or deform horizontally under the applied loads. These structures are typically designed based on at-rest earth pressure condition. (ii) Yielding retaining walls, when the structure is allowed to displace or deform horizontally under the applied load. Mostly, these structures are designed for active earth pressure condition. Compressible geofoam inclusion is generally more beneficial for non-yielding (rigid) retaining walls.
Researchers [30][31][32][33][34][35] have shown that the magnitude of the lateral loads on retaining walls can be reduced by installing a material of lower stiffness between the backfill soil and the wall structure, without increasing the wall stiffness. Partos and Kazaniwsky [30] reported a reduction in lateral earth pressure on a non-yielding basement wall by placing a prefabricated expanded polystyrene beaded drainage board between the wall and the granular backfill. Karpurapu and Bathurst [35] used numerical modeling to demonstrate that installing a geofoam layer with thickness that is equal to 1% of the wall height can bring the lateral earth pressure to a minimum level. Hatami and Witthoeft [36] extended the geofoam application to geosynthetic reinforced soil walls. Numerical results confirmed that the total forces that cause external sliding and overturning decreased by about 30% and 25%, respectively. Trandafir et al. [37] concluded that the presence of EPS geofoam within the upper half of the wall can help with load reduction.
Ertugrul and Trandafir [34] studied the behavior of geofoam inclusion behind flexible walls. The results indicated that geofoam inclusion behind a retaining wall can cause a significant decrease in lateral thrust. This was attributed to the relative stiffness between the wall and the backfill material. Azzam and AbdelSalam [38] carried out a numerical analysis to investigate the performance of rigid yielding retaining walls with geofoam inclusion. The results showed that use of EPS significantly reduced the lateral pressure depending on the ratio between the EPS thickness and wall height. Chauhan and Dasaka [33] showed that geofoam inclusion could provide 8-42% reduction in lateral pressure for a surcharge of 10-50 kPa. Hasanpouri Notash and Dabiri [39] investigated the performance of geofoam in yielding and non-yielding retaining structures. The results showed that EPS geofoam performs better in non-yielding retaining walls. Yadav et al. [40] concluded that the presence of EPS geofoam reduces the pressure on retaining walls under both static and seismic condition.
It has been observed that limited numerical studies have been performed to simulate and expand the results of the available physical tests conducted on rigid non-yielding retaining walls under static conditions. In addition, in most of the reported studies, a few of the numerical results have been compared with measured data based on instrumented large-scale physical models. Therefore, this study is a step forward towards developing and validating a numerical model using experimental data and using the model to investigate the role of geofoam properties on the performance of rigid retaining structures.
This study focuses on the use of EPS geofoam in reducing lateral pressure on rigid non-yielding retaining walls. A two-dimensional finite element model is developed and validated using the experimental data reported by Ertugrul and Trandafir [41]. Then, a parametric study is performed to examine the role of relative thickness, stiffness of the EPS geofoam, and strength parameters of the backfill soil on the lateral earth pressure acting on rigid retaining wall. The results of this study are presented in the form of normalized charts covering a range of wall configurations and material properties.

Description of the Physical Model
Ertugrul and Trandafir [41] performed a series of experiments on a rigid non-yielding retaining wall with vertical EPS geofoam inclusion. A steel wall with dimensions 700 × 980 × 8 mm was firmly welded to a steel base 980 × 500 × 8 mm and hosted in a rigid container 2 × 1 × 1 m. A schematic of the test setup and the locations of the pressure transducers (P1 to P4) are shown in Figure 2. Dry, clean sand was used as backfill material. Triaxial compression tests conducted on sand samples of relative density, D r , of 70% indicated friction and dilatancy angles of 43.5 • and 22.5 • , respectively. Blocks of EPS15 geofoam (15 kg/m 3 ) were installed between the wall and backfill. The properties of the sand and the EPS geofoam materials are summarized in Table 1.
3.31 --Percent finer than #200 sieve 1.14 --Experiments were conducted on four different wall configurations. In the first set of tests, lateral earth pressures were measured against the rigid wall with no geofoam inclusion, whereas the other sets of tests involved three different geofoam thicknesses that corresponded to t/H ratios of 0.07, 0.14, and 0.28, where t is the geofoam thickness and H is the wall height. The geofoam was placed behind the rigid retaining wall and the earth pressures were measured using four pressure transducers installed vertically along the wall height at 20 cm spacing, as shown in Figure 2.

Numerical Analysis and Model Validation
A two-dimensional finite element model was developed based on the experimental setup reported by Ertugrul and Trandafir [41]. The analysis was performed using the commercial software PLAXIS 2015 [42]. The retaining wall and the backfill material were modeled using 15-node triangular elements, as shown in Figure 3. Several element sizes were tested to find a suitable mesh that maintained the balance between accuracy and computation time. Smooth rigid boundary conditions were specified along the vertical boundaries, whereas rough rigid boundary conditions were specified along the base of the model. The backfill and foundation soil were both modeled as Mohr-Coulomb elastoplastic materials, whereas EPS geofoam was modeled as linear elastic material. Given that the EPS15 geofoam used in the experiments was subjected to a maximum horizontal pressure of about 3.6 kPa, which is much smaller than the yield stress (σ y = 38 kPa), the use of elastic material model for the geofoam is considered acceptable. Table 1 summarizes the material properties of both the soil and the EPS geofoam used in validating the numerical model. The retaining wall and its base were modeled as plate elements. The material properties of the wall and the base are given in Table 2.
The soil-wall, geofoam-wall, and soil-geofoam interfaces were modeled using Mohr-Coulomb elastoplastic interface elements. The interface friction angles were back calculated to match the experimental data. For the geofoam-wall, soil-geofoam, and soil-wall interfaces, the interface friction angles were found to be 16 • , 8 • , and 5 • , respectively. The EPS15 blocks of similar dimensions and material properties to those used in the experiments were adopted in the model with three different geofoam thicknesses that corresponded to t/H ratios of 0.07, 0.14, and 0.28.
The steps taken in modeling the retaining wall can be summarized as follows: (1) the foundation bed is first generated, (2) the wall and the base are activated, (3) the EPS blocks are introduced against the wall, (4) the backfill material is added in layers to reach the target height.  Four cases were numerically investigated starting with the reference model where no geofoam blocks were installed (t/H = 0).
The model validation results are presented in Figure 4, where the lateral earth pressure values, σ h , are normalized with respect to the vertical pressure at the wall base (σ v-base = γH = 11.55 kPa) and are plotted against the wall depth. For the four investigated cases, the calculated lateral pressures are compared with the experimental data. The results show that the numerical model is generally able to capture the pressure distribution with depth. This is particularly true for the case where no geofoam is installed, as represented by the solid line in Figure 4. For the cases where geofoam is placed behind the wall, the difference between the measured and the calculated stresses near the wall base is attributed to the interface friction that develops at the soil base as well as at the geofoam base. This in turn can lead to significant reduction in lateral pressures near the base of the wall.
For all investigated wall models, EPS geofoam experienced maximum horizontal stress, σ hmax , of 2.5, 2.0, and 1.3 kPa for t/H of 0.07, 0.14, and 0.28, respectively. This is found to be within the elastic limit of the used EPS15 geofoam, which has a compressive resistance of about 19 kPa at 1% strain [41]. This also confirms the assumption of linear elastic response for the EPS geofoam under the applied loading. It is also found that the presence of geofoam inclusion behind the non-yielding retaining wall resulted in an arching effect within the lower half of the wall, which helped to absorb the majority of the lateral earth pressure acting on the wall. The total thrust on the wall was calculated by computing the area under the pressure profile. The difference between the experimental data and the numerically calculated total lateral thrust was found to range between 5% and 9% for the four investigated cases.

The Effect of Geofoam and Backfill Properties on the Lateral Earth Pressure Acting on the Wall
A parametric study was conducted using a full-scale model to investigate the effect of geofoam density, thickness, and backfill friction angle (ϕ) on the lateral earth pressure acting on the wall. A 2.80 m high retaining wall model was developed using 15-node plane strain triangular elements. A wall (2800 × 3920 × 32 mm) with rigid base (3920 × 2000 × 32 mm) was modeled using plate elements. The numerical parameters used to model the wall and the base are summarized in Table 2. The foundation and backfill soils were modeled using Mohr-Coulomb elastoplastic material model. In addition, Mohr-Coulomb elastoplastic interface elements were used at the wall-soil, wall-geofoam, and soil-geofoam interfaces. For the wall-geofoam, geofoam-soil, and wall-soil interfaces, a range of friction angles between 8 • and 22 • was used in the analysis to cover wide possibilities of contact properties.
Before conducting the parametric study, a convergence analysis was performed using different mesh sizes for three cases involving geofoam panels with relative thickness of 0.1, 0.2, and 0.3 with respect to the wall height. The calculated displacements at a depth of 2.4 m are presented in Figure 5. The results indicate that, depending on the thickness of the geofoam, the sensitivity of the response to the mesh size becomes insignificant when the number of nodes reaches about 28,000, 30,000, and 33,000 for t/H of 0.1, 0.2, and 0.3 respectively. The properties of finite element meshes used in this study are summarized in Table 3.
Three different geofoam materials, namely EPS22, EPS29, and EPS39, of different relative thicknesses with respect to the wall height (t/H = 0, 0.05, 0.1, 0.15, 0.2, 0.25, and 0.3) were considered. The results of the uniaxial compressive strength tests performed on the three geofoam materials are depicted in Figure 6. For the three geofoam materials, with densities of 22, 29, and 39 kg/m 3 , the reported compressive strength results at 1% strain are found to be 70, 94, and 192 kPa, respectively. Given the expected range of lateral pressure (from 3 to 23 kPa), the geofoam materials are subjected to pressure levels that are within the linear elastic range. Material properties of both the soil and geofoam used in this parametric study are summarized in Table 4. It is worth noting that the material properties, aspect ratio, and boundary conditions used in the numerical model of the large-scale wall are similar to those used in the model validation shown in Figure 3.

Results and Discussion
The results of the numerical analysis are presented in this section by comparing the changes in horizontal pressure with depth for the benchmark case (no geofoam) with those obtained when geofoam blocks of different thicknesses and densities are installed. The calculated lateral earth pressure distributions with depth are presented using four sets of charts grouped in Figure 7a

Benchmark Case (No Geofoam Inclusion)
By inspecting the results in Figure 7a-f, it is evident that for backfill materials with lower friction angle values, 30 • and 35 • , the calculated earth pressure distributions were generally found to linearly increase with depth consistent with that of the theoretical at-rest earth pressure (J. AREP) line. However, for backfill with higher friction angles of 40 • and 45 • (Figure 7g-l) the lateral pressure distributions change from linear to nonlinear near the middle of the wall.
As the friction angle increased from 30 • to 45 • , the at-rest earth pressure coefficient decreased resulting in a decrease in lateral pressure from 22.4 to 13.4 kPa. This can be attributed to the frictional forces developing at the interface between the backfill and the base plate supporting the wall, which differs from the theoretical solution obtained for semi-infinite soil medium.

Effect of Geofoam Density
By inspecting Figure 7a-l, it is evident that for a given geofoam thickness and backfill properties, the density of the geofoam panel has significant effects on the lateral pressure acting on the wall. It was found that, for a given type of backfill material, geofoam of lower density absorbs more lateral movement and can bring the soil into an active or near active state as compared with higher density geofoam. For example, for t/H = 0.1 and ϕ soil = 30 • , the maximum pressure decreased from 12.4 kPa to 10.8 kPa as the geofoam density decreased from 39 kN/m 3 to 22 kN/m 3 . This is consistent with the fact that geofoam of lower density compresses more under the same applied lateral pressure resulting in more soil movement and, consequently, less pressure on the wall. This observation confirms that, for the investigated wall configuration and applied pressure levels, low density geofoams are considered to be effective in absorbing static lateral pressure and enhance the overall performance of the system.

Effect of Geofoam Thickness
For a given geofoam density and backfill material, the effect of geofoam thickness is examined in this section by plotting the lateral pressure results for different relative geofoam thickness (t/H), as depicted in Figure 7a-l. Using geofoam panels of smaller t/H value is found to absorb less soil movement as compared with the cases when thicker geofoam of the same density is used. For example, for a geofoam density of 22 kg/m 3 (EPS22) and ϕ soil = 30 • , the maximum earth pressure decreased from about 10.8 to 8.4 kPa as the relative thickness of the geofoam (t/H) increased from 0.1 to 0.3. This is attributed to the fact that thicker geofoam can compress more under the same applied lateral pressure, which in turn produces more soil movement and less pressure on the wall. However, as increasing the thickness of the geofoam can result in adding more costs to the project, a balance is needed between the additional cost of the geofoam material and the design benefits associated with the reduction in lateral earth pressure on the wall.

Effect of Friction Angle of Backfill Soil
The magnitude of lateral earth pressure is found to be also influenced by the friction angle of the backfill soil, ϕ soil . Soil with a high friction angle value exerts less pressure on the retaining wall due to a smaller coefficient of active earth pressure. As shown in Figure 7a-l, soils with smaller friction angle values induced more lateral pressure as compared to those with higher friction angle values for the same geofoam density and thickness. For example, for geofoam with density of 22 kg/m 3 (EPS22) and t/H = 0.1, soil with ϕ soil = 30 • corresponded to 38% reduction in pressure while soil with ϕ soil = 45 • corresponded to a 28% reduction in lateral earth pressure. This indicates that the use of EPS geofoam inclusion is effective in soils that have lower friction angle values.
It is important to mention that although the presence of geofoam inclusion behind rigid retaining walls caused a reduction in lateral earth pressure, there exist a limiting combination of geofoam density and relative thickness of the geofoam (t/H) to achieve active state, which depends on the frictional properties of backfill material, as depicted in Figure 7a-l. For example, for backfill with ϕ soil = 40 • , active condition can be achieved using t/H = 0.1 for EPS22, t/H = 0.2 for EPS29, or t/H = 0.3 for EPS39. Therefore, depending on the frictional properties of the backfill material and the tolerable pressure on the retaining wall, a suitable geofoam density along with required t/H value may be selected to achieve active or near active conditions in the backfill.

Isolation Efficiency (I E )
Another way to evaluate the performance of EPS inclusion is by expressing the results in terms of isolation efficiency. Isolation efficiency (I E ) is defined as "the ratio of the difference between the total lateral thrust on the wall before and after introducing the geofoam (T o − T EPS ) divided by T o " and can be calculated as follows: The total lateral thrust acting on the wall can be computed using the following equation: where T is the total lateral thrust acting on the wall, σ x is the lateral earth pressure, and H is the wall height. Figure 8 shows the effects of density and relative thickness of the geofoam and the frictional properties of the soil on the isolation efficiency. The results show that for the investigated four soil types, the isolation efficiency increases with an increase in the relative geofoam thickness. It was also found that the isolation efficiency increased with a reduction in the friction angle of the backfill material. For example, for geofoam density of 22 kg/m 3 (EPS22) and t/H of 0.1, the isolation efficiency decreased from 38% to 28% as the friction angle of the backfill soil increased from 30 • to 45 • . This is due to the fact that higher friction angles (ϕ soil ) correspond to a smaller at-rest earth pressure coefficient, which induces smaller lateral pressures on the geofoam and the wall. The maximum isolation efficiency is achieved with EPS22 for backfill soil with ϕ soil = 30 • , whereas the minimum value of isolation efficiency is obtained with EPS39 for backfill soil having ϕ soil = 45 • . This response is consistent with an increase in the material stiffness associated with an increase in the geofoam density [44][45][46].

Lateral Earth Pressure Coefficient Ratio
The lateral earth pressures obtained from the numerical models were used to back calculate the coefficient of lateral earth pressure K FEM based on following equation: where γ is the unit weight of soil, σ x is lateral earth pressure, and H is the wall height. Then, the K FEM values were normalized using Rankine active earth pressure coefficient, K a , and presented in Figure 9.   . A K FEM /K a ratio of less than 1 means that the lateral earth pressure values are less than the corresponding Rankine's active earth pressure. In addition, for a given soil friction angle, a decreasing trend in K FEM /K a ratio with an increase in t/H indicates a better geofoam performance. For example, for ϕ soil = 30 • , maximum performance is achieved with EPS22 at t/H = 0.3 where the K FEM /K a ratio is minimum.
It is important to note that an increasing trend in K FEM /K a ratio with an increase in friction angle should not be interpreted as "K FEM increases with an increase in friction angle (ϕ soil )". This is due to the fact that K a is not constant for the four investigated backfill soils (ϕ soil = 30 • , 35 • , 40 • , and 45 • ) as K a decreases with an increase in friction angle.

Horizontal Displacement in Backfill Soil
Figure 10a-d present the effect of geofoam density, relative thickness, and backfill properties on the horizontal displacement obtained in the backfill soil. It is found that for a given backfill soil, the horizontal displacement increases with an increase in t/H ratio and decreases with an increase in geofoam density. Moreover, for backfill with different frictional properties, horizontal displacement decreases with an increase in friction angle. For example, for a soil with ϕ soil = 30 • , geofoam density of 22 kg/m 3 , the horizontal displacement decreased from 1.34 mm to 0.23 mm as t/H decreased from 0.3 to 0.05. This is due to the fact that thicker geofoam inclusion allows for more mobilization of the soil strength as compared with geofoam of reduced thickness. The maximum horizontal displacement (1.34 mm) is achieved for the case of EPS22 and ϕ soil of 30 • , whereas the minimum horizontal displacement (0.24 mm) is achieved for the case of EPS39 and ϕ soil of 45 • .

Practical Implications
The results of this parametric study suggest that the isolation efficiency, I E , associated with a case of geofoam inclusion behind a rigid wall depends on the used geofoam density and thickness and the frictional properties of the backfill material. Therefore, the normalized charts presented in this study can help for the preliminary assessment of the various design options that could guide engineers in their design. The charts presented in Figure 8 can be easily adapted and used to select a suitable geofoam density and thickness to achieve a specific isolation efficiency. To illustrate this procedure, an example is given below as follows: For H = 4.0 m, ϕ soil = 35 • , I E = 40%, and EPS22 geofoam, the thickness, t, of the geofoam can be found as follows: For I E = 40% Therefore, the nominal EPS geofoam thickness (t) that satisfies 40% reduction in pressure is 47 cm.

Conclusions
In the present study, the effect of EPS geofoam inclusion on a rigid non-yielding retaining wall is numerically investigated using plane strain finite element modeling. Experimental data obtained using controlled laboratory tests were used to validate the numerical model. Then, a parametric study was performed to examine the effects of EPS geofoam density and relative thickness and the properties of the backfill soil on reducing the lateral earth pressure acting on a non-yielding retaining wall. The investigated parameters included three different geofoam densities and thickness interacting with four different backfill soils. The following conclusions can be drawn from this study:

1.
Geofoam inclusion placed vertically behind rigid non-yielding retaining walls can allow the backfill soil to move towards the wall. This deformation helps in mobilization the soil shear strength leading to a reduction in lateral earth pressure acting on the wall. 2.
The response of the granular backfill soil in these applications can be reasonably predicted using a Mohr-Coulomb elastoplastic material model, whereas a linear elastic model is found to be suitable for the geofoam material.

3.
Relative thickness and density of the EPS geofoam and the frictional properties of the backfill material are found to play major roles in the magnitude of the isolation efficiency. It is also found that low density geofoam can provide better performance as compared with higher density material. For the same geofoam density, the thickness of the geofoam inclusion is found to have an effect on the geofoam compression. It is also found that EPS geofoam inclusions are more effective in soils with relatively low friction angles.

4.
It is noted that possible long term creep effect of the geofoam is not considered in this study and the results are applicable to rigid non-yielding retaining walls supporting dry backfill. Informed Consent Statement: Not applicable.

Data Availability Statement:
The data that support the findings of this study are available on request from the corresponding author.