3.1. Mechanical Model of the GRS Segmental Wall
In GRS retaining walls with segmental block facings, the integral thrust behind the wall facing is theoretically counterbalanced by the reinforcement connection loads and toe resistance if the facing is considered a beam. Thus, the mechanical model of the GRS segmental wall can be assumed as the following:
where
Fe = the integral thrust behind the facing, Σ
Tcon = the total reinforcement connection load, and
Ft = the horizontal toe load. Experimental and numerical studies have demonstrated that toe restraint condition has a significant impact on the loads carried by the wall toe and reinforcement layers [
9,
19,
20,
21,
30,
32,
33].
The accuracy of Equation (3) and the impact of toe restraint on this equation was quantitatively analyzed using a numerical approach. The numerical model of the SR-18 wall was modified to have a more typical geometry (
Figure 7) and more typical material parameter values (
Table 4) for this analysis. The embedment soil at the toe was not included in the baseline numerical model, as shown in
Figure 7, considering that the passive soil resistance in front of the toe is usually ignored in the conventional design method [
15]. This is because the passive soil resistance may be lost unexpectedly and unpredictably by scouring and excavation during a long period of service. Nonetheless, a comparison of the connection loads in the SR-18 walls with and without the embedment soil is presented to demonstrate the effects of the embedment soil.
Figure 8 shows that the influence of the embedment soil on the connection loads in the SR-18 wall can be ignored. Bathurst and Naftchali [
45] reported the range of creep stiffnesses at a 2% strain for HDPE geogrids was approximately 330–1120 kN/m. The average value of this range (i.e., 725 kN/m) was selected for this numerical model. The value of the interface friction angle between reinforcement layers and backfill
φsr, as shown in
Table 4, was determined via the formula
φsr = tan
−1 (2/3 × tan
φ), where
φ = the backfill friction angle [
41]. It is noted that parameters with the same values as those of the SR-18 wall (as shown in
Table 1,
Table 2 and
Table 3) were not listed in
Table 4.
The toe restraint condition was quantified using the interface friction angle between the lowest facing block and the leveling pad. The shear strength of the leveling pad–foundation soil interface was not considered in this analysis as a result of its small effect on the capacity of the wall toe to carry load in cases where the leveling pad was embedded in the foundation soil. Zhang and Chen [
46] found that, for GRS walls with embedded leveling pads, only the shear resistance at the facing block–leveling pad interface acts as the toe resistance to counterbalance a portion of the horizontal earth load because the passive soil resistance in front of the leveling pad inhibits the development of shear stress and displacement at the base of the leveling pad. In cases of an exposed leveling pad, it is the leveling pad–foundation soil interface that works to carry the earth load because the wall is more likely to slide along this weaker interface. However, even if the leveling pad becomes exposed as a result of scour and soil erosion at the wall toe, the original leveling pad–foundation soil interface can be regarded as the facing block–leveling pad interface of a wall with the height of an added leveling pad, and this increase in wall height (usually 0.2 m) can be ignored. Hence, only the restraint condition at the facing block–leveling pad interface needs to be investigated. In this paper, the term “toe interface” refers to the interface between the facing block and the leveling pad. The selected values for the toe interface friction angle are presented in
Table 5. A change in the interface friction angle influenced the interface shear stiffness, as shown in
Figure 4. The interface shear stiffness values corresponding to the selected interface friction angles, determined via Equations (1) and (2), are also shown in
Table 5.
Figure 9 shows the loads carried by the wall toe and reinforcement layers and their sums for the different toe interface friction angles. The figure also shows the active earth pressure on the wall facing, calculated using the Coulomb earth pressure equation:
where
Ea = the active earth pressure,
Ka = the active earth pressure coefficient,
γ = the unit weight of the backfill, and
H = the wall height.
Ka is expressed as the following:
where
φ = the friction angle of the backfill and
ω = the facing batter. Equation (5) is a reductive Coulomb active earth pressure coefficient, which is recommended by FHWA standard [
15] for the internal stability analysis of GRS walls. The plane strain friction angle was used for the soil models in the numerical simulations of this paper, whereas a triaxial friction angle should be introduced for
Ka [
28]. Thus, a triaxial value of 38° was used for
φ in Equation (5), according to the relationship between the plane strain and triaxial friction angles proposed by Lade and Lee [
42]. It can be seen that the sum of the reinforcement and toe loads (i.e., Σ
Tcon +
Ft) is closer to the calculated active earth pressure
Ea than the total reinforcement connection load Σ
Tcon, but it is still smaller than
Ea. The difference between Σ
Tcon +
Ft and
Ea may be attributed to a small portion of the earth pressure that was counterbalanced by the friction resistances between the facing blocks and between the facing column and backfill soil. The resistance between the facing blocks was not considered in the mechanical model, as shown in Equation (3), for the convenience of application, which may lead to some conservativeness for the predictions of connection loads. It is also noted that the value of Σ
Tcon +
Ft is relatively small under toe interface friction angles of no more than 10°. This may be attributed to local soil failure occurring within the reinforced soil mass, resulting in the earth load acting on the wall facing less than the active earth pressure, in the two weaker toe restraint cases. Local soil shear failure developing in GRS walls with poor toe restraint was also observed in the numerical modeling reported by Huang et al. [
19].
Figure 10 shows the ratios of the toe load (
Ft) and reinforcement load (Σ
Tcon) to the total load (Σ
Tcon +
Ft) under different toe restraint conditions. The toe load ratio increased with the toe interface friction angle, whereas the reinforcement load ratio decreased. The toe load ratio was 14.0% for a toe interface friction angle of 5° and increased to 29.7% when the toe interface friction angle was 15°. When the toe interface friction angle overpassed 15°, the toe load ratio increased slowly, reaching only 33.5% for a toe interface friction angle of 45°. This may have been caused by the influence of toe restraint on the horizontal facing displacements, as shown in
Figure 11. When the toe restraint was weaker, a larger shear displacement between the lowest facing block and the leveling pad led to larger horizontal displacements in the bottom part of the wall. The larger, horizontal facing displacements resulted in high reinforcement connection loads and, thus, smaller toe loads. As the toe interface friction angle increased from 5° to 15°, the significant decrease in the facing displacements induced a quick increase in the toe load ratio. When the toe interface friction angle was in the 15–45° range, the negligible change in the facing displacements led to a slow increase in the toe load ratio.
3.2. Influence Factors of the Toe Load Ratio
The capacity of the wall toe and reinforcement layers for carrying a load is expected to be associated with various factors. For example, a variation in wall height will induce a change not only to the earth load on the facing column but also to the shear stiffness of the toe interface, which may influence the distribution of the total load between the toe and reinforcement layers. To study the influence of different factors on the toe and reinforcement load ratio, we changed the factors of wall height, facing batter, reinforcement spacing and stiffness, and backfill friction angle based on the baseline numerical wall model presented in
Figure 7.
Table 5 and
Table 6 provide the factor values and other parameter values affected by these factors. The reinforcement tensile strength in
Table 6 was determined using the linear relationship between the creep stiffness values and the ultimate tensile strength for HDPE geogrids, proposed by Bathurst and Naftchali [
45]. When one factor was investigated, only the values of this factor and the involved parameters were changed, while the rest of the parameters were set to their baseline values.
Figure 12 shows the toe and reinforcement load ratio versus the toe interface friction angle in walls with different heights. Since the variation characteristics of the reinforcement load ratio are opposite to those of the toe, hereafter, only the toe load ratio is discussed.
Figure 12 shows that the toe load ratio decreased with wall height under the same toe restraint condition. For example, the toe load accounted for approximately 51% and 22% of the total load in the 4 m and 10 m high walls, respectively, when
φt = 45°. This is attributed to the larger reinforcement connection loads motivated by the larger facing displacements and, thus, the smaller earth load transmitted to the toe interface in the taller walls. As the wall height increased, there was a diminishing influence of wall height on the toe load ratio.
Figure 13 shows the effects of the facing batter on the ratio of the earth load carried by the toe and reinforcement layers. The figure shows that the toe load ratio increased with the facing batter under the same toe conditions. As the facing batter increased, the earth load acting on the facing column decreased, resulting in a decrease in the horizontal facing displacements. Lower facing displacements reduced reinforcement connection loads and, thus, the contribution of the reinforcement layers to carrying the earth load, which led to an increase in the toe load ratio. However, the facing batter had little effect on the toe load ratio if it was less than 5°.
Figure 14 shows the influence of reinforcement stiffness on the relative contribution of wall toe and reinforcement layers to load capacity. The toe load ratio decreased as the reinforcement stiffness increased at the same toe interface friction angles, but at a diminishing rate. This is because the larger reinforcement stiffness led to a higher capacity of reinforcement layers for carrying loads, resulting in a lower toe load ratio.
Figure 15 shows that reinforcement spacing had less influence on the toe load ratio when
φt ≤ 10°. For the range of
φt = 15–45°, the toe load ratio increased with the reinforcement spacing. An increase in reinforcement spacing reduces global reinforcement stiffness [
16], which induced a decrease in the fraction of the earth load carried by the reinforcement layers and an increase in the toe load ratio.
The effects of the backfill friction angle is presented in
Figure 16. The toe load ratio increased with the friction angle of the backfill under the same toe conditions. This is because of a larger normal stress at the toe interface, induced by an increase in the downward friction between the backfill and the wall facing when the backfill friction angle increased. The larger toe interface normal stress resulted in a further increase in the horizontal toe load.
The toe load ratio plotted in
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16 increased rapidly with toe interface friction angles in the 5–15° range and then changed only slightly in all cases. This indicates that the involved factors had little influence on the distribution shape of the toe load ratio with the toe interface friction angle. Moreover, at the lower toe interface friction angle of 5°, the toe load ratio was less than 20% in all the cases. In a field wall, the friction angles of the interfaces at the wall toe would change with the surface roughness of the facing block and leveling pad and the foundation soil properties. For GRS walls constructed along rivers or seashores or in mountains, the soil at the wall toe and even the toe itself may be scoured away by wave actions, flooding, or debris flows. Tatsuoka et al. [
47] reported a practical case where the toe of a geosynthetic railway embankment in the mountains of Kyushu, Japan experienced strong scour and erosion caused by flooding. Tarawneh et al. [
48] reported that about 13% of 339 reinforced soil walls supporting bridge abutments inspected in the state of Ohio, USA experienced soil erosion at the wall toe. In these extreme cases, the shear strength of the interfaces at the wall toe must have reduced sharply, resulting in a significant decrease in the toe resistance. Hence, for GRS walls constructed in mountains and by rivers and seashores, it is recommended to ignore the toe resistance to increase margin of safety against reinforcement overstressing. The typical friction angle of the toe interface can be considered as in the range of 25–45°, because the relatively constant ratio of toe load to the integral thrust behind the facing (almost no more than 50% in any condition) is observed in this range, as shown in
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16. However, in a series of 3.6 m high GRS test walls on a rigid foundation, it was observed that the toe carried nearly 80% of the earth load acting on the facing [
9]. This indicates that a rigid foundation may magnify toe load capacity. In addition, the changes in the fractions of toe and reinforcement load with wall height, facing batter, reinforcement stiffness, and spacing in this study are consistent with the observations of other studies [
19,
21,
31].