Introduction

The construction of heavy structures like bridge abutments, transmission line towers, high raised buildings, retaining walls, and dolphins near natural or man-made slopes has increased due to rapid urbanization. Foundations for these types of structures required to support not only heavy axial loads but also the large lateral loads caused by wind, traffic, wave action, and earthquakes. As a solution, pile foundations are the most preferred type of foundations for supporting the heavy vertical and lateral loads coming from these superstructures.

The behavior and lateral load-carrying capacity of a pile resting on level ground encompass the interaction between the pile and the surrounding soil/or water. Standard methods, design charts, and empirical formulas for determining the lateral load-carrying capacity of piles on level ground are available in the literature [1,2,3,4,5,6]. Pile–soil interaction is generally a complex phenomenon, and it becomes more complicated when it comes to the sloping ground due to the additional parameters (i.e., soil slope and edge distance) affecting the lateral load behavior of the pile.

Poulos [7] was one of the initial studies, which examined the behavior of long flexible piles on clayey slopes by performing small-scale laboratory tests and then derived a mathematical model using the elastic continuum theory. He reported that after a edge distance of 5 times the diameter of a pile, the effect of slope on pile–soil interaction is negligible. After Poulos [7], a lot of research has been carried out in clayey soil slopes using model-scale testing [8, 9], field testing [10, 11], numerical simulations [12,13,14], and analytical methods [15,16,17]. However, limited studies are available for the laterally loaded pile behavior resting on cohesionless soil slopes.

Gabr and Borden [18] developed analytical equations for the ultimate lateral load-carrying capacity of rigid piers constructed on both cohesive and cohesionless soil slopes using the stress wedge approach [19]. They validated these equations by conducting a few field tests. Mirzoyan [20] conducted field tests on fully instrumented steel piles driven in cohesionless soil for both level and sloping ground conditions. They developed a mathematical model to predict the ultimate lateral load-carrying capacity of a pile by analyzing field test results using LPILE software. Several laboratory studies have been carried out on piles located at or near slope crest in cohesionless soils (e.g., [21,22,23,24,25,26,27]). They studied the variation of the lateral load-carrying capacity of a pile by considering the effects of slope inclination, loading direction (i.e., toward and away from the slope), pile L/D ratio, and edge distance. They also developed p-y curves for the laterally loaded pile located on the sloping ground by modifying existing p-y curves available on level ground.

Limited numerical studies are also available on laterally loaded piles located in cohesionless soil slopes using the three-dimensional finite element method [28, 29]. Chae et al. [28] studied the lateral resistance behavior of a single and group of short piles located near cohesionless soil slopes using the finite element method. They compared these results with available laboratory and field test results. Muthukkumaran and Gokul [29] developed an equation to determine the depth of fixity for a single pile located at the crest of cohesionless soil slope subjected to surcharge loading. The developed equation considered the effects of slope inclination, pile L/D ratio, and relative density of soil.

Few researchers studied the effect of edge distance on the lateral load-carrying capacity of a pile for determining critical edge distance of a pile [30,31,32,33]. Terashi et al. [30] found that after an edge distance of 2.5D, the effect of slope on the lateral load of a pile is negligible. Both Bouafia and Bouguerra [31] and Mezazigh and Levacher [33] performed centrifuge testing to study the effect of slope and edge distance on the lateral load-carrying capacity of the long flexible piles located in cohesionless soil slopes. Bouafia and Bouguerra [31] reported a critical edge distance of 15D for dense sands of 2H: 1 V slope, whereas Mezazigh and Levacher [33] reported a critical edge distance of 8D and 12D for 2H: 1 V and 3H: 2 V slopes, respectively. Mezazigh and Levacher [33] stated that critical edge distance values are independent of the shear strength of the sand. Bouafia and Bouguerra [32] performed centrifuge testing to study the effect of edge distance on short piles located in cohesionless soil slopes. They reported that the effect of slope inclination is found to be ineffective on the load-carrying capacity after an edge distance of 6.5D.

As seen from the past studies, there is no clear consensus over the critical edge distance of laterally loaded piles resting on cohesionless soil slopes. The critical edge distance values presented by different studies show a discrepancy with each other. Further, all these studies are experimental and conducted by considered only limited soil and pile parameters. In this study, these issues have been addressed by performing comprehensive parametric analysis using the three-dimensional finite element method. The effect of edge distance on the behavior of pile located in cohesionless soil slopes is studied by varying several influencing parameters. A comparison of critical edge distance results obtained from this study with various past studies has also been presented for a better understanding of the present finite element model.

Finite Element Modeling

A finite element software package PLAXIS 3D is used for modeling a single pile located on or near the crest of the cohesionless soil slopes subjected to lateral loading. The schematic diagram of a pile placed near the sloping ground is shown in Fig. 1, and the typical finite element model of pile–soil with interfaces is shown in Fig. 2. In this study, the soil domain size is selected such that the stresses and displacements generated in the model would not be affected by the extent of boundaries for any loading condition. The behavior of dry cohesionless soil is modeled using a linearly elastic-perfectly plastic model (Mohr–Coulomb Model). In this model, Hooke’s law was used to relate stress rates up to yield point after that Mohr–Coulomb yield condition [34] was used for the stress–strain relationship. In the present study, a non-associated plasticity theory was considered to include the dilatancy effects of medium and dense sands. The behavior of a concrete pile is modeled as linearly elastic material using Hooke’s law. An interface is modeled in between the concrete pile and surrounding cohesionless soil with a small virtual thickness to consider the pile–soil interactions.

Fig. 1
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Schematic diagram of a single pile located near the sloping ground

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Fig. 2
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Typical finite element model of a pile resting on sloping ground: a soil slope with standard boundary conditions; b pile with the interface

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Both pile and surrounding soil are modeled as volume elements using ten noded tetrahedral elements, whereas the interface between the pile and surrounding soil is modeled using 12 noded interface elements. The material properties for interface elements are considered the same as surrounding soil but with a strength reduction factor (Rinter) to consider the reduction of the strength of interface. Boundary conditions play an essential role in any finite element analysis. In the present study, all vertical sides (boundaries) are considered normally fixed (Ux = 0 in X-axis and Uy = 0 in Y-axis), and the bottom side of the model is assumed completely fixed (Ux = Uy = Uz = 0), as shown in Fig. 2.

Mesh convergence studies are performed before parametric analysis to study the effect of element size by considering different element sizes (i.e., by changing the coarseness in PLAXIS 3D). From mesh convergence studies, it is found that medium-mesh size with soil coarseness factor of 0.75 is giving better results for all the cases considered in the present study. Numerical analysis of a single pile on cohesionless soil slope is carried out in 3 phases. In the first phase, only initial soil stresses were calculated by using the gravity loading procedure. In the second phase or construction phase, structural elements (i.e., pile and interfaces) are activated. Before starting calculations in the final phase (loading phase), displacements from the construction phase are made to zero, so that absolute deformations can be obtained using applied loading. A prescribed lateral movement of 20%D is applied at the pile top in the direction of the slope. The corresponding load obtained at the pile top is considered as the lateral load-carrying capacity of the pile.

Validation of Numerical Modeling with Experimental Studies

The accuracy of the numerical modeling is validated by comparing the results from the numerical analyses with the centrifuge testing results. Mezazigh and Levacher [33] carried out an extensive centrifuge testing on a single pile resting on cohesionless soil subjected to lateral loading. They considered a hallow aluminum pile having an outer diameter of 18 mm and a wall thickness of 1.5 mm for model testing. Centrifuge testing was carried out on the model pile at an acceleration of 40 g. As per the similitude laws, model test pile simulation was equivalent to the prototype pile of an outer diameter 720 mm with a wall thickness of 60 mm. This prototype of laterally loaded pile resting on cohesionless soil was modeled using the finite element method. Soil domain of size 10 m × 10 m × 20 m was considered in the numerical analysis, such that stresses generated in the model will not be affected by the extent of the boundaries. A hollow aluminum pile of outer diameter 720 mm with a wall thickness of 60 mm and an embedment length of 12 m was modeled in the numerical simulation to represent the prototype model. Table 1 provides the material properties of both cohesionless soil and hollow pile, considering in the numerical analyses. In the analysis, the lateral load was applied at the top of the pile in terms of prescribed displacement.

Table 1 Properties of soil and pile considered in the analysis [33]
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Results extracted from the numerical analysis and centrifuge test results are shown in Fig. 3. From Figures (a–c), it can be seen that the depth versus deflection curves of the present study has been closely matching with centrifuge results, whereas bending moment and lateral soil pressure curves follow the same trend. Still, there are slight deviations in bending moment and soil pressure results. Because they used a software described by Deny [35] to smoothing the quintic spline equation fitted for bending moment curves. In this study, deflections along the pile depth obtained directly from the analysis. By using this data, a best-fit regression equation has been developed, and then a direct differentiation is used to get bending moment and soil pressure data. Overall, it can be observed that results from the finite element analysis are in reasonable agreement with the experimental results.

Fig. 3
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Comparison of numerical results with past experimental results for a single pile located on cohesionless soil: a depth versus deflection profile; b depth versus bending moment profile; c depth versus soil pressure profile

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Results and Discussions

Parametric analyses are carried out using the finite element method by varying the several influencing parameters like slope inclination, L/D ratio of a pile, pile diameter, soil parameters, and edge distance, which affects the behavior of a pile resting on the sloping ground. A dry cohesionless soil with the properties given in Table 2 and having a Poisson’s ratio of 0.30 was considered for the analysis. The dilatancy angle of cohesionless soil was considered by using the following equation, ψ = ϕ − 30° [28, 36]. A solid circular concrete pile made of M25 grade (minimum grade of concrete suggested by IS 2911:2010 Part I [37]) with a Poisson’s ratio of 0.15 was considered. Different material and physical parameters considered in this study are mentioned in Table 2. Interface elements are modeled in between the pile and surrounding soil with a strength reduction factor of 0.67 [38]. In the present study, the ultimate lateral load-carrying capacity is considered as the load at which displacement equals to 20% of pile diameter.

Table 2 Properties of the soil and pile used in the parametric study
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Typical numerical simulation results obtained for a single pile (D = 0.6 m, L/D = 15) resting on 40° cohesionless soil slope (γ = 19 kN/m3, ϕ = 45°) are shown in Fig. 4. Plots 4(a–e) show lateral load-carrying capacity and deformation profile obtained for different edge distances of the pile. From deformation profiles, it can be seen that the effect of slope on the laterally loaded pile is decreasing as the pile moves away from the slope crest. The effect of slope inclination on the deformation profile and load-carrying capacity completely vanishes after a certain edge distance. The edge distance, after which pile on sloping ground behaves similar to the level ground condition, is considered as critical edge distance. Figure 4(f) shows the effect of edge distance on the lateral load-carrying capacity of a pile resting on cohesionless soil slope. It can be seen that the lateral load increases with an increase in edge distance and becomes constant after an edge distance of 12.5D. So, the critical edge distance for a given soil slope configuration is 12.5D. The effect of each parameter on the edge distance and lateral load-carrying capacity are discussed separately in the following sections.

Fig. 4
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Effect of edge distance on the behavior of a pile resting on cohesionless soil slope (ϕ = 45° & β = 40°): a ED = 0D; b ED = 5.0D; c ED = 7.5D; d ED = 10.0D; e ED = 15.0D; f lateral load capacity versus edge distance profile

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Effect of Slope Inclination

The effect of slope inclination on the lateral capacity and normalized edge distance (NED) of a pile (D = 0.6 m) located in cohesionless soil slope having a unit weight of 19 kN/m3 and angle of internal friction of 45° for different L/D ratios of a pile is shown in Fig. 5(a–c). It can be seen that the lateral load capacity decreases with an increase in the slope inclination. This is due to a decrease in the initial soil confinement and passive resistance of the soil. The lateral load capacity is lowest when a pile is situated at the crest of the soil slope. The lateral load of a pile resting on sloping ground decreases significantly, even up to 85% as compared to level ground.

Fig. 5
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Effect of slope inclination on the lateral load capacity of a 0.6 m diameter piles: a L/D = 5.0 m; b L/D = 10.0 m; c L/D = 20.0 m

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From Fig. 5, it can be seen that the lateral load capacity increases with an increase in the edge distance, up to a particular range (critical edge distance) for any slope configuration after that, it becomes constant. Because after critical edge distance, the resistance and confinement offered by passive zone will not be affected by slope inclination (i.e., similar to level ground condition). The critical edge distance of a given pile–soil system is also increasing with an increase in slope inclination for a given situation. It can be noticed that the critical edge distances of a pile are 5.0D, 7.5D, and 10.0D for slope inclinations of 20°, 30°, and 40°, respectively.

Effect of Relative Density

Variation of lateral load-carrying capacity with normalized edge distance of a pile (L/D = 12.5) for different relative densities of soil is shown in Fig. 6(a, b). It can be seen that the lateral load increases as the relative density of soil increases irrespective of any soil-pile configuration. With an increase in relative density, the shear strength of soil increases, which further increases the passive resistance of soil in front of the pile. From the figure, it can be seen that the lateral load initially increases irrespective of the relative density with an increase in edge distance up to critical edge distance for any given soil condition, after that, it remains constant. It can also be seen that the relative density of particular soil has a subtle effect on critical edge distance as compared to slope inclination.

Fig. 6
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Effect of relative density on the lateral load-carrying capacity of piles having different diameters: a D = 0.3 m; b D = 0.6 m

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Effect of Pile Length (L/D Ratio)

Figure 7(a–c) shows the variation of lateral load capacity with the normalized edge distance of a pile for different L/D ratios having a pile diameter, D = 0.5 m. It can be seen that load-carrying capacity increases with increase in the L/D ratio of a pile, up to a certain value after that, it remains constant for a given condition. As the length of the pile increases, the lateral load-carrying capacity increases up to a certain extent after that, it remains constant. The pile length up to which lateral load-carrying capacity increases is called an effective length of the pile. It mainly depends on pile diameter, slope inclination, and relative soil-pile stiffness. From the curves, it can also be seen that the lateral load-carrying capacity curves become flat after an edge distance of 7.5D for the given condition. As compared to the level ground, decrease in pile lateral load-carrying capacity is found to be varying in the range of 85–60%, 70–50%, and 50–30% for slope inclinations of 40°, 30°, and 20°, respectively. From the results, it is observed that critical edge distance is not significantly affected by the length to diameter ratio of the pile.

Fig. 7
figure 7

Effect of L/D ratio on lateral load of a pile located in cohesionless soils (γ = 17 kN/m3 & β = 30°): a ϕ = 35°; b ϕ = 40°; c ϕ = 45°

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Figure 8(a–c) shows the variation of deflection along with the depth of a pile for different lengths of a pile for a given soil and pile diameter. It can be observed that the pile having L/D ratios less than 10.0D exhibits rigid pile behavior (i.e., short piles, shows rotation only) for any given diameter of a pile. Piles having L/D ratios greater than 20.0D exhibit flexible pile behavior (i.e., long piles, deflection occurs), and L/D ratios between 10.0D and 20.0D behave as intermediate piles.

Fig. 8
figure 8

Depth versus deflection curves for different L/D ratios of a pile resting on sloping ground: a β = 20°; b β = 30°; c β = 40°

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Effect of Pile Diameter

The effect of pile diameter on lateral load capacity and critical edge distance of a pile (L/D = 15.0) located on cohesionless soil (γ = 17 kN/m3, ϕ = 40°, E = 30 MPa) for different slope configurations are shown in Fig. 9(a, b). It can be seen that the load-carrying capacity of a pile resting on cohesionless soil slope increases continuously with an increase in the diameter of a pile. As the diameter of the pile increases, lateral resistance of the pile increases due to the increase in the contact area between the pile and surrounding soil interface. From the figures, it can be observed that the effect of diameter on the normalized critical edge distance of the pile is also insignificant as compared to the slope inclination of the surrounding soil.

Fig. 9
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Effect of the pile diameter on the lateral load-carrying capacity of a pile on sloping ground: a β = 25°; b β = 35°

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Critical Edge Distance

From past literature available on critical edge distance, it is observed that all these studies are experimentally carried out with limited parameters. It can also be seen that different authors proposed different values for the critical edge distance of a pile. These past studies mainly determined the critical edge distance for a particular relative density of soil and a slope (26.6°). In the present study, an extensive parametric study has been carried out to determine the critical edge distance of pile by considering the effects of variation of the slope inclination, unit weight, angle of internal friction, modulus of elasticity of soil, pile length, pile diameter. A total of more than 20,000 3D analyses have been conducted to determine the critical edge distance of a pile located on cohesionless soil slope. From these overall results, the effect of each parameter on the edge distance has been clearly studied. The comparison of the critical edge distance of the present study with various past studies is shown in Fig. 10.

Fig. 10
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Comparison of critical edge distance results of the present study with past studies

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Based on the detailed parametric analyses, the critical edge distance is determined for different parameters affecting the behavior of laterally loaded pile located in cohesionless soil. The critical edge distance of a single pile resting on cohesionless soil is presented in Table 3. It is observed that the critical edge distance of a pile is significantly affected by the slope inclination as compared to the other material properties of surrounding soil and pile.

Table 3 Critical edge distance (CED) of a single pile resting on cohesionless soil slope
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Conclusions

In this paper, extensive numerical analyses have been carried out using the finite element method by varying several influencing parameters to determine the critical edge distance of a laterally loaded pile located in the cohesionless soil slopes. Important conclusions drawn from the study are given below:

  • As compared to the level ground, the lateral load-carrying capacity of a pile resting on sloping ground is found to be decreasing significantly, even up to 85% based on the slope inclination.

  • The lateral load-carrying capacity of a single pile increases continuously with an increase in the diameter of pile, unit weight, angle of internal friction, and modulus of elasticity of the soil.

  • Lateral load-carrying capacity is found to be increasing with an increase in length to diameter ratio of the pile up to a certain ratio, after that, it attains a constant value. This ratio is found to be different for different slopes. However, it is observed that irrespective of soil and slope characteristics, the piles having an L/D ratio of less than 10 behave as a short pile and piles having an L/D ratio greater than 20 behave as a long pile. Piles having L/D ratios between 10 and 20 behave as intermediate piles.

  • With an increase in slope inclination, the lateral load-carrying capacity decreases almost linearly with edge distance up to critical edge distance after that, it becomes constant.

  • It is observed that the critical edge distance significantly affected by the slope inclination. Results show that the maximum critical edge distance of a single pile for any soil slope configuration is 15.0D.

  • Ranges of critical edge distance of a pile located in a given cohesionless soil are 5.0–7.5D, 7.5–10.0D, and 10.0–15.0D for slope inclinations of 20°, 30°, and 40°, respectively.

  • The effect of geometrical parameters of a pile like L/D ratio and pile diameter on the critical edge distance was negligible for any given cohesionless soil slope configuration.

  • The effect of modulus of elasticity of soil on critical edge distance has a significant impact as compared to the angle of internal friction of soil.

  • The soil slope inclination effect on critical edge distance is found to be more important as compared to the individual soil parameters.

  • Overall, it is observed that the critical edge distance of a pile slightly increases with an increase in the relative density of the cohesionless soil irrespective of slope inclination.