Pile-Spacing Calculation of Anti-Slide Pile Based on Soil Arching Effect

Key Laboratory of Geological Hazards on ree Gorges Reservoir Area (China ree Gorges University), Ministry of Education, Yichang, Hubei Province, China National Field Observation and Research Station of Landslides in ree Gorges Reservoir Area of Yangtze River, Yichang, Hubei Province, China Division of Energy, Resources and Infrastructure, Department of Sustainable Development, Environmental Science and Engineering, KTH-Royal Institute of Technology, Stockholm, Sweden Institute of Information Science and Technology, China ree Gorges University, Yichang, Hubei Province, China


Introduction
Many people worldwide suffer from serious destructions caused by landslides annually [1][2][3][4]. In landslide control, anti-slide piles are regarded as essential [5][6][7]. In anti-slide pile design, pile-spacing is a crucial parameter. If the pilespacing is designed to be excessively large, soil between the anti-slide piles will slip away; if the pile-spacing is designed to be excessively small, the investment would be high, and the construction process would be difficult.
In 1884, Roberts first discovered the granary effect, which is also known as the soil arching effect. In 1943, Terzaghi first investigated and defined the soil aching effect through a trap-door test. Subsequently, the soil arching effect was investigated comprehensively and further developed by many scholars. Atkinson and Potts [8], Li and Yang [9], and Huang et al. [10] studied the effect of soil arching on tunnel stability. Lee et al. [11] determined the boundaries of arching zones for both single and parallel tunneling. Bossecher and Gray [12] discovered that the soil arching effect occurred as soil attempted to move through the fixed piles in piled walls. Many engineering practices and laboratory experiments have shown that in landslide control with antislide piles, the soil arching effect occurred.
Pile-spacing is typically determined by a single pile's load capacity or engineering experience. In recent years, many studies have focused on the relationship between pilespacing and the soil arching effect, and researchers have attempted to calculate pile-spacing based on the soil arching effect. Sadrekarimi and Abbasnejad [13] discovered that pile clear spacing significantly affected the formation of a stable soil arch in a piled embankment. Yang et al. [14] discovered that for a certain embankment height, the soil arch height increased with the clear spacing of the pile cap within the range from 1 to 2 m. Durrani et al. [15] proposed a limit on pile-spacing along a row where the arching of soil between adjacent piles was maintained during pile-soil movement. Such a limit would significantly affect the design of "discrete pile walls" used to prevent soil movement in potentially unstable slopes.
An effective measurement to detect soil arch form has not been developed yet. Pile-spacing calculation methods based on the soil arching effect have been established according to different hypotheses. Different hypotheses of soil arch form result in different pile-spacing calculation methods. We discovered that different pile-spacing calculation methods based on different hypotheses of soil arch form applied in the same project yielded different results and that some differences were large, e.g., the Zi Yang landslide that occurred on the north bank of Hanjiang, Hubei, China. Zhao et al. [16] considered an inclined soil arch and calculated the pile-spacing to be 7 m. Jia et al. [17] proposed a pile-spacing calculation method based on assuming that soil arches between and behind piles function simultaneously; furthermore, the pile-spacing of the Zi Yang landslide was calculated to be 37.9 m [16].
We comprehensively studied the pile-soil interaction mechanism. In this study, combining the analysis of practical engineering phenomena and laboratory experimental results, we established a soil arch force analysis model and derived a new pile-spacing calculation equation. We compared the results obtained by applying the equation to calculate the pile-spacing of slopes controlled using antislide piles. e calculated results were similar to those of practical engineering designs, in which the difference was within 10%.

Pile-Soil Interaction Process.
e soil arching effect has been widely accepted in the pile-soil contact. During pilesoil interactions, soil arches behind and between piles generate, develop, and fracture. Different understanding regarding soil arches resulted in different pile-spacing calculation methods. e soil arches are shown in Figure 1. e processes of soil arch generation, development, and fracture can be described as follows: (1) e landslide thrust and relative displacement of soils between piles are small, and a local compressive zone is generated in the pile-back. (2) As the landslide thrust increases, the local compressive zone of the pile-back enlarges, and the compressive deformation of soil increases accordingly. e soil between the piles generates a horizontal displacement relative to the piles. Because the friction effect exists between the pileside and the soil, a relative horizontal displacement will be hindered. Hence, a soil arch is generated between the pile-sides. e stability of the soil arch between the pile-sides relies on the frictional resistance generated between the soil and pile-side.
(3) As the landslide thrust continues increasing, the displacement of soil between the pile-sides increases relative to the piles accordingly. Owing to the limited bearing capacity of the soil arch between the pilesides, during its displacement development, the pileback soil arch generates and resists the landslide thrust to compensate for the insufficient bearing capacity of the soil arch between the pile-sides. (4) As the landslide thrust increases steadily, the density of soil within the soil arch between the pile-backs increases owing to compression, and the displacement of soil between the pile-sides increases relative to the pile accordingly. Owing to the small bearing capacity of the soil arch between the pile-sides, after a relatively large displacement, the soil arch between the pile-sides will generate fissures and hence breakage, and the soil arching effect of the soil arch between the pile-sides wears off or even disappears. For a cantilever pile, at this time, the soil between the pile-sides may begin to fall off. (5) As the landslide thrust increases to the ultimate bearing capacity of the soil arch between the pilebacks, the deformation of soil within the soil arch foot area enlarges, fissures are generated, and the displacement of soil between the pile-sides continues to increase. (6) Finally, as the landslide thrust increases, shear failure toward the pile-front will occur in the soil arch between the pile-backs. e soil surrounding the piles will be extruded, and the soil arching effect will fail.
Based on the soil arching effect introduced above, we conclude that the destruction of soil arches between pilebacks signifies the failure of landslide control using anti-slide piles. In other words, the soil arch between the pile-backs governs the soil arching effect in pile-soil interactions. is has been proven directly through laboratory tests and practical engineering phenomena. For example, the laboratory test, as illustrated in Figure 2(a), shows that the soil arch between the pile-sides was destroyed, whereas the pile-  back soil remained steady [18]; the same was observed in practical engineering, as illustrated in Figure 2(b) [19].
According to the analysis above, we propose pile-back soil arch ultimate force analysis. Based on the latter, we derived the following pile-spacing calculation equation.

Pile-Back Soil Arch Form.
Soil arch generation can be explained as follows: in a soil under an external force, a nonuniform displacement is produced, the soil adjusts the shear strength to resist the external force, and then a soil arch is generated. Soil arch generation is resulted from an optimized adjustment under an external force. Hence, the soil arch can develop a high bearing capacity. Additionally, neither shear force nor bending moment existed in the soil arch, only axial force.
To simplify the soil arch mode, we formed the following hypotheses [20][21][22]: (1) e pile-back soil arch form is a reasonable arch axis curve, and a reasonable arch axis curve is a parabola. (2) e landslide thrust is equally distributed between the anti-slide piles. e soil arch plane graph is shown in Figure 3. e antislide pile cross section is a rectangle, the soil arch rise is h, and the soil arch span is 2l.
According to the introduction above, the soil arch curve passes three points: A(− l, 0), B(l, 0), and C(0, h). e soil arch curve is expressed as

Pile-Spacing Calculation.
After analyzing the pile-soil interaction and the soil arch geometrical characteristics, we calculate the pile-spacing. Considering the complexity of the soil arching effect and to aim for a concise calculation process, we formed the following hypotheses: (1) Only the pile-back soil arch is considered, i.e., the pile-side soil arch is neglected. (2) e soil arch distributes equally along the anti-slide pile length, and no changes in form occur. (3) Pile-soil friction is much larger than soil-soil friction; therefore, the soil arch will not break along the pilesoil contact face.
Considering the symmetry characteristic of the soil arch, we selected the left part for analysis, as shown in Figure 4. Based on equilibrium conditions, the forces can be calculated as follows: erefore, the soil arch axial force F of point K is expressed as follows: Equation (4) shows that the soil arch axial force increases with the distance from point K to the y-axis. erefore, in the arch foot section, the largest axial force appeared. In other words, the arch foot section is the most susceptible to damage. Furthermore, we will not consider the control section of the midspan in the following limit analysis.
At the soil arch foot site, the forces can be calculated similarly as equations (2)-(4), as follows:

Advances in Civil Engineering
(7) Figure 5 shows the pile-back soil arch force, where ΔADE is the arch foot compressive zone, AE is the arch foot section, and a is the pile-back width.
According to limit equilibrium theory, the angle between the major principal stress-acting face and fracture face is 45°+ (φ/2). erefore, at point A(− l, 0), the soil arch curve starting point, i.e., its tangent slope, is For the soil arch, the arch foot section is the most susceptible to damage. According to the Mohr-Coulomb rule of uniaxial compression, the following equation can be obtained: where t is the soil arch thickness, which can be calculated using trigonometric functions, as follows: Substituting equations (7) and (10) into equation (9) yields ql ah Finally, substituting equation (8) into equation (11) yields l � ac q erefore, the pile-spacing L can be expressed as L � 2ac q

Illustration Results
To verify the proposed pile-spacing calculation method effect, we investigated practical projects controlled using anti-slide piles to perform a comparison.
3.1. Case 1. Piles were used to stabilize an 8-m-high railway embankment of weald clay at Hildenborough, Kent, UK [23]. Bored concrete piles measuring 0.6 m in diameter were constructed at a spacing of 2.4 m. e cohesion and friction angle of the weald clay embankment fill were 20.9 kPa and 25°, respectively [24]. Smethurst and Powrie [23] estimated the soil driving (shear) force required to achieve the desired safety factor and transferred by the pile to be 60 kN. e critical failure surface passed through the proposed pile location at a depth of 4 m below the slope surface.
As the pile was round, we first transformed it to a rectangle, of which the width is According to equation (13), the pile-spacing can be obtained as follows: e calculated pile-spacing was 2.3 m, which was smaller than that of the actual project, i.e., 2.4 m. Smethurst and Powrie [23] reported that the actual restoring force must ensure that the stability of the slope is smaller than the estimated value; therefore, if the actual restoring force is adopted, the calculated pile-spacing will be larger than 2.3 m, which is approximately or larger than the actual project.

Case 2.
A highway cut slope located in north Sichuan, China, adopted cantilever piles for control. e soil cohesion and friction angle behind the pile were 50 kPa and 28°, respectively. e pile cross section width was 2 m, the length of the entire pile was 22 m, and the cantilever measured 11 m. e calculated unit width landslide thrust was 1050 kN/m [25].
Substituting the project parameters introduced above into equation (13), we obtain L � 2 × 2 × 50 (1050/11) e calculated pile-spacing was 5.5 m. e actual pilespacing was 6 m; the slope remained steady since the completion of the project. erefore, it is indirectly proven that the calculation equation is reasonable.

Discussion and Conclusion
Based on the assumption that the pile-back soil arch resists only the landslide thrust at the limit, we derived an equation to calculate the pile-spacing. e equation expresses the relationship among the pile-spacing, pile-back soil shear strength parameters, pile cross section, and landslide thrust. e pile-spacing increased accompanied with increased soil cohesion and friction angle; however, it decreased with increasing landslide thrust. For the calculation based on the hypothesis that the soil arch covers the entire pile-back, which aims to simplify the relationship between the pile cross section width and the soil arch thickness, the pile cross section is regarded as a definite value to some extent. In fact, the soil arch may not cover the entire pile-back if the pile cross section width continues increasing; therefore, this aspect must be investigated further.
e case studies showed that the calculated values were smaller than those from engineering practices. is may be attributed to the following reasons. First, the calculation method was based on the soil arching effect, but the landslide thrust adopted in the equation was still calculated to the pilesite, when it should be calculated to the arch rise. erefore, the landslide thrust will be smaller, and according to the equation, the pile-spacing will be larger. Furthermore, the equation derived was based on the plane soil arching effect; as the sliding surface had an angle of inclination at the pilesite, the force on the soil arch should be the horizontal component of the landslide thrust. In the case studies, parameters related to the tilt angle were not considered; we directly adopted the landslide thrust instead, which might result in a smaller calculated pile-spacing. e pile-spacing calculation equation derived is simple and direct.
e calculated values were smaller than the values obtained from engineering practices within 10%, thereby directly proving the steadiness and effectiveness of the calculation method.

Conflicts of Interest
e authors declare that they have no conflicts of interest.