Probabilistic pile reinforced slope stability analysis using load transfer factor considering anisotropy of soil cohesion

A probabilistic limit equilibrium framework combining empirical load transfer factor and anisotropy of soil cohesion is developed to conduct pile‐reinforced slope reliability analysis. The anisotropy of soil cohesion is determined conditioned on that the thrust force direction is parallel to the major principal direction and it is easily combined with load transfer factor, which are related with soil parameters, and pile parameters. The proposed method is illustrated against a homogeneous soil slope. The sensitivity studies of pile parameters on factor of safety (FS; calculated at respective means of soil parameters) and β demonstrated that the anisotropy of soil cohesion tends to pose significant effect on reliability index β than on FS. The effect of anisotropy of soil cohesion on FS is found to be slightly different under different pile locations, whereas its effect on β is observed to be least if piles are drilled at the middle part of slope and more significant effect is observed when piles are drilled at the lower and upper part of slope. The plots from the sensitivity studies provide an alternative tool for pile designs aiming at the target reliability index β. The proposed method contributes to the pile‐reinforced slope stability within limit equilibrium framework.

The soil-pile interactions have been studied using numerical methods with strength reduction strategy 9,[24][25][26] where the interactions between piles and soils can be rationally simulated via proper contact model. 4,27Although the numerical methods can provide much insight into pile-reinforced slope stability, the disadvantage of these methods lies in the demanding computational effort especially for probabilistic pile-reinforced slope stability analysis where a large number of implementations of numerical methods are involved.Hence, the limit equilibrium method still finds its place in the pile-reinforce slope stability analysis.The load transfer factor (LTF) introduced by Li and Liang 28 quantifies the resultant reinforcement from piles to the slope stability in a smart manner.The LTF-based limit equilibrium method facilitates the research on pile-reinforced slope stability.
0][31][32][33][34][35] For example, the results from 1D shear test of consolidated soil have revealed a phenomenon wherein the vertical cohesion tends to surpass its horizontal counterpart. 29The dismiss of anisotropy in engineering computations tends to yield conservative decisions.Therefore, how to conduct pile-reinforced slope stability considering the uncertainties and anisotropy in soil properties and the soil arching effect as well remains an open question.
The present study starts with the brief introduction of LTF-based limit equilibrium method combined with imbalance thrust force method, followed by the definition of anisotropy of soil properties.Subsequently, a reliability framework for piles reinforced slope stability using empirical LTF considering anisotropy of soil properties is described.The proposed method is applied to a soil slope to investigate the influences of anisotropy of soil properties, pile diameter, pile openings, and pile location on the slope reliability.Finally, some notes and insights are provided.

Load transfer factor in piles reinforced slope stability
Figure 1 shows a slope reinforced with one row pile.It is seen from Figure 1 that the critical circular sliding surface with the minimum factor of safety (FS) is determined by SLOPE/W (https://www.geoslope.com).The sliding mass is discretized into n slices with slice 1 at the top right side and slice n at the down left side.The pile with diameter D and opening S is located between slice i − 1 and slice i. x p is the x-coordinate of the pile center.LTF is introduced by Li and Liang 28 to quantify the contribution of piles to the slope stability.It is defined as the ratio of the thrust force at downslope side slice (i) to that at upslope side slice (i − 1).Refer to Figure 2, let L i−1 denote the resultant thrust force acting at upslope side slice i − 1 from pile and R i is the resultant thrust force acting at downslope side slice i from pile. i is the inclination angle of slice i base,  i-1 is the inclination angle of slice i − 1 base.A LTF  is defined as the ratio of R i to L i-1 , that is: F I G U R E 1 A slope reinforced with stabilizing piles.

F I G U R E 2
Forces acting on the pile and its adjacent slices.
L i−1 and L i−1 ′ are two counterparts of the interaction between upslope side slice i−1 and pile.Similarly, R i and R i ′ are two counterparts of the interaction between downslope side slice i and pile.The unbalanced forces and moments are countered by the pile itself.Hence, the piles are assumed to behave as expected and no failures occur regarding the piles.In addition, group pile effect is neglected and the LTF  is independent from each other if more than one row of piles are adopted. 28he determination of  is very complicated and difficult since many factors as soil properties, pile diameter, pile openings, and pile locations.An empirical LTF has been proposed in Li and Liang 28 where a large number of numerical simulations on homogeneous slopes have been investigated and a regression Equation (2) has been developed.It is noted that the piles are assumed to be drilled into bedrock with sufficient depth. = −0.272C where C h is soil cohesion on the horizontal plane (see the next section);  is friction angle;  is slope angle;  p is relative indicator of pile location and it is defined as: where x toe = the x-coordinate of slope toe; x crest = the x-coordinate of slope crest.

Anisotropy of soil cohesions
It is fairly well acknowledged that soils exhibit remarkable anisotropic behavior (see Figure 3) owing to the factors of deposition process, stress history, consolidation pressure, and cutting action as well. 36The anisotropy of soil cohesions indicates that the soil cohesions vary depending on the site-specific principal stresses.Consider the slope failure along the sliding surface as shown in Figure 3. Take the cohesion for example, let C h denote the cohesion along the horizontal direction and C v denote the cohesion along the vertical direction.The cohesion at the slice i base is determined as: 29 where C i is the cohesion at the slice i base. i represents the angle between vertical direction and the direction of major principal stress  1 .In accordance the study of Wang et al., 29 the direction of R i is identical to the direction of major principal stress in the current study.The internal friction angle exhibits non-linearity rather than anisotropy, therefore, only the anisotropy of soil cohesions is considered in this study.F I G U R E 4 Forces acting at slice i.
An anisotropy coefficient K is defined as the ratio of C v to C h .

LTF-based limit equilibrium method for pile reinforced slope stability
Following the assumptions in unbalanced thrust force method of slices, 27,37 a modified limit equilibrium method for pile reinforced slope stability is developed incorporating LTF and anisotropy of soil properties.The key assumption in the unbalanced thrust force method of slices lies in that the direction of left side thrust force is parallel to the slice base while the direction of right side one is parallel to the preceding (upslope side) slice base.As shown in Figure 4, the thrust force R i was assumed to be parallel to the base of slice i−1 (i.e.,  i−1 ), and the thrust force L i was assumed to be parallel to the base of slice i (i.e.,  i ).
Consider the equilibrium of forces acting on slice i along the tangential and normal directions to the slice base: where G i is weight of slice i, U i is the pore pressure at the base of slice i, N i is the force normal to the base of slice i, T i is the force parallel to the base of slice i.L i is the thrust force acting on the left side of slice i, R i is the thrust force acting on the right side of slice i,  i is the inclination angle of slice i base,  i-1 is the inclination angle of slice i − 1 base.Follow the Mohr-Coulomb strength criterion: where Fs is the factor of safety for the pile reinforced slope stability, and  is the internal friction angle of the soil at the slice base.Substituting Equations ( 6), (7), and (8) into Equation ( 9), it is obtained: Equation ( 10) is reformulated as: When a pile is drilled between slice i and i−1, Equation ( 1) is applicable, that is, R i is determined as: It is noted that if there are no piles drilled between slice i and i−1, Equation ( 14) still holds on with a constant  = 1.Therefore, a unified equation is reformulated to incorporate the cases of with piles and without piles between two adjacent slices: where  i−1 is the LTF attributed to the pile located just between upslope side slice i−1 and downslope side slice i and it can be determined using Equation (2). i−1 is equal to 1 when there is no pile drilled just between upslope side slice i−1 and downslope side slice i. Substituting Equation (15) into Equation ( 11), one gets: Equation (11) relates the left thrust force and right thrust force for a given slice i. and Equation ( 16) relates the left thrust forces for two consecutive slices (e.g.L i = f (L i−1 ) for slices i and i − 1).They are combined to derive the relationship between L n and R 1 as Equation ( 17): The factor of safety for pile reinforced slope stability, Fs, can be calculated using Equation (17) in an iterative procedure in conjunction with the boundary conditions described in Equation (18).

PROBABILISTIC PILE REINFORCED SLOPE STABILITY ANALYSIS CONSIDERING ANISOTROPY OF SOIL PROPERTIES
The probabilistic slope stability analysis for slopes without reinforcements have attract worldwide attention [38][39][40][41] and it has been well recognized that both the reliability index or failure probability serve as proper indicator for the probabilistic slope stability.The deterministic pile reinforced slope stability method described in the previous section can be easily extended to probabilistic one where the soil properties are modeled by random variables with specified statistics (e.g., mean, standard deviation, and distribution).
Consider a homogeneous slope model with height of H, slope angle of , cohesion of c, and friction angle of .H and  are taken as deterministic constants, whereas the c and  are modeled by lognormally distributed random variables.Each set of sampled values of c and  is transferred to deterministic pile reinforced slope stability method to obtain a Fs.All the transferred sampled values of c and  yield a group of Fs which can be reanalyzed to obtain the reliability index or failure probability.The proposed methodology is demonstrated in Figure 5.It is noticed that the proposed methodology consists of two parts, that is, part I: Deterministic analysis and part II: Reliability analysis.Two sub-parts are involved in Part I, that is, deterministic slope stability analysis without piles to locate the critical failure surface under means of soil properties and deterministic slope stability analysis with piles to calculate the FS with piles under specific soil parameters, which can be transferred from Part II.In the Part II, a finite number (10,000 in this study) of random samples are generated and transferred to Part I to obtain the respective FSs, which can be reanalyzed to get the reliability index or failure probability.

Deterministic slope stability analysis without piles
Figure 6 shows a homogeneous soil slope with slope height H = 12 m and slope angle  = 30.96• .The phreatic line is indicated using real blue line.The soil unit weight has a deterministic value of 18 kN/m 3 .The cohesion C h and internal friction angle  follow lognormal distributions.The respective mean and standard deviation of C h are 8 and 2 kPa.The mean and standard deviation of  are 20 • and 2 • , respectively.The correlation coefficient between C h and  is zero.Following the procedures demonstrated in Figure 5, the deterministic slope stability analysis without piles is conducted using the commercial software SLOPE/W (https://www.geoslope.com)to obtain the critical sliding surface with minimum FS.The C h and  occupy the respective means (i.e., C h = 8 kPa and  = 20 • ) and Bishop method is selected to calculate the FS for a potential circular sliding surface.The minimum FS is found to be 1.10 at K = 1 and the critical circular sliding surface is shown in Figure 6.

Deterministic slope stability analysis with piles
In accordance with the specifications in technical code for building slope engineering (GB50330-2013), the studied slope is classified into class II slope and its required minimum FS is 1.3, the current slope should be furtherly stabilized.Although the critical sliding surface of pile reinforced slope differs from that of slope without piles, the critical sliding surface of slope without piles is furtherly adopted in pile-reinforced slope stability analysis focusing on the global slope failure instead of local failures.Assume that single row of piles is adopted to reinforce the slope stability.Six pile locations are considered, Under x p = 26 m, the piles are drilled near the upper zone of sliding mass.The upslope side slice adjacent to the drilled pile is easily distinguished and it is denoted as m.The respective FSs with piles at  m = 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 are 1.11, 1.14, 1.16, 1.20, 1.24, 1.28, 1.33, 1.39, 1.47.It is noted that as  m decreases from 1.0, the FS with piles increases dramatically.Since a smaller  m implies a lager transfer reduction of thrust force from upslope side slice, the slope stability level is consequently enhanced.The similar trends have been observed under other pile locations.It is noted that there exist susceptible pile locations, where the most significant influence of  m is found.In this study, x p = 18, 20 m are the susceptible pile locations.If these locations are preferred, efforts must be spent on the drilled works to obtain a smaller  m and finally to obtain a larger FS.
Since the determination of  m is so complicated that the empirical Equation ( 2) reported in Li and Liang 28 is used in the following studies.Refer to Equation ( 2), the  m s under different pile locations can be easily determined.from Figure 8 that as x p ranges between 10 and 20 m,  m decreases significantly from 1.0 to 0.52 and after x p is greater than 20 m, it increases from 0.52 to 0.75.To investigate the effect of K on the FS without and with piles, six K values (K 0.3, 0.5, 0.7, 1.0, 1.5, and 2.0) are assumed and the cohesion at each of slice base can be determined using Equation ( 6) and finally the  m is updated in accordance with Equation (2). Figure 9 plots the variation of FS without and with piles versus K values.It is noted from Figure 9 that the FS without piles (case x p = 10) increases as K ranges between 0.3 and 2.0.For the cases of with the similar variation trend has been noticed.This observation agrees well with those in Wang et al. 29 and Ning et al. 42 As for the variation trend of FS versus pile locations, consider K = 1 for example, as x p increases from 10 to 16 m, the FS increases slightly from 1.113 to 1.175.As x p increases from 16 to 20 m, the FS increases significantly from 1.175 to 1.336.The FS under x p = 22 m is quite similar to that under x p = 20 m.After x p is greater than 22 m, the FS decreases considerably from 1.335 to 1.186.This variation trend can be attributed to that of  m in Figure 8.Let R a denote the ratio of the FS under a given K to that under K = 1. Figure 9A can be transformed to Figure 9B.
It is noted that x p = 10 in the x-axis means the case of FS without piles.For cases of K > 1.0, consider K = 2.0 for example, the R a under x p = 10, 16, 18, 20, 22, 24, 26 m is 1.061, 1.056, 1.054, 1.043, 1.040, 1.041, and 1.045, respectively.It is noted that as x p ranges between 10 and 18 m, the R a varies slightly around 1.05, when x p ranges between 18 and 20 m, it decreases slightly from 1.054 to 1.043, and when x p ranges between 20 and 26 m it varies slightly around 1.04.Similar variation trend has been observed under K = 1.5.In the cases of K < 1.0, take K = 0.3 for example, the respective R a is 0.957, 0.961, 0.960, 0.969, 0.972, 0.971, 0.969 under x p = 10, 16, 18, 20, 22, 24, 26 m.It is noticed that the variation trends of R a at K = 0.3 and at K = 2.0 are approximately symmetrical to the K = 1.0 line.The range of R a under a given x p represents the influence of K on the FS.Therefore, it is readily found that although the effect of K on the FS is the least pronounced under x p = 20 m, only slight difference regarding the influence of K on FS is noticed at different pile locations.

Reliability analysis
In this section, the reliability index rather than FS of the slope is calculated and is based on to conduct sensitivity studies of D and S of drilled piles.The total number of samples in Monte Carlo simulation is 10,000 which is proven to be a sufficiently large number.As a reference and benchmark, the reliability index is 1.394 for the case of without piles (i.e., case x p = 10).The s under different pile locations for a combination of D = 1 m, S = 4 m, and K = 1 are calculated to demonstrate the influence of pile locations on .As shown in Figure 10, the variation of  with pile locations is similar to that of FS with pile locations, which has been demonstrated in Figure 10.The difference between the two variations is that the enhancement percentage of FS is smaller than that of reliability index .Take K = 1 in Figure 9 for example, the  For a specific D, for example D = 1.0 m, the  is 1.39, 1.39, 1.83, 2.53, and 3.50 for S = 8, 7, 6, 5, 4 m, respectively under x p = 20 m.It is noticed that as S varies from 8 to 4 m, the  increases gradually from 1.39 to 3.50 The similar variation trend is observed for cases of D = 1.2, 1.4, 1.6, 1.8, and 2.0 m.The alternative use of the plots in Figure 11 is to perform pile design aiming at a target reliability .As an illustration, the target  = 2.5 is considered, six tentative designs are provided by finding the intersections of line  = 2.5 with S line.The six designs are listed in Table 1.The optimal design can be achieved through balancing the D, S, and x p as well aiming at the easiness in construction and construction cost.
To quantify the effect of D and S on ,  D , and  S are introduced.Let  D denote the discrepancy in  resulting from D for a specific S and  S denote the discrepancy in  resulting from S for a specific D.  D and  S are calculated as: this variation trend still holds on for other cases of x p = 16, 18, 22, 24, and 26 m.The reason for this trend lies in that a larger D is more likely to yield soil arching between two adjacent piles for the specified S ranges(4-8 m), which contribute to slope stability in a positive manner.

4.3.2
The effect of K on The effect of K on  is investigated in a manner similar to that used in Figure 9B.Let R b denote the ratio of  under a given K to that under K = 1.Consider D = 1 m and S = 4 m for example, Figure 13 shows the variation of R b versus pile locations.For cases of K > 1.0, consider K = 2.0 for example, the R b under x p = 10, 16, 18, 20, 22, 24, 26 m is 1.47, 1.28, 1.18, 1.13, 1.11, 1.15, and 1.30, respectively.It is found that as x p ranges between 10 and 20 m, the R b decreases considerably from 1.47 to 1.11, and when x p ranges between 20 and 26 m it begins to grow significantly from 1.11 to 1.30.Similar variation trend has been observed under K = 1.5.In the cases of K < 1.0, take K = 0.3 for example, the respective R b is 0.60, 0.76, 0.86, 0.90, 0.90, 087, 0.77 under x p = 10, 16, 18, 20, 22, 24, 26 m.It is seen that as x p ranges between 10 and 20 m, the R b increases significantly from 0.60 to 0.90, and when x p ranges between 20 and 26 m it begins to drop from 0.90 to 0.77.It is noticed that the variation trends of R b at K = 0.3 and K = 2.0 are nearly symmetrical to the K = 1.0 line.The range of R b under a given x p represents the effect of K on .It is demonstrated that the effect of K on  is the most significant under x p = 10 (without piles).The least significant effect of K on  is found at x p = 20 and x p = 22 m (piles are drilled at middle part of slope).K exhibits more significant effect if piles are drilled at the lower and upper parts of slope than they are at middle part of slope.The reason for this phenomenon lies in that the  of the reinforced slope when piles are drilled at middle part of slope with K = 1 (without considering anisotropy) is so high that the incremental or decremental quantity of the slope reliability is limited when the anisotropy is considered.

F I G U R E 3
Stress state at different locations along the sliding surface.

F I U R E 5
The flowchart probabilistic pile-reinforced slope reliability analysis.that is, x p = 16, 18, 20, 22, 24, 26 m.The tentative values of D and S are 1.0 and 4.0 m, respectively.K is assumed to be 1.0.The respective means of C h and  are considered, that is C h = 8 kPa and  = 20 • .A set of  values are considered to investigate the effect of  on the FS with piles.Figure7plots the variations of FS with piles with  m (m may be different owing to the variable pile locations).
Figure 8 plots the variation of  m with pile locations under a combination of D = 1, S = 4 m, C h = 8 kPa,  = 20 • , and K = 1.It is observed F I G U R E 6 Illustrative slope model.F I G U R E 7 Variations of FS with piles with  m .F I G U R E 8 Variations of  m with pile locations.

(
A) Variations of FS with pile locations (B) Effect of K on FS under different pile locations F I G U R E 9 The effect of K on factor of safety with piles under different pile locations (D = 1.0 m, S = 4 m).

F I G U R E 10
Variation of  with pile locations under D = 1 m and S = 4 m.FS is increased from 1.113 (x p = 10) to 1.336 (x p = 20 m), whereas the  is enhanced from 1.393 (x p = 10) to 3.5 (x p = 20 m).This may be attributed to the uncertainties in soil properties, which has been accounted for in reliability index.4.3.1 Effect of p , D and S on the reliability index  The sensitivity studies of x p , D and S on  are performed.Six x p s (x p = 16, 18, 20, 22, 24, 26 m), six Ds (D = 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 m) and five Ss (S = 4, 5, 6, 7, and 8 m) are designed and for each set of combinations, the proposed method is implemented to obtain 10,000 FSs and finally to get the  for each set of combinations.Figure 11 plots the variations of  with Ds under different pile locations and spacing.Under x p = 20 m and S = 4 m, the respective  is 3.45, 3.74, 3.91, 4.20, 4.36, and 4.70 under D = 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 It is seen that as D increases from 1.0 to 2.0 m, the  increases considerably from 3.499 to 4.360 at an approximate linear rate.Similar trends have been noticed for cases of S = 5, 6, and 7 m.Under S = 8 m, the  remains almost unchanged as D increase from 1.0 to 2.0 m.This phenomenon can be explained by the fact that soil arching does not occur between adjacent piles under the condition of S = 8 m.

) 12
Figure 12 plots the variations of  D and  S under different pile locations.Refer to Figure 12A, under x p = 20 m, the respective  D is 34, 32, 34, 30, and 0% at S = 4, 5, 6, 7, and 8 m.It is noticed that from Figure 12A that  D tends to decrease as S increases.Negligible effect of D on  is observed due to the inability to trigger soil arching at S = 8 m.Similar variation trends have been noticed for cases of x p = 16, 18, 22, 24, and 26 m.It must be noted that the most significant effect of D on  is found at S = 4 and 5 m under x p = 26 m.Turn to Figure 12B, under x p = 20 m, the  S is 151%, 168%, 180%, 201%, 212%, and 236% at d = 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 m, respectively tends to increase as D increases from 1.0 to 2.0 m and