J. Cent. South Univ. (2019) 26: 1719-1734

DOI: https://doi.org/10.1007/s11771-019-4128-1

Stability analysis for nonhomogeneous slopes subjected to water drawdown

SUN Zhi-bin(孙志彬)¹, SHU Xing(疏醒)¹, Daniel DIAS²

1 School of Automotive and Transportation Engineering, Hefei University of Technology,Hefei 230009, China;

2. Laboratory 3SR, Grenoble Alpes University, CNRS UMR5521, Grenoble F-38000, France

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Comparing with the homogeneous slope, the nonhomogeneous slope has more significance in practice. The main purpose of the present study is to provide a preliminary idea that how the nonhomogeneity influences the stability of slopes under four different water drawdown regimes. Two typical categories of nonhomogeneity, identified as layered profile and strength increasing with depth profile, are included in the paper, and a nonhomogeneity coefficient is defined to quantify the degree of soil properties nonhomogeneity. With a modified discretization technique, the safety factors of nonhomogeneous slopes are calculated. On this basis, the variation of safety factor with the nonhomogeneity coefficient of friction angle and the water table level are investigated. In the present example, safety factor correlates linearly with friction angle nonhomogeneity coefficient from a whole view and the influences of the water table level on safety factor is basically similar with that in homogeneous condition.

Key words: upper bound; discretization technique; non-homogeneous slope; water drawdown

Cite this article as: SUN Zhi-bin, SHU Xing, Daniel DIAS. Stability analysis for nonhomogeneous slopes subjected to water drawdown [J]. Journal of Central South University, 2019, 26(7): 1719-1734. DOI: https://doi.org/10.1007/ s11771-019-4128-1.

1 Introduction

Slopes located nearby a reservoir are more susceptible to landslides, which entails casualties and property losses of residential areas. The risk of slope collapse will increase significantly when the slope suffering the process of water table drawdown. Sufficient knowledge of the stability condition is vital to the protection design of such a slope. In order to provide effective stability evaluation tools, scholars have proposed numerous types of method. More details can be found in the pieces of literature by MORGENSTERN [1], PINYOL et al [2], VIRATJANDR et al [3], where the analysis approaches can be grouped into three categories: the limit equilibrium, the numerical approach, and the limit analysis.

Limit equilibrium is a widely adopted approach for the analysis of slope stability with pore pressure. MORGENSTERN [1] used such a method to evaluate the capability of the slope during the rapid water drawdown, where the pore pressure was defined by the pore-pressure coefficient. With the help of the stability charts proposed by the study, engineers can figure out the slope safety factor directly. However, the limit equilibrium method didn’t cover the stress-strain relation of soils which makes the solutions derive an approximate value rather than an exact one [4].

Alternately, the numerical simulation was also employed to estimate the slope stability affected by water drawdown. LANE et al [5] utilized the finite element method to investigate the influence of the water level drop rate and the initial position of the reservoir water level on homogeneous slopes. Lately, considering the effect of water on the soil strength, PINYOL et al [2] incorporated the influences of pore pressure into the finite element method to assess the stability of slopes subjected to both rapid and slow drawdown. The numerical simulation can provide comprehensive information of slope, such as stress, strain, and displacement, rather than only the stability condition. However, such approach is puzzled by the complication and time-consuming in the modeling process and the accuracy of the calculation more depends more on the user’s experience.

Apart from the two above methods, another attractive approach is limit analysis [6-10], which is based on the plastic mechanics rigidly. Such a method includes two different approaches: the kinematic approach and the static method. The kinematic approach, which gives a true upper bound of the limit solution, is more widespread due to the more conveniences. Through the multiple reliable validations by experiments or numerical calculations, this kinematic approach has become an effective tool for slope stability assessment [11-13].

Including the effect of the pore pressure in a kinematic mechanism, the kinematic analysis can also be employed to analyze the influence of water on a slope [14-17]. As to a slope subjected to the water drawdown process, VIRATJANDR et al [3] assumed four different water drawdown processes and then gave a closed-form solution for the slope safety. The stability results were presented as stability charts, therefore, the variation of the safety factor during the drawdown process can be obtained directly without the process of iterative calculation.

The previous researches involved with the slope subjected to water drawdown were limited to the slope with uniform soil properties, i.e., the homogeneous slope, which is not always in accordance with the real case. The non- homogeneity in soil properties has a significant influence on the stability condition of slopes [18-20]. In order to offer a more general guidance to the practical engineering, it is of practical significance to extend the reservoir slope analysis to the nonhomogeneous condition. However, an insurmountable difficult will emerge in the prediction of the velocity discontinuity surface due to the requirement of normal condition when the frequently adopted log-spiral kinematic mechanism is implemented for a non-homogeneous slope.

The presented study adopts a new type of the kinematic analysis technique, so-called the discretization technique, to overcome such shortcomings. This novel technique was originally proposed by MOLLON et al [21-24] for the face stability issue of a heterogamous tunnel and then was introduced into the slope analysis by SUN et al [19], LI et al [25], QIN et al [26]. This technique is characterized by the concept of a sliding surface consisting of a series of segments rather than being a continuous log-spiral curve, which makes the normal condition easily satisfied in a nonhomogeneous case by adjusting the location of each segment.

Aiming to provide some useful descriptions about the influence of non-homogeneity on the slopes subjected to water drawdown, this paper investigates the nonhomogeneous slope stability under four different water drawdown regimes as shown in Figure 1. Case 1 corresponds to the rapid drawdown from completely submerged condition to fully drained, similarly, Case 2 expresses the rapid drawdown with the reservoir been fully drained, Case 3 denotes the slow drawdown with the water level in slope been equal to the reservoir, and Case 4 represents the slow drawdown with the water level in slope and reservoir keep a constant.

This paper is organized as follows: First, a brief introduction of the discretization mechanism and the calculation flow are presented. Later, a general method of the computation of the weight work rate is proposed to extend to the application of the discretization to the density varying condition. With the improved approach, the safety factor of two typical nonhomogeneous reservoir slopes is solved and then the influence of soil strength parameters, geometry, and water drawdown condition is analyzed. The results are supposed to give a brief characteristic that how the non- homogeneity influences the stability of the reservoir slope.

2 Pore pressure in kinematic analysis

As mentioned, this study uses the kinematic approach to the stability evaluation of soil slopes subjected to water drawdown. The concept of taking the water effect into account in the framework of kinematic analysis is on the basis of transforming the work rate of the pore pressure into those of seepage force and buoyancy force [16]. Grounding on that and using Guess theorem, the work rate of the pore pressure can be expressed as:

                  (1)

where is the volumetric strain rated defined as ; u denotes the pore pressure; V denotes the volume of the potential failure block; S is the surface bounding the volume V; v_i corresponds to the velocity along the falling blocks at failure, and n_i is a unit vector normal to the surface S.

Figure 1 Four different water drawdown processes (adapted from Ref. [3]):(H: slope height; L₁: vertical distance from the water level in the reservoir to the slope crest; L₂: vertical distance from the water level in the slope to the slope crest)

The first integral of the right side of Eq. (1) is the work rate of pore pressure on the volumetric strain of the skeleton, and the second integral is the work rate of pore pressure on the boundary S of the structure.

Thus, when the upper bound analysis is employed, the work rate balance equation including the effect of the pore pressure can be written as:

                             (2)

where σ_ij and denote the stress tensor and strain rate in the admissible velocity field; T_i is the external load on the surface S; X_i is the internal force inside the area V.

Equation (2) is a complete form of the upper bound theorem under the action of pore pressure. With the equality of the work rates on the left and right sides in an admissible velocity field, the final limit state can be obtained and then the upper bound solution can be determined after an optimization process. The following analysis is based on this principle.

3 Discretization collapse mechanism generation

A suitable failure mechanism was necessary for the upper bound kinematic analysis to define a reliable admissible velocity field. The failure surface of a log-spiral shape was dominant in the conventional kinematic analysis, however, such a surface had limitations in analyzing slopes with various soil properties. Considering the need for non-homogeneous slope analysis, we adopt the discretization technique to generate the slope potential collapse surface herein. And the main principle for the generation of this mechanism will be presented in this section.

Figure 2 illustrates a reservoir slope with a discretization mechanism, where H and β define the height and angle of the slope. It is assumed that sliding surface AC passes through the slope toe and can be identified by two parameters, r₀ and θ₀, where r₀ is the length of OC and θ₀ is the angle between the x-axis direction and the line OC. The rigid sliding block ABC, rotating around the point O with an angular velocity ω, is postulated as a rigid body. The part below the surface AC is also rigid and unmovable.

In a discretization mechanism, the velocity discontinuity surface of the slope consists of a series of the segment P_iP_i+1 which is determined by the discretization points P₀, P₁, …, P_i, P_i+1, …, P_n. In order to achieve geometric consistency, the angle between two adjacent radical line OP_i and OP_i+1 is supposed to be the same and is denoted as δθ.

The value of δθ determines the accuracy of the discretization, and an adequately small value of δθ makes the discretization sliding surface close to the standard log-spiral surface for a homogeneous case. According to the research of SUN et al [19], δθ=0.1° is adopted in this study, which is recommended as a great combination between accuracy and time cost.

To obtain all the location of the discretization points, a “point to point” technique is used in this study, which means that each point is derived from the preceding one and the distribution of all discrete points is quasi-uniform.

The key to the process is to find the recursive formula to solve the coordinates of the discretization point. Thus, two points P_i and P_i+1 are taken for an example in Figure 3 to illustrate the derivation of such formula. For simplification, a coordinate system with slope toe as the origin is established.

and are two vectors starting from the point O. The angle between and the negative direction of the x-axis is θ_i; n_i is a unit vector that perpendicular to In order to satisfy the flow rule of the upper bound theorem, the angle between the tangent velocity and the real velocity is required to be the friction angles φ_i. This means that the angle between the vector and the velocity v_i of the point P_i is (π-φ_i).

is the velocity vector of a point P_i, whose direction is perpendicular to the vector of and can be written as

Figure 2 Discrete failure mechanism of slope

Figure 3 Illustration of generation of velocity discontinuity surface

                              (3)

where ω can be set to 1.

Then decompose orthogonally to the x and y axes and can be written as

                     (4)

The angle between the and horizontal direction isthus the horizontal and vertical components of the vector are

                        (5)

Assumewhere λ_i is the module of and is a unit vector. Besides, the angle between and the horizontal direction is π-θ_i, thus the horizontal and vertical components are

                       (6)

is a unit vector which is perpendicular to and the slip surface, namely

                             (7)

It can be obtained by adding or subtracting vectors:

                        (8)

Substituting Eq. (7) into Eq. (8) and entering the coordinate operation of the vector, the can be expressed as

   (9)

where θ_i+1 is the angle between OP_i+1 and x-axis.

Then, the coordinates of the point P_i+1 can be denoted by P_i as

                    (10)

where x₀ and y₀ are the coordinates of the rotating center O, respectively.

Starting from the point C with the recursion Eq. (10), all the discrete points on the velocity discontinuity surface can be deduced. The generation of the slip surface terminates when the ordinate y_i of the point P_i is greater than (or equal to) H, then stop calculation and vector discontinuity surface is generated. Besides, if y_i>H, the linear interpolation technique should be used to adjust y_i, and make sure the last discrete point just fall on AB, in other words y_i=H.

4 Work rate calculations

Based on the upper bound theorem of limit analysis, it is necessary to establish a virtual work rate equation as Eq. (11) to provide an upper bound solution.

W=D                                  (11)

where W is the work rate of external forces and D is the internal energy dissipation rate. Normally, W is only defined by the work rate of soil weight W_γ.

However, in order to account for the effect of water in the present study, an additional work rate W_p done by pore pressure should also be included in W. The statement of the calculation of the work rates is described in more details as below.

4.1 Work rate of soil weight

This section aims to illustrate the solution process of W_γ in the discretization approach. It should be noticed that a novel calculation flow is proposed here especially for soils with varying density, which is considered as an improvement to the original discretization method that only applies in the uniform density case [19].

To offer a better description, one typically layered slope with n soil layers is represented in Figure 4(a) as an example, where AC is a potential sliding surface generated by discretization technique. The dividing line between two adjacent soil layers and is denoted as l_n(x), and γ_n, φ_n, c_n are the unit weight, friction angle and cohesion of the nth layer soil, respectively.

Notice that the potential collapse block ABC is divided into a series of triangular elements by the discretization points on the sliding surface.

Thus, the total weight work rate of the whole mechanism can be achieved by the summation of all the weight work rate of each element:

                               (12)

where W_γ is the weight work rate of the collapse mechanism, W_i is the weight work rate of the i-th element.

To define the calculation process of W_i,Figure 4(b) describes a single triangular element BP_iP_i+1 with the three vertexes being B, P_i, P_i+1, respectively.

Notice that one element contains m (m≤n) soil blocks which are denoted by b₁, b₂, …, b_k … b_m, each of them belongs to different soil layers with different unit weight. Thus, the weight work rate of the element BP_iP_i+1 is the sum of each block:

                             (13)

where W_bk is the weight work rate of the k-th block.

The shape of the block can be a triangle or a quadrangle in one element, thus the equations of W_bk are categorized.

The weight work rate of the first triangular block BS₁T₁ can be expressed as

                       (14)

where γ₁ is the unit weight of the first layer soil, ω is the angular velocity, S₁ is the area of the block BS₁T₁, x_m is the barycentric coordinates of the block BS₁T₁ and x₀ is the abscissa of the point O.

The general formula of weight work rate for the rest blocks can be expressed as

        (15)

where γ_k is the unit weight of the m-th layer soil, S_k is the area of the triangular element BS_kT_k, x_k is the barycentric coordinates of the triangular element BS_kT_k. When the subscript of changes to k-1, γ_k-1,S_k-1 and x_k-1 represent the corresponding index of the block k-1.

Figure 4 A layered soil slope (a) and element of failure block (b)

The expression of S_k and x_k can be compiled as

         (16)

                    (17)

where p is the half of the perimeter of the triangular element SB_kT_k; x_B, x_Sk and x_Tk are the abscissa of the points B, S_k, T_k.

4.2 Work rate of pore pressure

To determine the work rate of pore pressure, the spatial distribution of pore pressure should be clarified during the process of water table drawdown. However, as pointed out by MORGENSTER [1] and VIRATJANDR et al [3], the premise of obtaining an exact solution of the pore pressure distribution only involved with, from a theoretical view, the specific hydraulic and geometry conditions with a given slope. It is difficult to obtain the generalized results for such so-called transient seepage problem.

With the purpose of achieving the pore pressure solutions with sufficient accuracy and convenience, several simplifying assumptions have been made in the pieces of literature, which are also adopted in the present study: 1) The assumption of vertical equipotential within the slope is made as shown in Figure 5; 2) Neglecting seepage, and there is no dissipation occurring after water drawdown; 3) The slope soil below the free-surface is fully saturated.

The value of the pore pressures can be expressed as

                               (18)

where u is the pore pressure; h_w is the vertical distance from the point to the water line; and γ_w is the unit weight of water.

As shown in Figure 6, the acting surface of the pore pressure of a reservoir slope can be divided into two parts: 1) on the velocity discontinuity surface and 2) on the slope surface of the submerged area. The pore pressure in the former part is induced by the underground water in slope, while the second part is caused by the water in the reservoir.

Figure 5 Details of assumption of pore pressure distribution

Accordingly, the computation of the work rate of the pore pressures is done in such parts as well.

                          (19)

where W_p1 is the work rate of the pore pressure on the failure surface; W_p2 is the work rate of the pore pressure on the slope surface of the submerged area. W_p1 is related to the height of the water table in the slope and the shape of the sliding surface, namely the location of all discretization elements P_iP_i+1.

Figure 6 Details of water effect on velocity discontinuity surface and slope surface of submerged area

The expression of W_p1 can be presented as

                      (20)

where u₁ is the pore pressure on the element P_iP_i+1; v_i is the velocity of the midpoint of P_iP_i+1; L_i is the length of P_iP_i+1 and φ_i is the friction angle of soils at the midpoint of P_iP_i+1.

The expressions of L_i can be compiled as

               (21)

where x_i and x_i+1 are the x-coordinate of the point P_i and P_i+1, y_i and y_i+1 are the y-coordinate of the point P_i and P_i+1.

The pore pressure caused by the reservoir water acts on the surface CE, where the point C is the slope toe and the point E is the intersection between the water table and the slope surface. Thus, the work rate of the pore pressure for this part can be calculated by integration and can be presented as

                         (22)

where u₂ is the pore pressure to the slope surface of the submerged area; v_i is the velocity of the surface and n_i is the unit vector perpendicular to the surface CE.

4.3 Work dissipation rate

Owing to the rigid block assumptions mentioned in previous, the internal energy dissipation only occurs on the critical sliding surface AC. Like the computation of W_γ, the work dissipation rate D can be achieved by the summation of the work dissipation rate per discrete segment P_iP_i+1, which can be expressed as

                      (23)

where c_i and φ_i represent the cohesion and friction angle at the midpoint of P_iP_i+1; R_i is the distance from O to the midpoint of P_iP_i+1. The expressions of R_i can be compiled as:

       (24)

5 Safety factor calculation

With the various initial values of θ₀ and r₀, a series of potential collapse surfaces with different safety factors can be generated. The strength reduction method is conducted to obtain the true value F_S that reflects the steady state of the slope which is the minimum of these safety factors. The method is based on the idea that adopting discounted shear strength to make slope on the verge of failure, which specific details can be seen in Eq. (25).

                   (25)

Therefore, the calculation of the minimum safety factory in this study is realized through the computer program, and the mathematical optimization problem is considered in the process. The specific calculation steps are shown in Figure 7.

6 Comparison and parametric analysis

6.1 Comparison of homogenous slopes with others results

In order to prove the accuracy of the discretization-based method, we calculate the safety factor of a homogeneous slope in the four water drawdown regimes mentioned in the introduction section and conduct the validation the discretization-based results with the previous achievement may be due to the two diverse assumptions of the plastic behavior of soils in different approaches: associated flow in limit analysis and non-associated flow in numerical simulation, as pointed out by VIRATJANDR et al [3]. During the validation, the stability factor c/γH of the slope is set to 0.05 and the slope inclination is 1:2.

As listed in Table 1, the comparison shows that the maximum difference between the presented results and those of VIRATJANDR et al [3] is only 4.25%, which is within a reasonable limit. The good agreement illustrates that the proposed method can give a reasonable solution to the slope stability in drawdown conditions.

Moreover, LANE et al [5] adopted the numerical simulation to investigate the slope stability issues in two different water drawdown conditions: the rapid drawdown process and the slow drawdown process, namely the Case 1 and Case 3 in the presented paper.

To make the presented study more convincing,another validation was further carried out with the results by LANE et al [5]. It can be observed that, as shown in Table 2, the safety factors by LANE et al [5] and the present study are quite close with the maximum difference 6.9%, which indicates that the solutions derived from this present study are accurate and reasonable for the stability of slope subjected to water drawdown. The major reason of the differences in comparisons may be due to the two diverse assumptions of the plastic behavior of soils in different approaches: associated flow in limit analysis and non-associated flow in numerical simulation, as pointed out by VIRATJANDR and MICHALOWSKI [3].

Figure 7 Computation and optimization flow of safety factor

Table 1 Comparison of safety factor with traditional kinematic analysis method

Table 2 Comparison of results with finite element method

6.2 Parametric analysis of non-uniform soil slope

Due to the complex geology and environment effect in the formation history of soils, the non- homogeneity of soil exists in various forms [18, 27]. Roughly the non-homogeneity types of slopes can be divided into two categories: the soil strength distributing linearly throughout the slope [6, 26] and the soil strength being non-uniform for different slope layers [28-30].

Practically, cohesion and friction angle can illustrate a variation in the soil space. Most previous researches were only involved with the effect of the variation of cohesion and/or friction angle.

Considering the unique merit of analyzing the friction angle varying case, and in order to especially grasp the influence of the variation of friction angle on slopes subjected to water drawdown, the focus of this paper was on the case where only the friction angle shows non- homogeneity. Moreover, for the purpose of describing the non-uniformity of slope intuitively, a parameter named the non-uniform coefficient of the friction angle λ_φ is introduced, which is able to quantify the degree of the non-homogeneity of friction angle of slope to a certain extent.

6.2.1 Stratified soil slope profile

To give a typical example of the layered slope, a simple two-layered soil slope is discussed as sketched in Figure 8, which has two soil layers with the same height. The geometry parameters of the slope and the detail strength parameters of soils are also presented in the figure.

Figure 8 Schematic diagram of a two-layered soil slope under pore pressure

The friction angle of the second layer φ₂ is treated as a variable with the range from 20° to 40° and it can be expressed by the conduct of the non-uniform coefficient of the friction angle λ_φ and the friction angle of the first layer φ₁, namely φ₂=λ_φφ₁. In addition, two different slope inclinations (1:2, 1:3) are considered here.

Four sets of stability charts in Figures 9-12 are plotted to provide the safety factors of the slope under rapid and slow drawdown processes. Specially, Figures 9-12 correspond to the regimes indicated in Figures 1(a)-(d), respectively, and each curve in these charts represents the change in safety factor F_S with the λ_φ. The first and second sets of stability charts (Figures 9 and 10) are given for two rapid drawdown processes, where Figure 9 (Case 1) represents a completely submerged condition to a fully drained one (L₁=0). Different from Case 1, Figure 10 (Case 2) is presented for a drawdown of a slope with the reservoir has been fully drained (L₁=H).

Generally, it can be found that in these two charts the variation of F_S with the non-uniform coefficient λ_φ is roughly similar. It is indicated that Case 1 and Case 2 have the comparable stability condition under non-homogeneity in friction angle.

Comparing Figures 9(a)-12(a) with Figures 9(b)-12(b) in all charts, we can easily find that the stability of the slope decreases with the increase of slope inclination. For instance, in Figure 9, an increase of λ_φ from 4/6 to 8/6 leads to a 29%-34% (slope inclination 1:2) increase in the safety factor F_S, while with the same variation range of λ_φ, an increase of F_S around 34%-38% is exhibited in the slope inclination 1:3 case.

Observing the variation of F_S with the changing of water level in Case 1 and Case 2, two opposite patterns can be noticed.

In Case 1, as shown in Figure 9, the safety factor decreases with the increase of the water drawdown ratio L₁/H. For instance, F_S is decreased by around 35% (slope inclination 1:3) and 42% (slope inclination 1:2) for L₁/H varying from 0 to 1.0, yet the F_S increases around 37% (slope inclination 1: 3) and 40% (slope inclination 1:2) in Figure 10 (Case 2) with the same level of water drawdown.

Figure 9 Safety factor for a rapid drawdown from completely submerged condition to fully drained (L₂=0):

Figure 10 Safety factor for rapid drawdown with reservoir has been fully drained (L₁=H):

Figure 11 Slow drawdown with water level in slope is equal to reservoir (L₁=L₂):

This phenomenon is caused by the different effects of water in reservoir and slope on stability. Generally speaking, the pore pressure from the reservoir, which acts on the slope surface of the submerged area, is equivalent to a reverse pressure load to benefit slope stability. Thus, the water level drawdown of the reservoir in Case 1 will lead to the decline of the safety factor. Meanwhile, it is a general phenomenon that the pore pressure inside the slope is not conducive to the slope stability. Hence, the increase of L₂/H in Case 2, in other words, the drawdown of the water level inside the slope, will inevitably lead to the increase of safety factor F_S.

Figure 12 Safety factor for slow drawdown with water level in slope and reservoir keep a constant (L₁-L₂=0.2H):

Furthermore, the water level also affects the variation characteristic of F_S with the non- uniformity coefficient λ_φ. To give a whole view, the variation rate of F_S and λ_φ decreases with the changing of the water table, no matter in Case 1 and Case 2.

The third and fourth sets of stability charts (Figures 11 and 12) are given for two slow drawdown processes, where Figure 11 (Case 3) indicates that the water level inside the slope is always equal to that in the reservoir (L₁=L₂), and Figure 12 (Case 4) is for the drawdown process with the water level in the reservoir and in the slope keeping constant (L₁-L₂=0.2H).

As shown in Figures 11 and 12, these two sets of charts have an unquestionable commonality with Case 1 and Case 2 in the variation of F_S with λ_φ. In Figure 11, F_S is decreased by 27%-39% (slope inclination1:2) and 34%-41% (slope inclination 1: 3) for λ_φ varying from 4/6 to 8/6. In addition, in the slow drawdown process changing of safety factor with drawdown ratio L₁/H has a similar regularity, but is different from the rapid drawdown case. Specifically, in Case 1 and Case 2, as L₁/H increases, F_S always varies monotonically, however, in Case 3 and Case 4, the varying trend of F_S is not always the case. During a slow drawdown process, the value of F_S has its maximum at the beginning of the drawdown process and the minimum safety factor is reached when a certain water table before the drain process is done. The characteristic of the variation of water table is consistent with the conclusions of LANE et al [5], VIRATJANDR et al [3] and GAO et al [31].

GAO et al [31] proposed the possible explanation that such phenomenon was caused by the trade-off between the work rate which done by the external pore pressure (by reservoir water benefit the stability) and the internal pore pressure (by water in slope prejudice the stability). At the beginning of water drawdown, the impact of internal pore pressure on the slope stability is greater than the external pore pressure, which leads to the reduction of the safety factor F_S. However, when the water level drops to a certain extent, the external pore pressure plays a dominant role, so the safety factor F_S has a small rise, ultimately.

6.2.2 Linearly increased strength profile

As shown in Figure 13, the soil friction angle is distributed linearly throughout the slope. H is the slope height, φ₀ represents the friction angle on the horizontal surface passing the slope toe, the friction angle at the depth with a distance z from the slope crest can be expressed as:

                         (26)

where λ_φ is the non-uniform coefficient of the friction angle for the case where the rate of the friction angle varies linearly with depth. The value of stability factor c/γH is 0.05 and two slope inclinations 1:2 and 1:3 are also included. In addition, the detail strength parameters of the slope are also shown in the diagram.

Figure 13 Schematic diagram of slope which friction angle varying with depth

Similar to the layered slopes, four sets of stability charts are also demonstrated in Figures 14-17 to reflect the relationship between the safety factor F_S and the λ_φ under four special drawdown regimes.

Figure 14 Safety factor for rapid drawdown from completely submerged condition to fully drained (L₂=0):

Figure 15 Safety factor for rapid drawdown with reservoir been fully drained (L₁=H):

Figure 16 Slow drawdown with water level in slope equal to reservoir (L₁=L₂):

Figure 17 Safety factor for slow drawdown with water level in slope and reservoir keep a constant (L₁-L₂=0.2FH):

The first and second sets of stability charts (Figures 14 and 15) are for the rapid drawdown processes. Without a doubt, the increase in the non-uniform coefficient of the friction angle λ_φ does help to enhance the slope stability. The relationship between the non-uniform coefficient λ_φ and the safety factors F_S is basically linear.

Observing the variation of the safety factor F_S with the water drawdown ratio L₁/H, there is a similarity between this case and the layered one. The safety factor F_S is decreased by 40%-44% (slope inclination 1:3) and 40%-43% (slope inclination 1:2) with the water drawdown ratio L₁/H varying from 0 to 1 in Figure 14. Conversely in Figure 15, variation of L₁/H from 0 to 1 will increase F_S by 39%-43% (slope inclination 1:3) and 39%-45% (slope inclination 1:2).

The third and fourth sets of stability charts (Figures 16 and 17) are given for slow drawdown processes, where the water in the reservoir and inside of slope always drains with equivalent rates.

It is not difficult to draw from Figures 16 and 17 that the safety factor F_S exhibits a linear increase along with the coefficient of non-uniform of friction angle λ_φ. When it turns to the influences of L₁/H on the safety factor F_S, the value of F_S increases first and then decreases with the L₁/H increase. These conclusions are consistent with the layered slope.

7 Conclusions

Based on the kinematic approach of limit analysis, the discretization technique is utilized to investigate the stability of the nonhomogeneous slopes subjected to water drawdown. The sliding surfaces of slopes are generated not with a single line but a series of non-standard curves. From the perspective of plastic theory, such a discretization failure mechanism is more reasonable.

The discretization-based results are compared with the conventional limit analysis and numerical approach as well. As expected, the present results have a good consistency with those of the two approaches, which means that the proposed mechanism is effective.

In order to better understand the influences of the non-uniformity of friction angle on slope stability, the results of the safety factor F_S varying with λ_φ are presented in the form of stability charts. A basically linear relationship can be observed between F_S and λ_φ, of which the change rate is affected by slope inclination and water table.

During the slow drawdown process (Case 3 and Case 4), the slope safety factor is minimized when the water table reaches a certain level before drained fully. The level of the critical water table is influenced by the value of λ_φ when other parameters are given as constant. According to the example presented in the paper, the critical water level fluctuates between 20% and 40% of the original water level height. A preliminary conclusion can be drawn that that the critical water table for homogeneous slopes has a certain guiding significance for the nonhomogeneous slope with the variation of friction angle. However, to give a better understanding of the impact of the slope non-homogeneity on the critical water table, a more comprehensive parameter analysis will be conducted in our future research.

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(Edited by HE Yun-bin)

中文导读

水位下降条件下非均质边坡稳定性分析

摘要：与均质边坡相比，非均质边坡对于实际工程中更具意义。本研究着重讨论土体非均质性在四种水位下降模式下对边坡稳定性的影响。在考虑分层与强度随深度线性增加两类典型非均质性的基础上，定义了非均匀性系数以量化土体强度的非均质性程度。采用改进的离散上限分析法，计算了不同工况下非均质边坡的安全系数F_s，研究了F_s随摩擦角非均匀系数以及水位高度的变化规律。整体上看F_s与摩擦角非均匀系数呈线性相关，而水位对F_s的影响与均质边坡基本相似。

关键词：上限法；离散化技术；非均质边坡；水位下降

Foundation item: Project(51408180) supported by the National Natural Science Foundation of China

Received date: 2018-12-24; Accepted date: 2019-02-22

Corresponding author: SUN Zhi-bin, PhD, Associate Professor; E-mail: sunzb@hfut.edu.cn; ORCID: 0000-0002-3607-8825