#### Abstract

In the strict framework of limit analysis, an analytical approach is derived to obtain the upper bound solutions for three-dimensional inhomogeneous slopes in clays under undrained conditions. Undrained strength profiles increasing linearly with depth below the crest of the slope and below the outline surface of the slope are assumed representative of cut slopes and natural slopes, respectively. Stability charts are produced for the cut slopes and the natural slopes under both static and pseudostatic seismic loading conditions. The presented charts are convenient to assess the preliminary and short-term stability for 3D slopes in practical applications, such as rapid excavation or buildup of embankments and slopes subjected to earthquakes. Compared with the available results from the finite element limit analysis method, a better estimate of the slope safety is obtained from the analytical approach.

#### 1. Introduction

Stability charts for slopes provide an efficient tool for the preliminary assessment of slope safety and for the calibration of any sophisticated numerical models that are ultimately used for solving more complex slope stability problems. The development of stability charts has been the subject of many investigations since the pioneering work of Taylor [1]. One of the most intensively studied issues is the stability charts for slopes in clays under short-term undrained conditions. These charts are usually used to obtain the stability for rapid excavation or buildup of embankments and slopes subjected to earthquakes. Based on different assumptions about the undrained strength profiles, failure mechanisms, and loading conditions, a series of stability charts has been derived using the traditional methods for total stress analysis of slope stability such as the limit equilibrium method, the limit analysis method, and the finite element method.

For homogeneous slopes, stability charts have been derived for two-dimensional (2D) slope failures [1–6] and three-dimensional (3D) slope failures [7–12]. For inhomogeneous slopes, as encountered in most practical situations, an undrained strength profile increasing linearly with depth below the crest of the slope, as shown in Figure 1(a), is usually assumed representative of cut slopes, as generally expressed bywhere *c*_{u} is the undrained strength of soil at a given depth; *z* is the depth below the crest of the slope; *c*_{u0} is the undrained strength at the crest of the slope; and *ρ* is the gradient of undrained strength with respect to the depth. The limit equilibrium (LE) method [13–15] is employed to stability analyze of cuttings in normally consolidated clays. Koppula [16] conducted seismic stability analysis of undrained slopes using the pseudostatic approach. Yu et al. [17] also adopted the finite element limit analysis method to obtain the least upper bound solutions, which are compared with traditional LE results. Based on the presented methods, the stability charts for such defined cut slopes have been derived for 2D slope failures. For more realistic 3D failures of cut slopes, Li et al. [18] derived a set of stability charts by using the finite element methods for upper bound and lower bound limit analyses developed by Lyamin and Sloan [19, 20] and Krabbenhoft et al. [21]. Furthermore, by using these methods, Li et al. [22] derived another set of stability charts for 3D failures of natural slopes by assuming an undrained strength profile increasing linearly with depth below the outline surface instead of the crest of slope, as shown in Figure 1(b), where *z* is the depth below the surface of the slope and *c*_{u0} is the undrained strength at the surface of the slope. These two sets of the stability charts provide the upper and lower bounds to the safety factors for cut slopes and natural slopes. Nevertheless, their accuracy may be further refined by using a rigorous analytical method for slope stability analysis to avoid being affected by the artificial boundary conditions and mesh sizes specified in the adopted numerical methods.

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In the strict framework of limit analysis, Michalowski and Drescher [23] proposed a class of kinematically admissible 3D rotational failure mechanisms for slopes in clays under both undrained and drained conditions. Also, the proposed is the corresponding analytical approach for the upper bound limit analysis of stability of homogeneous slopes. Recently, their work has been extended by Gao et al. [24] to include face-failure and base-failure mechanisms. And an optimization technique proposed by Chen [25] had been adopted and shown more effective in finding the critical failure surfaces and the least upper bounds to the critical heights of homogeneous slopes.

In the present paper, the analytical approach originated by Michalowski and Drescher [23] and extended by Gao et al. [26] is further exploited to derive the stability charts for inhomogeneous cut slopes and natural slopes (as shown in Figure 1) under both static and pseudostatic seismic loading conditions [27–29]. The stability charts are plotted for a range of parameters wider than those presented by Li et al. [18, 22]. A comparison is made between the calculated results from this study and from the study of Li et al. [18, 22] to illustrate the effectiveness of the adopted failure mechanisms in finding the least upper bounds to the slope safety factors.

#### 2. Analytical Procedures for Limit Analysis of Slope Stability

##### 2.1. 3D Failure Mechanism

The 3D rotational failure mechanism proposed by Michalowski and Drescher [23] was extended to involve the failure surface passing below the toe by Gao et al. [24]. Figure 2(a) shows the extended failure mechanism for slopes in clays under undrained conditions. The failure mechanism is generated by rotating a circle of diameter *R* (shaded area in Figure 2(a)) about an axis *r*_{m} passing through point *O* outside the circle:where the radii *r* and *r′* are shown in Figure 2(a). The failure surface passes through the toe when angle *β′* equals the slope angle *β* (where angle *β′* can be found in Figure 2(a) and *H* is the slope height). Figure 2(b) shows 3D drawings of the failure surface for slopes with finite width *B* modified with a plane insert of width *b*, to allow the transition to the 2D failure mechanism of Chen [30] as *b* approaches infinity. For details of the construction of the 3D admissible rotational failure mechanism, see the source references [23, 24]. The failure mechanism is kinematically admissible for both homogeneous slopes and inhomogeneous slopes under undrained conditions.

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##### 2.2. Stability Analysis

Based on the abovementioned 3D mechanism, an upper bound to the stability number *N*_{ρ} = *γHF*/*c*_{u0} (where *F* is the factor of safety and *γ* is the unit weight of soil), which has been analyzed numerically by Li et al. [18, 22], can be determined by equating the rate of wok *W*_{γ} done by soil weight to the rate of internal energy dissipation *D*. To account for the effect of horizontal seismic forces on slope stability, an additional rate of work *W*_{s} done by the pseudostatic seismic forces is counted into the energy balance equation. In general, the balance equation is given as follows:where *k*_{h} is the horizontal seismic acceleration coefficient; the superscript “curve” denotes the work rates for a section of the curvilinear cone at the two ends of the failure mechanism and the superscript “plane”’ relates to the plane insert in the center of the mechanism. The expressions of , , and *D*^{plane} for the plane insert can be found elsewhere [30, 31]. The expressions of , , and *D*^{curve} for cut slopes and natural slopes can be derived following the same procedures and similar symbols for obtaining the expressions for homogeneous slopes as presented by Michalowski and Drescher [23].

Since the work rate done by soil weight is independent of the undrained strength of soil, the expressions of and for a cut slope are identical to those for a natural slope with the same failure mechanism. For the extended failure mechanism, the work rates and are derived aswhere *ω* is the angular velocity and variables *a*, *d*, *e*, *θ*_{B}, and *θ*_{C} are obtained from the geometrical and trigonometric relations in Figure 2(a) as

Unlike the work rates and , the rate of internal energy dissipation *D*^{curve} is, however, dependent on the undrained strength profile of soil. To account for the effect of the assumed linearly increasing undrained strength with depth, a dimensionless parameter *λ*_{cρ}, called cohesion ratio and defined by Koppula [15], is used here, as it gives

For a cut slope, the energy dissipation rate *D*^{curve} is derived aswhere

For a natural slope, it becomeswhereand variables *θ*_{M} and *θ*_{N} are obtained from the trigonometric relations in Figure 2(a) as

According to the balance equation (3), the least upper bounds to the stability number *N*_{ρ} = *γHF*/*c*_{u0} can be derived in terms of *λ*_{cρ} from the optimization scheme of Chen [25]. For a slope of given values of *β*, *λ*_{cρ}, *k*_{h}, and relative width *B/H*, independent variables in the optimization process include (c.f. Figure 2(a)) angles *θ*_{0}, *θ*_{h}, and *β*′, ratio *r*′*/r*, and relative width of the plane insert *b/H*. Similar results for a homogeneous slope (i.e., *ρ* = 0) can be also derived by applying *λ*_{cρ} = 0 to the above expressions.

#### 3. Results and Discussions

Figures 3 and 4 show the upper bounds to the stability number *N*_{ρ} for cut slopes and natural slopes under static conditions, respectively. They are plotted against the cohesion ratio *λ*_{cρ} for ratios of *B*/*H* ranging from 1.5 to 10.0 and for the 2D case. According to the practical experiences of Hunter and Schuster [14], Koppula [15], and Zhang et al. [32, 33], the value of *λ*_{cρ} is selected in the range of 0.0–5.0. Each chart in Figures 3 and 4 illustrates the results for one inclination angle of the slope.

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It can be seen from Figures 3 and 4 that the stability number *N*_{ρ} increases almost linearly with the cohesion ratio *λ*_{cρ}. For a given value of *λ*_{cρ}, the stability number *N*_{ρ} decreases with the increasing ratio of *B*/*H*, and the value of *N*_{ρ} for 3D failure is greater than that for 2D failure. Obviously, the constraint *B* on the width of the slope has a significant effect on the stability number.

In the numerical limit analysis of Li et al. [18, 22], collapses of 3D cut slopes and natural slopes are limited by a rigid base at a depth *d* below the crest of the slope (Figure 1). When the slope is gentle and the depth *d* is small, the rigid base makes the slope more stable, as expected. The static solutions in Figures 3 and 4 do not take into account the effect of rigid base on the slope stability due to a conservative estimate on the slope stability. Besides, the maximum depths of slope failure surfaces are less than 2*H* for 3D slopes with *β* ≥ 15°, and then, the rigid base at the depth *d* = 2*H* has no effect on the critical values. For this reason, a comparison can be made between the critical values of *N*_{ρ} calculated from this study and from the numerical limit analysis method by Li et al. [18, 22] for cut slopes and natural slopes with depth factor *d*/*H* = 2.0 and a ratio of *B*/*H* = 5.0, as shown in Figure 5. It can be seen that the stability number *N*_{ρ} of this study is always less than that in the numerical results. Figure 6 shows the comparisons of the analytical upper bound results derived from this study and the numerical upper and lower bound solutions presented by Li et al. [18, 22] for various slope angles. It can be seen that the analytical upper bound is closely bracketed by the numerical upper and lower bounds. The upper bound result of this study is obviously close to the numerical lower bound solution rather than the numerical upper bound. Therefore, the best estimate of the upper bound to the critical value of *N*_{ρ} has been obtained by the analytical approach performed in this study.

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For cut slopes and natural slopes subjected to seismic excitation, a set of stability charts are presented in Figures 7–12 for horizontal acceleration coefficients *k*_{h} of 0.1, 0.2, and 0.3. It should be noted that, when the slopes with the small value of slope angle *β* are subjected to stronger seismic excitation, the critical value of *N*_{ρ} will tend to zero as the 2D homogeneous slopes in Michalowski [3]. A more rational value is obtained by limiting the depth of the failure mechanism to a realistic value *d* below the crest. The depth factor of *d*/*H* = 2.0 is adopted in the mechanism for gentle slopes under seismic conditions. As expected, the stability number *N*_{ρ} reduces with increasing magnitude of the horizontal acceleration.

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##### 3.1. Applications

According to the above-derived stability charts (Figures 3 and 4 for static conditions and Figures 7 to 12 for seismic conditions), the factor of safety *F* can be easily obtained for a given 3D slope.

For comparison purposes, the same example slope as analyzed numerically by Li et al. [18, 22] is adopted here. The calculation parameters for the example slope are as follows: the slope height *H* = 12 m, the slope angle *β* = 60°, the unit weight of soil *γ* = 18.5 kN/m^{3}, the undrained strength at the crest of slope *c*_{u0} = 40 kPa, and the gradient of undrained strength with respect to the depth *ρ* = 1.5 kPa/m. To determine the safety factor of the cut slopes and the natural slopes, *N*_{ρ}/*λ*_{cρ} = *γ*/*ρ* = 18.5/1.5 = 12.33 is first obtained. From the charts for *β* = 60° in Figures 3 and 4, a straight line passing through the origin with a gradient *N*_{ρ}/*λ*_{cρ} = 12.33 is plotted. This straight line intersects the curves for the results of 3D and 2D slope failures. The safety factor *F* can be easily calculated by reading *N*_{ρ} for a given ratio of *B*/*H* and dividing the value of *γH*/*c*_{u0}. Table 1 shows the calculated results for various ratios of *B*/*H* and for the 2D case, together with the corresponding data retrieved from Table 1 of Li et al. [18, 22]. Compared with the results derived from the finite element upper bound limit analysis, a better estimate of the upper bounds to safety factor is obtained from the analytical limit analysis proposed in this study, with the unexpected exception of the 2D case in the natural slope. However, a careful check against Figure 7(b) of Li et al. [18] shows that the reading value of *N*_{ρ} for the 2D natural slope is approximately 6.6 rather than the number 6.3 presented in Table 1 of Li et al. [22]. Therefore, the only exception is doubted.

Alternatively, an analytical approximation of the curves in the stability charts can be made aswhere coefficients *A* and *B* are determined by a linear fitting technique. It should be noted that the curves have a slight curvature for natural slopes with small value of *λ*_{cρ}. Nevertheless, the goodness of fit measured by the statistical coefficient *R*^{2} can reach 0.99 for natural slopes and even exceed 0.999 for cut slopes. The coefficients *A* and *B* are given in Tables 2 and 3 for cut and natural slopes, respectively. Thus, the safety factor *F* can be more easily derived from equation (12), and it can avoid the error by reading the stability charts.

For the above slope example, the factor of safety *F* is obtained from equation (12) and presented in Table 4. Not surprisingly, the factor of safety for cut slope or natural slope decreases with increasing magnitude of the horizontal acceleration *k*_{h} and with increasing ratio of *B*/*H*. Moreover, using the results from the 2D analysis underestimates the stability of 3D cut slopes and natural slopes. It can be found from Table 4 that the difference in the safety factors between 3D and 2D analysis increases with increasing magnitude of horizontal acceleration *k*_{h} and reducing ratio of *B*/*H*. Typically, the difference can exceed 50% when the slope is constrained to a narrow width of *B/H* = 1.5.

#### 4. Conclusions

Based on the 3D kinematically admissible rotational failure mechanism, an analytical approach is derived for the upper bound limit analysis of the stability of cut slopes and natural slopes under short-term undrained conditions. Compared with the finite element limit analysis method adopted by Li et al. [18, 22], the proposed analytical approach gives the better estimate of the upper bounds to the stability number *N*_{ρ}. A set of stability charts is presented for both cut slopes and natural slopes under static and pseudostatic seismic loading conditions. The safety factor can be easily obtained from the charts to evaluate the stability of cut slopes and natural slopes. Furthermore, the results indicate that using 2D solutions to evaluate the stability of 3D slopes will underestimate the factor of safety. The difference between 2D and 3D factors of safety increases with the reducing ratio of *B*/*H* and increasing horizontal acceleration coefficient *k*_{h}.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.