Factor of safety of strain-softening slopes

Hossein Rafiei Renani, C. Derek Martin

a Klohn Crippen Berger Ltd., Vancouver, Canada

b Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Canada

Abstract Stability analysis of strain-softening slopes is carried out using the shear strength reduction method and Mohr-Coulomb model with degrading cohesion and friction angle.The effect of strain-softening behavior on the slope factor of safety is investigated by performing a series of analyses for various slope geometries and strength properties. Stability charts and equations are developed to estimate the factor of safety of strain-softening slopes from the results of traditional stability analysis based on perfectly-plastic behavior. Two example applications including an open pit mine in weak rock and clay shale slope with daylighting bedding planes are presented. The results of limit equilibrium analysis and shear strength reduction method with perfectly-plastic models were in close agreement.Using perfectly-plastic models with peak strength properties led to overly optimistic results while adopting residual strength properties gave excessively conservative outcomes. The shear strength reduction method with a strain-softening model gave realistic factors of safety while accounting for the process of strength degradation.

2020 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Slope stability analysis Factor of safety ratio Shear strength reduction Perfectly-plastic behavior Open pit slope Clay shale slope

1. Introduction

Stability of soil and rock slopes has long been a subject of study in geotechnical engineering with applications in civil and mining projects.The most common indicator of slope stability is the factor of safety(FOS)defined as the value by which the shear strength of the slope material must be divided in order to bring the slope to the point of failure. Limit equilibrium analysis is a simple and commonly used method for determination of the FOS from the driving and resisting forces and moments acting on a critical sliding mass(e.g.Morgenstern and Price,1965;Fredlund and Krahn,1977;Zhou and Cheng, 2013; Rafiei Renani and Martin, 2020). However,displacements are not taken into account in such analysis assuming that driving and resisting forces are independent of deformation.This corresponds to a perfectly-plastic behavior in which strength remains unchanged after failure.

In reality, a wide range of soils and rocks exhibit a reduction in strength properties with excessive deformation, damage and failure. This characteristic is referred to as strain-softening behavior.The combination of strain-softening behavior and non-uniform distribution of stresses can lead to progressive failure of slopes(Terzaghi and Peck, 1948; Skempton, 1964). Strain-softening behavior can be incorporated in stress analysis of slopes using the finite difference method (FDM) or finite element method (FEM).Adopting the shear strength reduction(SSR)method,the slope FOS can be obtained from stress analysis (e.g. Zienkiewicz et al.,1975;Griffiths, 1980; Smith and Griffiths,1988; Zhou and Cheng, 2015;Rafiei Renani et al., 2019).

Despite its significance, the effect of strain-softening behavior on slope stability has been explored in relatively few studies.Early attempts were made by Lo and Lee (1973) who used the FEM to explore the effect of strain-softening on slope stability. Nonveiller(1987) discussed the impact of strain-softening on the progressive failure of the Vajont reservoir slope. Griffiths (1989) used the von Mises criterion with cohesion softening for stability analysis of an undrained clay slope. Chen et al. (1992) used strain-softening models to back-analyze the progressive failure of the Carsington Dam. A similar approach was taken by Potts et al. (1997) to investigate the delayed collapse of cut slopes in London Clay. Troncone(2005) implemented a strain-softening model to back-analyze the Senise landslide. Mohammadi and Taiebat (2016) used the FEM to investigate the evolution of deformation and failure in strainsoftening slopes. More recently, Zhang and Zhou (2018) incorporated the SSR method in a discrete particle model to study the behavior of a strain-softening slope. Most of these studies have focused on in-depth analysis of specific slopes limiting the relevance of the results to other projects.

Explicit incorporation of strain-softening behavior in practical slope design remains rare,largely due to the additional complexity and computation cost involved. In addition, there are no preliminary design tools for strain-softening slopes such as the slope stability charts frequently developed based on perfectly-plastic behavior (e.g. Hoek and Bray, 1981; Leshchinsky and Mullet,1988; Michalowski, 2002; Li et al., 2008; Steward et al., 2011;Shen et al., 2013).

This study aims at providing practical tools for preliminary design of slopes in strain-softening material. A comprehensive series of slope stability analyses has been carried out on slopes with a wide range of geometries and strength characteristics. The results have been used to develop new stability charts and equations to estimate the impact of strain-softening on the slope FOS.The level of simplicity maintained in the stability analyses render the findings useful for a wide range of slope conditions.Two examples are presented to illustrate the application of strain-softening models in slope stability analysis.

2. Strain-softening behavior in soils and rocks

The process of deformation and failure of geomaterials under various loading conditions has been well documented.For example,Wawersik and Fairhurst (1970) investigated the stress-strain behavior of Tennessee marble using a series of triaxial compression tests.Fig.1a shows the strain-softening behavior of Tennessee marble especially at lower confinement.

Levels. As effective confining stress increases, the amount of post-peak stress drop decreases and perfectly-plastic behavior may be observed at relatively high confinement levels. However, instability and failure of geostructures such as slopes and tunnels typically occur in regions with little to no confinement where strainsoftening behavior is most pronounced. This underlines the importance of considering strain-softening behavior in geotechnical analysis and design.

Strain-softening behavior has also been observed during tests on overconsolidated clays and dense sands (Bjerrum, 1954;Skempton, 1964; Hettler and Vardoulakis, 1984). For example,Fig.1b shows the response of Yellow clay from the Carsington Dam under shear tests(Skempton,1985).Similar to many types of rock,a significant decrease in post-peak strength can be observed in strain-softening soils.

Using a series of triaxial compression or shear tests at different confinement levels, peak and residual strength envelopes can be obtained. Fig. 2a shows the peak and residual strength envelopes obtained from triaxial compression tests on Buchberg sandstone(Kovari,1977). It can be observed that residual strength is consistently and significantly lower than peak strength. However, the slopes of peak and residual strength envelopes are similar, indicating that strain-softening behavior of Buchberg sandstone is mainly due to loss of cohesion while friction angle is largely unaffected.

Peak and residual strength envelopes of Walton’s Wood clay obtained from shear tests (Skempton,1964) are shown in Fig. 2b.Similar to the Buchberg sandstone, complete cohesion loss contributes significantly to the strain-softening behavior.However,the slope of residual strength envelope is also lower than that of peak strength envelope, indicating a reduction in friction angle of Walton’s Wood clay in the residual state due to microstructural reorientations and alignment of clay platelets in the direction of shear displacement.

3. Modeling of strain-softening behavior in slopes

Shear strength of soils and rocks is composed of two main components;cohesive strength resulting from intergranular bonds and cementation, and frictional strength resulting from frictional resistance during shearing which is directly related to confining stress.The most common strength criterion for geomaterials is the Mohr-Coulomb criterion in which frictional strength is a linear function of effective normal stress:

Fig.1. Stress-strain response of(a)Tennessee marble at various levels of effective confining stress 3 (Wawersik and Fairhurst,1970), and(b) Yellow clay under different effective normal stresses, n (Skempton,1985).

Fig. 2. Peak and residual strength envelopes of (a) Buchberg sandstone (Kovari,1977) and (b) Walton’s Wood clay (Skempton,1964).

In a perfectly-plastic Mohr-Coulomb model, shear strength parameters remain constant regardless of the extent of plastic deformation.This causes the strength to remain constant even after failure. In order to capture the post-peak stress drop observed for strain-softening materials,strength parameters can be degraded as plastic strain increases.Linear and nonlinear variations of strength parameters have been used for modeling brittle failure of rocks around underground excavations under high confinement(Vermeer and de Borst, 1984; Hajiabdolmajid et al., 2002; Rafiei Renani and Martin, 2018a; b). For simplicity, linear degradation of cohesion and friction angle is typically assumed in slope stability analysis (e.g. Potts et al.,1997; Troncone, 2005; Conte et al., 2010;Mohammadi and Taiebat, 2013). Hence, the Mohr-Coulomb criterion with linearly degrading strength parameters was adopted in this study to model the strain-softening behavior slope material(Fig. 3).

Fig.3. Linear degradation of effective cohesion and friction angle from peak to residual values in the strain-softening model.

Table 1Material parameters for the slope subjected to displacements at the crest.

The plastic shear strain threshold εp*controls the rate of strainsoftening and may be considered as a measure of material brittleness. Results of tests on brittle soils and rocks indicate that postpeak strength drop occurs quite rapidly. When possible, the stress-strain curves obtained from laboratory experiments may be used to estimate this parameter(Skempton,1985;Chen et al.,1992;Rafiei Renani and Martin,2018a;b).In other cases,values reported in the literature for similar materials may be used as a first estimate.In the absence of relevant experimental and empirical evidence,the conservative assumption of instantaneous softening may be adopted.It is typically assumed that εp*is independent of confining stress. Although this may be a simplification, the stress-strain curves obtained with this assumption can be quite realistic (e.g.Vermeer and de Borst, 1984; Skempton, 1985; Chen et al., 1992;Rafiei Renani and Martin,2018a; b).

In order to illustrate the effect of strain-softening behavior on slope stability, it is useful to analyze a simple homogeneous slope with vertical displacements applied over a finite area of its crest.Although this is not a typical loading,it was chosen to illustrate the impact of strain-softening behavior on the load-displacement response of the slope. In this case, a slope with a height,H, of 10 m and slope angle,, of 60was analyzed. The material parameters of the slope including unit weight,,Young’s modulus,E,Poisson’s ratio,, peak and residual cohesion,cpandcr, peak and residual friction angles,pandr,and plastic strain threshold,εp*,are given in Table 1. Non-associated plasticity with zero dilation angle was assumed. A perfectly-plastic model with peak strength parameters and a strain-softening model with partial cohesion loss and constant friction angle were compared.

The FLAC3D code (Itasca Inc., 2017) based on explicit FDM was used to analyze the slope response. A uniform mesh with 50 quadrilateral elements across the height of the slope was used to discretize the slope (Fig. 4). The boundary conditions include horizontal restraints on the sides and full fixity at the base of the model. The initial state of the slope was obtained by bringing the model to equilibrium under gravity and resetting displacements.Subsequently a vertical displacement with a rate of 1 mm/s was applied over a 3 m strip of the crest. Development of failure and evolution of vertical displacement and induced vertical stress on the strip were monitored during loading.

Fig. 4. Homogeneous slope subjected to vertical displacement dv at the crest.

The mechanism of development and final position of the failure surface in perfectly-plastic and strain-softening models were similar, starting above the toe and extending upward towards the crest.Fig.5a shows the corresponding vertical displacements when failure reaches different points along the slip surface obtained using the strain-softening model. At different stages of analysis,normalized failure length was calculated as the ratio of the current length of failure to the final length of slip surface.Fig.5b shows the relationship between the normalized failure length, induced vertical stress and vertical displacements at the loading strip. The failure length in perfectly-plastic and strain-softening models followed a similar trend and accelerated as loading progressed.However, initiation and propagation of failure occurred sooner in the strain-softening model.In both models,slip surfaces were fully developed immediately after the maximum vertical stresses were reached.The perfectly-plastic model showed no meaningful stress drop after reaching the maximum vertical stress while the strainsoftening model captured the post-peak reduction in load bearing capacity.

4. Application of the shear strength reduction method

The slope FOS is commonly determined using limit equilibrium analysis.In this approach,the sliding mass is divided into slices,and equations of force and/or moment equilibrium of slices are satisfied.Beside the assumptions regarding the interslice forces and the necessity of using secondary search algorithms to find the critical slip surface with the lowest FOS, this approach does not take into account the interrelationship between stress, strength, and displacement (Krahn, 2003). One implication of ignoring displacements is the inherent assumption that strength is independent of deformation,i.e.perfectly-plastic behavior.As a result,limit equilibrium analysis may not capture the progressive failure of strain-softening slopes (e.g. Chen et al.,1992).

In order to enhance the method by which the FOS is calculated,Zienkiewicz et al. (1975) introduced the shear strength reduction(SSR) method which was later developed by Dawson et al. (1999)and Griffiths and Lane (1999). In this approach, a series of stress analyses are carried out in which shear strength is progressively reduced to bring the slope to the point of failure where numerical convergence is no longer possible. According to Eq. (1), scaling of shear strength is possible by applying a strength reduction factor(SRF) to shear strength parameters:

wherectrialandtrialare the trial values of cohesion and friction angle for a given SRF, respectively. Hence, by changing SRF and using corresponding trial values of cohesion and friction angle in numerical analysis, it is possible to find the critical SRF which corresponds to the state of limiting equilibrium.The critical SRF in the SSR method has the same definition as the FOS in limit equilibrium analysis. Similar results have been obtained from limit equilibrium analysis and SSR method for slopes with simple geometry and perfectly-plastic behavior (Dawson et al., 1999;Griffiths and Lane,1999; Cheng et al., 2007).

Fig.5. Slope response to applied displacements:(a)Development of failure surface and(b)Vertical stress and normalized failure length in perfectly-plastic(PP)and strain softening(SS) models.

Fig.6. Application of the strength reduction factor to perfectly-plastic(PP)and strainsoftening (SS) models.

In the shear strength reduction approach, the constitutive relationship between stress and strain as well as strain compatibility of the continuum is satisfied. In addition, the mechanism of failure and associated slip surface emerge naturally during stress analysis. Another benefit is that realistic material models can be incorporated in numerical stress analysis and SSR method.

The strength reduction method was originally applied to perfectly-plastic models in which strength parameters were independent of deformations. In this study, application of the strength reduction method was extended to strain-softening models. This was achieved by applying the SRF to the values of cohesion and friction angle which were functions of plastic strain rather than constant value. Fig. 6 illustrates the application of the SRF in perfectly-plastic and strain-softening models. Original strength parameters (SRF1) are constant in the perfectly-plastic model and variable in the strain-softening model. Reduced strength parameters(SRF>1)are obtained by applying the SRF to the original strength parameters according to Eqs. (2) and (3). These reduced strength parameters are also constant in the perfectly-plastic model and variable in the strain-softening model.

To illustrate the process of obtaining the FOS using the SSR method, a 10 m high slope with an angle of 60and parameters given in Table 1 was analyzed under gravitational forces. Two uniform meshes,referred to as the medium and fine meshes,were used with 50 and 77 elements across the height of the slope,respectively.A series of analyses with progressively increasing SRF values was carried out to reach the state of limiting equilibrium.

Fig.7a shows the relationship between the total displacement at the crest and SRF using perfectly-plastic and strain-softening models. As expected, increasing SRF led to increased displacements in all cases. Perfectly-plastic models with medium and fine meshes showed very similar responses with no numerical convergence beyond SRF of 1.55. This compares closely with the FOS of 1.56 obtained from limit equilibrium analysis with the Morgenstern and Price(1965)method and non-circular slip surface search using the SLIDE software (Rocscience Inc, 2018a).

The critical values of SRF for the strain-softening models were 1.22 and 1.19 using the medium and fine meshes, respectively.Hence, the effect of element size on the FOS in this range of mesh resolution is less than 3%which is acceptable in most geotechnical applications. Fig. 7b shows the slip surface obtained using the strain-softening model and the fine mesh.

5. Factor of safety of strain-softening slopes

In order to develop practical tools for preliminary design of slopes in strain-softening materials, a comprehensive series of slope stability analyses was carried out using the SSR method with perfectly-plastic and strain-softening models. The analyses were based on typical simplified assumptions to keep the findings relevant for general slope conditions.Gravity loading of the completed slope with horizontal restraints on the sides and full fixity at the base of the model was adopted (e.g. Zienkiewicz et al., 1975;Griffiths, 1989; Griffiths and Lane, 1999; Cheng et al., 2007).Quadratic triangular elements were adopted in numerical analyses with the FEM using the RS2 software (Rocscience Inc, 2018b). The height and angle of slope as well as shear strength parameters were varied to cover a wide range of potential conditions. In stability analyses using the perfectly-plastic model, peak shear strength parameterscpandpwere used.

Fig. 7. (a) Relationship between strength reduction factor and total displacement at the crest using perfectly-plastic (PP) and strain-softening (SS) models and (b) Displacement vectors indicating the predicted sliding mass using the strain-softening model and medium mesh.

Fig. 8. FOS ratio for the first scenario obtained using the shear strength reduction method (data points) and the proposed Eq. (5) (solid lines).

In all stability analyses using the strain-softening model,instantaneous and complete loss of cohesion after failure was assumed(cr0).This is in keeping with the fact that the residual strength envelopes of most geomaterials pass through the origin(Fig. 2). Two scenarios were considered for the post-peak variation of friction angle. In the first scenario, the friction angle remained unchanged after failure (rp). This may be considered as an upper bound for residual strength and is consistent with the behavior of materials whose peak and residual strength envelopes are approximately parallel (Fig. 2a). The second scenario represents a crude lower bound for residual strength in which the friction angle is reduced by half after failure (rp=2). This is consistent with the behavior of materials whose peak strength envelope is significantly steeper that the residual strength envelope (Fig. 2b).

It is well known that associated flow rule considerably overestimates the dilation of frictional materials, resulting in frequent use of non-associated plasticity with zero dilation angle in modeling the behavior of geomaterials(e.g.Roscoe,1970;Griffiths,1981,1989; Vermeer and de Borst,1984; Griffiths and Lane,1999;Troncone, 2005). This was also adopted in all the analyses presented in this study.However,it has been shown that using higher values of dilation angle has minimal influence on the slope FOS(e.g.Zienkiewicz et al.,1975; Chen et al.,1992; Cheng et al., 2007). To illustrate the effect of strain-softening behavior on the slope FOS,the results of stability analyses were presented in terms of the FOS ratio defined by

whereFOSPPis obtained using the perfectly-plastic model with peak strength parameters andFOSSSis obtained from the strainsoftening model with similar peak strength parameters and degraded residual strength parameters.

Fig. 8 shows the FOS ratios obtained for the first scenario in whichcr0 andrpin the strain-softening model. In this scenario, the difference between perfectly-plastic and strainsoftening models was due to cohesion loss only. For the case of zero peak cohesion,there was no cohesion to be lost and therefore the perfectly-plastic and strain-softening models coincided giving an FOS ratio of unity. By increasing peak cohesion, the amount of cohesion loss during strain-softening increased and the FOS ratio decreased and approached an asymptotic value. The FOS ratio in thse first scenario can be approximated using

It should be emphasized that k1is derived from strain-softening models with zero residual cohesion and similar peak and residual friction angles.

Fig. 9 shows the FOS ratios obtained for the second scenario in whichin the strain-softening model.In this case, the perfectly-plastic and strain-softening models differed in terms of both cohesion and friction angle.In the case of zero peak cohesion,the FOS was controlled by friction angle alone which was reduced by half in the strain-softening model,resulting in the FOS ratios of about 0.5.In cases with peak cohesion of greater than zero,the results were affected by cohesion loss leading to a brief increase in the FOS ratio followed by a continuous reduction towards an asymptotic value.Because cohesion plays a more significant role in the stability of steeper slope,the observed variation of the FOS ratio is more pronounced for steeper slopes and is negligible for slopes at an angle of 30. The FOS ratio in the second scenario can be estimated using

Fig.10. Cross-section of the open pit slope.

As indicated earlier,k2is obtained from strain-softening models with zero residual cohesion and residual friction angles that are half the peak friction angle. Note that for 30slopes, similar to steeper slopes,bothFOSPPandFOSSSincreased with increasing peak strength parameters andFOSSSalways remained lower thanFOSPPdue to strain-softening.However,the ratio ofFOSSStoFOSPPvaried over such a narrow range which justified approximation with a constant value.

Fig. 9. FOS ratio for the second scenario obtained using the shear strength reduction method (data points) and the proposed Eq. (6) (solid lines).

Table 2Material parameters for the open pit slope.

Fig.11. Factors of safety of the open pit slope obtained using the SSR method with the perfectly-plastic (PP) and strain-softening (SS) models.

It is worth emphasizing that Figs.9 and 10 and Eqs. (5)and (6)were derived to provide a first estimate of the likely range of FOS reduction due to strain-softening behavior for a wide range of slope geometries and strength properties. To maintain generality, simplifications were made in the analysis such as using idealized and completed geometry for the slopes. When dealing with a specific slope, however, the actual geometry, construction sequence and coupled hydro-mechanical processes may be modeled.In addition,the results were obtained from analysis of dry slopes and, strictly speaking,are only applicable to slopes where the water table is low enough, naturally or by using drainage measures, to not interact with the slip surface. However, if the presence of pore pressure causes a similar percentage of reduction in the FOS of perfectlyplastic and strain-softening slopes, then by definition, the FOS ratio will not be affected by the presence of pore pressure and the equations remain applicable.

6. Example applications

Capturing the fundamental characteristics of geomaterials is necessary in engineering design and analysis. As discussed previously, strain-softening behavior associated with the post-peak reduction of load bearing capacity has been observed in a wide range of geomaterials.Yet,such fundamental characteristic is rarely accounted for in practical slope stability analyses. In this section,two examples are presented to illustrate the application of strainsoftening model in slope stability analysis using the SSR approach.

Fig.12. Displacement vectors for the open pit slope indicating the predicted sliding mass using the strain-softening model SS1 and water table WT2 with(a)Mesh 1 and(b)Mesh 2.

6.1. Open pit slope in weak rock

To demonstrate the application of the presented FOS ratios, an open pit slope with a height of 200 m and slope angle of 45excavated in a weak rock was considered. Stability analyses were carried out assuming dry condition as well as using three water tables designated as WT1,WT2 and WT3 in Fig.10.In addition to a perfectly-plastic model (PP), two strain-softening models with instantaneous reduction of cohesion to zero were used.In the first strain-softening model (SS1), the residual friction angle was equal to the peak friction angle whereas in the second strain-softening model (SS2), it was half the peak friction angle (Table 2). Limit equilibrium analysis of the open pit slope was carried out using the Morgenstern and Price(1965)method and non-circular slip surface search in the SLIDE software (Rocscience Inc, 2018a). The SSR approach with FEM was also adopted using the RS2 software(Rocscience Inc, 2018b). Two uniform meshes, referred to as the Mesh 1 and Mesh 2,were used with 30 and 50 quadratic triangular elements across the slope height, respectively.

Fig. 11 shows the results of the SSR method for different scenarios using the Mesh 1 and Mesh 2.In all scenarios,the values of FOS obtained from the Mesh 1 and Mesh 2 were very similar. This suggests that for the range of mesh resolutions adopted here, the slope FOS is not significantly affected by mesh resolution. This is consistent with the observations of Griffiths(1989)and the results presented in Section 4.The slip surfaces obtained using the Mesh 1 and Mesh 2 were also in close agreement. As an example, Fig.12 shows the sliding mass obtained using the strain-softening model SS1 and intermediate water table WT2.

The values of FOS obtained using limit equilibrium analysis and SSR method with the Mesh 2 are given in Table 3.As expected,the results of limit equilibrium analysis and SSR method using the perfectly-plastic model were similar.In addition,incorporating the strain-softening behavior significantly reduced the FOS. Although the absolute values of FOS decreased with increasing water table,it is interesting to note that the ratio of FOS from a given strainsoftening model to that from a perfectly-plastic model remained almost unchanged. This suggests that increasing pore pressure causes almost the same percentage reduction in calculated values of FOS obtained from perfectly-plastic and strain-softening models.

The ratios of FOS obtained from the strain-softening models SS1 and SS2 to those obtained using the perfectly-plastic model varied over 0.61e0.63 and 0.51e0.53,respectively.These values compared closely with the corresponding FOS ratios of k10.62 and k20.53calculated using Eqs. (5) and (6), respectively. Multiplying these FOS ratios by the values of FOS from perfectly-plastic models gave the estimated FOS values given in Table 3. It can be observed that the values of FOS obtained for strain-softening slopes using Eqs.(5)and (6) are in close agreement with those obtained directly from the SSR method.

Table 3Factor of safety of the open pit slope using the perfectly-plastic (PP) and strainsoftening(SS) models.

Table 4Material parameters for the clay shale slope.

This example demonstrated that although FOS ratios given by Eqs. (5) and (6) were based on systematic stability analysis of dry slopes, they remained almost unchanged in the presence of pore pressures caused by the water levels, as shown in Fig. 10. It also illustrated that Eqs. (5) and (6) can be effectively utilized to find a likely range for the FOS of strain-softening slopes from traditional slope stability analysis based on perfectly-plastic behavior.

6.2. Clay shale slope along Canadian river valley

Stability of a slope excavated in the clay shale deposits typically found along river valleys of Western Canada was considered.River down cutting in these formations has caused valley rebound and subsequent shearing displacement along weak bedding planes.The process of induced shearing on weak planes was described by Ferguson(1967)and Matheson and Thomson(1973)and its impact on slope stability was investigated by Soe Moe (2008), Bromhead(2013) and Rafiei Renani and Martin (2018c). These clay-rich rocks are susceptible to gradual loss of cohesion due to the mechanism of softening first described by Terzaghi (1936). Progressive loosening of the slope due to valley rebound can also contribute to cohesion degradation. Hence, failure mechanism in these bedded clay shales is a combination of shearing along weak planes and failure through strain-softening rock.

Fig.13. (a) Slip surfaces of the clay shale slope obtained using limit equilibrium analysis with peak strength (solid line) and residual strength(dashed line), and (b)Displacement vectors indicating the predicted sliding mass using the shear strength reduction method with strain-softening model.

The clay shale slope analyzed in this study contained two subhorizontal bedding planes daylighting into the slope face. Parameters of the clay shale and bedding plane material typical of river valley clay shales of Western Canada(Cornish and Moore,1985)are given in Table 4.

Limit equilibrium analysis of the clay shale slope was carried out using the Morgenstern and Price (1965) method and non-circular slip surface search in the SLIDE software (Rocscience Inc, 2018a).The SSR approach and FDM were adopted for stability analysis of the clay shale slope using the FLAC3D code (Itasca Inc., 2017).Mechanical behavior of the clay shale was represented using a perfectly-plastic model with peak strength, a perfectly-plastic model with residual strength and a strain-softening model.

The slip surfaces obtained from limit equilibrium analysis and SSR method are shown in Fig. 13. Failure on the lower bedding plane was predicted from limit equilibrium analysis. In addition,the slip surface obtained using residual strength was closer to the slope face than that predicted using peak strength.The SSR method with strain-softening model also predicted a slip surface similar to that obtained from limit equilibrium analysis with peak strength parameters.

The factors of safety obtained using different approaches are given in Table 5. It can be observed that the results of limit equilibrium analysis and SSR method with perfectly-plastic models were in close agreement.The FOS obtained using peak strength was quite high and acceptable in most projects. However, the strength of clay shale along river valleys may fall to residual levels due to stress relaxation,moisture uptake and weathering.Therefore,using peak strength in traditional stability analysis can lead to dangerously misleading results. On the other hand, adopting residual strength in stability analysis led to an alarmingly low FOS. This is because the peak strength that must be overcome before reaching residual strength is ignored in such analysis. The strain-softening model, while allowing for the process of strength degradation,gave a reasonable FOS which was in between those obtained using peak and residual strength properties.

Table 5Factor of safety of the clay shale slope using perfectly-plastic (PP) and strainsoftening (SS) models.

7. Conclusions

Strain-softening is a fundamental characteristic of a wide range of geomaterials which cannot be incorporated in traditional slope stability analysis using the limit equilibrium approach.In this study,application of the shear strength reduction method was extended to determine the FOS of slopes in strain-softening materials. A Mohr-Coulomb model was adopted in which cohesion and friction angle were degraded by increasing plastic shear strain. A comprehensive series of finite element analyses was carried out on slopes with various geometries and shear strength properties.The results were used to develop new stability charts and equations for estimating the effect of strain-softening behavior on the slope FOS. It was shown that using peak strength in stability analysis of slopes in strain-softening material can lead to unsafe design.

Two examples were presented to illustrate the application of the strain-softening model in slope stability analysis.The first example involved an open pit slope excavated in a weak rock with different levels of pore pressure.As expected,the results of limit equilibrium analysis and SSR method showed that higher pore pressures lead to lower values of FOS.However,the ratio of FOS from strain-softening model to that from a perfectly plastic model was found to be almost identical for various levels of pore pressure considered in this example. In addition, the proposed equations provided accurate estimates of the FOS for the strain-softening slope.

The second case involved a typical clay shale slope in the river valleys of Western Canada with sub-horizontal bedding planes daylighting into the slope face. The mechanism of instability consisted of failure through the strain-softening clay shale and slip on the bedding plane.The results of limit equilibrium analysis and SSR method with perfectly-plastic models were in close agreement.The values of FOS obtained using peak and residual strength properties were overly optimistic and excessively conservative, respectively while adopting the SSR method with a strain-softening model gave realistic results.

In traditional slope stability analysis based on perfectly-plastic behavior, the designer is faced with a dilemma as using peak strength typically results in overly optimistic results, while using residual strength gives excessively conservative values of FOS which are below unity in many cases.Yet,it is well-established that the strength of slope material and therefore the slope FOS can vary between these extremes. The SSR approach with strain-softening model adopted in this study provides a solution to this dilemma and properly accounts for strain-softening behavior of geomaterials and its impact on slope stability.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts of interests associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

This work was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC: RES0014117).