A COMPARATIVE STUDY OF LEM-BASED DETERMINISTIC AND PROBABILISTIC APPROACHES
6.7. INFLUENCE OF CORRELATION LENGTH AND SPATIAL VARIATION OF SOIL SHEAR STRENGTH PARAMETERS
Fig. 6.6 Line diagram showing the variation of probability of failure with correlation coefficient (ρ) for varying extent of toe cutting
6.7. INFLUENCE OF CORRELATION LENGTH AND SPATIAL VARIATION OF SOIL
presented by Vanmarcke (1983) and El-Ramly et al. (2002, 2005), as discussed in Section 5.4 of Chapter 5. Although few studies, as mentioned above, has been carried out incorporating 1-D spatial variability of soil properties (mostly on hydraulic conductivity) for rainfall-induced landslide analysis, yet no such studies incorporating 1D spatial variability are reported in the context of hillslope instability induced by toe-excavations.
Therefore, this section reports the influence of one dimensional spatial variation of soil properties on the cut slope stability. For different horizontal extent of vertical toe cutting, Fig. 6.7 shows the three-dimensional (3D) variation of the probability of failure and reliability index in terms of the dimensionless correlation length ( =/L, where L is the length of slope in the horizontal direction as shown in Fig. 6.1). Very small correlation length (represent extremely erratic random field) and very large correlation length (representing sufficiently homogeneous field) have lesser practical significance, while the intermediate range of correlation length is of prime interest to the geotechnical engineers (Griffiths et al., 2007). Therefore, in this section, a one-dimensional random field is generated for each MCS by incorporating spatial variation of soil properties in the horizontal direction using correlation length. In this study, the dimensionless correlation length (Θ) is varied from 0.1 to 1. For better clarity, for the virgin slope, the variation of probability of failure with dimensionless correlation length has been explicitly highlighted for specific values of the horizontal extent of toe cutting (Fig. 6.8) for various magnitudes of CoV (0.2, 0.3 and 0.4). It is observed that the failure probability increases rapidly (reliability index decreases) with increase in correlation length. The failure probability is negligible when the correlation length is very small.
This is because, for small correlation lengths, the soil properties tend to take their mean values.
Now, as the virgin slope is safe for its mean value of shear strength parameters, the failure
probability is very less (negligible). The failure probability is highest when dimensionless correlation length is equal to unity, indicating the entire soil domain as homogeneous for all the MCS iterations, with different values of shear strength in different iteration. Hence, it is seen that ignoring spatial variability may result in underestimation or overestimation of failure probability depending on its correlation length. This trend is observed for all the cut slopes as shown in Fig.
6.7.
Based on the results from several analyses, the following can be stated regarding toe excavation for the slope considered in the present study:
(a) For CoV = 0.2, a maximum horizontal extent of 7 m can be excavated without leading to slope failure if Θ ≤ 0.2, beyond which toe excavation is not recommended (for Θ > 0.2).
(b) For CoV = 0.3, a maximum horizontal extent of 5 m can be excavated without leading to slope failure if Θ ≤ 0.1, beyond which toe excavation is not recommended (Θ > 0.1).
(c) For, CoV = 0.4, toe excavation is not recommended at all, as the virgin slope itself has a high probability of failure.
The observed results clearly indicate that depending upon the actual CoV and correlation length, the stability analysis of the same slope would yield different outcomes. Hence, the outcomes clearly highlight that it is extremely important to conduct elaborate field investigation to ascertain the spatial variation of the soil shear strength parameters. This would be useful for specific cases of assessment of slope stability on a probabilistic framework, in which a meaningful site-specific correlation length can be considered for the practical purpose.
(a1)
(a2)
(b1)
(b2)
(c1)
(c2) Fig. 6.7 Variation in the probability of failure and reliability index with dimensionless correlation length (Θ = θ/L) for various horizontal extent of vertical toe cutting (a1, a2) CoV = 0.2 (b1, b2) CoV = 0.3 (c1, c2) CoV=0.4
Fig. 6.8 Variation in the probability of failure with dimensionless correlation length (Θ = θ/L) for the virgin slope section chosen for the present study
It is worth mentioning that the present study is limited to only assigning 1D random field by sampling the soil domain for a specific distance in horizontal direction. However, it is well known that soil exhibits inherent variation in both the horizontal and vertical direction. Therefore, characterizing soil domain with 1D spatial variation may not simulate the actual field condition appropriately. Nonetheless, this approach can be essentially considered applicable for those residual hill-slopes that are mostly characterized by predominantly single soil layer with spatially distributed properties in horizontal direction and insignificant spatial variation in the vertical direction. To cater the probabilistic study of effect of toe excavation of hill slopes comprising variation of soil properties in both horizontal and vertical, a 2D random field can be generated to
theory. A detailed study of the effect of toe cutting on hill slope stability using RFEM is further presented in Chapter 8 of this thesis.
Although probabilistic analysis in various fields of geotechnical engineering have taken huge leaps, yet it is very much necessary that simplistic methods are adopted to gain knowledge and understanding to the application of the same in design practices. The work presented here is simplified enough to get a good understanding of the outcomes of toe excavation of hill-slopes form a probabilistic framework. Deterministic analysis of assumed homogeneous slopes might bring misfortune in real cases of expansion of transportation networks in hilly terrains. Hence, this study provides a good insight into the chances of failure of toe-excavated hillslopes even when the deterministic analysis may adjudge them to be complete safe. Such scenarios are quite common in the hillslopes of north-eastern India, and such studies give a sense of caution and necessitate the requirement for better design practices.
Furthermore, although the present study has primarily addressed the hilly terrains of north-eastern India, given the range of parameters considered in the present study, the developed methodology for assessing the instability due to the excavation of the hillslope toe is equally applicable for any residual and lateritic soil slope situated anywhere in India or even around the world. It is to be noted that in the present study, the considered slope geometry is much simplified representation of the complex geometries of soil slope commonly encountered in practice. Most of the slopes in practice will comprise undulating geometry with noticeable changes in the slope inclinations, formations of concave, straight or convex curvatures, and presence of convergent, parallel or divergent plan shapes (Sabzevari and Noroozpour, 2014). It is worth mentioning that the
fundamental methodology and the proposition of the LEM-based probabilistic approach presented in this chapter for slopes with simplified geometry yet remains valid for the slopes with complex geometries as long as the geometry is properly represented in the numerical model. The presence of complex geometries might yield stability outcomes of different magnitude based on the toe- excavations, but nonetheless the proposed concept and methodology remains effective for natural slopes having undulating curvatures and plan shapes.
It is also worth mentioning that along with the uncertainties involved with the shear strength parameters, the uncertainties in slope geometry and water table location should also be considered as important factors affecting the cut slope instability in hilly terrains. The uncertainty in slope geometry primarily encompasses the uncertainty in slope inclination and presence of complex slope profiles. Based on the fundamentals of slope stability, given the interdependency of the soil shear strength parameters and slope inclination, a steeper slope is always expected to develop instability. In this study, although the uncertainties in the slope inclination is not directly incorporated, the effect of the slope inclination on both deterministic and probabilistic studies are presented with expected outcomes. The presence of water table in a hillslope leads to the saturation of soil below the phreatic line. Further, the inclination of the water table governs the severity of the seepage forces which the hillslopes may experience. Hence, given that both the stated factors are detrimental to the stability of a hillslope, the uncertainties associated with the location of intersection of the water table with the slope face and the inclination of the water table are supposed to influence the probability of failure of a hillslope. On a deterministic basis, the influence of the above two factors are well illustrated in earlier literature (Chakraborty and Dey, 2016 a,b,c). Thus, the uncertainties associated with the slope geometry and the location of water table are supposed
to influence the probability of slope failure. However, the present study presumed the slope geometry to be deterministic and the slope to be dry and devoid of the presence of water table.
Accordingly, the uncertainties involved only with the shear strength parameters are considered in the present study, leaving behind the future scopes to incorporate the uncertainties in complex slope geometries and hydraulic conditions commonly encountered in the hillslopes.