Discussion of “New Perspective on Seismic Slope Stability Analysis” by Changbing Qin and Siau Chen Chian

The original paper proposes a graphical method associated with the $g$-line method to evaluate seismic slope stability. In the paper, the authors state: “The new graphical approach proposed by Klar et al. (2011) offers such convenience. … In this approach, its principle lies in determining the $g$ line, which depicts the relationship between $c/\gamma H$ and $\tan \phi$ at the limit (equilibrium) state ($F_s=1$).” This attribution is erroneous since the first time a graphical relationship between $c/\gamma H$ and $\tan \phi$ at the limit (equilibrium) state ($F_s=1$), here described as the $g$ line, was conceived by Utili and Nova (2007). In Fig. 8 of Utili and Nova (2007), two charts are plotted to show that the slope factor of safety is equal to the ratio of the segment joining the point P to the origin, PC in Fig. 1(a) and $l_1$ in Fig. 1 of the original paper, over the segment joining the intersection of PC with the $g$ line, $l_c$ in Fig. 1 of the original paper. In Utili and Nova (2007), two relationships between $c/\gamma H$ and $\tan \phi$ at the limit state are provided, one for slopes of planar shape and one for slopes of logarithmic shape. Also, the authors appear to claim that the novelty of their paper is in producing the $g$ line by employing the limit analysis upper-bound method by stating: “The kinematic solution of $c/\gamma H$ is formulated as a function of $\tan \phi$; hence, it easy to establish the $g$ line with the upper-bound analysis considering seismic effect. This approach has not yet been developed in the existing literature.” But in Fig. 8 of Utili and Nova (2007), reprinted here as Fig. 1, the $g$ line was achieved by deriving the energy balance equation of the limit analysis upper bound and seeking the least upper bound by optimization of the analytical expression obtained with respect to the angles describing the slope failure mechanisms.

The second point of critique of the paper concerns the proposal of using Eq. (4) to calculate the probability of failure of a slope. According to the authors, such a probability could be calculated as the ratio of two areas, namely A and A1 in Fig. 1 of their paper. First I note that no theoretical justification is provided by the authors to support the proposed use of the ratio of areas as a suitable approximation of the slope probability of failure. In the discusser’s opinion, the proposed formula is erroneous since it implies that all the $c/\gamma H$ and $\tan \phi$ values enclosed by the areas A and ${\mathrm{A}}_1$ exhibit the same probability of occurrence, which is obviously not the case since the probability of occurrence of the $c/\gamma H$ and $\tan \phi$ values enclosed by the areas A and ${\mathrm{A}}_1$ will in general exhibit significant variations. Also, the probability of occurrence of any $c/\gamma H$ and $\tan \phi$ value is a function of the uncertainty due to their experimental determination and of ground variability. This uncertainty is different from slope to slope since it depends on the quality of the site investigation employed, e.g., the better the site investigation the less uncertainty, and on the variability of the ground.

In the section “Stability Charts,” the authors report that “Moreover, it can be concluded that the downward (positive) seismic force (acceleration) is unfavorable to slope stability because the $g$ line spreads outward with the increase in ${\mu }_v$.” This result appears erroneous since Utili and Abd (2016) ran an extensive parametric analysis based on the same method employed in the paper under discussion, i.e., an objective function to be optimized obtained from the energy balance equation of the limit analysis upper-bound method, where it is shown that the downward seismic force can be favorable for some combinations of the slope inclination angle and of $\tan \phi$. Moreover, the upward force can be unfavorable for some combinations of these values [Figs. 6(b and d) in Utili and Abd (2016)]. This finding is confirmed by another independent analysis of Shukha and Baker (2008) employing limit equilibrium methods. Utili and Abd (2016) also show that the presence of tension cracks always reduces the stability of uniform $c$, $\phi$ slopes subject to seismic action, in some cases significantly, and therefore the possibility of tension cracks forming in the slope cannot be ignored in the assessment of the factor of safety or probability of failure of $c$, $\phi$ slopes.

Another statement in the introduction of the paper seems erroneous: “The relationship between the safety factor $F_s$ and $c/\gamma H$ was established with the use of upper-bound analysis and presented in the form of figures for practical use (Michalowski 2002, 2010; Michalowski and Martel 2011).” But this relationship was already introduced by Bell (1966). Also later on the paper states that “The procedure using the $g$-line method allows the safety factor to be determined without complicated iterations.” This statement seems to imply that it is the introduction of the $g$-line method that has eliminated the need for iterations, but the calculation of the safety factor without iterations appears already in the charts published in Michalowski (2002).

Finally, the statement “For a specific combination of ($c/\gamma H$, $\tan \phi$), a greater factor of safety is produced in the case of the outermost $g$ line” seems erroneous to the discusser since for an outermost $g$ line a lower factor of safety is expected instead.

Overall, while the paper is an interesting addition to the literature, the discusser hopes that the authors will be willing to respond to the concerns raised and believes greater care should have been taken on the background literature search to ensure that this, and the future body of work, correctly builds on the foundations of this subject.