*3.2. Numerical Approach*

The numerical code used to perform the FoS calculations is the FLAC/Slope v. 8.0. In contrast to other "limit equilibrium" programs, which test several assumed failure surfaces (method of the slices) thus choosing the one with the lowest FoS, FLAC/Slope uses the procedure known as the "strength reduction technique", a method commonly applied with the Mohr–Coulomb failure criterion in FoS calculations by progressively reducing the shear strength of the material in stages until the slope fails [45–50]. The main advantage of a slope stability analysis performed with FLAC/Slope is the possibility to determine a broad variety of failure mechanisms with no prior assumptions concerning their type, shape or location. Moreover, FLAC/Slope is able to combine slip along joints with failure through intact material, thus offering clear advantages in the modelling of jointed rock masses.

The material failure can be defined by either the "Mohr–Coulomb", the "Modified Hoek–Brown" or the "ubiquitous-joint" plasticity models.

The Mohr–Coulomb model is the conventional model used to represent shear failure in soils and rocks. It assumes that failure is controlled by the maximum shear stress which in turn depends on the normal stress. The solution is obtained by plotting Mohr's circle for states of stress at failure in terms of the maximum and minimum main stresses: the Mohr–Coulomb failure line is the best straight line tangent to the Mohr's circles.

The Mohr–Coulomb criterion can be written as

$$
\pi = \mathfrak{c} + \sigma\_{\mathfrak{n}} \tan \phi \tag{1}
$$

where *τ* is the shear stress, *c* is the cohesion of the material, *σn* is the normal stress (negative in compression) and *φ* is the angle of friction.

The ubiquitous joint in FLAC/Slope is an anisotropic plasticity model that includes the presence of an oriented weak plane (such as weathering joints) in a Mohr–Coulomb model; failure may occur in either the intact rock, along the weak plane or both and depends on the stress state, the material properties of the rock and weak plane and the orientation of the latter. The failure of the weak plane (ubiquitous joint) may occur by shear, for which the envelope criterion is:

$$
\pi = c\_{\dot{l}} + \tan \varphi\_{\dot{l}} \sigma\_{\text{ll}} \tag{2}
$$

or by tension, for which the criterion is:

$$
\sigma\_3 = -T\_{\dot{\jmath}} \tag{3}
$$

with:

$$T\_{\hat{\jmath}} \le \frac{c\_{\hat{\jmath}}}{\tan \varrho\_{\hat{\jmath}}} \tag{4}$$

where *τ* and *σn* are shear and normal stresses respectively, and *σ*3 is the minimum principal stress. *cj*, *ϕj* and *Tj* are the cohesion, friction angle and the tensile strength of the ubiquitous joints, respectively.

Both the Mohr–Coulomb and the ubiquitous-joint models require another parameter, the dilation angle ψ, usually assumed as a fraction of the friction angle and ranging between *φ*/4 (very good quality rocks) and 0 (very poor quality rocks) [51–53].

The Hoek–Brown failure model for jointed rock masses is defined by the following:

$$
\sigma\_1 = \sigma\_3 + \sigma\_\varepsilon \left( m\_b \frac{\sigma\_3}{\sigma\_\varepsilon} + s \right)^a \tag{5}
$$

where *σ*1 and *σ*3 (Pa) are the maximum and minimum stresses at failure respectively. Concerning the other parameters, *mb* is the value of the Hoek–Brown constant for the rock mass, *s* and *a* are constants that depend upon the characteristics of the rock mass and *σc* (MPa) is the uniaxial compressive strength of the intact rock.

The Modified Hoek–Brown model, sometimes referred to as the "Mhoek model" [54] includes a tensile yield criterion, similar to that used by the Mohr–Coulomb model and can specify a dilation angle ψ. Compared to the original version, the Mhoek model provides a simplified flow rule for both tensile and compressive regions.

The value of *σc* is usually obtained by laboratory analyses even though several field estimates exist in literature (e.g., Table 1 in [51]).

The constants *mb*, *s* and *a* are usually calculated starting from the evaluation of the geological strength index (*GSI*) of the rock mass [55–59]. The *GSI* is a system of rock-mass characterization particularly suitable for use in engineering rock mechanics and input into numerical analysis; through a visual assessment of the geological characters of the rock material, it allows the prediction of the rock-mass strength and deformability.

The *GSI* estimation is carried out using specific charts (see Tables 4 and 5 in [51]): once the index has been evaluated, the constants *s*, *a* and *mb*, can be derived with the following equations:

$$s = \exp\left(\frac{GSI - 100}{9}\right) \quad \text{and } a = 0.5 \quad \text{for } GSI > 25\tag{6}$$

$$s = 0 \quad \text{and } a = 0.65 - \frac{GSI}{200} \quad \text{for } GSI < 25 \tag{7}$$

And

$$m\_b = m\_i \exp\left(\frac{GSI - 100}{28}\right) \tag{8}$$

where *mi* is the Hoek–Brown constant for intact rock pieces estimated using *GSI*, *σc* and the chart of Figure 7 in [51].

The Mhoek model properties can be entered in FLAC/Slope in two different ways; the *s*, *a* and *mb* constants can be input, or *GSI*, *mi* and ψ can be entered, and the Hoek–Brown strength properties are calculated automatically from the software.

The FoS calculation model for each cross-section was performed choosing a proper resolution and failure criterion (Mohr–Coulomb or Hoek–Brown) for each numerical mesh. In the case of the arenaceous-conglomeratic body of Mt. Falcone, the ubiquitous-joint model, which considers the characteristics (orientation, tensile strength, cohesion and friction angle) of the previously described joint systems, was used.

Taking into account the type of landslides and the fact that no significant erosion phenomena at the expense of the clayey bedrock are known in the area, the initial state stress, in contrast to other models [59], was assumed as lithostatic as a first approximation. Concerning the presence of water, as it was impossible to implement a discontinuous water table in the models (i.e., to simulate a real perched aquifer), a static water table close to the topographic surface was applied.

The input parameters of the geological formations (Table 1) were partly obtained from laboratory analyses, performed by professional geologists and provided privately, and partly from direct observations in the field (through *GSI* evaluation) (Figure 6); a minor number of data were obtained from literature or neighboring territories [51,57,60–64]. Uniform geotechnical properties were assumed throughout the slope, and all the parameters were then used individually or in association with the simulations; in the case of friction angle and cohesion, linked in every failure envelope, the software takes into account only the pair of values specified in the table.

**Figure 6.** Arenaceous-conglomeratic bedrock outcropping on the eastern side of the relief of Mount Falcone.

The estimation of the Mohr–Coulomb geotechnical parameters starting from the GSI evaluation was performed by means of the open-source software RocLab v.1.0 (Rockscience Inc., Toronto, ON, Canada) which can plot the Hoek–Brown and the Mohr–Coulomb failure envelopes on the same x–y plane. The results of these calculations are shown in Figure 7.


**Table 1.** Material properties, related to different geological units and implemented within the numerical simulations.

*Land* **2021**, *10*, 624



#### *Land* **2021**, *10*, 624

**Figure 7.** Shear vs. normal stress plots of Hoek–Brown and Mohr–Coulomb failure envelopes, obtained for each lithotechnical unit and different range of values.

A shear vs. normal stress plot was obtained for each bedrock unit and for each class of values (min, mean and max).

The range of values (minimum, mean and maximum) was also used for the development of numerical models for each geotechnical cross-section.
