Landslides, one of the natural disasters, have resulted in significant injury and loss to human life and damaged property and infrastructure throughout the world (Varnes, 1996; Parise and Jibson, 2000; Dai et al., 2002; Guha-Sapir et al., 2004; Crozier and Glade, 2005; Khan, 2005; Toya and Skidmore, 2007; Raghuvanshi et al., 2015; Girma et al., 2015). Normally, heavy rainfall, high relative relief and complex fragile geology with increased manmade activities have resulted in increased landslide (Gutiérrez-Martín, 2015). It is essential to identify, evaluate and delineate landslide hazard-prone areas for proper strategic planning and mitigation (Bisson et al., 2014). Therefore, to delineate landslide susceptible slopes over large areas, landslide hazard zonation (LHZ) techniques can be employed (Anbalagan, 1992; Guzzetti et al., 1999; Casagli et al., 2004; Fall et al., 2006).

Landslides are a result of intrinsic and external triggering factors. The intrinsic factors are mainly geological factors or the geometry of the slope (Hoek and Bray, 1981; Ayalew et al., 2004; Wang and Niu, 2009).

The external factor which generally triggers landslides is rainfall (Anderson and Howes, 1985; Collison et al., 2000; Dai and Lee, 2001). Several LHZ techniques have been developed in the past, and these can be broadly classified into three categories: expert evaluation, statistical methods and deterministic approaches (Wu and Sidle, 1995; Leroi, 1997; Guzzetti et al., 1999; Iverson, 2000; Crosta and Frattini, 2003; Casagli et al., 2004; Fall et al., 2006; Lu and Godt, 2008; Rossi et al., 2013; Raia et al., 2014; Canili et al., 2018; Zhang et al., 2018). Within these categories, we want to highlight the empirical models that are based on rainfall thresholds (Wilson and Jayko, 1997; Aleotti, 2004; Guzzetti et al., 2007; Martelloni et al., 2011). Each of these LHZ techniques has its own advantage and disadvantage owing to certain uncertainties on account of factors considered or methods by which factor data are derived (Carrara et al., 1995). Limit equilibrium types of analyses for assessing the stability of earth slopes have been in use in geotechnical engineering for many decades. The idea of discretizing a potential sliding mass into vertical slices was introduced in the 20th century. During the following few decades, Fellenius (1936) introduced the ordinary method of slices (Fellenius, 1936). In the mid-1950s Janbu and Bishop developed advances in the method (Janbu, 1954; Bishop, 1955). The advent of electronic computers in the 1960s made it possible to more readily handle the iterative procedures inherent in the method, which led to mathematically more rigorous formulations such as those developed by Morgenstern and Price and by Spencer (Morgenstern and Price, 1965; Spencer, 1967).

Until the 1980s, most stability analyses were performed by graphical methods or by using manual calculators. Nowadays, the quickest and most detailed analyses can be performed using any ordinary computer (Wilkinson et al., 2002). There are other types of software based on the modelling of the probability of occurrence of shallow landslides LHZ, in more extensive areas using geographic information system (GIS) technology and DEM (digital elevation model), as is the case of deterministic models like the software TRIGRS, SINMAP, R-SHALSTAB, GEOtop or GEOtop-FS, and r.slope.stability, among others (Montgomery and Dietrich, 1998; Pack et al., 2001; Rigon et al., 2006; Simoni et al., 2008; Baum et al., 2008; Mergili et al., 2014a, b; Michel et al., 2014; Reid et al., 2015; Alvioli and Baum, 2016; Tran et al., 2018). These are widely used models for calculating the time and location of the occurrence of shallow landslides caused by rainfall at the territorial level, some even in three dimensions, in order to obtain a probabilistic interpretation of the factor of safety. Currently other approaches and/or theoretical studies for landslide prediction are used (for triggering and/or propagation; Martelloni and Bagnoli, 2014; Martelloni et al., 2017). One of the achievements of the presented study is to discretize the potential slip mass in the critical profile of the slope, once unstable areas have been detected through the LHZ (landslide hazard zonation) programs. The terrain stability (TS) calculation tool is not limited to shallow landslides and debris flows but allows analysis of deep and rotational landslides, which other models often do not account for. We use the hydrological variable ru of Spencer in our algorithm to consider the infiltration of rainfall in the calculation of stability of the considered slope.

Limit equilibrium types of analyses for assessing the stability of earth slopes have been in use in geotechnical engineering for last years. Currently, the vast majority of stability analyses using this method of the equilibrium limit are performed with commercial software packages like SLIDE V5, SLOPE/W, Phase2, GEOSLOPE, GALENA, GSTABL7, GEO5 and GeoStudio, among others (González de Vallejo et al., 2002; Acharya et al., 2016a, b; Johari and Mousavi, 2018). Currently there are other slope stability models based on the theory of limit equilibrium that are still in analysis and testing, as is the case with the SSAP software package (Borselli, 2012), but in this case a general equilibrium method model is applied. Secondly, sometimes for commercial models, the introductions of parameters to perform calculations are not very interactive. For the stability analysis, different approaches can be used, such as the limit equilibrium methods (Cheng et al., 2007; Liu et al., 2015), the finite element method (Griffiths and Marquez, 2007; Tschuchnigg et al., 2015; Griffiths, 2015) and the dynamic method (Jia et al., 2008), among others. Limit equilibrium methods are well known, and their use is simple and quick. These methods allow us to analyse almost all types of landslides, such us translational, rotational, topple, creep and fall, among others (Zhou and Cheng, 2013). For the stability analysis, different approaches can be used, such as the limit equilibrium methods (Zhu et al., 2005; Cheng et al., 2007; Verruijt, 2010; Liu et al., 2015), the finite element method (Griffiths and Marquez, 2007; Tschuchnigg et al., 2015; Griffiths, 2015) and the dynamic method (Jia et al., 2008), among others (SSAP 2012, Slide V5, 2018). Also, limit equilibrium methods can be combined with probabilistic techniques (Stead et al., 2006) or with other models, like stability analysis of coastal erosion (Castedo et al., 2012). However, they are limited in general to 2-D planes and easy geometries. Numerical methods – finite element methods – give us the most detailed approach for analysing the stability conditions for the majority of evaluation cases, including complex geometries and 3-D cases. Nevertheless, they present some problems, such as their complexity, data introduction, the mesh size effect, and the time and resources they require (Ramos Vásquez, 2017).

The above-mentioned software packages provide useful tools for determining the stability through the Fs (safety of factor) and for giving the most probable breakage (shearing) surfaces. This technique is fast and allows the field or emergency engineer to make timely decisions. Although this methodology is only available in some current software (Slide V 5.0, STB 2010, GEOSLOPE) and is based on limit equilibrium methods, it is highly recommended because of its reliability for representing real conditions in the field (Chugh and Smart, 1981). This rain infiltration produces a substantial reduction of cohesion (a key soil parameter for stability) that cannot be reproduced by actual software, and then several real situations cannot be predicted.

Delft University of Technology has developed a well-known and free software program to analyse landslides, the STB 2010 (Verruijt, 2010). This program is based on a limit equilibrium technique, using a modified version of Bishop's method to calculate the Fs only for circular failures. It is a user-friendly tool, but it does not allow the calculation of water infiltration on a hillside. This is a critical point, as it is well known that rainfall infiltration is one of the main causes of landslides worldwide (Michel et al., 2015). Reviewing these issues, a new solution must be developed for cases where landslides are linked to heavy rainfall. In this study, we developed a new model and programed it using MATLAB. The primary result of this model was a stability index, namely the minimum Fs, based on the limit equilibrium technique, in this case Bishop's method. The model also provides a possible failure curve and surface area, including the infiltration effects, which can be used to coincide with analysis of the actual event as tested with field data. Topographical data can also be introduced into the model from the DEM in a GIS.

In the model we developed, the TS model, we used the limit equilibrium technique for its versatility, calculation speed and accuracy. An analysis can be done studying the whole length of the breakage (shearing) zone or just small slices. Starting with the original method of slides developed by Fellenius (1936), some methods are more accurate and complex (Spencer, 1967; Morgenstern and Price, 1965) than others (Bishop, 1955; Janbú, 1954). Using Spencer's method (Spencer, 1967; Duncan and Wright, 1980; Sharma and Moudud, 1992) here would mean dividing our slope into small slices that must be computed together. This method is divided into two equations, one related to the balance of forces and the other to momentum. Spencer's method imposes equilibrium not only for the forces but also for the momentum on the surface of the rupture. If the forces for the entire soil mass are in equilibrium, the sum of the forces between each slice must also be equal to zero. Therefore, the sum of the horizontal forces between slices must be zero as well as the sum of the vertical ones (Eqs. 1 and 2): 1∑[Qcos⁡θ]=0,2∑[Qsin⁡θ]=0. In this equation, Q is the resultant of the pair of forces between slices, and θ is the angle of the resultant (Fig. 1). From this, it can be stated that the sum of the moments of the forces between slices around the critical rotation centre is zero, conformed to Eq. (3): 3∑[QRcos⁡(α-θ)=0]. When the R is the radius of the curvature, α is the angle of the slope referred to each slice. This takes into account that the sliding surface is considered circular, so the radius of the curvature is constant.

These equations must be solved to get the Fs and tilt angles of the forces among the slices (θ). To solve these equations, an iterative method is required until a limiting error is reached. Once Fs and θ are calculated, the remaining forces are also obtained for each slice. Spencer's method is considered to be very accurate and suitable for almost all kinds of slope geometries and may be the most complete equilibrium procedure. It may also be the easiest method for obtaining the Fs (Duncan and Wright, 2005). Depending on the type of slope analysed, this model is able to establish the failure curve following the typical rotational circle, among other uses (Verruijt, 2010).

The Fs, classically defined as a ratio of stabilizing and destabilizing forces, determines the stability of a slope as follows: 4FS=∑(Forcesstandingagainst/opposesliding)∑(Forcesthatinduceslidingt). According to limit equilibrium methods, the two equilibrium conditions (forces and moments) must be satisfied. Taking into account these elements, the Fs is then obtained from the following expression (Spencer, 1967): 5Fs=1∑Wsin⁡α∑[c′bsecα+tan⁡ϕ′(Wcos⁡α-ubsecα)], where ϕ′ is the friction angle at the fracture surface, u is the pore pressure at the fracture zone, c′ is the soil cohesion, α is the angle at the base of the slice, W is the external vertical forces and b is the width of the slice. According to Eqs. (4) and (5), the slope can be considered unstable if its value of the safety factor Fs is lower than 1 or stable if it is equal to or higher than 1. It should be noted that, when applying the factor in the engineering and architecture fields, the limiting value tends to be higher than 1, with common values being 1.2 or even up to 1.5 (Burbano et al., 2009) and security coefficients that include the European technical regulations, specifically the technical regulations of Spanish application (Table 2.1 of the DB-C of the CTE, or technical code of the building), among others. This is just a confidence measure for your calculations. The Fs can also be defined as the ratio between the shear strength (τ), based on the cohesion and the angle of friction values, and the shear stress, based on the cohesion and the internal friction angle required to maintain the equilibrium (τmb).

As mentioned, the minimum Fs for considering a slope stable is equal to 1. However, several authors (Yong et al., 1977; Van Westen and Terlien, 1996) suggest that the angle of a slope would have to be defined by a value of the Fs superior to the unity to take into account the exogenous factors of the slope. Following Jiménez Salas (1981), a value of FS≥1.3 can be considered stable by most standards.

To analyse the slope using Spencer's method, a set of equations must be solved to satisfy the forces and momentum equilibrium and to obtain the Fs. The values of Fs and θ are the unknowns that must be solved. Some authors suggest that the variation in θ can be arbitrary (Morgenstern and Price, 1965), although the effect of these variations in the final value of Fs is minimal. The variation in the angle depends on the soil's ability to withstand only a small intensity of the shear stress.

With that being said, if we assume that the forces between slices are parallel (in other words, that θ is constant), Eqs. (1) and (2) become the same, resulting in 6∑Q=0. The assumption that the forces between slices are parallel gives optimal results for the calculation of the critical safety coefficients in Eq. (5) (Spencer, 1967). To solve these equations, we used the FSOLVE function of the MATLAB software, giving an initial Fs and angle. The FSOLVE function is a tool inside the optimization toolbox from MATLAB that solves systems of non-linear equations. When using this tool, an initial value must be provided to start the calculation.

When solving the normal and parallel forces at the base of the slice of the five acting forces, we obtain (Q), resulting from the forces between slices: 7Q=c′bFsecα+tan⁡ϕ′FWcos⁡α-ubsecα-Wsin⁡αcos⁡(α-θ)1+tan⁡ϕ′Ftan⁡(α-θ). In this expression, u is the pore pressure (permanent interstitial pressure) at the base of the slice and the weight of the slice is determined by W. If we assume that the soil is uniform and its density (γ) is also uniform, the weight of a slice of height h and width b can be written as follows: 8W=γbh. The application of a homogeneous pore-pressure distribution (permanent interstitial pressure) has been included in the model (Bishop and Morgenstern, 1960). In this case, the permanent interstitial pressure on the base of the slice was determined by the following expression: 9u=ruγh. In this expression, u is the pore pressure (permanent interstitial pressure) at the base of the slice, γ is the density of soil, h is the mean height of slice (if the height is not constant) and the weight of it affects the W evaluation.

The pore pressure will be hydrostatic, defined by the following: u=γw(h-hw), where γw is the saturated density of soil, and h and hw are the difference between saturated and dry height. The calculation of the infiltration factor is calculated with the following equation: 10ru=uγh. The factor ru is a coefficient of pore pressure (interstitial pressure coefficient), which determines the rain infiltration factor on the slopes. It is well known that the water that infiltrates the soil may produce a modification of the pore pressure, affecting its resistant capacity. This factor may vary from 0 (dry conditions) to 0.5 (saturated conditions). In the article of Spencer (Spencer, 1967), assuming a homogeneous pore-pressure distribution as proposed by Bishop and Morgenstern (1960), the mean pore pressure on the base of the slice can be written like Eq. (7).

This equation is used in our proposed model for calculating the safety factor (substituting the expression of u in Eq. 5).

Figure 2 shows the results of applying the terrain stability model to an irregular slope, including the initial and final points of the first failure circle (shown in yellow). This circle corresponds with the initial value introduced by the user into the FSOLVE function. The points of the slope are extracted from a DEM model in ArcGIS 10 (Glennon et al., 2008). The slope height is equal to 15 m, and the soil is considered to be uniform with the following nominal properties: γ=19500 N m-3, φ=22∘, c=15000 N m-2 and u=0 N m-2. For the application example of our algorithm in this section, we have used geotechnical data of a cohesive soil of the flysch type of Gibraltar (González de Vallejo et al., 2002).

The code works as follows: the initial circular failure curve is plotted using the FPLOT tool, as shown in Fig. 2 (yellow line). In this example, the centre coordinates are equal to xc=7 m, yc=14 m, and the lower cut has slope coordinates (P1 point) equal to xt=0 m, yt=0 m. The Fs obtained was 1.6, which is, in principle, a stable slope. It must be taken into account that the mass susceptible to sliding must be divided into a sufficient number of slices. This value is entered into our code through the parameter N. In the application example of our algorithm, the sliding mass was divided into N=500 slices; this value of N is entered into the code by the user, who decides the value of that parameter. The greater the number of slices in which we divide the sliding mass, the more accurate the calculation will be. For the N=500 slice, we consider it to be a balanced value for an optimal calculation, which relates two fundamental parameters (computer calculation capacity and capacity accuracy).

The next step is to apply Spencer's method to the different breakage surfaces until the curve with the lowest Fs is found, and that will be the critical surface susceptible to a circular slip. To determine the minimal Fs using this model, the algorithm calculates the displacement of the lower cut-off point of the critical slip from the slope as well as the position of the centre of rotation of the critical failure curve. In addition, the user must enter a series of possible circular faults. Then, the user introduces the following constraints into the program: the initial or lower point of the failure curve (P1) in its intersection point with the slope, which may or may not match the origin of the slope analysed. Another restriction is the centre of the failure circle (Xc, Yc), which should initially cut the slope, i.e. the breaking curve must be within the feasible sliding region. With these data, the program automatically draws a first curve, in this case the yellow line in Fig. 3, and calculates the safety coefficient Fs for that initial curve.

On the basis of this first curve (yellow line in Fig. 2), the program enforces new restrictions:

The curve passes through the origin of slope P1=(0,0).

To complete the second phase in the TS model operation, the effect of rain infiltration must be introduced by the coefficient of the pore-pressure factor ru. In this example, the infiltration factor was introduced at the base of each slice to account for the infiltration and pore pressure at the base of the break surface of the slope. If ru increases, the cohesion of the soil mass of the slope decreases, directly affecting the reduction of the slope's Fs. The result is that a dry slope has FS=1.45, but if including the ru parameter equal to 0.3, the Fs decreases to a value of FS=0.95, which means that an Fs is below the unity, so an unstable circular failure appears (see Fig. 4). Entering the infiltration factor, ru, in Spencer's method to introduce the infiltration effects in slopes, the geotechnical cutting elements of the analysed soil are reduced, also reducing the values of the Fs, both for the initial yellow curve and the optimum green curve (Fig. 3). Note that the initial curve in the run shown in Fig. 4 is different from the one in Fig. 3, as it depends on the data introduced.

We can determine that if this infiltration factor value is small enough, taking into account the safety coefficients, the design may still be adequate, but critical information was missing to calculate this parameter.

To clarify the procedure employed in the suggested algorithm, the flow chart (block diagrams) presented in Fig. 5 demonstrates the calculation and iteration process as implemented in our software.

Our algorithm (software) is more versatile compared to the STB 2010; the model developed here can analyse slope from right to left and vice versa, and the STB 2010 only allows the analysis from right to left. Other software programs, like the STB 2010, use a modified version of Bishop's method, a less accurate methodology than Spencer's method. A modified version of Bishop's method solves only the equilibrium in momentum, while the Spencer method also considers the equilibrium in forces.

Another improvement made by the TS code, in comparison with others, is that the use of Spencer's method allows us to analyse any type of slope and soil profile. In this procedure, we calculated the worst breaking curve by modifying the calculation points.

In the TS model, from the first slip rotational circle obtained in MATLAB, many circles were then calculated using the fmincon function, with some user restrictions. However, other models, like the STB 2010, require the definition of a quadrangular region (to look for the centres of rotational failures) and a point (namely five; see Fig. 9) to define the curve as where the failure must pass. Also, the number of circles that the STB 2010 model can analyse for their minimum value is limited to 100.

The TS model can detect relevant earth movements derived from rainfall infiltration, both translational and rotational types (Stead et al., 2006), such as those that usually occur in regions like India, the US, South America and the UK, among other places. The programs that do not contemplate this option will overestimate the Fs, potentially with great errors.

The TS model has an additional advantage: it also offers the opportunity to incorporate, in the same code, the stability analysis and the effect of the infiltration factor in the rainfall regime. This is a step forward from open-access programs, such as STB 2010, and also from alternative payment software, such as Slide.

In 2010, La Viñuela, Málaga, (Spain) experienced torrential rainfall. The main consequence was a devastating landslide with serious personal and material losses, as shown in Fig. 6. The coordinates where this event occurred were in degrees (36.88371409801, -4.204982221126).

The terrain stability (TS) analysis defines the critical surface to landslides in 2-D of each profile of the analysed slope and the safety slip factor (Fs) fairly well. We developed this model due to the need for a useful tool to predict landslides, especially when heavy rains occur.

The TS model we developed uses Spencer's method, which is more precise than the modified Bishop method; this model is used by other software such as the case of the STB 2010, so it differs in the results it provides for the Fs. It also takes into account the factor of water infiltration due to critical rains, which other software programs do not consider. A failure surface can be determined by constraints using the MATLAB function fmincon. The data needed to run the model include soil and climate properties that may vary in space and time. The exit indices of the analysis (Fs) should be interpreted in terms of relative risk. The methods implemented in the TS model are based on data structures, which are based on the data entry of the elevation model (DEM), so we obtain a topographic map, a key element in obtaining the topographic profile to be studied with our algorithm.

In the case study analysed, the slope was initially stable and was determined by the analysis performed with the STB 2010 model. However, the slope became unstable due to the heavy rains of that hydrological period, which called for the application of the pore-pressure coefficient ru. For analysing cases of heavy rain, this model is a powerful tool for determining slope stability. In addition, thanks to the great versatility of this model, it is applicable to any analysis in other parts of the world, based on the methods of limit equilibrium (Spencer, 1967). The TS model can also be used in combination with GIS software, SINMAP, the TRIGRS model and aerial photographic analysis, and mapping techniques, or even as a part of other models like the coastal recession models (Castedo et al., 2012).