A BBO-based algorithm for slope stability analysis by locating critical failure surface

Abstract

Determination of the critical failure surface is performed in stability evaluation process for road cut slope, embankments, dam, excavations, retaining walls and many more. Finding the critical failure surface in a rock or soil slope is very cumbersome and becomes a difficult constrained global optimization problem. Due to existence of discontinuous function and strong multiple local minima points, researchers are facing difficulties to employ trial-and-error methods in a large search space. Thus, classical optimization techniques fail to converge to a valid solution. In this study a stochastic method called biogeography-based optimization algorithm was adopted for analyzing the factor of safety. Based on the finding from the implementation and quantitative evaluation, it was found that the proposed method for locating critical failure surface in homogeneous soil slope acquires more efficient results over other implemented methods such as grid search and genetic algorithm. The validation and effectiveness of the proposed method are investigated by solving two benchmark case studies from the literature, while the simulation design for slip surfaces is carried out using ‘Rocscience slide’ software tool for comparing the results.

1 Introduction

Slope stability evaluation is a classical problem in the field of civil, mining, hydraulic and geotechnical engineering. The recognition of safe design of human-made or natural slopes is achieved by assessing the stability analysis. The stability analysis includes the study of a wide range of ground movements or slope failures influenced by gravity. These failures may arise due to earthquake, floods, landslides and many other geohazards. These failures can often catastrophic and sometimes involve the extensive loss of economy, social and environment damage. In recent years, numerous methods have been developed to access the failure mechanism and to understand the instability due to geological, hydrological, seismology and geotechnical exploration. But due to existing lots of imprecision and uncertainties and many complex decision-making problems encountered in the actual scenario make the slope stability as a very challenging task. Many stochastic and fuzzy-based techniques [1,2,3,4] have been rapidly increased to solve such problems. Locating the critical slip surface associated with least factor of safety value is traditionally performed to estimate the slope stability. It is an NP-hard, unconstrained global optimization problem due to its discontinuous function and several local minima points available in the search space. In this concern, the slip surfaces with its corresponding forces such as the forces tending to make slope down, restored forces which stabilized the slope mass are essential to evaluate the available safety margin. Various approaches such as finite element method, rigid element method, limit equilibrium method, limit analysis method, distinct element method, probabilistic analysis methods and others have been used for such analysis. But, the most widely analytical technique used for geotechnical analysis is limit equilibrium [5]. In limit equilibrium analysis (LEM), some methods of slice, such as Fellenius method [6], Bishop method [7], Morgenstern and Price [8], Spencer method [9], Janbu method [10] and others, derive factor of safety based on some assumption criteria [11]. The essence of these methods is to divide the whole sliding slope mass into a finite number of vertical slices and evaluate the slip surfaces to calculate the factor of safety margin in terms of different acting forces. To evaluate the slip surfaces in a large search space made the safety factor margin big and prove to be a difficult problem in the field of geotechnical engineering. Ching and Fredlund [12] discussed some encountered problems associated with the limit equilibrium methods of slices. Baker and Garber [13] also demonstrated the problem of limiting equilibrium in terms of the variational calculus, and then, it is proven that the minimal factor of safety must occur on slip surfaces with a special geometric property. Trial-and-error methods are also not found suitable due to lack of engineer’s experience or high solution search space [14]. The factor of safety function is often highly multimodal, and it is certainly nonsmooth due to varying the soil parameters, ground condition and external loads. In the presence of several local minima points in the search space and high computational complexity environment, classical techniques fail to give a valid solution. In the presence of these disadvantages, stochastic techniques could efficiently approximate the optimal solution to such kind of problems. Yamagami and Ueta [15] used some stochastic approach such as BFGS and DFP method to investigate the factor of safety of the different slopes. In a similar way, an extensive number of stochastic approaches [16–19,17,18,] also received a lot of attention, because of its elegance and efficiency. Many researchers have successfully employed meta-heuristic optimization algorithms for slope stability problems, such as: Greco [14] used Monte Carlo technique; Chen and Morgenstern [20], McCombie and Wilkinson [21], Das [22], Zolfaghari et al. [23], Jianping et al. [24] and Sengupta and Upadhyay [25] used different search techniques like grid method and genetic algorithm. Kahatadeniya et al. [26] used ant colony optimization. Ahangar-Asr et al. [27] applied evolutionary polynomial regression (EPR)-based approach for analyzing the slope stability of soil and rock slopes. Khajehzadeh et al. [28, 29] have implemented a new metaheuristic approach GSA based on the law of gravity and motion. And Kashani et al. [30] used ICA-based search for slope stability problems.

In the present study, the biogeography-based optimization (BBO) algorithm is employed to locate the margin of the safety factor. This algorithm has been widely adopted due to its ease of implementation and ability to solve highly complex, nonlinear optimization problems. The Fellenius method from LEM’s methods of the slice is defined as an objective function to calculate the factor of safety. The effectiveness of the proposed method is investigated by solving two benchmark problems from the literature, where the simulation design for slip surface model is compared using ‘Rocscience slide’ software tool for validating the results. From the implementation result and quantitative evaluation, it is clearly visible that the proposed method for locating critical failure surface for homogeneous soil slope acquires more efficient results over implemented methods.

2 Slope stability analysis

The slope stability analysis is done through many conventional methods such as numerical methods, kinematics methods, probabilistic methods, limit equilibrium methods, imperial methods [27, 31]. The most widely analytical technique used for geotechnical analysis is limit equilibrium [28,29,34], which calculates the safety factor based on the Mohr’s coulomb criteria. In this study, the procedure based on the method of slices is to run a large number of iterations to locate the critical failure surface, where the center coordinate points and radius of the circle are varied until the slip surface for which the factor of safety would be minimum. The essence of this method is to divide the whole sliding slope mass into a finite number of vertical slices so that the comparison between the total available shear strength along the slip surface with total forces acted in the slice’s face required to be in equilibrium state could easily derive. In the proposed methodology, a finite number of valid slip surfaces are only computed, whereas the invalid surfaces which yield the unreasonable FoS values are ignored. The methodology for evaluating the failure surface is illustrated in Fig. 1.

Fig. 1

Transition workflow diagram for slope stability analysis

2.1 Formulation of slip surface

A 2-dimensional slip surface is formulated here. The failure direction and the x–y plane where the search procedure for slip surface is to made must be specified. In this study, the prime objective is to locate the critical failure surface—a surface along which the rock mass or soil is more likely to fail. This ‘likelihood-to-fail’ is quantified by the factor of safety, which is associated with a unique slip surface. This slip surface is described by a polyline of 2-dimensional-N-points {$P_0$, $P_1$, $P_2,\ldots ,P_n-2$, $P_n-1$}. It must satisfy all its kinematic and geometric constraints [24]. Here, the slip surface is considered as a function F(P).

$$\begin{aligned} \varvec{P} = { \begin{bmatrix} \varvec{P_{0}} = {\mathrm{slope}}(t_{0}) = (x_{0},y_{0})\\ \varvec{P_{1}} = (x_{0},y_{0}) \\ \varvec{\ldots }\\ \varvec{P_{i}} = (x_{i},y_{i}) \\ \varvec{\ldots }\\ \varvec{P_{n-2}} = (x_{n-2},y_{n-2}) \\ \varvec{P_{n-1}} = {\mathrm{slope}}(t_{n-1}) = (x_{n-1},y_{n-1})\\ \end{bmatrix}} \end{aligned}$$

(1)

where positional vector matrix (P) is a solution vector and represents a polyline that describes a failure surface. The function F(P) shown in Eq. 1 consumes n-points of a polyline as inputs while outputting a scalar factor of safety value associated with a failure surface [35, 36]. However, F(P) function is often highly discontinuous as well as multimodal when its all geometric and kinematic constraints are being satisfied. Sometimes, multimodality may be the results from the multilayering of the soil or mineral lenses in the potential mass. In the research of Chen and Shao [37], it is clearly demonstrated that there may be several local minima presented in the solution space. Hence, the function F(P) is here certainly nonsmooth and being regarded as an NP-complete problem. These all pertain to make the considered problem as a global optimization problem [36, 37].

2.1.1 Dimension of the problem

The dimension of the problem will be dictated by the number of vertices which describe a slip surface from bottom to top. Each inner vertex of the slip circle contributes to two extra dimensions, while single parametric values are needed to describe the entry and exit vertex surface since both must lie on the slope surface. Thus, a failure surface of n vertices (i.e., including entry and exit points) will translate to $2\times (n-1)$ a dimension problem [35, 38].

2.1.2 Constraints and Bounds

Let us consider, a Cartesian, X–Y plane shown in Fig. 2, the function $y = S(x)$ represents slip surface, while the bedrock surface is described by $y = R(x)$. In order to satisfy all kinematic and geometric conditions, for a potential failure surface, some geometric constraints are very necessary to bind to the surface to make in equilibrium. These constraints will define the domains for each $2{n-2}$ control variables [39]. The slope surface is first parameterized from right to left between 0 and 1, and as such, ${t_{0},t_{n-1}|t_{0}>t_{n-1}\in [0,1]}$ define the entry and exit point of the failure surface. This generates points $p_{0}$ $=$ slope $(t_{0}) = (x_{0}, y_{0})$ and point $p_{n-1}$ $=$ slope $(t_{n-1}) = (x_{n-1}, y_{n-1})$.

Fig. 2

Control variables for an admissible slip surface

2.1.3 Bounds for x-coordinates

The failure surface of n vertices can be divided equally from $x_{0}$ and $x_{n-1}$ to generate $n-2$ slices. These slices bound the set of x-coordinates such that-

$$\begin{aligned}&x_{i}> x_{j} \mid j > i\quad \forall \; x_i = 1,2,3,\ldots ,n-2,\; {\mathrm{and}}\quad x_j = 2,3,4,\ldots n-1 \\&\quad {{\text {where}}} \; x_{i}\in \bigg (x_{i-1},x_{i-1}-\dfrac{x_{n-1}-x_{0}}{n-2}\bigg ) \end{aligned}$$

(2)

Since, the distance between each vertices ($P_{0},P_{1},\ldots P_{n-1}$) are equal. That is,

$$\begin{aligned} (x_{i}-x_{i+1}) = (x_{i+1}-x_{i+2}) = \cdots = \bigg (\dfrac{x_{n-1}-x_{0}}{n-2}\bigg ), \quad {{\text {where}}}\;{i\in [0,n-1]} \end{aligned}$$

(3)

2.1.4 Bounds for y-coordinates

The control variables $y_{0}$ and $y_{n-1}$ directly depend on slope height and base of the slope which can be determined easily from the slope geometry, while remaining $2n- 2$ variables i.e., $\{x_0, (x_1, y_1), \ldots , (x_i,y_i),\ldots ,(x_{n-2}, y_{n-2}), (x_{n-1})\}$ will be derived as per the coordinate system for slip surface. Cheng [36] proposed in his study that if the value of $x_{1}\;{{\mathrm{to}}}\;x_{n-1}$ are defined, then corresponding lower bounds and upper bounds to $(y_{2},\ldots , y_{n})$ are then calculated by utilizing the geometry and bed rock [40, 41]. The bounds on the control variable should be dynamic for being an admissible surface. The bounds for $y_{0}$-coordinate will be-

$$\begin{aligned} y_{0}\;\in \;\big ((R_{x}),(S_{x})\big ) \end{aligned}$$

(4)

As the line connecting points $p_{n-1}$ and $p_{1}$, intersects with the line $x = x_{2}$ at point H with y-coordinate $y_{H}$ and a line passing through $p_{0}$ and $p_{1}$ which is extended to interact with the line $x = x_{2}$ at point G with y-coordinate $y_{G}$ determine. Hence $\{y_{2min},y_{2max}\}$ can be determined as $y_{2_{{\mathrm{min}}}} = {\mathrm{max}}(y_{G},R(x_{3}))$ and $y_{2_{{\mathrm{max}}}} = {\mathrm{min}}(y_{H},y_{0}(x_{3}))$. Similarly, the bounds on y-coordinate of each point on the slip surface, i.e., $\{y_{{\mathrm{1min}}},y_{{\mathrm{1max}}},\ldots , y_{n-1_{{\mathrm{min}}}},y_{n-1_{{\mathrm{max}}}}\}$ shown in Fig. 2, can be determined by the relation,

$$\begin{aligned} y_{i}\in&\bigg (\big ({\mathrm{Max}}(R_{(x_{i})}),Y_{i_{{\mathrm{min}}}}),\big ({\mathrm{Min}}(S_{(x_{i})}), Y_{i_{{\mathrm{max}}}}\big )\bigg ) \end{aligned}$$

(5)

$$\begin{aligned}&{{\text {where}}}\quad Y_{i_{{\mathrm{min}}}} = y_{i-1} + \bigg (\dfrac{y_{i-1}-y_{i-2}}{x_{i-1}-x_{i-2}}\bigg )({x_{i} -x_{i-1}}) \end{aligned}$$

(6)

$$\begin{aligned}&Y_{i_{{\mathrm{max}}}} = y_{i-1} + \bigg (\dfrac{y_{n-1}-y_{i-1}}{x_{n-1}-x_{i-1}}\bigg )({x_{i}-x_{i-1}}) \end{aligned}$$

(7)

2.2 Modeling the objective function

In this study, the development of the objective function is made for employing the BBO algorithm. The factor of safety equation derived using Fellenius method [6, 32] is defined as the objective function or fitness function for the algorithm. In Fellenious method, the slope failure is considered to be in circular shape, while the slide side forces E₁, E₂ and X₁, X₂ shown in Fig. 3 are assumed to be negligible. In this study, the objective function is chosen here to be express a safety factor of a circular slip surface, where it would be derived in terms of center points (a, b) and radius (R). In regard to proceed this, a circular surface is drawn within its slope geometry by random selection of a center point (a, b) and radius (R). The intersection points that meet the circular surface and slope boundary are derived. The circular arc path (AB is shown in Fig. 3) is assumed to be a failure surface with center ‘O’ and radius ‘R.’ The whole slope mass (ABCD) is divided into a finite number of slices which are depicted in Fig. 3 so that the total available shear strength (resisting force $F_r$) and total shear stress (deriving force $F_d$) can be easily calculated. The Fellenius factor of safety equation is expressed as

$$\begin{aligned} F = \sum _{i=1}^{n}\frac{F_{r}}{F_{d}} = \dfrac{ \sum _{i=1}^{n}[c'_{i}l_{i} + (W_{i}\cos \alpha _{i}-\mu _{i}l_{i})\tan (\phi '_{i})]}{\sum _{i=1}^{n}W_{i}\sin \alpha _{i}} \end{aligned}$$

(8)

Finally, the working of BBO algorithm is mentioned for minimization of the objective function. The center coordinates and radius are varied in every iteration until the slip surface whose corresponding factor of safety (FoS) is found minimum.

Fig. 3

Slip surface of having finite vertical slices with FBD

2.2.1 Derivation of failure slip surface in terms of (a, b, R)

To employ the BBO algorithm, the factor of safety (FoS) is defined as the fitness function, which must be derived in terms of the center point (a, b) and radius (R) of the circular surface. In the next stage, the development of objective function is built. First, we determine the intersection points that meet the circular path with the sloped boundary and then the distance from top to bottom of the circular failure surface is calculated for finding the width of the slices. Finally, the point of intersection with the middle of the slice and boundary of slip surface is calculated for deriving the base angle ($\alpha$) of the slice. Let $(x_1,y_1 )(x_2,y_2 ),\ldots ,(x_n,y_n)$ are the coordinates of intersection points of the failure slip surface with the middle of n-slice in the direction from foot to top of the slope, where the height of the slope is $y=h$.

The equation of circle is:

$$\begin{aligned} (x-a)^2 + (y-b)^2 = R^2 \end{aligned}$$

(9)

The intersection points $(x_l,y_l)$ and $(x_U , y_U)$ of a circle and a line of the form $y = mx + d$ are as follows:

$$\begin{aligned} x_{l,u}=\frac{a + bm-dm\pm \sqrt{\partial } }{1+m^2} \end{aligned}$$

(10)

where $\partial = r^2 (1+m^2)-(b - ma -d)^2$

$$\begin{aligned} y_{l,u}=\sqrt{(R^2-(x_{l,u}-a)^2)}+b \end{aligned}$$

(11)

If the intersection points are not in slope region, it will be corrected using geometry so that an admissible failure surface is created with valid intersection points on slip surface. Now, the width of slices can be determined as

$$\begin{aligned} {{\rm {Width\;of\;each\;slice\;(b)}}} =\frac{|x_u-x_l|}{n} \end{aligned}$$

(12)

where n $=$ number of slices divided in the region. Now, using geometry the angle of slice base ($\alpha$) with the tangent at intersection point ($x_{mi},y_{mi}$) is derived. A free body diagram (FBD) of $i\mathrm{th}$- slice and angle of slice base is shown in right of Fig. 3.

$$\begin{aligned} {{\rm {Angle\;of\;slice\;base}}}(\alpha ) = \sin ^{- 1}\bigg (\frac{x_{mi}}{R}\bigg ) \end{aligned}$$

(13)

where $x_{mi}= x_{i}- b/2 \,\,{\mathrm{and}}\,\, y_{mi} = \sqrt{(R^2-(x_{mi}-a)^2)}+b$

Now, other terms that come into play in the derivation for factor of safety (FoS) will be calculated as follows:

$$\begin{aligned} {{\text {Length\;of\;Slice}}}(L)= & {} \frac{b}{\cos {\alpha }} \end{aligned}$$

(14)

$${{\rm{Tengential\;force\;at\;each\;slice\;}}}(T)= w *\sin (\alpha ) = \gamma *h*b *\sin (\alpha )$$

(15)

$${{\rm{Normal\;force \;at \;each\; slice \;}}}(N)= w *\cos (\alpha ) = \gamma *h*b *\cos (\alpha )$$

(16)

Thus, all the terms are calculated here. Now, by substituting all these terms associated with each slice in Eq. 8, the factor of safety (FoS) value corresponding to a failure surface is derived. Algorithm 1 illustrates the pseudocode for deriving factor of safety for slope surface.

3 Working principle of biogeography-based optimization algorithm

Biogeography-based optimization (BBO) is a stochastic random search algorithm inspired from island biogeography [42]. It presents the robustness of the optimum solution to many complex-type engineering problems. BBO algorithm has been widely adopted due to its ease of implementation and the ability to solve highly nonlinear problems. With these advantages, BBO algorithm has been used in many applications recently, such as classification [43, 44], sequence alignment [45], TSP problems [46] and other global optimization problems. The algorithm examines the factors that affect the distribution of biological species among neighboring islands [47, 48]. The algorithm is implemented typically with a population of possible solutions and uses probabilistic transition rules instead of deterministic rules, whereas the classical optimization techniques such as linear programming and integer programming work with a single solution [49]. The algorithm starts with the initialization of the population (called habitats). Each habitat represents a possible solution of the problem under consideration. Every solution or habitat is a collection of suitability index variables (SIVs), where SIVs indicate the independent variables of the habitat (features of the solution) that represents suitability of the habitat to which it belongs. Similarly, another habitat suitability index (HSI) relates the dependent variables associated with goodness of habitat solution. The BBO approach describes the immigration and emigration of species between habitats. High HSI of the habitat is analogous to good solutions and will be occupied by a large number of species, so it is having a high emigration rate and low immigration rate (since the habitat is nearly saturated with species), while low HSI habitat is analogous to poor solution and will be having a small number of species. Through the migration operation, high HSI solutions share a lot of features with poor solutions and poor solutions can accept a lot of features from good solutions [47]. This probabilistic operation (migration) modifies SIVs of the habitats based on the immigration rate ($\lambda _i$) and the emigration rate ($\mu _i$), where both $\lambda _i$ and $\mu _i$ are the functions of a number of species in $i\mathrm{th}$ habitat ($H_i$). For mathematical convenience, it is assumed that the immigration rate ($\lambda _i$) and emigration rate ($\mu _i$) are linear functions of the number of species. The linear migration model for $i\mathrm{th}$ habitat is formulated using Eqs. 17 and 18.

$$\begin{aligned} \lambda _i= I * (1-n_i)/n \end{aligned}$$

(17)

$$\begin{aligned} \mu _i= E * n_i/n \end{aligned}$$

(18)

where I and E represent the maximum possible immigration and emigration rate, ‘n’ indicates the maximum number of species, and $n_i$ is the number of species in the $i\mathrm{th}$ habitat.

The relationship between species count (S), maximum immigration (I) and emigration rate (E) is illustrated in Fig. 4, where ‘$S_0$’ is the equilibrium number of species and $S_{{\mathrm{max}}}$ is the maximum species count. The decision to modify each solution is taken based on the immigration rate of the solution. After following migration operation habitats go through the mutation process for maintaining the diversity among the population. In this process of mutation, a change is made by replacing an HSI from the habitat with another randomly generated HSI with a very low probability [47]. Each candidate solution is associated with a mutation probability defined by Eq. 19.

$$\begin{aligned} M(s) = \frac{M_{{\mathrm{max}}} (1-Ps)}{P_{{\mathrm{max}}}} \end{aligned}$$

(19)

Here, $M_{{\mathrm{max}}}$ is a user-defined parameter, Ps is the species count of the habitat, $P_{{\mathrm{max}}}$ is the maximum species count. Finally, the whole process is run for a consecutive number of generations until to have a minimum objective function value is attained. Figure 5 represents the workflow diagram for the whole BBO process.

Fig. 4

Species model for species count, immigration rate and emigration rate relationship

Fig. 5

Flowchart of BBO algorithm

4 Implementation of BBO algorithm in failure slip surface analysis

To implement the BBO algorithm, the factor of safety (FoS) function is defined as fitness function which is derived in terms of the center coordinates (a, b) and radius (R) of the circular surface. The Fellenius method [6] from limit equilibrium methods is used to derive the factor of safety (FoS). The analysis of the problem can be considered in two stages. In the first stage, development of objective function is built, which is demonstrated in Sect. 2.2, while in a second stage the working of BBO is mentioned for minimization of the fitness function using migration and mutation operation. Further, the population passes through elitism mechanism, where a habitat with best fitness value is chosen to pass in the next generation for preserving the solution quality. Real values are coded for the variables in the derivation of the fitness function, while the parameters of the algorithms such as population size, emigration rate (E), immigration rate (I), mutation rate ($M_{{\mathrm{max}}}$) are fine-tuned with the best values in the implementation. The proposed BBO algorithm has the following steps:

4.1 Generating population

In the first step of implementation, a number of habitat population are generated, where each population constitutes with no. of variables and represents a solution in search space. The variables (a, b, R) in the population are chosen randomly within its range, and then intersection points $(x_l,y_l)$, $(x_u,y_u)$ of circular surface are derived. Thus, a typical habitat design with 7 variables like $a|b|R|x_{l}|y_{l}|x_{u}|y_{u}$ computes a slip surface. During the implementation process, the real values are used for each variable, while the range of variables is tuned according to the problem under consideration. The encoding representation of a slip surface from a habitat is shown in Fig. 6.

Fig. 6

Pictorial view of a habitat structure to represent a failure slip surface

4.2 Migration and mutation

The migration operation in BBO is used to diversify the solution space. It generally explores the whole solution search space. In this process, a pair of habitat is picked up on the basis of migration rate and emigration rate and then exchanges the variables (features) of high HSI habitat to low HSI to improve the solution quality [50]. The probability of the habitat ($H_i$) to be modified is proportional to its immigration rate $(\lambda _i)$, while the probability of exchange variables from the source habitat ($H_j$) is proportional to the emigration rate $(\mu _j)$. This migration between variables of habitats can be expressed using Eq. 20.

$$\begin{aligned} H_i(SIV) = H_j(SIV) \end{aligned}$$

(20)

The pseudocode for the whole migration process is depicted in Algorithm 2. After cross-exchange of variables (migration) between habitats, the FoS values for all possible habitats are calculated. The habitat having lower fitness value is chosen for participating in next further process, while habitats associated with invalid solutions (belong to invalid surfaces) are ignored. This process can result a better habitat which is being added to the population. With this new value of habitat, the algorithm may able to produce a better solution.

This whole migration process is followed by mutation operation. The mutation operator generally intensifies the solution search space by altering one or more features in a habitat and gives a new solution chance to improve better fitness value. In this process, a probability is assigned to each habitat. High probability means the solution is very near to an optimum solution. If it is low, the solution is far away from the optimum solution and shows a high chance for mutation operation. Algorithm 3 shows the pseudocode for mutation process.

5 Validation using different numerical benchmark studies

In this section, effectiveness and validation of the proposed approach are examined by solving two different numerical benchmark problems from the literature. These problems are considered for homogeneous soil slope, where the Fellenius method is considered to be used as fitness function of the algorithm. At first, the safety factor is examined using grid search (GS) method, where the center point of a failure surface falls into a specified rectangular area, whose position is determined on the basis of solid engineering experiences. In this rectangular grid, each point represents a center of slip surface on a certain radius range. Finally, on each grid point, the factor of safety is calculated to examine the minimum safety factor [51, 52]. In the similar way to validate the methodology, genetic algorithm (GA) is also implemented. In this approach, a binary-coded chromosome in the population is encoded using 24 bits (00111000|10010010|01000010) associated with its variables (a, b) and R. Each chromosome in the population described a possible solution in the search space. Finally, the minimization of the fitness function is achieved by evaluating the chromosome selected over consecutive generations where the performance has been improved by using the application of crossover, mutation operators. The remaining of the implementation have been employed the same as in the literature of Sengupta and Upadhyay [25]. In the next stage, the BBO algorithm is adopted to locate the critical failure surface associate with optimal factor of safety. The findings of this algorithm indicate that the BBO algorithm produces the minimum result more efficient with fast convergence rate. The statistical analysis of the experimental results also confirms the higher stability analysis of the algorithm.

5.1 Case study 1

The geometry of this homogeneous soil slope shown in Fig. 7 is taken from the study of Yamagami and Ueta [15]. The geotechnical properties of the soil material are taken as effective internal base friction $(\phi )$ is 10$^{\circ }$, effective cohesion (c) is 9.8 KPa, pore pressure $(\mu )$ is 0, unit weight $(\gamma )$ is set 17.64 kN/m$^{3}$. In the proposed analysis, the above algorithms (grid search, GA and BBO) are rigorously run a consecutive number of iterations. As can be seen from Fig. 8, it has been observed after successive iterations that the factor of safety obtained by grid search and the GA is stabilized at 1.265 and 1.237, which shows the local minima. BBO algorithm has little better performance as it produces the minimum factor of safety at 1.224, which represents global minima in the search space. The tuning parameters of the algorithm which are rigorously found as best parameters to fine-tune the result are depicted in Table 1.

Table 1 Tuned BBO parameters for case study 1

Fig. 7

Geometry and properties of slope model for case study 1

Fig. 8

The variation of FoS with no. of generation on  presented methods. a Factor of safety versus generation graph and b factor of safety versus different methods

The critical slip surfaces located by the above algorithms over different iterations are shown in Fig. 9, whereas its associated factors of safety are tabulated in Tables 2, 3 and 4. The Tables 5, 6 and 7 illustrate the result with all slice data values for global minimum factor of safety using the above approaches (GS, GA and BBO). From the comparative result of present method’s and former studies summerized  in Table 8, it is observed that the critical slip surface associated with a minimum factor of safety obtained by grid search, GA and BBO methods tends to near with previously computed values.

Fig. 9

Critical slip surfaces determined with Fellenious method over different generations. a Minimum FoS at generation 10, b minimum FoS at generation 20, c minimum FoS at generation 50, d minimum FoS at generation 70, e minimum FoS at generation 100, f minimum FoS at generation 150, g minimum FoS at generation 200 and h minimum FoS at generation 250

Table 2 Results for minimum FoS achieved by grid search over the iterations

Table 3 Results for minimum FoS achieved by GA over different iterations

Table 4 Results for minimum FoS achieved by BBO over different iterations

Table 5 Global minima safety factor computation using grid search

Table 6 Global minima safety factor computation using genetic algorithm

Table 7 Global minima safety factor computation using BBO algorithm

Table 8 Comparative summary of factor of safety’s results for case study 1

5.2 Case study 2

This example problem is abstracted from the literature by Fredlund and Krahn [31]. The analysis is done for homogeneous cohesive soil slope, whereas the geotechnical properties of the material are taken as; unit weight ($\gamma$) is 120 pcf, friction angle is 20$^{\circ }$, and effective cohesion is set at 29 KPa. The geometric view of the slope model is depicted in Fig. 10.

This problem is also examined over grid search, GA and BBO algorithm with a similar setup and objective function. The minimum safety factor has been analyzed over successive iterations. The variation of factor of safety derived by present algorithms on various consecutive generations are depicted in Fig. 12, whereas the critical failure surface associated with minimum FoS over different iterations are illustrated in Tables 9, 10 and 11. The parameters which involved in deriving the optimal factor of safety corresponding to critical failure surfaces are tabulated in Tables 12, 13 and 14.

Fig. 10

Geometry and properties of slope model in case study 2

Fig. 11

The variation of FoS with no. of generation over presented methods. a Factor of safety versus generation graph and b factor of safety versus different methods

Fig. 12

Critical failure surfaces determined with Fellenius method over  different generations. a Minimum FoS at generation 10, b minimum FoS at generation 20, c minimum FoS at generation 50, d minimum FoS at generation 70, e minimum FoS at generation 100, f minimum FoS at generation 120, g minimum FoS at generation 150 and h minimum FoS at generation 200

Table 9 Results for minimum FoS achieved by grid search over the iterations

Table 10 Results for minimum FoS achieved by GA over different iterations

Table 11 Results for minimum FoS achieved by BBO over different iterations

Table 12 Global minima safety factor computation using grid search

Table 13 Global minima safety factor computation using genetic algorithm

Table 14 Global minima safety factor computation using BBO algorithm

From the result shown in Fig. 11, it is clearly understood that the grid search and GA approach would have assumed 1.928 and 1.921 to be the lowest factor of safety. But in BBO algorithm, upon increasing the number of generations and fine-tune the parameters of the algorithm, the final factor of safety stabilizes at 1.916, which represents the global minima. The tuning parameters which are rigorously found to be best for fine-tune the result are depicted in Table 15.

Table 15 Tuned parameters of BBO algorithm in case study 2

The comparison of results and effectiveness in terms of a minimum safety factor of the proposed method with the previous methods is summarized in Table 16.

Table 16 Comparative summary of factor of safety’s results for case study 2

6 Result discussion and comparison

As can be seen from the above results of the case studies, the factor of safety obtained by BBO algorithm is stabilized at lowest value as compared with grid search and GA. From the quantitative evaluation and implementation result, it is clearly visible that the BBO outperforms over these methods with stable convergence and local minima avoidance. From Fig. 8 of case study 1, it has been illustrated that grid search and GA achieved 1.265 and 1.237 factor of safety values. But in BBO, upon increasing the number of generations and suitable tuning parameters, the factor of safety stabilizes finally at 1.224 on 250 generations. In a similar way, from Fig. 11 of case study 2, it appears that the minimum FoS value is stabilized at 1.928 and 1.921 with grid search and GA, whereas the BBO acquired at 1.916 as global optima on 200 generations. The variation of the factor of safety with the number of generations and its corresponding comparison graphs as determined by grid search, GA and BBO methods is demonstrated in Figs. 8, 9, 11 and 12. In order to obtain the reliability of the solution and validation of the proposed approach, the above results of the case studies have been validated through a Rock-science slide software tool [55]. This geotechnical analysis tool includes the probabilistic analysis capabilities that statistically compute the factor of safety based on Monte Carlo or Latin hypercube simulation techniques [1, 14, 51]. In the present study, each slip surface associated with its FoS value obtained by stochastic approaches (GS, GA and BBO) over the different generations is validated over slide tool. An error between these methods is calculated to show their superiority. The maximum percentage errors between stochastic approaches with the calculated values from slide are summarized in Table 17. From this table, it is seen that for a given slip surface associated with optimum FoS value obtained in case study 1, the maximum error (%) is nearly 9.8% which is a minimum error as compared to computed value 10.11 and 11.05% by grid search and GA, whereas in case 2, the maximum error (%) for the optimum factor of safety value by BBO method is nearly 2.45%, which is also found least as compared to 4.3 and 4.9% by grid search and GA. In addition to this, the present approach has been easily implemented without additional computational complexity. It utilizes better solution quality with stable convergence rate and local minima avoidance. The algorithm provides a viable tool that allows users to gain information about safety factor in a large search space. Based on these advantages the BBO algorithm acquired the highest stability analysis.

Table 17 The maximum error (%) analysis between the presented methods and slide tool evaluation

7 Conclusion and future work

The BBO algorithm provides a viable tool to solve complex and nonlinear-type geotechnical problems. The proposed analysis allows the user to gain information about stability evaluation in a large solution space. From the detail discussion of this paper, it is found that the BBO algorithm is a suitable technique for employing to locate critical failure surface. Based on the findings, it appears that the stability analysis procedure makes BBO approach more efficient and applicable to achieve the reasonable factor of safety. In addition, the procedure followed for stability analysis eliminates invalid slip surfaces, which yields an unreasonable factor of safety. Hence the valid surfaces under the geometric and kinematic constraints were evaluated. The validation and performance of the algorithm were investigated by solving two different numerical benchmark case studies adopted from the literature. The findings of the study indicate that the BBO algorithm presents more consistent results with least percentage error as compared to previously developed methods such as grid search and GA. Nevertheless, the modifications to the proposed algorithm are still needed to continuously enhance the performance. Nonhomogeneous soil and rock slope models with water and load conditions in their stability analysis may be tried to explore more in future.