The moment the potential energy of a slope changes to a relative minimum, it will have a stationary value, which offers the mathematical method to obtain the global sliding direction of the slope. Therefore, the global sliding direction of the slope can be theoretically determined by the principle of minimum potential energy.

For a 2D simple slope (Figure 1), the force balance equation can be expressed as Eq. 1, here, b is just the unit weight of the simple slope and σ M is the plane stress at the point M on the slip surface. ∫ V b d v = ∫ L σ M d L (1)

When the slope is about to slide, it can only slide along the potential slip surface because of the restriction of bedrock. Assuming the global sliding direction is d, the displacement of the slope can be expressed as Eq. 2. d s = d − d n = d − d n n (2)

Therefore, when the displacement of the slope is d _s , the change of potential energy can be given by: Π = Π 0 + ∫ S σ M ⋅ d s d S (3)

Then, the first-order variation can be written as ∂ Π ∂ d = 0 (4)

Eq. 4 can also be considered as: ∂ ∂ θ ∫ S σ M ⋅ d s d S = 0 (5) where θ is the sliding angle defined anticlockwise from the axis X to the global sliding direction.

Finally, the global sliding angle θ can be deduced and simplified as Eq. 6 (Guo et al., 2019). tan ⁡ θ = ∫ l n x n y σ M x + ( n y 2 − 1 ) σ M y d l ∫ l n x n y σ M y + ( n x 2 − 1 ) σ M x d l (6) where σ M x and σ M y are the components of σ M on the axes X and Y, respectively. n x and n y are the components of the unit direction n on the axes X and Y, respectively. From Eq. 1, it can be concluded that the global sliding direction can be theoretically determined by the stress state and the shape of the slope.

Because of the strength decrease of the soil, the potential sliding body can gradually fail from the normal stress state, as shown in Figure 2. At the critical stress state, the total resisting and driving forces should be equal along the global sliding direction, and the force equilibrium equation can be established by Eq. 7. ∫ l ( σ τ c ′ + σ n c ′ ) • ( − d ) d l = ∫ l ( σ τ c + σ n c ) • ( d ) d l (7) where σ n c and σ τ c are the driving normal and shear stresses, respectively, at the critical state at any point on the slip surface, and correspondingly, σ ′ n c and σ ′ τ c are the resisting normal and shear stresses, respectively.

For the resisting stress on the slip surface, the Mohr–Coulomb yield criterion was utilized to determine the shear strength of the soil, which can be expressed as Eq. 8 ‖ σ τ c ′ ‖ = τ max F = c + ‖ σ n c ′ ‖ tan ⁡ φ F (8) where c and φ are the cohesion and the friction angle of slope material, respectively, and F is the safety factor.

Furthermore, the loads applied on the slope remain unchanged during the evolution process from the normal to the critical state, then, the assumption was made that the normal stress remains constant during the evolution process from the normal to the critical state of the slope.

Hence, the resisting forces along the slip surface along the sliding direction can be simplified as Eq. 9. ∫ l ( σ τ c ′ + σ n c ′ ) • ( − d ) d l = ∫ l ( τ max a F − σ n c ) • ( − d ) d l (9) where a stands for the unit direction of resisting shear stress at point M along a slip surface.

At the normal state of the slope, the macroscopic forces applied on the sliding body and micro forces along the slip surface should be equal to Eq. 10. ∫ A B ( σ τ + σ n ) d l = P H + P V (10) where P H and P V are the horizontal force and vertical force, respectively, applied on the potential sliding body belonging to macroscopic forces. σ n and σ τ are the normal and shear stresses, respectively, at any point on the potential slip surface belonging to micro forces.

Because of being considered invariable during the evolution process from the normal state to the critical state of the slope for macroscopic forces, the driving force remains constant for the potential sliding body, which can be expressed as Eq. 11. ∫ A B ( σ τ + σ n ) • ( d ) d l = ∫ A B ( σ τ c + σ n c ) • ( d ) d l (11)

Connecting Eqs 7–11, the force equilibrium equation can be simplified as ∫ A B ( τ max a F − σ n ) • ( − d ) d l = ∫ A B ( σ τ + σ n ) • ( d ) d l (12) That is, ∫ A B ( τ max a F ) • ( − d ) d l + ∫ A B ( σ n • d ) d l = ∫ A B ( σ τ • ( d ) ) d l + ∫ A B ( σ n • ( d ) ) d l (13)

Further simplification can be made as Eq. 14: ∫ A B ( τ max a F ) • ( − d ) d l = ∫ A B ( σ τ • ( d ) ) d l (14)

As an invariable along the slip surface, the safety factor can be finally expressed as: F f = ∫ l ( τ max a ) • ( − d ) d l ∫ l ( σ τ • d ) d l (15)

Eq. 15 demonstrates that the resisting and driving shear stresses have an effect on the F_S when the force equilibrium of a slope is considered along the sliding direction. Moreover, the safety factor (Eq. 15) can be directly obtained by integrals along the slip surface rather than iterative calculation, which is required in the LEM and SRM in computing the F_S.

At first, the initial population should be generated based on the geometrically feasible requirements of the individual, then, the corresponding fitness of each individual is also computed. After crossover, mutation, and selection operation to this population based on modifications to previous studies on the genetic algorithm, the next generation can be obtained as follows:

1) The parents are selected to be crossed to generate the offspring as the candidate of the next generation.

The aforementioned routine process is repeated until the requirement of convergence or a given number of generations is met. A real-coded GA for specific slope stability problems is explained in the following sections in detail.

According to previous studies on encoding the slip surface using polygons with N nodes denoted as V _i (i = 1 … … N), the kinematically feasible, upward-concave, trail slip surface is also taken into account, which can be completely defined by the coordinates of its nodes from the bottom to top. First, the extreme nodes (i = 1 or i = N) are randomly generated lying within specific intervals on the slope boundary. Then, similarly, the interior nodes must lie within an allowable search domain restricted by specific requirements of the feasible and upward-concave slip surface.

Figure 3 shows a feasible slip surface with N = 6 nodes from bottom to top, and the top to bottom generation has a similar process with obvious changes of nodes, which is not presented in detail in this study. As shown in Figure 3, the feasible slip surface with six nodes can be generated by fulfilling the following geometrical and kinematical compatibility constraints.

1) The extreme nodes V₁ and V₆ randomly located within their allowable intervals using a uniform probability distribution on the slope boundary have to be constrained by their corresponding lower and upper intervals, (V_1a V_1b) and (V_6a V_6b) (Figure 3).

A feasible slip surface within the slope can be generated through the procedure mentioned previously, then, the initial population with certain individuals can be produced by repeating this procedure.

For the fitness of an individual of the population, the factor of safety (F_S) can be naturally defined as the fitness of this individual. The critical slip surface of the slope we are trying to search corresponds to the global minimum F_S within the slope, that is to say, the lower the FS of a slip surface is, the better it will be as the search target. The F_S can be computed by the vector sum method in Section 3, therefore, the fitness of each individual can be determined as the basis of the selection schemes.

According to the fitness ranking (i.e., the individual can be ranked with a fitness from low to high), the selecting probability of an individual is computed by Eq. 16 P k = c q ( 1 − q ) k − 1 (16) where P k means the selecting probability of the individual with the kth fitness ranking, c is a parameter meeting all P k must add up to one, and q is a user-defined parameter between zero and one.

Based on previous studies on GA operators to work with the slip surface, the heuristic and arithmetic crossover operators are used in this study.

Before this crossover operator is performed, two parents must be selected with a specified probability (r_crs). Given two feasible slip surfaces, S _i and S _j , with factors of safety F_S (S _i ) ≤ F_S (S _j ), a new polygonal slip surface can be produced as the offspring using the heuristic crossover operator (Figure 4), which is denoted as S’ computed on the basis of Eq. 17. S ′ = S i + ξ ( S i − S j ) (17) where ξ is a random number between 0 and 1. The new offspring S′ has to be tested whether it is valid or not. Under some conditions, the new offspring cannot meet the requirement of geometrical constraints described in Section 3.2. If it is valid, the slip surface S′ can be considered as the candidate of the next generation, or else, it will be rejected and a new trial offspring is generated till a user-defined number of trials come to the end.

Figure 5 shows the description of the arithmetic crossover operator, and two auxiliary polygonals, S′ and S″, can be produced by the given two parents, S _i and S _j , as the offspring (Eqs 18, 19). S ′ = ξ S i + ( 1 − ξ ) S j (18) S ′ ′ = ξ S i + ( 1 − ξ ) S j (19)

Similarly, the offsprings S′ and S″ also need to be individually tested through geometrical constraints described in Section 4.2, and if they are valid, they will be taken as the candidateS of the next generation. If no valid offspring is produced after a user-defined number of trails, the arithmetic operation of the parents, S′ and S″, returns no offspring.

Based on the custom mutation operators, four types are mainly used in this study, that is, uniform mutation, non-uniform mutation of single nodes, total non-uniform mutation, and extreme vertices mutation.

According to the description in Section 3, the search process of the real-coded GA to locate the critical slip surface is shown in Figure 8.

This example analyzes a homogenous benchmark slope organized by ACADS (Chen, 2003), and the standard F_S of the slope is 1.0, that is, the slope is a critical slope.

These material properties of the homogenous slope are shown in Table 1. For computing conditions, the elastic model is adopted because this example is a critical slope, for which the standard F_S is 1.00. For boundary conditions, the bottom is fixed and the lateral boundaries are normally restricted. The size of the slope is shown in Figure 9 and the calculating model of the finite element analysis is shown in Figure 10, and the total element number is 636.

The critical slip surface of this example can be searched using the real-code GA in Section 3 and the vector sum method in Section 2. Figure 11 shows the evolution with the number of generations using the real-code GA, of the minimum Fs with the vector sum method, which demonstrates the search process of the GA. Figure 12 shows the geometry and locations of the critical slip surfaces using the proposed approach and rigorous Morgenstern–Price method for comparison. From Figure 11 and Figure 12, it can be seen that either the location of the critical slip surface or corresponding minimum F_S is in good agreement with the results obtained with the rigorous Morgenstern–Price method by the commercial software Geo-slope. Moreover, for this example, the minimum Fs by the proposed method or M–P method is almost equal to 1.0, which revealed that the vector sum method can reliably assess this example using this real-code GA.

This example is also initially organized by ACADS [Chen, 2003; Giam and Donald, 1989;], and used to test the ability of the proposed procedure for a non-circle slip surface. This slope includes a weak layer located between two strong layers, which is shown is Figure 13. Table 2 presents the properties of two types of soil layers. The factor of safety published by ACADS was equal to 1.26 for the specified slip surface, and for the critical slip surface, the FOS was organized to be 1.24. For boundary conditions, it is similar to that in example 1, that is, the bottom is fixed and the lateral boundaries are normally restricted.

Moreover, this case was also studied with different optimization algorithms (Gandomi et al., 2017). Figure 14 shows the locations of critical slip surfaces and the corresponding F_S by different optimization approaches, in which BBO means biogeography-based optimization and DE stands for differential evolution algorithm. From this figure, the exits of the critical slip surfaces with different approachs are almost located near the slope toe, and the entrances are located in a little different place on the top boundary of the slope. For the minimum F_S, F_S = 1.275, 1.214, 1.221, and 1.339, is obtained by GA with the vector sum method, DE with the M–P method, BBO with the M–P method, and GA with the M–P method, respectively. Except for F_S = 1.339, the rest are consistent with the standard answer 1.24 organized by ACADS for this example. Figure 15 demonstrates the evolution process with the generations of the minimum F_S, in which the global minimum FS can be obtained through almost 60 generations in the proposed GA (Figure 15).