Reliability Analysis Based on Artificial Bee Colony (ABC) and Its Application in Geotechnical Engineering

2.1. Form Method

The reliability index is a measurement of the uncertainty of engineering system that takes into account the inherent uncertainty of input variables. The hasofer-lind index is widely used in reliability analysis [14]. The Hasofer-Lind index can be defined as the minimum distance in units of directional standard deviations from the mean-value point of the random variables to the boundary of the limit state surface. The compact form of the Hasofer-Lind index for correlated normal can be expressed as Eq 1:

(1)

Where X is a vector representing the set of random variables x_i (i=1,2,…,n), μ the vector of mean values; C the covariance matrix; and F is the failure domain.

Low and Tang presented an alternative interpretation of the Hasofer-Lind index based on the perspective of an expanding ellipsoid in the original space of the basic random variables [19], and it was given as Eq 2:

(2)

where R is the correlation matrix; and σ_i is the standard deviation of random variable x_i.

For correlated non-normals, the ellipsoidal perspective remains valid if Eq 3 is expressed as:

(3)

where μ_i^N and σ_i^N are the equivalent normal mean and equivalent normal standard deviation of random variable x_i, respectively.

Based on the reliability index, the probability of failure can be calculated by Eq 4:

(4)

where Ф() is the cumulative distribution function of the standard normal variable.

To obviate the computations of equivalent normal means and equivalent normal standard deviations, Low and Tang proposed a new efficient algorithm for the FORM by varying dimensionless number n_i [19], Eq 5 can be expressed as:

(5)

where n is a column vector of n_i (i=1,2,…,m).

When the value of n_i is varied during strained optimization, the corresponding value of x_i is automatically calculated by Eq 6:

(6)

2.2. Artificial Bee Colony (ABC) Algorithms

The Artificial Bee Colony (ABC) algorithm was originally developed by Karabogain 2005 [16]. In the ABC algorithm, the colony of artificial bees consists of three groups of bees: employed bees, onlooker bees and scouts bees. ABC algorithm requires a cycle of four phases: initialization phase, employed bee phase, onlooker bee phase and scout bee phase.

2.2.1. Initialization Phase

At the first step, the ABC generates a randomly distributed initial population of SN/2 solutions (bees) and calculates the fitness of each solution.

(7)

where x(i,j) is the candidate solution of problem; i = 1, 2,...,SN/2 and SN/2 denotes the size of population; j=1,2, …,D and D is the dimension number of each solution; rand(0,1) is a random number between [0, 1]; xⁱ_min and xⁱ_max are the upper and lower bound of each solution.

2.2.2. Employed Bee Phase

Once initialization is completed, employed bees search for a solution and calculate the fitness value. A candidate solution can be produced by the memory of bees, which is defined as:

(8)

where k used to be different from, i is the randomly chosen indexes from {1, 2,..., SN/2}, j is also randomly chosen indexes from {1, 2,..., D}, φ_ij is a random number in [-1,1] and controls the generation of neighbor food sources around x(i,j) and represents the comparison of two food positions seen by a bee.

2.2.3. Onlooker Bee Phase

Onlooker bees choose a solution based on the fitness value, determine the solution to be abandoned and allocate its employed bee as scout bees. The probability of being selected for each fitness value, p_i can be expressed as:

(9)

where fitness_i is the fitness value of the solution.

3.1. ABC-Based Reliability Analysis

Low and Tang proposed a new efficient algorithm for the FORM via varying dimensionless number (Equation 5). According to the FORM algorithm, one has to solve a constrained optimization Eq 10 that is:

(10)

where G(X_i) is the limit state function. In order to implement ABC to reliability analysis, the above-constrained optimization problem can be changed into the following non-constrained optimization form by penalty method.

(11)

where M is the penalty coefficient. The choice of the penalty coefficient M is crucial for the convergence and accuracy of Eq 11. In the algorithm of ABC, the fitness function is defined as:

(12)

3.2. The Procedure of Proposed Method

The flowchart of the proposed method is showed in Fig. (1). The algorithm of Low and Tang is used to compute reliability index in this study. Then, ABC algorithm is used to search the design point and determine the reliability index based on fitness function Eq 12.

4.1. Example 1

In this section, the serviceability reliability of a cantilever beam with a rectangular cross-section, distributed uniform load and linear elastic behavior is used to verify ABC- reliability analysis. Serviceability failure is considered as an occurrence when the maximum vertical displacement at the free end of the beam exceeds an allowable deflection limit. The allowable maximum deflection of the cantilever beam is defined as L/325, where L is the length of the beam. In this study, suppose the length L and the Young modulus E of the beam have been measured, they are 600 mm and E = 2.6 × 10⁴ MPa, respectively. The limit state function [20] is expressed as Eq 13:

(13)

where x₁ and x₂ are two independent normal random variables, are the intensity of the distributed uniform load and the height of the beam, respectively. The two variables are subject to independent normal distributions. Their statistical properties are listed in Table 1.

The control parameters SN/2 and the maximum cycle of ABC are 50 and 1000, respectively. The results obtained by the proposed method are shown in Table 2. The failure probability is 0.9607 × 10⁻² by using Monte Carlo Simulation [20]. The failure probability obtained by ABC-reliability analysis is 0.9879 × 10⁻². The relative error is less than 3%. The reliability index and design points are listed in Table 2 using Excel Solver. The results of the proposed method are in well agreement with the results obtained by Excel Solver. The relationship between reliability index and cycle number is shown in Fig. (2). Fig. (2) shows that the convergence of the proposed method is fast.

4.2. Example 2

In this section, a nonlinear example [20-22] around design point is used to verify the ABC-reliability analysis. The limit state function is given as:

(14)

where x₁ and x₂ are two independent normal random variables, the parameters of random variables are given in Table 3.

The control parameters of ABC are the same as Example1. The ABC-reliability analysis is used to calculate reliability index and design point, results are listed in Table 4. Fig. (3) shows the variation of design points. The ABC algorithm converges at the 200th cycle. the results are similar to Example1. Results show that ABC-reliability analysis can be used to compute the reliability index and design point.

5.1. Reliability Analysis of Finite Beam on Winkler Medium

To verify the robustness and accuracy of ABC-based reliability analysis, it was applied to finite beam on elastic foundations with non-normal correlated distribution variables and complex nonlinear limit state function Fig. (4). Beams on elastic foundations can be analyzed using various analytical, numerical simulation [23, 24] and reliability methods. The length and width of the beam are l and b, respectively. The concentrated loads W act at a from the left end of the beam. The bending moment of the beam can be calculated by Eq. 15-25 [24].

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

where I_b is the moment of inertia (in m⁴) of beam, E_b is Young’s modulus (MN/m²) of the beam, k_s is the subgrade modulus (MN/m³), l is the length of beam (in m), x is the distance from the left end of beam, a is the position which load acts from the left end, W is the load.

In this study, the beam is a length of 15 m, a width of 1 m. Three concentrated loads W₁, W₂, and W₃, act at 1.5, 7.5, and 13.5 m from the left end of the beam, respectively. Six random variables are considered: I_b, E_b, k_s, W₁, W₂, and W₃. The statistical parameters of random variables are listed in Table 5. Failure is considered to occur if the bending moment reaches 1 MNm anywhere along the beam. For multiple loads W₁, W₂, and W₃, the moment M can be obtained by the superposition method. So, the limit state function is g(x) =1-M.

For BetaDist: Para1,Para2,Para3 and Para4 are α, λ, minimum value and maximum value.

ABC based reliability analysis method was adopted to analyze the finite beam on elastic foundation. The control parameters SN/2 and the maximum cycle of ABC are 50 and 1000, respectively. The proposed method in this study is used to calculate the reliability index and design point (Table 6), the reliability index is 2.3834, which is basically equal to 2.3833 obtained by Excel Solver. The relationship between reliability index and the cycle is shown in Fig. (5). Fig. (6) shows the relationship between fitness value and cycle number. Figs. (5 and 6) show the convergence is quick and ABC-reliability analysis can effectively compute reliability index for the complex, nonlinear limit state function and non-normal and correlated random variables.

5.2. Circular Tunnel with Rock Bolt

The reliability was evaluated by the proposed method for a radius circular tunnel reinforced by grouted rock bolts. The stability analysis, mechanism of rock bolt and analytical solution are presented in detail by Carranza-Torres [25]. The parameters of tunnel, rock mass and rock bolt are shown in Fig. (7). The elastic modulus (E) of the rock mass the support parameters of rock bolt (rock bolt length (L_b), a distance of rockbolts out of plane(D) and the number of rock bolts) were the random variables (Table 7). The internal pressure pf represents the support provided by the tunnel face at the time the rock bolts are installed. Poisson’s ratio is 0.25. The diameter of all rock bolts is 25 mm; elastic modulus is 210 GPa. The displacement criterion of the tunnel wall was investigated. In this study, the value of ε_L is taken to be 0.005. So, the limit state function is g(x)=0.005-d (d is the displacement of tunnel wall). The reliability index is 3.1869 based on ABC algorithm (Fig. 7). This is very good agreement with the FORM solution (3.1869). The searching process is shown in Fig. (8).