Optimization of retaining wall design using recent swarm intelligence techniques

Engineering Structures 103 (2015) 72–84

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Optimization of retaining wall design using recent swarm intelligence techniques Amir H. Gandomi a,b,⇑, Ali R. Kashani c, David A. Roke b, Mehdi Mousavi c a

BEACON Center for the Study of Evolution in Action, Michigan State University, East Lansing, MI 48824, USA Department of Civil Engineering, The University of Akron, Akron, OH 44325, USA c Department of Civil Engineering, Arak University, Arak, Iran b

a r t i c l e

i n f o

Article history: Received 27 January 2015 Revised 26 August 2015 Accepted 26 August 2015

Keywords: Swarm intelligence techniques Retaining wall Meta-heuristic algorithms Accelerated particle swarm Firefly algorithm Cuckoo search

a b s t r a c t In this paper, cantilever retaining wall design is studied as an important optimization task in civil engineering. The current study explores the efficiency of some recent swarm intelligence techniques: accelerated particle swarm optimization (APSO), firefly algorithm (FA), and cuckoo search (CS). These algorithms are verified using two benchmark case studies. In order to better determine the proficiency of the utilized algorithms, they are benchmarked with the particle swarm optimization (PSO) algorithm, a classical swarm intelligence algorithm. To that end, a code is developed to model retaining wall design based on the ACI 318-05 procedure. In this study, continuous variables are used for wall geometry and discrete variables are used for steel reinforcement to optimize the structural design. Moreover, the sensitivity of the proposed algorithms to surcharge load, base soil friction angle, and backfill slope are investigated with respect to the geometry and design parameters. Though CS and PSO reached nearly identical lowest cost and lowest weight designs of the wall under two case studies, CS has lower values for standard deviation, mean, and worst design, and therefore may be a better optimization algorithm for engineering design. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction A retaining wall is a structure used to enhance the stability of masses of earth that are unstable in their natural slopes. These soil slopes occur frequently in the construction of railways, highways, bridges, and other civil engineering projects; therefore, minimum cost design of reinforced concrete retaining walls is an important design optimization task because of its frequent application in civil engineering. Design of retaining walls must satisfy geotechnical, structural, and economic requirements. A trial and error approach is typically necessary to design retaining walls: designers must develop an initial trial design for the wall to reach a proper final design that satisfies all the requirements. However, there is no guarantee that the final design will be an economical design. Optimal retaining wall design has been the subject of many studies in the past (e.g., [1–8]). Mathematical modeling of the wall design procedure as an objective function for optimization will be an efficient method to reach the optimal design. Recently, several ⇑ Corresponding author at: BEACON Center for the Study of Evolution in Action, Michigan State University, 1450 BPS, 567 Wilson Road, East Lansing, MI 48824, USA. E-mail addresses: [email protected], [email protected] (A.H. Gandomi). URL: http://gandomi.beacon-center.org (A.H. Gandomi). http://dx.doi.org/10.1016/j.engstruct.2015.08.034 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

researchers have attempted to utilize various metaheuristic optimization techniques for retaining wall design; for example, Ahmadi-Nedushan and Varaee [9] and Khajehzadeh et al. [10] used particle swarm optimization, Khajehzadeh et al. [11] used modified particle swarm optimization, Khajehzadeh and Eslami [12] used a gravitational search algorithm, Ceranic et al. [13] and Yepes et al. [14] utilized simulated annealing, Villalba et al. [15] applied CO2 optimization, Ghazavi and Bonab [16] applied ant colony optimization, Kaveh and Abadi [17] adopted harmony search, Kaveh and Behnam [18] utilized the charged system search algorithm, Sheikholeslami et al. [19] used the hybrid firefly algorithm, and Camp and Akin [20] applied Big Bang Big Crunch. Furthermore, despite limited research on concrete retaining wall optimization, there are numerous studies on structural and geotechnical engineering optimization problems, including Sahab et al. [21], Pezeshk and Camp [22], Gholizadeh and Barati [23], Bekdas [24], Das [25], Das and Basudhar [26], Kashani et al. [27] and Khajehzadeh et al. [28,29]. Metaheuristic algorithms are techniques that can be used to solve complex problems like retaining wall design optimization. Metaheuristic algorithms are computational methods that use iterative improvement of a candidate solution by some predetermined rules to optimize a problem. These algorithms need no initial

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solution and are capable of searching a large possible solution space. Due to the stochastic performance of these algorithms, however, there is no guarantee that the final solution is the global optimum solution. Therefore, it is necessary to adopt a wide range of metaheuristic algorithms to a specific problem to find a robust algorithm that outperforms the other techniques. Metaheuristics are generally inspired by nature (e.g., bio-inspired [30] techniques) or by art (e.g., [31]). Metaheuristic optimization algorithms can be broadly classified into two categories: evolutionary algorithms and swarm intelligence algorithms [32]. This paper focuses on applications of swarm intelligence algorithms. The particle swarm optimization (PSO) algorithm is the classical swarm intelligence algorithm and has been used in many structural optimization problems (e.g., [33]). Accelerated PSO (APSO) is a recent variant of PSO that has been successfully applied to structural engineering problems (e.g., [34]). Among other new proposed swarm intelligence techniques, the firefly algorithm (FA) and cuckoo search (CS) were the subject of structural optimization studies by Yang and Deb [35,36], Gandomi et al. [37–40], Talatahari et al. [41], and Kaveh and Bakhshpoori [42]. These algorithms have also been applied to slope stability analysis by Gandomi et al. [43]. Therefore, in this study, four swarm intelligence optimization algorithms (classical PSO, APSO, FA, and CS) are applied to retaining wall design optimization. Designs will be conducted using a program developed in MATLAB software based on ACI 318-05 [44] to minimize the weight and cost of the retaining wall. To explore the efficiency of the utilized algorithms, two numerical examples are considered from Saribasß and Erbatur [4]. Moreover, sensitivity of the proposed algorithms to surcharge load, base soil friction angle, and backfill slope are investigated through a parametric study.

2. Designing of retaining wall Fig. 1 shows a retaining wall modeled by 12 design variables: width of the base (X1), toe width (X2), footing thickness (X3), thickness at the top of the stem (X4), base thickness (X5), the distance from the toe to the front of shear key (X6), shear key width (X7), shear key depth (X8), the vertical steel reinforcement in the stem (R1), the horizontal steel reinforcement of the toe and heel (R2 and R3, respectively), and the vertical reinforcement of the shear key (R4). Variables X1 to X8 determine the wall geometry, and variables R1 to R4 represent the steel reinforcement. For X1 to X8,

continuous variables are used, whereas for R1 to R4, a set of discrete values are considered, as shown in Table 1. A total of 223 reinforcement combinations were used to represent between 3 and 28 evenly spaced 10–30 mm diameter bars. Retaining wall design is divided into two phases: geotechnical design and structural design. In the geotechnical design phase, the wall must be checked for the overturning, sliding, and bearing capacity failure modes. In the structural design phase, the wall must be checked for shear and moment failure at the stem, heel, toe, and shear key. A brief review of the geotechnical and structural design procedure is presented in this section. All the effective forces on the wall are shown in Fig. 2, where WC is the combined weight of all the sections of the reinforced concrete wall; WS is the weight of backfill acting on the heel of the wall; WT is the weight of soil on the toe of the wall; q is the distributed surcharge load (Q is the resultant surcharge load); PA is the force resulting from the active earth pressure; Pk and PT are the forces resulting from passive earth pressure on the base shear key and front part of the toe section, respectively; and PB is the force resulting from the bearing stress of the base soil. The active and passive earth pressure coefficients are evaluated using Rankine theory [45] using Eqs. (1) and (2), respectively:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 b  cos2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ka ¼ cos b cos b þ cos2 b  cos2 h

ð1Þ

  h kp ¼ tan2 45 þ 2

ð2Þ

cos b 

where b is the backfill slope and h is the friction angle of the backfill slope. Three geotechnical and six structural failure modes are considered to design a retaining wall based on Das [45] and Camp and Akin [20] as follows: I. Geotechnical stability requirements: The over turning factor of safety of the wall computed using Eq. (3):

P MR FSO ¼ P MO

ð3Þ

P where MR is the sum of resisting moments against overturning P and MO is sum of applied overturning moments. The sliding factor of safety is defined by Eq. (4):

P FR FSS ¼ P FD

ð4Þ

P where FR is the sum of horizontal resisting forces against sliding P and FD is the sum of the horizontal sliding forces, defined by Eqs. (5) and (6), respectively. Table 1 Steel reinforcement properties for design variables R1 to R4. Index number (g)

Quantity

Fig. 1. Design variables for general retaining wall.

Total As (cm2)

Reinforcement

1 2 3 4 5

3 4 3 5 4

. . . 221 222 223

. . . 16 17 18

Bar size (mm) 10 10 12 10 12 . . . 30 30 30

2.356 3.141 3.392 3.926 4.523 . . . 113.097 120.165 127.234

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where / is the nominal strength coefficient (equal to 0.75), fc is the specified compressive strength of the concrete, and b is the width of the section. 2.1. Toe slab moment and shear force design demands As shown in Fig. 3, two types of loading affect the toe slab: one due to the weight of the soil, concrete, and surcharge load above the toe slab; and the other due to the earth pressure under the toe slab. The critical section for moment is at the junction of the stem and the toe slab, and the critical section for shear force is at a distance dt from the front face of the stem (dt = X5-CC), where CC is the concrete cover. Therefore, the moment and shear force design demands are evaluated using Eqs. (12) and (13), respectively:

Fig. 2. Forces acting on a retaining wall.

X X

FR ¼

X

   2/base 2Bcbase þ W wall tan þ Pp 3 3

ð5Þ

FSB ¼

ð6Þ

qu qmax

ð7Þ

P



 6e V ¼ 1 B B

ð8Þ

P where V is the sum of the vertical forces (resultant of weight of the wall, soil above the base, and surcharge load); B is the width of the base; and e is the eccentricity of the resultant force system, expressed as follows:

e¼

h q þ q  i max V t ¼ 1:7 dt  0:9ðcc X5 þ cs DÞ  ðltoe  dtÞ 2

ð13Þ

where q2 is the soil pressure intensity at the junction of stem with toe slab, qmax is the maximum soil pressure intensity, cc is the concrete unit weight, cs is the soil unit weight, D is the depth of soil in front of the wall, ltoe is the length of the toe slab, and qdt is the soil pressure intensity at a distance dt from the junction of the stem with the toe slab.

B  2

P

As shown in Fig. 1, the forces acting on the heel slab consist of the weight of soil above the heel slab and the surcharge load acting downward, and the force due to earth pressure acting upward. The critical section for moment is at the junction of the stem with the heel slab, and the critical section for shear is at a distance dh from the back face of the stem (dh = X5-CC). The resultant moment and shear force are computed using Eqs. (14) and (15), respectively.



Mh ¼

where qu is the soil foundation ultimate bearing capacity and qmax is the maximum bearing pressure, determined using Eq. (8):

max

ð12Þ

2.2. Heel slab moment and shear force design demands

F D ¼ Pa cos b

P where Wwall is the total weight of the wall, /base is the internal friction of the base soil, B is the total width of the base slab, cbase is the adhesion between the soil and the base slab, Pp is the passive force, and Pa is the active force. The bearing capacity factor of safety is expressed using Eq. (7):

q min

h q i q  2 Mt ¼ 1:7 2 þ max  0:9ðcc X5 þ cs DÞ  ltoe 6 3

P

MR  MO P V

ð9Þ

II. Structural strength requirements: The moment and shear capacity of each section of the retaining wall must be greater than or equal to the moment and shear force design demands. Based on ACI 318-05 [44], flexural strength can be computed using Eq. (10):

 a M n ¼ /As f y d  2

ð10Þ

where / is the nominal strength coefficient (equal to 0.9), As is the area of the reinforcing steel, fy is the reinforcing steel yield strength, d is the distance from the compression surface to the centroid of the tension steel, and a is the depth of the stress block. The shear strength is calculated using Eq. (11):

qffiffiffiffi V n ¼ /0:17 f c bd

ð11Þ

   1:7q þ 1:4cc X5 þ 1:4cs H 1:4W bs q þ 2qmin 2 þ  1  lheel 2 3 6

ð14Þ   W bs þ W bsdh q þ qmin  0:9 dh V h ¼ 1:7q þ 1:4cc X5 þ 1:4cs H þ 1:4 2 2  ðlheel  dhÞ ð15Þ where q is the surcharge load, Wbs is the maximum resulting load from the backfill soil weight, q1 is the soil pressure intensity at the junction of the stem with the heel slab, lheel is the length of the heel slab, Wbsdh is the resulting load of the backfill soil weight at the distance dh from the junction of the stem with the heel slab, qdh is the soil pressure intensity at a distance dh, and qmin is the minimum soil pressure intensity. 3. Optimization Generally, optimization problems minimize an objective function f(x), subject to the following constraints:

g i ðxÞ 6 0

i ¼ 1; 2; . . . ; p

hj ðxÞ ¼ 0

j ¼ 1; 2; . . . ; m

Lk 6 X k 6 U k

k ¼ 1; 2; . . . ; n

ð16Þ

where g(x) are inequality constraints, h(x) are equality constraints, and L and U are boundary constraints. This section describes the objective function and constraints used in this study.

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Fig. 3. Effective design forces acting on base slab.

respectively. These capacities are subject to the reinforcement ratio limitations shown in Eqs. (24) and (25) [44]:

3.1. Objective function In order to reach an optimal design using meta-heuristic techniques, it is necessary to define an objective function f(x). In this paper, two objective functions, cost and weight, as proposed by Saribasß and Erbatur [4], are used. The cost minimization consists of minimizing the cost of concrete and reinforcing steel as follows:

f cost ¼ C s W st þ C c V c

ð17Þ

where Cs and Cc are unit costs of steel and concrete, respectively; Wst is the weight of the steel; and Vc is the volume of concrete. The second objective function is defined based on the weights of the materials:

f weight ¼ W st þ 100V c cc

ð18Þ

where cc is the concrete unit weight, which is scaled by a factor of 100 as proposed by Saribasß and Erbatur [4]. 3.2. Constraints The design of safe and stable retaining walls based on ACI 31805 [44] requires satisfying certain conditions related to stability, capacity, and geometry of the wall [20].  Stability constraints Eqs. (19)–(21) define the stability of the retaining wall [45]:

FSO P FSOdesign

ð19Þ

FSS P FSSdesign

ð20Þ

FSB P FSBdesign

ð21Þ

where FSOdesign, FSSdesign, and FSBdesign are prescribed factors of safety against overturning, sliding, and bearing capacity, equal to 1.5, 1.5, and 3, respectively.  Capacity constraints Each section of the retaining wall must have the capacity to resist the effective loads. Two criteria, shown in Eqs. (22) and (23), are proposed in ACI 318-05 [44].

qmin ¼ 0:25

pffiffiffiffi 1:4 fc P fy fy

qmax ¼ 0:85b1

fc 600 f y 600 þ f y

ð24Þ ! ð25Þ

where qmin and qmax are the minimum and maximum reinforcement ratios, respectively; fc is the compressive strength of the concrete; fy is the yield strength of the steel, and b1 is calculated based on ACI 318-05 as follows:

(

f c 6 30 MPa b1 ¼ 0:85 ðf c  30Þ P 0:65 f c > 30 MPa b1 ¼ 0:85  0:05 7

ð26Þ

 Geometry constraints Geometry constraints consist of boundary constraints and inequality constraints defined to produce practical designs. For this study, based on the results of Camp and Akin [20], a set of continuous values were considered for variables X1 to X8 (as indicated in Fig. 1), and a set of discrete values were considered for variables R1 to R4 (as indicated in Table 1). For the examples considered in this study, variable limitations are tabulated as boundary constraints (discussed in Section 4). Inequality constraints, which prevent infeasible retaining wall dimensions [20], are shown in Eqs. (27) and (28).

X1 P X2 þ X3

ð27Þ

X1 P X6 þ X7

ð28Þ

4. Swarm intelligence techniques 4.1. Particle swarm optimization

Mn P Md

ð22Þ

The particle swarm optimization (PSO) algorithm is a natureinspired algorithm developed by Kennedy and Eberhart [46]. This algorithm mimics the social behavior of bird flocks and fish schools. In this algorithm, a population of individuals forage the search space to find the best solution. Each particle has a specific displacement and velocity; therefore, the new position of each particle is defined as follows:

Vn P Vd

ð23Þ

X tþ1 ¼ X ti þ V tþ1 i i

where Md and Vd are the moment and shear demands, respectively, and Mn and Vn are the moment and shear nominal capacities,

ð29Þ

where X tþ1 is the updated position of the ith particle, X ti is the curi rent position, and V tþ1 is the velocity (change in position). i

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Each individual seeks its own best solution based on its velocity, and the global best solution is the best individual solution. The solution for the ith particle is improved based on its velocity, which is updated as follows:

To make convergence faster, a simpler equation could be considered for updating the position:

V tþ1 ¼ xV ti þ C 1 r 1 ðPi  X ti Þ þ C 2 r 2 ðPg  X ti Þ i

The velocity term does not appear in Eq. (33); therefore, there is no need to initialize the velocity vector. Moreover, APSO reduces randomness through iteration using Eq. (34):

ð30Þ

where V tþ1 and V ti are the new velocity and old velocity of the ith i particle, respectively; Pi and Pg are the best position of each particle and best position among all the particles, respectively; C1 and C2 are stochastic weighting values within [0, 2]; r1 and r2 are random numbers within [0, 1]; and x is the inertia weight within [0, 1.2]. For more detail, a flowchart is presented in Fig. 4. 4.2. Accelerated particle swarm optimization Accelerated particle swarm optimization (APSO) is a simplified version of PSO proposed by Yang [47]. This algorithm concentrates on faster convergence and simpler application than the original PSO algorithm. The original PSO algorithm uses the particles’ best solutions to provide diversity; however, this is only necessary for highly nonlinear problems. Therefore, in APSO, the particle best position term is removed to accelerate the convergence of the algorithms, and the velocity vector generator equation is as follows:

V tþ1 ¼ V ti þ aen þ bðP g  X ti Þ i

ð31Þ

where V tþ1 and V ti are the updated velocity and previous velocity of i the ith particle, respectively; en is a random number with normal distribution between 0 and 1, N(0, 1), to replace the removed term from Eq. (30); a varies between 0.1L and 0.5L, where L is the scale of each variable; Pg is best position among all particles; X ti is the current position and b is between 0.1 and 0.7. The updated position may be evaluated as in Eq. (32).

X tþ1 ¼ X ti þ V tþ1 i i

ð32Þ

where X tþ1 is the updated position of the ith particle. i

X tþ1 ¼ ð1  bÞX ti þ bPg þ aen i

a ¼ a0 ect or a ¼ a0 ct ð0 < c < 1Þ

ð33Þ

ð34Þ

where the initial value of randomness a0 is between 0.5 and 1, t is the iteration number, and c is a control parameter [48]. A flowchart for APSO is depicted in Fig. 5. 4.3. Firefly algorithm The firefly algorithm (FA), which has proposed by Yang [49] at Cambridge University, is inspired by firefly behavior. Fireflies search for food and find mates using bioluminescent communication. All fireflies prefer to move toward brighter positions. This algorithm has been adopted for non-convex problems by Yang et al. [36]. In this algorithm, three idealizing rules are considered [36]:  All the fireflies are unisex, and they tend to move toward brighter fireflies regardless of their sex.  Attractiveness is related to brightness and the inverse of distance from other fireflies. If there is no brighter individual, each firefly will move randomly.  The brightness of fireflies is associated to the objective function; by moving toward the minimum objective function value, the brightness will be increased. Each firefly has its distinctive attractiveness b, which is determined as follows:

bðrÞ ¼ b0  expðcr 2 Þ

Initialize the population size, number of iterations (maxGen), and necessary parametersc1, c2 and ω

Initialize random solution and velocity

Evaluate fitness function and find the global best Pg

Update velocity and current position

Evaluate objective function and update the global best Pg and individual best Pi

K
Post-process results and visualization Fig. 4. Particle swarm optimization (PSO) flowchart.

ð35Þ

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Initialize the population size, number of iterations (maxGen), and necessary parameters α, β and γ

Randomly initializesolution Evaluate fitness function and find the global best Pg

Update randomness factor’s value using Equation (18)

Update velocity and current position

Evaluate objective function and update the global best Pg

K
Post processing and visualization Fig. 5. Accelerated particle swarm optimization (APSO) flowchart.

where b0 is the maximum attractiveness value (r = 0) and c is the light absorption coefficient (varying between 0.1 and 10). The Cartesian distance between two fireflies at positions Xi and Xj is representing by r as follows:

r ij ¼ kX i  X j k

ð36Þ

Each firefly moves to a new position as follows:

xitþ1

¼ xti þ b0 ðxtj  xti Þ  expðcr2 Þ þ ae

ð37Þ

where xtþ1 represents the updated position of firefly i; the first term, i xti , is the current position of firefly i; the second term represents brightness; and the third term is used for randomizing movement, where a is a random parameter between 0 and 1 and e is a vector of random numbers generated to follow a Gaussian distribution. A flowchart of FA is presented in Fig. 6. 4.4. Cuckoo search The basic idea of the cuckoo search (CS) algorithm, developed by Yang and Deb [50], is to simulate the breeding strategy of a certain species of cuckoo. These birds lay their eggs in the nests of other birds, or sometimes other species. Cuckoos are able to find recently-spawned nests and are specialized to produce an egg like the host birds’ eggs. After hatching, the cuckoochicks’ first instinctive action is to propel the other eggs out of the nest. The cuckoo chicks also mimic the host birds’ sounds. These actions increase the cuckoos’ hatching probability and share of food. If the host birds discover the cuckoo egg, they will discard it or abandon the whole nest. Animal and insect behavior shows that a random or quasirandom pattern is used to search for food. Animal foraging patterns are effectively random walks, since the next move is related to the

current location and the selected direction depends on probability, which can be modeled mathematically. Three idealized rules are considered in CS [39]:  Each cuckoo lays only one egg at a time, laying it in a randomlychosen nest.  The best nests (solutions) will carry over to the next iteration.  The number of available host nests is constant, and a host can discover an alien egg with a probability Pa. On discovery, the host bird can either throw the egg away or abandon the nest. In this case, every nest (solution) will be replaced by a new one with discovery probability Pa. The objective function is defined like in other optimization algorithms such as GA or PSO. Based on these three rules, CS updates solution as follows: ðtþ1Þ

xi

¼ xi þ a  Lev yðkÞ ðtÞ

ðtþ1Þ

ð38Þ ðtÞ

where xi is the updated position for the ith cuckoo, xi is the ith cuckoo’s current position, a is the step size (generally set to unity),  denotes entry wise multiplication, and the Lévy term provides a random walk with random steps that has infinite variance and an infinite mean:

Lev y  u ¼ t k ; ð1 6 k 6 3Þ

ð39Þ

Here the cuckoo steps form a random walk process following a power-law step-length distribution. A CS flowchart is presented in Fig. 7. 5. Numerical simulation In this section, the efficiency of the proposed algorithms is determined. For that purpose, each optimization technique is

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Initialize the necessary parameters: number of fireflies(n),β0, γ, α and maximum number of generations (maxGen)

Generate n initial solution in the feasible solution domain (between upper and lower bound)

Evaluatethe objective function Ii=f(xi)for all produced solutions

Initialize values: i=j=1 and k=0

If Ii>Ij,move firefly i toward firefly j, then obtain attractiveness and update new solution, j=j+1

j
i=i+1

i
Rank fireflies and find best, k=k+1

k
Post processing and visualization Fig. 6. Firefly algorithm (FA) flowchart.

coded in MATLAB software. The retaining wall design process is also modeled in MATLAB code to serve as the objective function. Two case studies from Saribasß and Erbatur [4] were redesigned to explore the performance of the algorithms presented herein. The first example is identical to the one analyzed in the original paper [4]. However, the second example presented in this study explores the effects of a base shear key. ACI 318-05 [44] requirements and discrete variables for steel reinforcement are considered in design. To account for the chaotic performance of the metaheuristic algorithms, each algorithm was run 100 times and all results are reported based on best, worst, mean, and standard deviation (SD). For each algorithm, population size and number of iterations are limited to 50 and 1000, respectively.

5.1. Example 1: 3 m-tall retaining wall design without a base shear key For the first example, a base shear key is not included in the retaining wall design. This example is designed twice, with two different objective functions: lowest cost and lowest weight. To perform a sensitivity analysis, these designs are analyzed under various values of surcharge load, backfill slope, and base friction angle. The utilized parameter values for Example 1 are indicated in Table 2. Design variables boundary limitations are indicated in Table 3. Table 4 compares the different optimization techniques for lowcost design and low-weight design, indicating best, worst, and mean designs, as well as the SDs. Tables 5 and 6 show the final design values for the low-cost and low-weight objective functions,

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Initialize the necessary parameters: number of Nests (n), discovery probability (Pa), maximum iterations (K)

Generate an initial solution withinthefeasible solution boundary

Evaluate objective function and find best Nest and its cost k=1

Update current population position by Levy flight and produce new Nest (solution)

Evaluate objective function and find best solution

k=k+1 Discover alien eggs with probabilityPa and produce new solution using random walk k=k+1

Evaluate objective function and find best solution

k
Post processing and visualization Fig. 7. Cuckoo search (CS) flowchart.

Table 2 Parameter values for Examples 1 and 2. Input parameters

Height of stem Yield strength of reinforcing steel Compressive strength of concrete Concrete cover Shrinkage and temperature reinforcement ratio Surcharge load Backfill slope Internal friction angle of retained soil Internal friction angle of base soil Unit weight of retained soil Unit weight of base soil Unit weight of concrete Cohesion of base soil Depth of soil in front of wall Cost of steel Cost of concrete

respectively, taken as the best designs from Table 4. Algorithm convergence rates are presented in Figs. 8 and 9 for low cost optimization design and low weight optimization design, respectively. For the low-cost objective function, the CS algorithm achieves the best results for Example 1. Though the final cost of the CS

Symbol

H fy fc Cc

qst q b / /0

cs c0s cc c D Cs Cc

Value

Unit

Example 1

Example 2

3.0 400 21 7 0.002 20 10 36 0 17.5 18.5 23.5 125 0.5 0.4 40

4.5 400 21 7 0.002 30 15 36 34 17.5 18.5 23.5 0 0.75 0.4 40

m MPa MPa cm – kPa ° ° ° kN/m3 kN/m3 kN/m3 kPa m $/kg $/m3

design is nearly identical to the costs of the PSO and APSO designs, its low standard deviation is preferable to the higher standard deviations of the PSO and APSO designs. For the low-weight objective function, the CS algorithm again achieved the best results. Though the lowest-weight CS design is

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the acceptable convergence rates, CS is the best algorithm for optimal design of Example 1. A sensitivity analysis of the utilized algorithms was explored based on the best results of a series of 100 runs. The low cost and low weight objective function design sensitivity analysis results are shown in Tables 7–9. These results indicate that APSO has low sensitivity to variation of surcharge load and backfill slope for both low-cost and low-weight design with 54.8% and 30.9% variation for low-cost designs, respectively, and 30.0% and 19.2% for low-weight designs, respectively. For the base soil friction angle, FA has the lowest sensitivity. Overall, the low-cost objective function designs are more sensitive to variation of the design parameters.

Table 3 Design variable boundary constraints for Examples 1 and 2 (from [4]). Design variables

Unit

X1 X2 X3 X4 X5 X6 X7 X8 R1 R2 R3 R4

m m m m m m m m – – – –

Example 1

Example 2

Lower bound

Upper bound

Lower bound

Upper bound

1.3090 0.4363 0.2000 0.2000 0.2722 – – – 1 1 1 –

2.3333 0.7777 0.3333 0.3333 0.3333 – – – 223 223 223 –

1.96 0.65 0.25 0.25 0.4 1.96 0.2 0.2 1 1 1 1

5.5 1.16 0.5 0.5 0.5 5.5 0.5 0.5 223 223 223 223

5.2. Example 2: 4.5 m-tall retaining wall design with and without a base shear key

nearly identical to PSO’s, CS achieves lower values for mean, worst design, and standard deviation. The poor performance of the APSO algorithm can be attributed to its fast convergence, as illustrated in Figs. 8 and 9. The fast convergence to the final design leads to high diversification. For the low cost optimization, shown in Fig. 8, CS and PSO converged after a similar number of iterations (though well after APSO converged), and the FA design converged last. For the low weight optimization, shown in Fig. 9, PSO converged after a similar number of iterations, and CS was the next to converge. As a result of its effectiveness and

For the second example, two different retaining wall design cases are considered: one with a base shear key (Case I), and one without a base shear key (Case II), for both low cost optimization and low weight optimization design. In this example, cohesionless soil is considered for the base, and other parameters are similar to the example proposed by Saribasß and Erbatur [4], as shown in Table 2. Table 3 shows the upper and lower bound values for the design variables. In the series of 100 runs, the best, the worst, mean, and SD values for Case I and Case II under low-cost and low-weight design are

Table 4 Desgin cost and weight values for Examples 1 and 2. Optimization algorithm

Cost ($/m)

Weight (kg/m)

Best

Worst

Mean

SD

Best

Worst

Mean

SD

Example 1 PSO APSO FA CS

73.06 73.06 73.16 73.06

74.22 84.31 79.25 73.15

73.12 75.76 74.52 73.07

0.136 2.279 1.053 0.014

2665.8 2668.0 2666.5 2665.8

2668.3 2759.5 2690.0 2665.8

2665.8 2687.6 2673.4 2665.8

0.356 12.918 5.245 0.004

Example 2 (Case I) PSO APSO FA CS

162.37 162.64 162.80 162.42

163.31 177.16 168.79 163.01

162.53 167.33 164.58 162.66

0.164 3.771 1.164 0.115

5550.3 5552.0 5566.3 5550.4

5578.0 5857.6 5688.3 5551.2

5551.7 5620.5 5597.4 5550.6

4.650 56.746 19.836 0.231

Example 2 (Case II) PSO APSO FA CS

162.37 165.53 165.86 164.10

163.31 190.97 173.76 166.13

162.53 174.12 169.16 164.91

0.164 5.411 2.14 0.362

5635 5688.6 5689.2 5637.4

5769.8 6069.2 5955.7 5648.8

5647.1 5817 5773.2 5640.6

30.967 81.793 52.096 1.703

Table 5 Final low cost optimization designs for Examples 1 and 2. Optimization algorithms

X1 (m)

X2 (m)

X3 (m)

X4 (m)

X5 (m)

X6 (m)

X7 (m)

X8 (m)

R1

R2

R3

R4

Example 1 PSO APSO FA CS

1.84 1.84 1.84 1.84

0.74 0.57 0.71 0.75

0.29 0.27 0.28 0.29

0.2 0.2 0.2 0.2

0.27 0.27 0.27 0.27

– – – –

– – – –

– – – –

33 40 36 33

(15 ⁄ 10 mm) (17 ⁄ 10 mm) (11 ⁄ 12 mm) (15 ⁄ 10 mm)

14 28 14 14

(9 ⁄ 10 mm) (13 ⁄ 10 mm) (9 ⁄ 10 mm) (9 ⁄ 10 mm)

14 17 14 14

(9 ⁄ 10 mm) (10 ⁄ 10 mm) (9 ⁄ 10 mm) (9 ⁄ 10 mm)

– – – –

Example 2 (Case I) PSO APSO FA CS

2.71 2.71 2.71 2.71

0.84 0.84 0.84 0.83

0.43 0.42 0.43 0.45

0.25 0.25 0.25 0.25

0.4 0.4 0.40 0.4

– – – –

– – – –

– – – –

76 77 76 71

(26 ⁄ 10 mm) (27 ⁄ 10 mm) (26 ⁄ 10 mm) (25 ⁄ 10 mm)

33 33 33 33

(15 ⁄ 10 mm) (15 ⁄ 10 mm) (15 ⁄ 10 mm) (15 ⁄ 10 mm)

40 45 45 40

(17 ⁄ 10 mm) (18 ⁄ 10 mm) (18 ⁄ 10 mm) (17 ⁄ 10 mm)

– – – –

Example 2 (Case II) PSO 2.70 APSO 2.7 FA 2.7 CS 2.71

0.94 0.93 0.90 0.97

0.42 0.45 0.46 0.41

0.25 0.25 0.25 0.25

0.4 0.4 0.40 0.40

2.50 2.35 2.36 1.96

0.2 0.25 0.23 0.2

0.2 0.25 0.21 0.2

77 71 67 82

(27 ⁄ 10 mm) (25 ⁄ 10 mm) (24 ⁄ 10 mm) (28 ⁄ 10 mm)

33 33 33 33

(15 ⁄ 10 mm) (15 ⁄ 10 mm) (15 ⁄ 10 mm) (15 ⁄ 10 mm)

33 33 36 33

(15 ⁄ 10 mm) (15 ⁄ 10 mm) (11 ⁄ 12 mm) (15 ⁄ 10 mm)

7 (6 ⁄ 10 mm) 20 (11 ⁄ 10 mm) 28 (13 ⁄ 10 mm) 7 (6 ⁄ 10 mm)

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A.H. Gandomi et al. / Engineering Structures 103 (2015) 72–84 Table 6 Final low weight optimization designs for Examples 1 and 2. Optimization algorithms

X1 (m)

X2 (m)

X3 (m)

X4 (m)

X5 (m)

X6 (m)

X7 (m)

X8 (m)

R1

Example 1 PSO APSO FA CS

1.84 1.84 1.84 1.84

0.69 0.73 0.68 0.69

0.2 0.2 0.20 0.2

0.2 0.2 0.2 0.2

0.27 0.27 0.27 0.27

– – – –

– – – –

– – – –

90 90 90 90

Example 2 (Case I) PSO APSO FA CS

2.70 2.70 2.70 2.70

0.91 0.89 0.9 0.91

0.27 0.27 0.27 0.27

0.25 0.25 0.25 0.25

0.4 0.4 0.40 0.4

– – – –

– – – –

– – – –

156 156 155 156

Example 2 (Case II) PSO 2.69 APSO 2.7 FA 2.71 CS 2.69

0.9 1.01 1.08 0.89

0.27 0.27 0.27 0.27

0.25 0.25 0.25 0.25

0.4 0.4 0.40 0.4

2.49 2.11 2.32 2.39

0.2 0.23 0.22 0.20

0.2 0.25 0.21 0.20

156 156 155 156

R2

(21 ⁄ 12 mm) (21 ⁄ 12 mm) (21 ⁄ 12 mm) (21 ⁄ 12 mm)

R3

R4

14 23 14 14

(9 ⁄ 10 mm) (12 ⁄ 10 mm) (9 ⁄ 10 mm) (9 ⁄ 10 mm)

14 14 14 14

(9 ⁄ 10 mm) (9 ⁄ 10 mm) (9 ⁄ 10 mm) (9 ⁄ 10 mm)

– – – –

(23 ⁄ 16 mm) (23 ⁄ 16 mm) (18⁄18 mm) (23 ⁄ 16 mm)

33 37 59 33

(15 ⁄ 10 mm) (16 ⁄ 10 mm) (22 ⁄ 10 mm) (15 ⁄ 10 mm)

47 50 82 47

(19 ⁄ 10 mm) (10 ⁄ 14 mm) (28 ⁄ 10 mm) (19 ⁄ 10 mm)

– – – –

(23 ⁄ 16 mm) (23 ⁄ 16 mm) (18 ⁄ 18 mm) (23 ⁄ 16 mm)

33 60 56 33

(15 ⁄ 10 mm) (7 ⁄ 18 mm) (21 ⁄ 10 mm) (15 ⁄ 10 mm)

47 (19 ⁄ 10 mm) 56 (21 ⁄ 10 mm) 114 (27 ⁄ 12 mm) 51 (20 ⁄ 10 mm)

7 (6 ⁄ 10 mm) 20 (11 ⁄ 10 mm) 36(11 ⁄ 12 mm) 12 (8 ⁄ 10 mm)

Table 7 Cost ($/m) and weight (kg/m) design variation under varying surcharge load, q, for Example 1. q (kPa)

PSO

APSO

FA

CS

Cost design 0 10 20 30 40 50

58.20 66.62 73.06 78.31 83.90 89.63

58.20 66.62 73.05 78.34 83.90 90.08

58.62 66.90 73.16 78.61 84.49 91.28

58.20 66.62 73.05 78.31 83.91 89.69

Variation (%)

54.0

54.8

55.7

54.1

Weight design 0 10 20 30 40 50

2357.46 2536.84 2665.80 2785.74 2907.69 3064.68

2358.58 2538.62 2668.01 2789.51 2911.52 3065.40

2358.54 2537.96 2666.53 2787.6 2909.49 3068.40

2357.46 2536.84 2665.80 2785.78 2907.82 3067.61

Variation (%)

29.99

29.97

30.09

30.12 Fig. 8. Convergence rate plot for low cost optimization design of Example 1.

Table 8 Cost ($/m) and weight (kg/m) design variation under varying base soil friction angle, /, for Example 1. / (°)

PSO

APSO

FA

CS

Cost design 28 30 32 34 36 38

84.40 81.10 78.12 75.54 73.06 70.62

84.68 81.11 78.17 75.62 73.05 70.71

84.50 81.3 78.42 75.71 73.16 70.74

84.46 81.10 78.12 75.54 73.06 70.62

Variation (%)

16.3

16.5

16.3

16.4

Weight design 28 30 32 34 36 38

2977.9 2887.10 2801.78 2728.23 2665.80 2606.46

2982.29 2891.43 2805.61 2729.71 2668.01 2609.07

2979.24 2889.88 2803.49 2729.65 2666.53 2607.8

2977.88 2887.25 2801.81 2728.23 2665.80 2606.46

Variation (%)

12.47

12.51

12.47

12.47

presented in Table 4. The final cost and weight designs for Case I and Case II are presented in Tables 5 and 6. From the results it is obvious that PSO and CS reached the lowest values for both cost design and weight design under Case I and Case II. However, CS exhibits lower values for mean, worst, and

Fig. 9. Convergence rate plot for low weight optimization design of Example 1.

standard deviation (indicating a steady performance with low variation), and seems more reliable than PSO. Note that the final design with a shear key (Case I) under a moderate loading condition is not very different from the case containing no base shear key (Case II), but in the more intensive loading

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Table 9 Cost ($/m) and weight (kg/m) design variation under varying backfill slope, b, for Example 1. b (°)

PSO

APSO

FA

CS

Cost design 0 5 10 15 20 25 30 Variation (%)

70.56 71.64 73.06 74.95 77.74 82.37 92.53 31.1

70.66 71.66 73.06 75.02 77.99 82.55 92.53 30.9

70.71 71.7 73.16 75.26 77.97 82.58 92.98 31.5

70.56 71.64 73.06 74.95 77.74 82.37 92.55 31.1

Weight design 0 5 10 15 20 25 30 Variation (%)

2632.3 2644.43 2665.80 2699.66 2772.35 2895.33 3137.18 19.18

2633.07 2646.52 2668.01 2702.31 2776.76 2897.72 3137.36 19.15

2633.74 2645.89 2666.53 2701.30 2774.78 2896.82 3138.25 19.15

2632.3 2644.43 2665.80 2699.66 2772.35 2895.41 3137.71 19.20

Table 10 Cost ($/m) and weight (kg/m) design variation under varying surcharge load for Example 2 (Case I). q (kPa)

PSO

APSO

FA

CS

Cost design 0 10 20 30 40 50

128.75 141.71 152.32 162.37 174.72 192.82

129.02 141.8 152.37 162.64 177.04 194.57

128.80 141.86 152.38 162.80 177.29 194.47

128.75 141.71 152.32 162.42 174.82 193.63

Variation (%)

49.76

50.81

50.98

50.39

Weight design 0 10 20 30 40 50

4835.34 5088.64 5307.62 5550.33 5824.86 6434.91

4845.02 5096.6 5311.64 5552.00 5834.82 6445.46

4843.15 5097.28 5321.03 5566.35 5842.91 6458.51

4835.34 5087.95 5307.63 5550.36 5826.18 6441.00

Variation (%)

33.08

33.03

33.35

33.21

Table 11 Cost ($/m) and weight (kg/m) design variation under varying base soil friction angle for Example 2 (Case I). / (°)

PSO

APSO

FA

CS

Cost design 28 30 32 34 36 38

219.58 200.91 184.20 171.33 162.37 156.02

221.43 201.62 186.38 171.84 162.64 156.16

222.97 202.53 185.91 172.37 162.80 156.15

220.55 201.58 184.31 171.39 162.42 156.03

Variation (%)

40.73

41.79

42.8

41.34

Weight design 28 30 32 34 36 38

7267.27 6628.97 6070.47 5760.44 5550.33 5381.76

7269.47 6634.43 6077.77 5765.45 5552.00 5384.59

7277.48 6646.41 6088.59 5774.12 5566.35 5399.45

7272.3 6630.96 6070.74 5760.46 5550.36 5381.77

Variation (%)

35.03

35.00

34.78

35.13

cases the inclusion of the shear key results in more optimal designs for both low-cost and low-weight. APSO and FA achieved the worst results for Example 2 under both Case I and Case II. From the sensitivity analysis shown in

Table 12 Cost ($/m) and weight (kg/m) design variation under varying backfill slope for Example 2 (Case I). b (°)

PSO

APSO

FA

CS

Cost design 0 5 10 15 20 25

162.37 166.77 174.53 190.05 220.53 290.99

162.64 167.28 175.47 191.29 222.27 292.21

162.80 167.50 175.83 192.11 222.55 292.73

162.42 166.82 174.64 190.32 220.85 293.43

Variation (%)

79.22

79.67

79.81

80.65

Weight design 0 5 10 15 20 25

5550.33 5639.32 5800.33 6178.90 6936.22 8612.61

5552.00 5650.89 5804.68 6187.38 6943.29 8618.94

5566.35 5654.12 5814.69 6195.08 6943.49 8628.13

5550.36 5639.33 5800.38 6178.97 6940.89 8659.61

Variation (%)

55.17

55.24

55.00

56.02

Table 13 Cost ($/m) and weight (kg/m) design variation under varying surcharge load for Example 2 (Case II). q (kPa)

PSO

APSO

FA

CS

Cost design 0 10 20 30 40 50

130.84 143.85 154.48 163.76 172.64 182.31

132.88 145.71 156.82 165.53 175.25 186.29

132.19 145.1 156.08 165.86 174.92 186.03

130.95 143.93 154.61 164.10 173.51 184.47

Variation (%)

39.34

40.19

40.73

40.87

Weight design 0 10 20 30 40 50

4916.44 5171.11 5391.30 5634.96 5878.58 6144.85

4949.43 5212.94 5421.62 5688.61 5916.27 6198.19

4946.65 5200.21 5441.75 5689.18 5920.98 6171.99

4916.28 5170.71 5392.07 5637.41 5883.92 6168.22

Variation (%)

24.98

25.23

24.77

25.46

Table 14 Cost ($/m) and weight (kg/m) design variation under varying base soil friction angle for Example 2 (Case II). / (°)

PSO

APSO

FA

CS

Cost design 28 30 32 34 36 38

192.22 184.23 176.53 169.52 163.76 158.19

196.55 190.08 180.1 171.35 165.53 161.37

196.18 185.82 177.93 171.16 165.86 159.77

193.77 185.66 177.59 170.17 164.10 158.55

Variation (%)

21.51

21.80

22.79

22.21

Weight design 28 30 32 34 36 38

6535.97 6265.05 6034.31 5831.41 5634.96 5466.36

6599.12 6304.93 6065.84 5878.11 5688.61 5516.13

6590.49 6326.15 6082.02 5871.62 5689.18 5518.09

6560.92 6278.11 6039.28 5834.03 5637.41 5467.96

Variation (%)

19.57

19.63

19.43

19.99

Tables 10–15, it can be inferred that PSO has the lowest (or close to the lowest) sensitivity to the variation of the soil parameters. Also, similar to Example 1, the low-weight design is less sensitive to parameter variation than the low-cost design.

A.H. Gandomi et al. / Engineering Structures 103 (2015) 72–84 Table 15 Cost ($/m) and weight (kg/m) design variation under varying backfill slope for Example 2 (Case II). b (°)

PSO

APSO

FA

CS

Cost design 0 5 10 15 20 25

163.76 166.48 170.17 176.13 185.01 204.86

165.53 169.6 171.60 178.53 189.38 211.80

165.86 168.12 172.45 177.49 186.93 207.66

164.10 166.97 170.46 177.1 186.01 207.33

Variation (%)

25.10

27.95

25.20

26.34

Weight design 0 5 10 15 20 25

5634.96 5710.9 5818.37 5995.52 6279.68 6773.42

5688.61 5742.02 5856.12 6037.42 6322.97 6827.68

5689.18 5747.19 5865.03 6027.66 6327.12 6819.69

5637.41 5714.43 5820.54 6001.96 6291.5 6804.75

Variation (%)

20.20

20.02

19.87

20.71

APSO exhibited fast convergence in Example 2 for both Case I and Case II. In this example, PSO’s convergence is faster than it had been for Example1, and is better than CS’s convergence. Overall, PSO is able to reach the best designs for Example 2 under both Case I and Case II. CS is close to PSO but shows nearly constant results in the series of 100 runs with low values for worst design, mean design, and standard deviation. Neither FA nor APSO reached a comparable design for Example 2. 6. Summary and conclusions In the present study, three swarm intelligence metaheuristic optimization algorithms (APSO, FA and CS) are applied to optimum design of reinforced concrete retaining walls. In the first aim of this study, automating an efficient optimum design process for cantilever retaining walls, these algorithms demonstrate efficient performance in satisfying geotechnical and structural stability simultaneously. In the second aim, finding a robust algorithm for optimization, the algorithms utilized in this study are investigated through two numerical case studies. Moreover, the effect of a base shear key on low-cost and low-weight designing is explored in the second case study. In order to better assess the algorithms’ performance, they are benchmarked with particle swarm optimization (PSO) as a well-known classical metaheuristic algorithm. Among the studied algorithms, CS reached identical values to PSO for both low-cost and low-weight design. However, CS, because of lower standard deviation, mean and worst design values, is the best algorithm over the two presented case studies. FA reached the worst results in the current study. For the third aim, the effect of design parameters variation were also investigated by changing surcharge load from 0 to 50 kPa, base soil friction angle from 28° to 38° and backfill slope from 0° to 30°. Designing based on low-weight is less sensitive to parameter variation than designing based on low-cost. Introducing a base shear key to the design caused lower values (greater efficiency) for both cost-design and weight-design in more intensive loading cases. In the moderate loading case there is no significant difference between the designs with and without the base shear key. For both cost-design and weight-design, surcharge load is the most sensitive parameter and base soil friction angle is the least sensitive parameter in shorter walls. However, in taller walls, backfill slope is the most sensitive parameter on the final design and base soil

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friction angle is the least sensitive parameter. Moreover, a base shear key makes the wall insensitive with respect to the backfill slope variation. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2015. 08.034. References [1] Rhomberg EJ, Street WM. Optimal design of retaining walls. J Struct Div 1981;107(5):992–1002. [2] Keskar AV, Adidam SR. Minimum cost design of a cantilever retaining wall. Indian Concr J 1989;63(8):401–5. [3] Dembicki E, Chi T. System analysis in calculation of cantilever retaining walls. Int J Numer Anal Methods Geomech 1989;13:599–610. [4] Saribasß A, Erbatur F. Optimization and sensitivity of retaining structures. J Geotech Eng 1996;122(8):649–56. [5] Basudhar PK, Vashistha A, Deb K, Dey A. Cost optimization of reinforced earth walls. Geotech Geol Eng 2008;26(1):1–12. http://dx.doi.org/10.1007/s10706007-9143-6. [6] Babu GLS, Basha BM. Optimal design of cantilever retaining walls using target reliability approach. Int J Geomech 2008;8(4):240–52. [7] Babu GLS, Basha BM. Inverse reliability based design optimization of cantilever retaining walls. 3rd international ASRANet colloquium 10–12th July 2006, Glasgow, UK; 2006. [8] Das SK, Das MR. Optimum design of RCC cantilever retaining wall using a simple optimization tool. Indian Roads Congr 2011;3(2):47–58. [9] Ahmadi-Nedushan B, Varaee H. Optimal design of reinforced concrete retaining walls using a swarm intelligence technique. In: Proceedings of the 1st international conference on soft computing technology in civil, structural and environmental engineering. Stirlingshire (UK): Civil-Comp Press; 2009. [10] Khajehzadeh M, Taha MR, El-Shafie A, Eslami M. Economic design of retaining wall using particle swarm optimization with passive congregation. Aust J Basic Appl Sci 2010;4(11):5500–7. [11] Khajehzadeh M, Taha MR, El-Shafie A, Eslami M. Modified particle swarm optimization for optimum design of spread footing and retaining wall. J Zhejiang Univ-Sci A (Appl Phys Eng) 2011;12(6):415–27. [12] Khajehzadeh M, Eslami M. Gravitational search algorithm for optimization of retaining structures. Indian J Sci Technol 2012;5(1). [13] Ceranic B, Fryer C, Baines RW. An application of simulate annealing to the optimum design of reinforced concrete retaining structures. Comput Struct 2001;79(17):1569–81. [14] Yepes V, Alcala A, Perea C, Gonz´alez-Vidosa F. A parametric study of optimum earth-retaining walls by simulated annealing. Eng Struct 2008;30:821–30. [15] Villalba P, Alcalá J, Yepes V, González-Vidosa F. CO2 optimization of reinforced concrete cantilever retaining walls. In: 2nd international conference on engineering optimization, September 6–9, Lisbon, Portugal; 2010. [16] Ghazavi M, Bonab SB. Learning from ant society in optimizing concrete retaining walls. J Technol Educ 2011;5(3). [17] Kaveh A, Abadi ASM. Harmony search based algorithm for the optimum cost design of reinforced concrete cantilever retaining walls. Int J Civ Eng 2010;9 (1):1–8. [18] Kaveh A, Behnam AF. Charged system search algorithm for the optimum cost design of reinforced concrete cantilever retaining walls. Arab J Sci Eng 2013;38:563–70. [19] Sheikholeslami R, Gholipour Khalili B, Zahrai SM. Optimum cost design of reinforced concrete retaining walls using hybrid firefly algorithm. IACSIT Int J Eng Technol 2014;6(6). [20] Camp CV, Akin C. Design of retaining walls using big bang-big crunch optimization optimum design of cantilever retaining walls. J Struct Eng 2012;138:438–48. [21] Sahab MG, Toropov VV, Gandomi AH. Traditional and modern structural optimization techniques – theory and application. In: Gandomi AH et al., editors. Metaheuristic applications in structures and infrastructures. Elsevier; 2013. p. 25–47 [chapter 2]. [22] Pezeshk S, Camp CV. Recent advances in optimal structural design. In: Burns Scott, editor. State of the art on the use of genetic algorithms in design of steel structures. ASCE; 2002. [23] Gholizadeh S, Barati H. A comparative study of three metaheuristics for optimum design of trusses. Int J Optim Civ Eng 2012;2(3):423–41. [24] Bekdas G. Harmony search algorithm approach for optimum design of posttensioned axially symmetric cylindrical reinforced concrete walls. J Optim Theory Appl 2014. http://dx.doi.org/10.1007/s10957-014-0562-2. [25] Das SK. Slope stability analysis using genetic algorithm. Electron J Geotech Eng 2005;10:429–39. [26] Das SK, Basudhar PK. Comparison study of parameter estimation techniques for rock failure criterion models. Can Geotech J 2006;43(7):764–71. [27] Kashani AR, Gandomi AH, Mousavi M. Imperialistic competitive algorithm: a metaheuristic algorithm for locating the critical slip surface in 2-dimensional

84

[28]

[29]

[30] [31] [32]

[33]

[34]

[35] [36] [37] [38]

A.H. Gandomi et al. / Engineering Structures 103 (2015) 72–84 slopes. Geosci Front 2014 [in press]. doi: http://dx.doi.org/10.1016/j.gsf.2014. 11.005. Khajehzadeh M, Taha MR, El-Shafie A, Eslami M. A modified gravitational search algorithm for slope stability analysis. Eng Appl Artif Intell 2012;25 (8):1589–97. Khajehzadeh M, Taha MR, El-Shafie A, Eslami M. Locating the general failure surface of earth slope using particle swarm optimization. Civ Eng Environ Syst 2012;29(1):41–57. Gandomi AH, Alavi AH. Krill Herd: a new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simul 2012;17(12):4831–45. Gandomi AH. Interior search algorithm (ISA): a novel approach for global optimization. ISA Trans 2014;53(4):1168–83. Gandomi AH, Yang XS, Talatahari S, Alavi AH. Metaheuristic algorithms in modeling and optimization. In: Metaheuristic applications in structures and infrastructures; 2013. p. 1–24. Talatahari S, Kheirollahi M, Farahmandpour C, Gandomi AH. A multi-stage particle swarm for optimum design of truss structures. Neural Comput Appl 2013;23(5):1297–309. Gandomi AH, Yun GJ, Yang XS, Talatahari S. Chaos-enhanced accelerated particle swarm algorithm. Commun Nonlinear Sci Numer Simul 2013;18 (2):327–40. Yang XS, Deb S. Engineering optimisation by cuckoo search. Int J Math Modell Numer Optim 2013;1(4):330–43. Yang XS, Deb S. Cuckoo search: recent advances and applications. Neural Comput Appl 2014;24(1):169–74. Gandomi AH, Yang XS, Talatahari S, Alavi AH. Firefly algorithm with chaos. Commun Nonlinear Sci Numer Simul 2013;18(1):89–98. Gandomi AH, Yang XS, Alavi AH. Mixed variable structural optimization using firefly algorithm. Comput Struct 2011;89(23–24):2325–36.

[39] Gandomi AH, Yang XS, Alavi AH. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Eng Comput 2013;29 (1):17–35. [40] Gandomi AH, Talatahari S, Yang XS, Deb S. Design optimization of truss structures using cuckoo search algorithm. Struct Des Tall Spec Build 2013;22 (17):1330–49. [41] Talatahari S, Gandomi AH, Yun GJ. Optimum design of tower structures using firefly algorithm. Struct Des Tall Spec Build 2014;23(5):350–61. [42] Kaveh A, Bakhshpoori T. Optimum design of steel frames using Cuckoo Search algorithm with Lévy flights. IJST Trans Civ Eng 2013;37(C1):1–15. [43] Gandomi AH, Kashani AR, Mousavi M, Jalalvandi M. Slope stability analyzing using recent swarm intelligence techniques. Int J Numer Anal Meth Geomech 2015;39(3):295–309. [44] American Concrete Institute. Building code requirements for structural concrete and commentary (ACI 318-05), Detroit; 2005. [45] Das BM. Principles of geotechnical engineering. Boston: PWS Publishing; 1994. [46] Kennedy J, Eberhart R. Particle swarm optimization. In: Proceeding of the IEEE on international conference on neural networks. Perth, Australia; 1995. p. 1942–8. [47] Yang XS. Nature-inspired metaheuristic algorithms. 2nd ed. Luniver Press; 2010. [48] Yang XS. Engineering optimization: an introduction with metaheuristic applications. John Wiley & Sons; 2010. [49] Yang XS. Nature-inspired metaheuristic algorithms. 1st ed. Frome: Luniver Press; 2008. [50] Yang XS, Deb S. Cuckoo search via Lévy flights. In: Proceedings of world congress on nature & biologically inspired computing. USA: IEEE Publications; 2009. p. 210–4.