Optimum Design of Retaining Walls Using Adaptive Firefly Algorithm ...

Optimum Design of Retaining Walls Using Adaptive Firefly Algorithm

1

Alper Akın and 2İbrahim Aydoğdu

1

Thomas & Betts Co., Memphis, TN; email: [email protected]

2

Akdeniz University, Department of Civil Engineering, Antalya, Turkey; email: [email protected]

ABSTRACT Optimum design of retaining walls is a complicated problem due to the fact that this optimization problem having discrete design variables and a large number of nonlinear constraints which are imposed by design codes. Finding the solution of this discrete optimization problem was not easy until the emergence of metaheuristic techniques. These techniques use specific rules that are generally inspired by natural behaviors of animals. Firefly algorithm is one of the recent additions to these techniques introduced in 2009. Firefly algorithm is based on the idealized behavior of flashing characteristics of fireflies. Although Classical Firefly Algorithm (CFA) has a satisfactory performance in many engineering problems, it does not work well for the presented problem. Stagnation and local convergence may occur in firefly movement phase of CFA. Therefore, CFA is improved by adding a function which changes randomness parameter of Firefly adaptively. Name of the new algorithm is defined as the Adaptive Firefly Algorithm (AFA). In this study, a procedure is developed for designing low-cost or low-weight cantilever reinforced concrete retaining walls with base shear keys using AFA. The objective of the optimization is to minimize the total cost or the total weight of the retaining wall. In the formulation of the optimum design problem the height and thickness of the stem, length of toe projection and the thickness of the stem at base level, the

length and thickness of the base, the depth and thickness of the key and the distance from the toe to the key are treated as design variables. The design constraints are implemented according to the requirements of the American Concrete Institute (ACI 318-05). The solution of the design problem is obtained by using AFA. Two retaining wall problems are presented as design examples which are previously used in literature studies. Obtained results of these examples are compared to the results of previous studies and CFA in order to demonstrate the efficiency and robustness of the algorithm presented. Keywords: Retaining walls, firefly algorithm, metaheuristic techniques; structural optimization. INTRODUCTION Retaining walls are used to provide stability for any material where conditions are such that the mass cannot be allowed to form a natural slope. Retaining walls are commonly used to hold earth slopes to a vertical or near vertical face, but they may also be used to retain ore, soil banks, grain, coal, or even water. They are also used as an abutment in bridge structures. One of the most commonly used shapes of geotechnical retaining walls is the reinforced concrete cantilever retaining wall. The analysis and design of retaining walls are implicit in the domain of the soil-structure interaction problem. The traditional practice in the design of reinforced concrete (RC) retaining walls involves; performing preliminary stability analysis based on assumed wall dimensions, and checking the adequacy of the wall against stability, strength, and other requirements that are imposed by the design codes. If the requirements given in the design codes are not satisfied, the wall dimensions are modified repeatedly until it satisfies the requirements in the codes. This iterative design process is performed without considering the relative costs of concrete and steel.

Many studies have been done based on the optimum design of retaining walls problems [1-7]. Traditional optimization methods are generally used in these studies. However, there are also applications of metaheuristic search techniques used for RC design problems such as genetic algorithm [8-10], simulating annealing [11-13], harmony search [14] and big bang big crunch [15]. The firefly algorithm is one of the recently developed metaheuristic optimization techniques [16]. This method is based on the idealized behavior of flashing characteristics of fireflies. Fireflies communicate, search for pray and breed using various flashing patterns. Firefly algorithm idealizes and mimics some of these flashing patterns in a numerical algorithm. Three rules that are applied in firefly algorithm are given in the following. (1) All fireflies are attracted to other fireflies because they are unisex. (2) Attractiveness is assumed to be proportional to the brightness of the firefly. A less bright firefly moves towards a brighter one. (3) The brightness of a firefly is assumed to be related to the objective function. The algorithm initiates the search for optimum solution by generating initial population of fireflies. Each firefly represents a potential solution to the design problem. The brightness of each firefly is determined by the corresponding value of the cost function. Fireflies are then moved towards each other depending on their brightness via L’evy flights. Attractiveness varies with distance r via where γ is light absorption coefficient. After the move, new solutions are evaluated and light densities are updated. Fireflies are ranked and current best one is determined. This process is continued until a termination criterion is satisfied. In some initial test, stagnation and local convergence occurred in firefly movement phase of Classical Firefly Algorithm (CFA) for optimum design of retaining wall problems. Therefore, CFA is improved by adding a function which changes randomness parameter of Firefly adaptively. Name of the new algorithm is defined as the Adaptive Firefly Algorithm (AFA).

In this study, the optimum detailed design of RC retaining walls, subject to ACI 318-05 [17] is presented. The distinctive feature of this paper is that not only the dimensions of the retaining wall, but also the reinforcement arrangements of the wall members are considered as design variables. In addition to geotechnical stability and strength constraints, the reinforcement arrangement constraints derived from ACI 318-05 [17] code specification are adopted for the optimization problem. The adaptive firefly algorithm (AFA) is used as a metaheuristic optimization method to obtain the optimum solution of the design problem and the numerical examples are presented to illustrate the performance of the algorithm developed. OPTIMUM DESIGN PROBLEM of CANTILEVER RETAINING WALLS Optimum design of retaining wall problems requires the selection of the values from available design variable pools or between lower and upper bounds for its design variables so that the retaining wall design provides safety and stability against failure modes and complies with concrete building code requirements. It is also necessary to consider the economy in this selection. This is achieved in the optimization problem by taking the cost of the retaining wall as an objective function. The objective function of the optimum design of retaining wall problems is formulated as minimization of the cost of the retaining wall which is expressed as: ( )= where,

+

(1)

is the vector which contains the sequence numbers of design variables, Cs is the unit

cost of steel, Cc is the unit cost of concrete, Wst is the weight of steel per unit length of the wall, and Vc is the volume of concrete per unit length of the wall. In this study, the unit cost values for concrete and steel are utilized from the work of Sarıbaş and Erbatur [5]. The unit cost values from that study are $40/m3 and $0.40/kg respectively for concrete and steel. Any unit cost includes material, fabrication, and labour.

Design variables of the retaining wall problem can be classified as cross-sectional dimensions and reinforcement of wall members. Thirteen design variables are considered in this study. The first eight design variables are related to the geometry of the cross-section and the last five design variables consider various steel reinforcement arrangements. These design variables are shown in Figure 1. Design Variables of Retaining Wall Eight design variables related to the cross-sectional dimensions of the wall are; the total base width (X1), toe projection (X2), bottom thickness of the stem (X3), top thickness of the stem (X4), thickness of the heel and the toe (X5), key distance from the toe (X6), thickness of the key (X7), and the height of key (X8). Lower and upper bounds of the first eight design variables are given in Table 1. The other design variables are related to the reinforcement of the wall members. These design variables are; the number and diameter of the first stem reinforcement together (R1), the number and diameter of the second stem reinforcement together (R2), the number and diameter of the toe reinforcement together (R3), the number and diameter of the heel reinforcement together (R4), and the number and diameter of the key reinforcement together (R5). The number and diameter of the bar for each wall member are selected as reinforcement design variables. In other words, the number and diameter of the bar for members are considered as one design variable as shown in Table 2. (n d n: number of bars, d: diameter of bars).

For the

reinforcement design variables, the design variable pool is created by combination of numbers and diameters of reinforcement bars. The number of reinforcement bars changes between 1 and 10 and the diameter of bars changes between 10 and 30 with an increment of 2 mm. Then, 111 candidate variables (combination of numbers and diameters of bars) are arranged for each reinforcement design variable. The design variable pool composed of reinforcement design variables is given in Table 2. The design philosophy of retaining walls covers the stability of the wall against failure modes, concrete design capacities, and reinforcement and geometry of the wall. We can classify the

constraints into four groups; stability, capacity, reinforcement arrangement, and geometric constraints. The constraints are expressed in a normalized form as given below. Feasible designs should satisfy the desirable factors of safety coefficients for overturning, sliding and bearing capacity failure modes (2-4). The “no tension” condition should also be satisfied (5). These stability constraints are formulated as;

( )=











( )=

( )= ( )=

−1≥0

(2)

−1≥0

(3)

−1 ≥0

(4)

≥0

(5)

The ultimate moment of resistance, Mu, at any critical wall section (stem, toe, heel, key) should be greater than the design moments, Md, arising out of system conditions (6). In the same way, shear capacities, Vu, at critical wall sections should be greater than the design shear forces, Vu (7).

( )=

,



,



( )=

−1≤0

(6)

−1 ≤0

(7)

Providing reinforcement areas for each wall member (for all critical sections) should satisfy the minimum and maximum reinforcement area conditions taken from the code (8).

( )=

− 1 ≤ 0,

( )=

−1 ≤0

(8)

Due to the geometric properties of the wall and the selection of the design variables, the following geometric constraints should be satisfied (9).

( )=

+

+

( )=

− 1 ≤ 0,

−1 ≤0 (9)



+

( )=

−1 ≤0

For all wall members, reinforcement bars should satisfy the minimum development length, Ld, condition or, if the hook is used; reinforcement bars should satisfy minimum hook development length, Ldh, and the minimum hook length (12

;

= diameter of the hooked bar). The

reinforcement bars should not go outside of the retaining wall. Then the wall dimensions should be satisfied for one of these development length conditions for suitable detailing of the reinforcement. For the stem, toe, heel, and key reinforcement;

( )=

− (2 )

−1≤0

(10)

or ( )(

)=

−

−

− 1 ≤ 0 (11)

and

( )(

)=

12 −1≤0 −

where c; clear cover in concrete section. For all wall members, the clear distance, Snet, between reinforcement bars should satisfy minimum clear spacing, Smin, and maximum clear spacing, Smax, conditions;

( )=

−1 ≤0

(12)

( )=

−1 ≥0

(13)

THE FIREFLY ALGORITHM The Firefly algorithm introduced by Yang [16] is one of the most recent stochastic search techniques. This optimization algorithm is improved by adopting the idealized behavior of the

characteristic of the fireflies. This natural behavior can be defined by using the three rules described as: 

All fireflies will be attracted to other fireflies due to being unisex.



Attractiveness of all fireflies is determined to be proportionate with their brightness.



The brightness of a firefly is assumed to be related to the objective function.

The steps of the optimum design algorithm developed for the retaining walls: Step 1: n fireflies start assigning values of design variables randomly from the design variables pool. n represents the number of fireflies in the swarm. At the end of this random selection process

retaining wall designs are obtained. The total costs of these retaining wall designs are

calculated by using equation (1). Step 2: The

retaining wall designs obtained in step 1 are analyzed and designed. At the end

of this process, total constraint values and penalized costs of these retaining walls are calculated as follows: ,

( )=

( ) ∗ (1 + ( ))

where, fp,cost(i) is the penalized weight of frame generated by

(14) firefly and  is penalty

coefficient. Step 3: Original light intensities (I0) of all fireflies, which are inverse proportional to the costs of these retaining walls, are calculated by using the following equation.

=

= 1,2, … ,

(15)

where, Fmin is the minimum total cost of the retaining wall obtained in step 1. After these calculations, original light intensities and design variables are sorted in descending order of their penalized cost.

Step 4: In the last step, all fireflies in the algorithm start moving to a better location. These movements are dependent on their attractiveness and attractiveness is proportional to the light density, which also varies with distance (d). Attractiveness of each firefly is calculated by following function. ( )=

(16)

where,  is attractiveness of the firefly at the original location, is the absorption coefficient and d is the distance calculated by following function.

=

(

−

(17)

)

where, Ndv is the total number of design variables in the optimization problem, x represents design variables in the optimization problem. Then, all fireflies select new values and generate new designs by using the formula given as:

=

+ ( )∙

−

+

∙

−

1 2

(18)

= 1,2, … , = 1,2, … , = 1,2, … , where, xik value of design variable selected by the firefly i for the kth firefly,  is the randomness parameter and rand is a random number generator uniformly distributed in [0,1]. After movements of all fireflies, one cycle is completed. The algorithm then goes back to the step 2 and evaluates these new n designs that are obtained due to movements. Steps 2 and 4 are repeated until a pre-assigned maximum number of cycles is reached.

ADAPTIVE RANDOMNESS PARAMETER STAGE In classical firefly algorithm, randomness parameter  is taken as static. Selecting randomness parameter as lower value, stagnation and local convergence can be seen in the large scale optimization problems. On the contrary, selecting the randomness parameter as a higher value could result in convergence issue in the optimization problems. In order to resolve these problems, the adaptive randomness parameter strategy is improved. In this strategy, the randomness parameter () changes dynamically as expressed in following equation;

=

−(

−

)∙

− −

(19)

Equation (18) is adopted from Coello [18]. In this equation, i represents randomness parameters at cycle i, max and min represent maximum and minimum randomness parameters defined in the algorithm respectively. Iimax, Iimin and Iimean represent maximum light density, minimum light density and mean value of light densities of all fireflies at cycle i respectively. DESIGN EXAMPLES Two examples are presented to demonstrate the effectiveness and efficiency of the design algorithm developed for RC cantilever retaining walls. The retaining walls supporting a backfill soil of 3.5 m and 5.2 m heights are considered as design examples. The design based on a 1.0 m wide strip of the retaining wall and the unit costs of concrete and steel are taken as $40/m3 and $0.40/kg respectively in these examples [6, 15, 16]. The input data used are shown in Table 3. The unit weight of concrete and steel are taken as 23.5 kN/m3 and 78.5 kN/m3 respectively. The lower and upper bounds for design variables and optimum design values are given in Table 4. Shrinkage and temporary reinforcement area are computed as 0.002% of cross-sectional area of retaining wall and the length of these bars is taken as 100 cm (per meter).

Design Example 1 The retaining wall with the 3.5 m height is considered as the first example. The compressive strength of concrete and yield strength of steel are taken as 21 MPa and 400 MPa respectively in this example. This problem is optimized by using both firefly and adaptive firefly algorithms. Obtained results is compared with harmony search solution [14]. For classical firefly and adaptive firefly algorithms, the following search parameters are used: Number of fireflies = 80, attractiveness at original location = 0.5, absorption coefficient 10 and maximum iteration number is 100,000. The randomness parameter in the classical firefly algorithm is taken as static

. However, the randomness parameter in adaptive firefly algorithm changes dynamically, the boundaries of which are taken as max 0.8 and min = 0.1. The optimum costs per meter of wall are obtained as $130.80 for the classical firefly algorithm and $114.46 for the adaptive firefly algorithm. The obtained optimum dimensional, reinforcement variables are illustrated in Table 4. The factor of safeties bearing capacities and the cost details are shown in Table 5. The design history of optimum design of retaining walls with HS algorithm is shown in Figure 2. It is apparent from the table that the adaptive firefly algorithm optimum solution has the lowest cost among these optimization algorithms. The cost obtained by adaptive firefly algorithm result is 2.93% lower than the Harmony Search and 14.26% lower than the classical firefly algorithm solutions. Design Example 2 The retaining wall with 5.2 m height is considered as the second example. The compressive strength of concrete and the yield strength of steel are taken as 30 MPa and 400 MPa respectively. For classical firefly and adaptive firefly algorithms, the following search parameters are used: Number of fireflies = 100, attractiveness at original location = 0.5, absorption coefficient =10 and maximum iteration number is 100,000. Randomness parameter in classical firefly algorithm is taken as static . However, randomness parameter in

adaptive firefly algorithm changes dynamically, boundaries of which are taken as max 0.8 and min. The optimum costs per meter of wall are obtained as $180.04 for classical firefly algorithm and $174.03 for adaptive firefly algorithm. The obtained optimum dimensional, reinforcement variables are illustrated in Table 6. The factor of safeties bearing capacities and the cost details are shown in Table 7. The design history of optimum design of retaining walls with all algorithms is shown in Figure 3. It is apparent from Table 6 that the best design having the lightest weight is obtained by the adaptive firefly algorithm ($174.03). Value of this cost is 3.45% lower than the weight of the best design obtained by the harmony search algorithm and 0.75% lower than the weight of the best design obtained by the classical firefly algorithm. DISCUSSION An optimum design algorithm is developed for reinforced concrete retaining walls based on one of the recent metaheuristic techniques called the firefly algorithm. Due to a stagnation problem, algorithm is enchased and its name is changed to adaptive firefly algorithm. In the formulation of the design problem, the objective function is considered as the cost of the wall. The constraints are the design requirements that are the behavioral and side constraints defined as lower and upper limits on the design variables. Two reinforced concrete retaining wall examples are designed by the algorithm presented. Results obtained from the adaptive firefly algorithm are compared to those attained by the classical firefly and harmony search algorithms [14]. The design algorithm performs effectively in finding the optimum values of the design variables. The optimization algorithm arrives at rational and realistic design solutions that are directly constructible. The proposed optimization algorithm is mathematically less complex and easier than those that employed by traditional optimization techniques.

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TABLES and FIGURES

Table 1. Cross-sectional Design Variables and Bounds Design Variables Lower Bound Upper Bound X1 0.4H 0.8H X2 0.1H 0.6H X3 0.20 m 0.50 m X4 0.20 m 0.40 m X5 0.20 m 0.3H X6 0.5H 0.8H X7 0.20 m 0.40 m X8 0.20 m 0.90 m

Table 2. Reinforcement Design Variables # Value # Value 1 No Bar 34 0 2 35 110  3 . 112  4 . . 114 . . 78 0 . . 79 10 . . . . 13 100 0 10 14 101 10 12 . . . . . . 109  23 110  0 24 111 0 10

Table 3. Input Parameters for Design Examples Input Parameter

Unit

Height of stem Yield strength of reinforcing steel Compressive strength of concrete Concrete cover Shrinkage and temporary reinforcement percent 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 Design load factor Depth of soil in front of wall Cost of steel Cost of concrete Factor of safety for overturning stability Factor of safety for against sliding Factor of safety for bearing capacity

m MPa MPa mm

Value H fy fc cc

Ex. 1 3.50 400 21 50

Ex. 2 5.2 400 30 50

-

st

0.002

0.002

kPa degree degree degree kN/m3 kN/m3 kN/m3 kPa m $/kg $/m3 -

q   ’ s bs c c LF D Cs Cc FSo FSs FSb

10 30 36 0 17.50 18.50 23.50 65 1.7 0.50 0.40 40.00 1.5 1.5 3.0

20 5 38 0 18.50 18.50 23.50 80 1.7 0.70 0.40 40.00 1.5 1.5 3.0

Table 4. Optimum Values of Design Variables for Example 1 Optimum Values Des. Var. CFFA AFFA HS X1 2.721 2.668 2.7 X2 1.596 1.597 1.6 X3 0.355 0.348 0.35 X4 0.2 0.2 0.2 X5 0.367 0.322 0.35 X6 2.433 2.491 1.9 X7 0.242 0.2 0.2 X8 0.823 0.56 0.6 R1 812 416 1010 R2 710 710 R3 214 410 10 R4 814 912 1012 R5 1010 710 710

Table 5. Cost Details and Factor of Safeties for Example 1 CFFA AFFA HS 3 3 VolumeConc. 2.17 m 1.93 m 2.03 m3 Weightst. 110.05 kg 93.24 kg 92.20 kg CostConc. $ 86.78 $ 77.17 $ 81.10 Costst. $ 44.02 $ 37.29 $ 36.88 FSoverturning 2.89 2.785 2.8 FSsliding 1.57 1.5 1.5 FSbearing 4.87 4.91 4.77

Table 6. Optimum Values of Design Variables for Example 2 Optimum Values Des. Var. CFFA AFFA HS X1 3.258 2.906 2.85 X2 1.473 1.461 1.4 X3 0.462 0.463 0.45 X4 0.2 0.2 0.2 X5 0.348 0.342 0.35 X6 2.88 2.607 2.65 X7 0.201 0.2 0.2 X8 0.204 0.644 0.75 R1 226 516 516 R2 816 816 916 R3 - R4 914 912 714 R5 710 710 710

Table 7. Cost Details and Factor of Safeties for Example 2 CFFA AFFA HS 3 3 VolumeConc. 2.89 m 2.85 m 2.84 m3 Weightst. 160.72 kg 150.27 kg 154.62 kg CostConc. $115.75 $113.93 $113.50 Costst. $64.29 $60.10 $61.85 FSoverturning 2.61 2.01 1.98 FSsliding 1.5 1.5 1.5 FSbearing 4.57 3.79 3.58

Figure 1. Design Variables of Retaining Wall

170 HS 160 CFFA 150 Cost($)

AFFA 140 130 120 110 100 0

20000

40000

60000

80000

Iteration

Figure 2. Search Histories for Example 1

100000

220 HS 215 CFFA 210

AFFA

Cost($)

205 200 195 190 185 180 175 170 0

20000

40000

60000

80000

Iteration

Figure 3. Search Histories for Example 2

100000