Ant Colony Optimization Method for Design of Piled-Raft Foundations (DFI 2013 Student Paper Competition Winner) Hessam Yazdani, PhD Candidate, School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK, USA; (405) 325-5218;
[email protected] Kianoosh Hatami, Associate Professor, School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK, USA Elahe Khosravi, Graduate Student, School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK, USA ABSTRACT In comparison to conventional piled foundations, piled-raft foundations provide a more economical solution to support high-rise buildings constructed on compressible soils. In this type of foundation, the bearing capacity of the underlying soil is taken into account in supporting the superstructure loads, and the piles are placed to control both the total and differential movements of the superstructure. Currently, there are no universally accepted methods to design piled-raft foundations including the selection of the piles locations and dimensions. Most piled-raft foundation designs are based on empirical methods and the experience of designers. However, piled-raft foundations are massive and expensive. Therefore, developing methodologies for their optimal design could significantly help minimize their otherwise high construction costs and would make them more feasible and common practice. This paper examines the capability of the ant colony optimization (ACO) algorithm to optimize piled-raft foundations. The soil-pile interactions are taken into account by modeling the side and tip capacities of the piles using the nonlinear p-y, t-z, and Q-z springs in the OpenSees platform. The soil-raft interaction is taken into consideration using the Winkler springs beneath the raft. The objective of the optimization problem is to minimize the volume of the foundation by taking the number, configuration, and penetration depth of the piles, as well as the thickness of the raft, as design variables. The side and tip forces of the piles, the pressure applied on the underlying soil, and the total and differential movements of the foundation under the serviceability limit state are the constraints adopted for the optimization problem. Results indicate that the ACO algorithm is a suitable method for optimal design of piled-raft foundations. Findings of the study also indicate that including soil nonlinearity in the analysis (as opposed to a linear elastic soil model) can lead to a more economical design for these foundation systems.
INTRODUCTION Piled foundations are best suited for sites where a shallow foundation may incur excessive movements (settlements) and may not provide adequate bearing capacity to carry structural loads. Current design guidelines primarily require that the piles should carry the entire structural load of a piled foundation and transfer it to deeper and more competent layers (de Sanctis and Mandolini, 2006; Sales et al., 2010). However, field monitoring of several piled foundations has revealed that the raft could significantly increase the overall bearing capacity of a piled-raft foundation system (Kakurai, 2003). Consequently, designing a piled foundation merely as a pile group to meet the
required factors of safety within the framework of the allowable stress design could often lead to overly conservative, and hence costly solutions (Poulos and Davids, 2005). Piled-raft foundations are an economical alternative to the conventional piled foundations when competent soil strata exist immediately beneath the raft. In contrast to piled foundations, structural loads supported by piled-raft foundations are mostly carried by the raft (Burland et al., 1977). The piles, known as the settlement-reducing piles, are therefore located strategically to enhance the bearing capacity of the raft besides controlling or minimizing the total and differential movements that may cause distortion and
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cracking of the superstructure (Randolph, 1994; Momeni et al., 2012; Yazdani et al., 2013). Such a design approach can significantly reduce the cost of the foundation without jeopardizing the safety and performance of the superstructure (Sales et al., 2010). Optimal design of piled-raft foundations could significantly help minimize their construction costs. The optimal design of piled-raft foundations includes the selection of type, number, configuration and penetration depth of the piles in addition to the thickness of the raft in conformance with the existing design and construction standards (Gates and Scarpa, 1984; Prakoso and Kulhawy, 2001). Several factors influence the design of piled-raft foundations including the structural loads, the properties of foundation soil and the negative skin friction on piles associated with long-term settlements of a foundation underlain by soft ground (Poulos and Davis, 1980). Optimization of piled and piled-raft foundations has been the subject of a few past studies. Chow and Thevendran (1987) carried out an optimization analysis on pile groups and concluded that the central and peripheral piles of a pile group need to be designed with different degrees of rigidity in order to minimize the differential movements of a group under a flexible pile cap, or the load differentials among piles under a rigid cap. Valliappan et al. (1999) used a theoretical optimization approach together with the finite element method to analyze and optimize piled-raft foundations. They investigated two cases of uniform and non-uniform pile lengths and concluded that using non-uniform pile lengths increases the contribution of the raft in supporting the structural load. Kim et al. (2001) used recursive quadratic programming to find an optimum configuration for the piles which would minimize differential movements of piled-raft foundations. However, their approach did not fully account for the interactions among the elements of the foundations (i.e. the raft, piles and the underlying soil). Wang et al. (2002) introduced an analytical method for the optimal design of piled foundations in nonlinear soils. However, their procedure included only a limited number of design parameters. Metaheuristic algorithms are approximate but efficient algorithms that can be used to explore a search space for optimal solutions by [18] DFI JOURNAL Vol. 7 No. 2 December 2013
combining basic heuristic methods in higher level frameworks (Blum et al., 2011). This class of algorithms includes swarm intelligence (imitating the processes of decentralized, self-organized systems such as ant colony optimization - ACO and particle swarm optimization - PSO), evolutionary computation approaches (which are non-gradient, populationbased algorithms such as genetic algorithms - GAs and evolutionary programming - EP), simulated annealing (SA), and tabu search (TS), among others. The efficiency and robustness of an optimization approach depend on the problem in hand, and therefore, no globally accepted approach has been proven to best fit all engineering optimization problems (Rajeev and Krishnamoorthy, 1992). Geotechnical systems in general and foundations in particular have not so far benefitted significantly from the existing metaheuristic algorithms for optimal design (Cheng et al., 2007). Chan et al. (2009) proposed a modified genetic algorithm, which showed promise in optimizing a series of pile groups. Khajehzadeh et al. (2011) proposed a modified particle swarm optimization technique for optimal design of spread footings and retaining walls. In this paper, an ACO algorithm is developed and used to optimize piled-raft foundations. The ACO approach was selected because it retains the memory of the entire colony from all generations to approach the optimal solution, as opposed to GA-based methods in which the search information is contained only in the current generation of a GA (Camp and Bichon, 2004).
ANT COLONY OPTIMIZATION TECHNIQUE Inspired by the foraging behavior of blind animals such as ants (see Deneubourg et al., 1990), Dorigo et al. (1996) proposed a population-based metaheuristic approach, called ant colony optimization (ACO), in which a colony of artificial ants is used to construct solutions guided by pheromone intensities and heuristic information. Pheromone is a substance ants deposit on the ground when carrying food from sources to the nest. Ants use this medium to indirectly communicate information regarding the shortest paths between feeding sources and the nest and use the intensity of pheromone to evaluate the potential of marked
paths. In other words, the probability of a path to be chosen by an isolated ant moving essentially at random, searching for food, is proportional to the pheromone intensity of the path. The greater the pheromone intensity marking the path (i.e., the greater the number of ants selecting the path), the stronger the stimulus for an ant to follow that path, thus reinforcing the trail with its own pheromone and increasing the probability that subsequent ants will follow this path (Blum et al., 2011). In the ACO technique, several terms are adopted from the graph theory. Graphs are mathematical structures used to model pairwise relations between objects. The objects in a graph are represented by vertices and the pairwise relations are established using edges connecting the vertices. In ACO, the optimization problem is projected on a graph, where the shortest path determines the optimal solution of the problem. Consider a combinatorial optimization problem projected to a multigraph (a graph which is permitted to have multiple edges, also known as pseudograph), defined over a set x = {xi | i = 1, …, n} of design variables, where n is the number of design variables (vertices), and a set E = { eij | i = 1, …, n; j = 1, …, Ji} of options (potential values) for variables, where Ji is the number of options for the ith design variable (number of edges departing from the ith vertex when moving forward). A subset ψ of edges represents a solution of the problem. Since each edge has the possibility of being selected or rejected, there are 2Jn feasible solutions for each problem, where J = ¦ J i Let C = { c | i = 1, …, n; j = 1, …, i =1
ij
Ji} is the set of costs (weights) corresponding to the potential values for variables. Given this, the major steps in an ACO algorithm, shown in Fig. 1, are as follows (Maniezzo et al., 2004; Uğur and Aydin, 2009; Moeini and Afshar, 2012): Initialization The size of the colony (number of ants), m, is chosen and a proper initial intensity of pheromone, τ 0, is assigned to all options, eij. Increasing the size of the colony can improve the quality of the final results achieved by ACO. However, colony size should be chosen such that a good tradeoff between solution quality and computational effort is guaranteed (Viana et al., 2008). Initial intensity of pheromone is usually assumed to be:
[FIG. 1] ACO Algorithm
τ0 =
1 nLnn
[1]
where Lnn is the length of the tour between n cities created by the nearest neighbor heuristic (i.e., the minimum value of the optimization problem which is obtained by assigning the smallest options to n design variables - Camp and Bichon, 2004). It does not matter whether this assignment satisfies the design constraints or not (Rajasekaran and Chitra, 2009; Aydoğdu and Saka, 2012). Construction: A set of concurrent and asynchronous agents (a colony of ants) travel between vertices and construct solutions to the problem. Starting from an arbitrary or pre-selected design variable, i, ant k applies a stochastic local decision policy to select one of the Ji available options for the design variable. Known as random proportional transition rule (also known as pseudorandom proportional rule Maniezzo et al., 2004; Dorigo et al., 2006), this decision policy is governed by two parameters, namely visibility (also known as attractiveness of the move) and trail level (also known as pheromone intensity). Visibility indicates a priori desirability of a move and is an artificial sight for selecting the shortest path among the options without experience or observation. Trail level, in contrast, can be interpreted as an adaptive artificial memory, and indicates a posteriori, the desirability of the move and reflects the experience acquired by the ants at this stage. The transition rule used by ant k to select one of the options is: DFI JOURNAL Vol. 7 No. 2 December 2013 [19]
α
β
ªτ ij (t ) º¼ ª¬ηij º¼ pij (k , t ) = Ji¬ α β ¦ ª¬τ ij (t )º¼ ª¬ηij º¼
[2]
j =1
where pij(k,t) is the probability that ant k selects option j of the ith decision point, eij, at iteration t; τ ij is the trail level on option eij at iteration t; ηij is the visibility of the ant representing the local cost of choosing option j at the ith decision point ( ηij =1/cij ); and α and β are two parameters regulating the relative importance of trail level versus visibility. An iteration comes to an end when m moves are carried out by m ants, each making one move, in the time interval (t, t + 1). After each iteration, a local update rule is applied to reduce the trail level relative to the options most recently chosen for design variables (the paths chosen by ants) in order to prevent premature convergence to a suboptimal solution. When an ant travels from city i to city j, an optimization rule is applied to adjust the intensity of trail on the path connecting these two cities by:
τ ij (t ) = (1 − ij )τ ij (t ) + ijτ 0
[3]
where ij (0 ≤ ij ≤ 1) is the coefficient of decay (Uğur and Aydin, 2009), representing the persistence of the trail. Ants incrementally construct m solutions for the problem by choosing paths to travel between decision points, visiting each point once, until all points have been visited and they arrive back at their starting points. When they return to their point of origin, the ants have completed a tour, and each has constructed its own trial solution, ψk. A cycle is complete when m ants complete their tours and construct m trial solutions. Evaluation: The objective function is evaluated for the trial solutions and their fitness values (global scores) are calculated. Modification: Once m ants have completed a tour and the objective function has been evaluated for the solutions constructed, a global update rule is applied to the options selected for design variables. Typically, the global update rule has a dual function in communication among the colony agents. First, it implements a [20] DFI JOURNAL Vol. 7 No. 2 December 2013
positive feedback from the ants constructing satisfactory solutions by reinforcing the trail level on the paths they have selected. Secondly, a negative feedback mechanism reduces the trail level on the selected paths to promote exploration of the search space. The rule corresponds to the evaporation of the substance in nature and helps prevent early stagnation, an undesirable situation in which all ants repeatedly construct the same solutions making any further exploration in the search space impossible. The amount of this trail reduction is kept low to guarantee overall solution convergence. ACO algorithms are different from each other with respect to the techniques adopted to update the pheromone level and to implement the random proportional transition rule. Interested readers are referred to a summary provided by Uğur and Aydin (2009). A variation of the ACO approach, called the rankedbased ant system (RBAS - Bullnheimer et al., 1997) is used in this study. In the RBAS, only λ top ranked ants, having the best designs, are selected in the global update scheme. As positive feedback, the RBAS increases the trail + level by Δτ ij , corresponding to the solution + found by the elite ant, ψ , as: Δτ ij+ =
1 f (ψ + )
[4]
where f (ψ ) is the value of the objective + function associated with the solution ψ . The change in the trail level of the path i-j, if chosen by the ant ranked μ (1 ≤ μ ≤ λ), is given by: +
Δτ ijμ =
R f (ψ μ )
[5]
μ where f (ψ ) is the fitness value of the solution made by the ant ranked μ, and R is a quantity regulating the contribution of the top ranked ants called pheromone reward factor (Moeini and Afshar, 2012). Camp and Bichon (2004) used R = λ - μ in their study, meaning the contribution of a top ranked solution is linearly proportional to its ranking. The path i-j may be selected by more than one ant. Therefore, the total increase in the trail level of the path is given by: λ −1
Δτ ijr = ¦ Δτ ijμ μ =1
[6]
Therefore, the updated trail levels at the end of a cycle (at time t + n) are a function of the trail levels at the beginning of the cycle, the tour constructed by the elitist ant, and the tours made by the top ranked ants, and they are calculated as:
τ ij (t + n) = (1 − ρ )τ ij (t ) + λΔτ ij+ + Δτ ijr
[7]
where ρ is an adjustable parameter in the range 0 ≤ ρ ≤ 1 so that (1 – ρ) represents the evaporation rate (Camp and Bichon, 2004). The trail level update rule is followed by a feasibility analysis of the solutions constructed. An analysis is carried out for each solution. If any constraints are violated, a penalty is applied to the objective function corresponding to the solution. The value of the penalty is proportional to the extent of the violation of constraints (Camp et al., 1998). Many constrainthandling techniques for evolutionary algorithms have been proposed in the literature (e.g., static penalties, dynamic penalties, adaptive penalties). Interested readers are referred to a comprehensive survey by Coello (2002). In this study, objective functions that violate the imposed constraints are penalized using a static penalty approach as follows:
f p (ψ k ) = f (ψ k ) [1 + Φ ]
ε
[8]
k where f p (ψ ) is the penalized objective function of the kth ant, Φ is the total penalty and the summation of the force and deflection penalties (described in “Formulation of the Design Problem”), and ε is a positive penalty exponent, which remains constant in the static penalty approach and is adjusted proportionally to the extent of the violation of the constraints in adaptive penalties (Camp and Bichon, 2004; Rajasekaran and Chitra, 2009; Aydoğdu and Saka, 2012).
ANALYSIS OF PILED-RAFT FOUNDATIONS In this study, the OpenSees (Open Source for Earthquake Engineering Simulation) finite element platform was used for threedimensional analysis of piled-raft foundations. The piles were assumed to be drilled piles, where the pile capacity is provided by a combination of soil-pile friction and end bearing resistance at the pile tip (Brown et al., 2010). In practice, the geological and local soil conditions govern the depth of the piles. However, the underlying soil was assumed to be a homogenous, dry medium dense sand for simplicity. The analysis was carried out using two different assumptions of linear and nonlinear material behavior. Piles were modeled using nonlinear beam-column elements. The soil-pile interactions were modeled using the nonlinear p-y, t-z, and Q-z springs (Fig. 2). Vertical nonlinear springs (t-z) were used to describe the relationship between mobilized soil-pile shear transfer and local pile deflection at any depth. The relationship between the lateral soil resistance and pile deflection was modeled using p-y springs (Boulanger et al., 1999; API, 2000). The load-displacement response of the tip resistance of the piles was modeled using vertical Q-z springs. The beam on nonlinear Winkler foundation (BNWF) framework (Raychowdhury and Hutchinson, 2010) was adopted to model the raft resting on the soil. BNWF includes a fine mesh of independently distributed, nonlinear inelastic springs placed vertically beneath the raft to capture its rocking and settlement movements. The mechanical response of the springs was assumed as linear elastic in the linear analysis.
Termination: The ant decision mechanism, steps 2-4, continues until either a maximum number of cycles has been completed or all ants construct the same solutions. Production: The outputs of the optimization process are obtained.
[FIG. 2] Soil-Pile-Raft Interaction Model
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design were not included in the objective function because they can be determined using pertinent design guidelines after the design variables are determined numerically using the optimization algorithm.
Table 1 summarizes the soil and other input parameters used in the analysis. [TABLE 1] Soil Properties and other Input Parameters
Design Constraints Model
p-y (API, 2000) t-z (API, 2000)
Model Parameters
Value
Initial unit weight (kN/m3)
18.0
Peak friction angle, ϕp (°)
35
Void ratio, e
0.55
Initial modulus of subgrade reaction, k, (MN/m3)
43.0
The friction angle of soil-pile interface, δ (°)
20
(z/D)max(1)
0.01
(z/D)max(2)
0.1
Q-z (pile tip) (API, 2000) Q-z (raft)
Elastic modulus of soil, (Raychowdhury Es (MPa) and Hutchinson, Stiffness ratio, Rk 2010) 1
z: Local pile deflection, D: Pile diameter
2
z: Axial tip deflection, D: Pile diameter
35.0 1
FORMULATION OF THE DESIGN PROBLEM Objective Function and Design Variables In this study, an ACO approach was used to minimize the material cost of a piled-raft foundation and meet design requirements. The objective function is formulated as: np
π
Minimize V (x) = ¦ d p2 L p + BLt p =1
4
[9]
where x is a vector containing n design variables; dp and Lp are the diameter and length of the pth pile (np piles overall); and B, L and t are the breadth, length, and thickness of the raft, respectively. For ease of construction as well as to reduce the pile-pile interaction, the piles were assumed to be located on the nodes of a regular latticework with a constant spacing of three times the average piles diameter (Brown et al., 2010). The total number of piles, their configuration and penetration depth and the thickness of the raft were taken as design variables. Details of the steel reinforcement [22] DFI JOURNAL Vol. 7 No. 2 December 2013
The structural and geotechnical capacities of the piles, and the total and differential movements of the foundation under the serviceability limit state were considered as the constraints for the optimization problem in this study. Depending on their type, the ultimate structural capacities of piles in compression c and tension, Ps −u and Pst−u , are governed by the compressive strength of concrete, yield strength of steel, and the contributions of concrete, steel reinforcing bars, and steel casing in the cross-sectional area of the piles. The ultimate c capacity of piles in compression, Pg − u, is the sum of the tip and side resistances mobilized at failure. In contrast, the ultimate uplift capacity t of piles, Pg − u, is governed by the self-weight of the piles and their side resistances, neglecting the weight of the wedge of the soil around the piles accompanying them in tension (Brown et al., 2010). The allowable values of the piles vertical movement, δ v − a , and pile-head lateral displacement, δ h − a , were set to 60 mm and 25 mm (2.4 in and 1.0 in), respectively (Budhu, 2007). The allowable differential movement of the piles is expressed in terms of angular distortion of the raft (defined as the ratio of differential movement between two adjacent columns/piles to the distance between them), βa, which was set to 1/500 (Chan et al., 2009). The piles are allowed to operate at 100% of their ultimate load capacity (Randolph, 1994). However, a margin of safety was applied on the ultimate bearing capacity of the raft, σu, using a safety factor of 3 (σa = σu/3, where σa is the allowable bearing capacity of the raft). Finally, the constraints of the problem and their associated penalties as well as the constraintshandling technique used in this study are as given below: •
The structural force penalty for pile p, Φ Pp − s , is calculated as (where compressive forces are taken to be positive):
Pi − Psc−,ut t c p ° Pi < Ps −u or Pi > Ps −u Φ P − s = Psc−,ut ® ° t c p ¯ Ps −u ≤ Pi ≤ Ps −u Φ P − s = 0
[10]
•
RESULTS AND DISCUSSIONS
The geotechnical force penalty for p each pile p, Φ P − g , is calculated as:
Pi − Pgc−,tu °° Pi < Pgt −u or Pi > Pgc− u Φ Pp − g = Pgc−,tu ® ° t c p °¯ Pg − u ≤ Pi ≤ Pg − u Φ P − g = 0
[11]
Hence, the total force penalty for the solution generated by ant k is: np
[12]
Φ kP = ¦ ª¬ Φ Pp − s + Φ Pp − g º¼ p =1
•
For pile p, the vertical movement penalty, Φδpv , and the pile-head lateral displacement p penalty, Φδ h, are stated as:
p δ vp − δ v − a p °δ v > δ v − a Φδ v = δ v−a ® °δ p ≤ δ Φ p = 0 v−a δv ¯ v p δ hp − δ h − a p °δ h > δ h − a Φδ h = δ h−a ® °δ p ≤ δ Φ p = 0 h−a δh ¯ h
•
[13]
Example Problem [14]
The angular distortion penalty between piles p,q p and q, Φ β , is:
β p,q − β a p,q °β p,q > β a Φ β = βa ° ® δ vp − δ vq ° p,q ≤ Φ = = 0; β β β , , β p q a p q ° Lp , q ¯
The capability of the ACO algorithm to optimize the piled-raft foundations is demonstrated using the following example. The optimization task was carried out using a mutual communication between MATLAB® and OpenSees. The algorithm was executed in MATLAB environment and the foundations generated by ants were analyzed in OpenSees platform. OpenSees results were imported in MATLAB to evaluate the objective function, fitness value, and penalties corresponding to each solution. This information was used to direct the next ant colony towards the optimal solution. The process continued until either a maximum number of cycles was met or all ants constructed the same solutions.
[15]
Consider a high-rise building with the structural loads shown in Fig. 3. A 20 m × 20 m (66 ft × 66 ft) reinforced concrete raft has been proposed to safely transfer the building loads to the underlying sandy soil, with the properties as given in Table 1. Assuming that the groundwater level is well below the ground surface, and the embedment depth of the raft is 1 m, the ultimate net bearing capacity of the raft using Meyerhof’s equation is calculated as (Buhdu, 2007):
where Lp,q is the distance between piles p and q. Therefore, the total deflection penalty for the solution generated by ant k is: np
np
np
p =1
p =1 q =1 q≠ p
Φδk = ¦ ª¬Φδpv + Φδph º¼ + ¦¦ Φ βp , q •
[16]
For the solution generated by ant k, if the bearing pressure of the raft, σ, exceeds its allowable bearing capacity, σmax, the following penalty is applied:
σ − σ max k °σ > σ max Φσ = σ max ® ° k ¯σ ≤ σ max Φσ = 0
[17]
Therefore, the total penalty for ant k, Φ k , in Equation 8 is given by:
Φ k = Φ kP + Φδk + Φσk
[18]
A static constraints-handling technique was considered in this study in which a constant positive penalty exponent of ε = 2 was used in Equation 8.
[FIG. 3] Structural Loads on Raft
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σ u = γ D f ( N q − 1) sq d q + 0.5γ B ' Nγ sγ dγ
[19]
where Nq and Nγ are bearing capacity factors that are functions of the peak friction angle, Φp; sq and sγ are shape factors; dq and dγ are embedment depth factors; and B' is the equivalent footing breadth. After the ultimate net bearing capacity is determined, the allowable bearing capacity can be calculated as:
σ max =
σu FS
+ γ Df
[20]
[TABLE 2] Input Parameters for the ACO Algorithm
Parameter
Value
Size of the colony, m
100
Evaporation rate, ρ
0.1
Coefficient of decay, φ
0.1
α
5
β
0.2
λ
10
Therefore, [TABLE 3] Potential Values for Design Variables
σ u = 18 ×1× 32.2 ×1.7 ×1 + 0.5 ×18 × 20 × 37.1× 0.6 ×1 = 4,991kPa 16 × 20, 000 σ= = 800 kPa 20 × 20 4991 + 18 ×1 FS = 6.4 800 = FS
Parameter
The vertical elastic settlement of the raft is given by:
δv =
σ B(1 −ν 2 ) E
Is
L I s = 0.62 ln( ) + 1.12 B
[21]
where σ is the surface stress and Is is a settlement influence factor. The settlement is calculated as:
I s = 0.62ln(1) + 1.12 = 1.12
δv =
800 × 20 × (1 − 0.32 ) ×1.12 = 485mm 30,000
Therefore, the raft provides a bearing capacity adequate to carry the working loads with a large safety factor. However, the raft will experience excessive settlements from the serviceability point of view. Therefore, settlement-reducing piles are required to control the building settlements, and to reduce the bending moments in the raft.
Value
Pile diameter, dp (m)
0.8, 1.0, 1.2
Pile length, Lp (m)
20, 25, 30
Raft thickness, t (m)
1.0, 1.2, 1.4
Grid spacing (m)
3
Fig. 4 shows the convergence history of the foundation design represented by the raft volume. The optimization process is initiated with a design generated by randomly chosen values for the design variables and evolves to an optimal design. Results in Fig. 4 indicate that the ACO algorithm yields an optimum solution in approximately 25 cycles using linear analysis and 31 cycles using nonlinear analysis. However, including the soil nonlinearity in the analysis results in a 5% more economical design based on the volume of the foundation raft. The 5% reduction in the volume of the raft could prove significant, or even critical, in cases where physical obstacles are present for the construction of the foundation.
The ACO algorithm is used to determine the optimal design of the foundation. The values used for the algorithm parameters are summarized in Table 2. A set of candidate values for the design variables are given in Table 3.
[FIG. 4] Foundation Design’s Convergence History
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The influence of soil nonlinearity on the optimum configuration of piles is depicted in Fig. 5. It can be observed that assuming a linear elastic behavior for the soil increases the load concentration of the peripheral piles and results in an overdesigned foundation system. In contrast, taking soil nonlinearity in the analysis reduces the share of the load carried by the peripheral piles and results in a redistribution of loads among the load carrying elements. As a result, it helps achieve a more favorable design in which all piles carry comparable loads as the load applied on the foundation increases. Results shown in Fig. 5 are in conformity with those reported in the literature (e.g., Basile, 2003).
CONCLUSIONS The capability of the ACO algorithm for optimizing piled-raft foundations was investigated. Soil-pile-raft interactions were included in the analysis using nonlinear p-y, t-z and Q-z Winkler springs in the OpenSees platform. The analysis of a piled-raft foundation system indicated that including soil nonlinearity in the analysis results in a more uniform distribution of predicted loads among the piles and hence a more economical design. Results shown in this paper illustrated that the ACO algorithm developed in this study is capable of finding an optimal solution for piled-raft foundation systems. Further work is underway to compare the performance of the ACO algorithm with other metaheuristic algorithms in order to identify faster and more effective methods for optimal design of foundations.
REFERENCES 1. API, (2000) “Recommended practice for planning, designing and constructing fixed offshore platforms—working stress design”, API Recommended Practices, American Petroleum Institute. 2. Aydoğdu, İ. and Saka, M. P., (2012) “Ant colony optimization of irregular steel frames including elemental warping effect”, Advances in Engineering Software, Vol. 44, No. 1, pp. 150–169. 3. Basile, F., (2003) “Analysis and design of pile groups”, in Numerical Analysis and Modelling in Geomechanics, J. W. Bull, Editor, Spon Press (Taylor & Francis Group Ltd), Oxford, pp. 278–315. 4. Blum, C., Puchinger, J., Raidl, G. R. and Roli, A., (2011) “Hybrid metaheuristics in combinatorial optimization: A survey”, Applied Soft Computing, Vol. 11, No. 6, pp. 4135–4151. 5. Boulanger, R., Curras, C., Kutter, B., Wilson, D. and Abghari, A., (1999) “Seismic soilpile-structure interaction experiments and analyses”, ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 125, No. 9, pp. 750–759.
[FIG. 5] Piles Configuration in Optimal Solutions; a) linear analysis, b) nonlinear analysis
6. Brown, D. A., Turner, J. P. and Castelli, R. J., (2010) “Drilled shafts: construction procedures and LRFD design methods”, National Highway Institute, U.S. Department of Transportation, Federal Highway Administration, Washington DC. DFI JOURNAL Vol. 7 No. 2 December 2013 [25]
7. Budhu, M., (2007) Soil Mechanics and Foundations. Wiley, Hoboken, NJ. 8. Bullnheimer, B., Hartl, R. F. and Strauß, C., (1997) “A new rank based version of the ant system - a computational study”, Central European Journal for Operations Research and Economics, Vol. 7, pp. 25–38. 9. Burland, J.B., Broms,, B.B. and de Mello, V.F.B. (1977) "Behaviour of foundations and structures, State of the Art Review, Proceedings IX ICSMFE, Tokyo, 2: pp. 495546, Rotterdam, Balkema
18. Deneubourg, J.-L., Aron, S., Goss, S. and Pasteels, J. M., (1990) “The self-organizing exploratory pattern of the argentine ant”, Journal of Insect Behavior, Vol. 3, No. 2, pp. 159–168. 19. Dorigo, M., Birattari, M. and Stutzle, T., (2006) “Ant colony optimization”, IEEE Computational Intelligence Magazine, Vol. 1, No. 4, pp. 28–39.
10. Burland, J. B., (1990) “On the compressibility and shear strength of natural clays”, Géotechnique, Vol. 40, No. 3, pp. 329–378.
20. Dorigo, M., Maniezzo, V. and Colorni, A., (1996) “Ant system: optimization by a colony of cooperating agents”, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol. 26, No. 1, pp. 29–41.
11. Camp, C. V. and Bichon, B. J., (2004) “Design of space trusses using ant colony optimization”, Journal of Structural Engineering, Vol. 130, No. 5, pp. 741–751.
21. Gates, M. and Scarpa, A., (1984) “Optimum penetration of friction piles”. Journal of Construction Engineering and Management, Vol. 110, No. 4, pp. 491–510.
12. Camp, C., Pezeshk, S. and Cao, G. (1998) “Optimized design of two-dimensional structures using a genetic algorithm”, Journal of Structural Engineering, Vol. 124, No. 5, pp. 551–559.
22. Kakurai, M., (2003) “Study on vertical load transfer of piles”, Ph.D. Thesis, Tokyo Institute of Technology, (in Japanese).
13. Chan, C., Zhang, L. and Ng, J., (2009) “Optimization of pile groups using hybrid genetic algorithms”, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 4, pp. 497–505. 14. Cheng, Y. M., Li, L. and Chi, S. C., (2007) “Performance studies on six heuristic global optimization methods in the location of critical slip surface”, Computers and Geotechnics, Vol. 34, No. 6, pp. 462–484. 15. Chow, Y. K. and Thevendran, V., (1987) “Optimisation of pile groups”, Computers and Geotechnics, Vol. 4, No. 1, pp. 43–58. 16. Coello, C. A., (2002) “Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art”, Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 11–12, pp. 1245–1287. 17. De Sanctis, L. and Mandolini, A., (2006) “Bearing capacity of piled rafts on soft clay soils”, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 132, No. 12, pp. 1600–1610.
[26] DFI JOURNAL Vol. 7 No. 2 December 2013
23. Khajehzadeh, M., Taha, M. R., El-Shafie, A. and Eslami, M., (2011) “Modified particle swarm optimization for optimum design of spread footing and retaining wall”, Journal of Zhejiang University SCIENCE A, Vol. 12, No. 6, pp. 415–427. 24. Kim, K. N., Lee, S.-H., Kim, K.-S., Chung, C.-K., Kim, M. M. and Lee, H. S., (2001) “Optimal pile arrangement for minimizing differential settlements in piled raft foundations”, Computers and Geotechnics, Vol. 28, No. 4, pp. 235–253. 25. Maniezzo, V., Gambardella, L. M. and Luigi, F. de., (2004) “Ant colony optimization”, New Optimization Techniques in Engineering, Studies in Fuzziness and Soft Computing, Springer, Heidelberg, Berlin, pp. 101–121. 26. Moeini, R. and Afshar, M. H., (2012) “Layout and size optimization of sanitary sewer network using intelligent ants”, Advances in Engineering Software, Vol. 51, pp. 49–62.
27. Momeni, M., Yazdani, H., Fakharian, K., Shafiee, A., Salajegheh, J. and Salajegheh, E., (2012) “Case study of a micropiled raft foundation design in soft calcareous sandy soil, Kerman–Iran.”, The 4th International Conference on Geotechnical and Geophysical Site Characterization, Porto de Galinhas, Pernambuco, Brazil, pp. 1063-1068. 28. Poulos, H. G. and Davids, A. J., (2005) “Foundation design for the Emirates Twin Towers, Dubai”, Canadian Geotechnical Journal, Vol. 42, No. 3, 716–730. 29. Poulos, H. G. and Davis, E. H., (1980) Pile Foundation Analysis and Design. John Wiley & Sons, Australia. 30. Prakoso, W. and Kulhawy, F., (2001) “Contribution to piled raft foundation design”, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 127, No. 1, pp. 17–24. 31. Rajasekaran, S. and Chitra, J. S., (2009) “Ant colony optimisation of spatial steel structures under static and earthquake loading”, Civil Engineering and Environmental Systems, Vol. 26, No. 4, pp. 339–354. 32. Rajeev, S. and Krishnamoorthy, C. S., (1992) “Discrete optimization of structures using genetic algorithms.” Journal of Structural Engineering, Vol. 118, No. 5, pp. 1233–1250. 33. Randolph, M. F., (1994) “Design methods for pile groups and piled rafts”, XIII fCSMFE, Rotterdam: Balkema, New Delhi, pp. 61–82. 34. Raychowdhury, P. and Hutchinson, T. C., (2010) “Sensitivity of shallow foundation response to model input parameters.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 136, No. 3, pp. 538–541.
35. Sales, M. M., Small, J. C. and Poulos, H. G., (2010) “Compensated piled rafts in clayey soils: behaviour, measurements, and predictions”, Canadian Geotechnical Journal, Vol. 47, No. 3, pp. 327–345. 36. Uğur, A. and Aydin, D., (2009) “An interactive simulation and analysis software for solving TSP using ant colony optimization algorithms”, Advances in Engineering Software, Vol. 40, No. 5, pp. 341–349. 37. Valliappan, S., Tandjiria, V. and Khalili, N., (1999) “Design of raft–pile foundation using combined optimization and finite element approach”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 23, No. 10, pp. 1043– 1065. 38. Viana, F. A. C., Kotinda, G. I., Rade, D. A. and Steffen Jr., V., (2008) “Tuning dynamic vibration absorbers by using ant colony optimization”, Computers & Structures, Vol. 86, No. 13–14, pp. 1539–1549. 39. Wang, S.-T., Reese, L. C. and Farmer, G., (2002) “Optimum design of threedimensional pile groups in nonlinear soil.” Deep Foundations 2002, M. W. O’Neill and F. C. Townsend, Eds., American Society of Civil Engineers, pp. 245–261. 40. Yazdani, H., Momeni, M. and Hatami, K., (2013) “Micropiled-raft foundations for high-rise buildings on soft soils - a case study: Kerman, Iran”, The 7th International Conference on Case Histories in Geotechnical Engineering, Chicago, US, Paper No. 2.40
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