optimization of pile groups using hybrid genetic algorithms

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Optimization of Pile Groups Using HybridGenetic Algorithms

C. M. Chan, M.ASCE1; L. M. Zhang, M.ASCE2; and Jenny T. M. Ng3

Abstract: This paper presents an automated optimal design method using a hybrid genetic algorithm for pile group foundation design.The design process is a sizing and topology optimization for pile foundations. The objective is to minimize the material volume of thefoundation taking the configuration, number, and cross-sectional dimensions of the piles as well as the thickness of the pile cap as designvariables. A local search operator by the fully stressed design �FSD� approach is incorporated into a genetic algorithm �GA� to tackle twomajor shortcomings of a GA, namely, large computation effort in searching the optimum design and poor local search capability. Theeffectiveness and capability of the proposed algorithm are first illustrated by a five by five pile group subjected to different loadingconditions. The proposed optimization algorithm is then applied to a large-scale foundation project to demonstrate the practicality of thealgorithm. The proposed hybrid genetic algorithm successfully minimizes the volume of material consumption and the result matches theengineering expectation. The FSD operator has great improvement on both design quality and convergence rate. Challenges encounteredin the application of optimization techniques to design of pile groups consisting of hundreds of piles are discussed.

DOI: 10.1061/�ASCE�1090-0241�2009�135:4�497�

CE Database subject headings: Pile foundations; Pile caps; Pile groups; Algorithms; Optimization; Limit states.

Introduction

Foundations are crucial to the safety and serviceability of sup-ported structures such as bridges and buildings. Any potentiallyunsafe foundation designs could spell disaster for human livesand economic losses. Unlike many structures of which the behav-ior is well known, foundation design involves much uncertaintyin soil properties and soil-pile interactions. Due to limited designtime and budget, engineers generally tend to provide unnecessar-ily conservative designs.

Much progress in numerical optimization has been made inrecent years and such an optimization approach has been widelyused in many civil engineering applications such as transportationand structural design of bridges and buildings. Very few attempts,however, have been made in developing an effective optimizationmethodology for foundation design. Truman and Hoback �1992�optimized steel pile groups using the optimality criteria �OC� ap-proach. Due to the limitation of the gradient-based method em-ployed, only the cross-sectional size and the orientation of thepiles were considered as design variables. Hurd and Truman

1Associate Professor, Dept. of Civil and Environmental Engineering,Hong Kong Univ. of Science and Technology, Clear Water Bay, HongKong, People’s Republic of China. E-mail: [email protected]

2Associate Professor, Dept. of Civil and Environmental Engineering,Hong Kong Univ. of Science and Technology, Clear Water Bay, HongKong, People’s Republic of China. E-mail: [email protected]

3Formerly, Research Assistant, Dept. of Civil and Environmental En-gineering, Hong Kong Univ. of Science and Technology, Clear WaterBay, Hong Kong, People’s Republic of China.

Note. Discussion open until September 1, 2009. Separate discussionsmust be submitted for individual papers. The manuscript for this paperwas submitted for review and possible publication on June 5, 2007; ap-proved on June 18, 2008. This paper is part of the Journal of Geotech-nical and Geoenvironmental Engineering, Vol. 135, No. 4, April 1,
2009. ©ASCE, ISSN 1090-0241/2009/4-497–505/$25.00.
JOURNAL OF GEOTECHNICAL AND

J. Geotech. Geoenviron. Eng

�2006� incorporated a “weightless optimality criterion” into theoriginal OC approach. This criterion was to handle design vari-ables that do not have a measurable effect on the objective func-tion, such as spacing and batter of piles. In these studies, theauthors also showed that, for problems with locally optimumpoints, the OC method does not guarantee that the final designrepresents the global optimum. Huang and Hinduja �1986� trans-formed a shape optimization problem for pile foundations to anunconstrained one and applied a quasi-Newton method to opti-mize the shape of a pile foundation. In order to determine thederivatives of the constraints and the objective function with re-spect to the design variables, a linear elastic force-deflection re-lationship was applied to the pile-soil system. Kim et al. �2002�studied the optimal location of piles in a piled raft foundationusing genetic algorithms �GAs� assuming linear elastic pile-soilinteraction.

Most previous research studies on numerical foundation de-sign optimization were based on traditional gradient methods�Truman and Hoback 1992�, which are restricted to design prob-lems that must be mathematically formulated and differentiable.As a result, the traditional design techniques are limited to prob-lems with continuous design variables and are not applicable todesign problems with discrete design variables. In this paper, anautomated optimal design method using a hybrid genetic algo-rithm is presented for pile group foundation design. A local searchoperator is incorporated into a conventional GA to enhance thelocal search capability of the algorithm. GAs are particularly suit-able for pile group foundation optimization, in which the designvariables are often discrete in nature and the relationships be-tween the objective function and design constraints cannot beeasily expressed mathematically in terms of design variables. Theeffectiveness and capability of the optimization algorithm are firstillustrated by a five by five pile group under combined gravityloads and wind-induced loads. The algorithm is then applied to
the design of a practical large scale foundation.
GEOENVIRONMENTAL ENGINEERING © ASCE / APRIL 2009 / 497

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Design Problem Formulation

Objective Function and Design Variables

In this study, an automated optimization algorithm is developed tominimize the total concrete volume V of a pile foundation sub-jected to various design considerations. The objective functioncan be expressed as

minimize V = �j=1

M�

4dj

2Lj + BWt �1�

where dj and Lj =diameter and length of the jth pile; and B, W,and t=breadth, width, and thickness of the pile cap, respectively.This study involves bored piles. The pile founding depths dependon geological conditions �Buildings Department 2004�. Thebreadth and width of the pile cap are preset according to thesuperstructure requirements. The pile spacing is usually in a rangeof 2.5–5 pile diam to satisfy constructability and structural re-quirements �O’Neill and Reese 1999�. Thus, the design variablesare the location, number, and diameter of the piles, as well as thethickness of the pile cap. In order not to overcomplicate the de-sign optimization problem, the details of steel reinforcement havenot been included as part of the design objective. Once the num-ber of piles and the element sizes of the pile group are estab-lished, the steel reinforcement details can then be determinedseparately according to code specified strength design criteria.

Design Constraints

Based on the design guidelines for large diameter bored pilegroups founded on rock in Hong Kong �Buildings Department2004� and the general design practice �Poulos et al. 2001; Geo-technical Engineering Office 2006�, several design constraints areimposed on the optimization formulation:1. The maximum end bearing pressure of pile is 5 MPa, assum-

ing that the bedrock is grade III rock;2. The maximum vertical settlement of the piles is 75 mm and

the maximum pile-head lateral displacement is 25 mm;3. The maximum differential settlement of the piles is limited to

a threshold value of 1 /500 for the angular distortion of thepile cap;

4. The structural capacity of piles is considered using biaxialinteraction, in which the pile axial force and moments areconsidered; and

5. The internal moment and shear in the pile cap are also lim-ited to be within the capacity of the pile cap.

Genetic Algorithms

Genetic algorithms are search algorithms that model the mechan-ics of natural selection and natural genetics �Goldberg 1989�. Thisstochastic optimization algorithm selects fitter individuals amonga population based on the principle of “survival of the fittest.”After generations of reproducing and selecting better individuals,the search is guided based on probabilistic rules toward a regionof the search space with likely improvement.

When compared with conventional gradient-based optimiza-tion methods, GAs operate differently in several fundamentalways and are regarded as more superior in two aspects as follows:1. Like a zero-order method, GAs require only function values

in the search space and, thus, the continuity or differentiabil-

498 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINE


ity of problem functions is neither required nor used in theoperation of the algorithms. Therefore, GAs have globalsearching capability and are applicable to general problemswith or without discrete design variables; and

2. With the gradient-free capability, GAs do not require a math-ematical formulation of both the objective function and thedesign constraints to be explicitly expressed in terms of de-sign variables.

One of the major drawbacks of GAs is that they are computa-tionally expensive, particularly for optimization of practical foun-dation designs that require nonlinear analysis computations anduse a relatively large number of piles. A genetic algorithm nor-mally comprises three parts: coding the design variables intostrings that represent the problem, evaluating the fitness of eachsolution string, and applying genetic operators to produce the nextgeneration of solution strings. In this paper, binary coded stringsare used to represent the pile group designs, and the simple ge-netic algorithm �SGA� proposed by Goldberg �1989� is adopted.SGA consists of three basic genetic operators, namely, reproduc-tion, crossover, and mutation.

Reproduction or Selection

In the reproduction process, individuals with higher fitness valuehave a higher probability of being selected and producing off-spring in the next generation. Among various types of selectionschemes, tournament selection is employed as it is less affectedby the definition of the fitness function. The tournament size is setas two as it often provides sufficient selection intensity on fitterindividuals �Goldberg 1989�. In a tournament selection, two indi-viduals are randomly chosen and the one with higher fitnessvalue will be placed in the mating pool for the use of offspringproduction.

Crossover

Crossover is a major genetic operator that produces new designsin the optimization process. Two individual strings �parents� cho-sen from the mating pool are combined to form a new design inthe search space. In this paper, a modified uniform crossover op-erator is introduced. Similar to a typical uniform crossover, acrossover mask is randomly created, but with a string length equalto the number of piles plus one, involving both the piles and thepile cap. The crossover will take place for pile i if the bit in thecrossover mask is 1 at position i. If two bits are used to representthe properties of a pile, the two bits will be swapped togetherduring the crossover operation �Fig. 1�. The modified crossoverscheme enhances the physical meaning of swapping the string bits

Fig. 1. Graphic representation of the modified uniform crossover

by in fact swapping piles with piles.

ERING © ASCE / APRIL 2009

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Mutation

During the reproduction and crossover operations, some poten-tially useful genetic materials may be lost. A mutation operator,which simply changes from 1 to 0 or vice versa, is an insurancepolicy to protect against the irrecoverable loss of important no-tions. It also explores solution space wherein designs are not ini-tially generated nor produced by a crossover operator.

Penalty and Fitness Functions

In GAs, the constrained optimization problem needs to be trans-formed to an unconstrained one by using penalty functions. Manydifferent forms of penalty functions were proposed in the past�Coello 2002�; however, no single function has been identified thebest. A general guideline on choosing a penalty function is thatthe penalty on infeasible solutions should not be too large suchthat the design process converges prematurely without a sufficientexploration of the search space, nor is it too small to have a veryslow convergence, which results in unnecessarily excessive com-putational efforts.

The penalized objective function of design i is as follows:

Vi� = Vi�1 + �i� �2�

where Vi’ and Vi=penalized volume and the actual volume of

material of a foundation system; �i=penalty function added tothe design objective function whenever a violation is found in anystress and displacement constraint.

The penalty function used in this study is as follows:

�i = �j=1

M

�i,j� + �

k=1

N

�i,k� �3�

where Fi,j� and Fi,k

� =penalty values for stress constraint j anddisplacement constraint k for design i, respectively. Stress con-straints are checked for j=1,2 , . . . ,M members and displacementconstraints are checked for k=1,2 , . . . ,N nodes.

The penalty values for the stress and displacement constraintsare defined, respectively, using bilinear functions as

�i,j� = �0 if ri,j � 1

k1ri,j if ri,j � 1� and �i,k

� = �0 if ri,k � 1

k2ri,k if ri,k � 1��4�

where k1 and k2=weighting factors that measure the importanceof the corresponding type of constraints; and r=normalized stressor displacement with respect to its corresponding limit

ri,j = ��i,j�/� j,allow and ri,k = ��i,k�/�k,allow �5�

In GAs, reproduction is performed based on a fitness valueassigned to each individual. Individuals with a higher fitnessvalue have a higher probability of being selected to survive in thenext generation. As in an optimal design problem with an objec-tive function of minimizing the volume of material consumption,the larger the volume, the less favorable is the design. Therefore,we define the individual fitness of a design as the reciprocal of itsvolume

Fi =Vmax�

Vi��6�

where Vmax� =maximum penalized volume in the generation.



Hybrid Genetic Algorithm

Local Search OperatorVarious optimization algorithms can be used to perform the localsearch. The local search operator can be generally classified intotwo types: a rigorously derived gradient-based method or an in-tuitive heuristic nongradient-based method. In this study, a localsearch operator based on an intuitive optimality criteria method isincorporated into the conventional genetic algorithm.

Optimality Criteria MethodsOptimality criteria �OC� methods are indirect methods that do notdirectly minimize or maximize the objective function; instead,they are required to satisfy a set of necessary conditions for theoptimal design. Basically, OC methods first derive a set of neces-sary conditions that the optimal design must satisfy; and thenapply a recursive algorithm to resize members for satisfying thenecessary optimality conditions.

The optimality criteria can be derived either intuitively or rig-orously. Although the rigorously derived OC method that satisfiesthe Karush-Kuhn-Tucker �KKT� necessary conditions is very ef-ficient to handle large-scale structural optimization �Chan 2001,2004�, it is not applicable to the pile group designs due to the lackof gradient information. Therefore, an intuitive OC method calledfully stressed design �FSD� is employed as the local search op-erator in this study.

Fully Stressed Design MethodThe FSD optimality criterion is intuitive in nature. It can be statedas follows: for the optimal design, each member of the structurethat is not at its minimum size limit is fully stressed under at leastone of the design load conditions. By stating this, it is suggestedto remove material from members that are not fully stressed un-less they are at their minimum size limit. However, there is animplicit assumption that adding or removing material affects onlythe stresses in that member. In indeterminate structures, forcedistribution is dependent on member sizes. Therefore, the fullystressed design technique involves the iterative application ofanalysis and member-resizing algorithm for indeterminate struc-tures. Moreover, such fully stressed design optimality criterion isonly approximate for indeterminate structures because the mini-mum weight design may not be necessarily fully stressed. In mostcases, a fully stressed design is close if not the same as the opti-mum point �Haftka and Gurdal 1992�.

The most common redesign rule of FSD is based on a stress-ratio resizing technique

Xinew = Xi

old ·�i

old

�iu �7�

in which the new member size Xinew is in terms of the current size

Xiold, the current stress �i

old, and the allowable stress �iu.

Most of the pile group foundations are statically indeterminate.As Eq. �7� does not consider the force redistribution due tochanges in member sizes, an iterative reanalysis-redesign proce-dure is needed �Fig. 2�.

Hybridization of Genetic Algorithm and Fully StressedDesign Method

Strategy of GA-FSD MethodIn the proposed hybrid GA method, a local search operator using
the fully stressed design �FSD� method is incorporated into a

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simple genetic algorithm. The genetic algorithm randomly gener-ates an initial population and then produces new designs by cross-over and mutation. The location and number of piles, whichdefine the topology of a pile group, pile size, and pile cap thick-ness are randomly changed by the genetic operators. The FSDoperator is applied to some randomly selected designs with fixedtopology to improve the size of the piles. The improved designthen replaces the original child design for the next generation.

The optimal topology of piles can be well explored by thegenetic operators because GA is good at global search. However,a large computational effort is generally required for determiningthe optimal size of individual piles under a fixed topology whenusing pure GA. On the other hand, the FSD operator exploits anefficient local search algorithm to modify the size of individualpiles under a fixed topology. Therefore, the sizing optimizationprimarily relies on the FSD operator whereas the topological op-timization relies on the GA.

Rate of FSD OperatorAlthough the local search operator greatly enhances the perfor-mance of the optimization algorithm, the FSD operator cannot beapplied to every design because this requires additional computa-tional effort and becomes impractical for large-scale projects.Similar to crossover and mutation, a parameter pFSD is used todefine the probability that a child design will undertake the FSDoperator. This FSD rate also controls the extent of exploitation ofefficient member resizing. The range of FSD rate for pile groupdesigns will be discussed in later sections.

Overall Design Procedure

The overall design procedure of the hybrid GA is presentedgraphically in the flowchart as shown in Fig. 3. The design pro-cedure using hybrid GA is very much similar to a conventionalGA. The only difference is the application of FSD operator afterthe mutation operator. When a child design is produced by cross-over and mutation, a random number is generated to determine ifthe child design undertakes the FSD operator. If the random num-ber is smaller than the FSD rate, finite-element analysis of thechild design will be performed to obtain the pile internal forcesfor member resizing. The improved design by the FSD operatorwill then replace the original child design and perform fitness

Fig. 2. Iterative reanalysis-redesign procedure for fully stressed de-sign method

evaluation with other new designs in the next generation.



Illustrative Examples

The proposed hybrid genetic algorithm has been applied to thedesign of a number of pile group foundations founded on rock inaccordance with the Hong Kong Code of Practice for Foundations�Buildings Department 2004�. Two sample applications are pre-sented in this paper. The effectiveness and capability of the opti-mization algorithm are first illustrated by a simple five by five pilegroup under combined gravity and wind loading conditions.While the optimization results match the engineering expectation,the hybrid GA is applied to a large scale foundation project todemonstrate the practicality of the algorithm.

The following example is to illustrate the capability of thealgorithm in searching for the optimal foundation design in termsof its total volume, by varying the location, number, and diameterof the piles, as well as the pile cap thickness. A 40 m by 40 m pilecap is modeled as a 5�5 grid of 10 m spacing �Fig. 4�. At eachnode, the algorithm is to determine whether a pile exists or notand what diameter it should be. The choices of pile diameters are2.0, 2.5 and 3.0 m. All the piles have the same length of 40 m.The thickness of the pile cap will also be determined by thealgorithm from the range of 1 to 4.5 m.

The ground consists of two layers, namely, a soil layer and thesound bedrock. The piles are founded on the bedrock. Accordingto the Hong Kong Code of Practice, the shaft resistance of suchlarge diameter, rock socketed piles can be ignored and the verticalloading is resisted solely by the end bearing of the bedrock. The

Fig. 3. Design optimization flowchart of hybrid genetic algorithms

Fig. 4. Plan view of the pile group and the pile cap


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toe resistance-displacement relationship is assumed to be elastic-perfectly plastic. The allowable unit toe resistance is assumed tobe 5 MPa for rocks of weathering grade III or better �BuildingsDepartment 2004� and the toe stiffness is based on a rock massmodulus of 2,875 MPa. The axial group effect can be ignoredwhen the pile capacity is derived from end bearing followingBuildings Department �2004�. Note that the ignorance of axialshaft resistance in soil and group effect in this optimization ex-ample is solely intended to reflect the design practice in HongKong. Axial shaft resistance and group effect are considered inthe practical design example that will be described later.

The lateral or moment load is resisted by the lateral resistanceof the soil layer. The nonlinear subgrade reaction-displacement�p-y� curve defined by Reese et al. �1974� is adopted to describethe lateral soil resistance of the soil ground, and the p-multipliersproposed by McVay et al. �1998� are used to represent the lateralpile-group interactions. The properties of the soil and rock, andthe model parameters are shown in Table 1. The pile and pile-capconcrete is assumed to be linear elastic, with its unit weight,compressive strength, and Young’s modulus being 25 kN /m3,30 N /mm2, and 26 GPa, respectively.

Typical gravity load and single x-direction wind induced shearand moment of a 40-story building are applied to the pile group.For this simple example, the intention is to illustrate the capabilityof the hybrid OC-GA method to produce a reasonable asymmetricpile configuration under combined gravity and single x-directionwind loading condition. The building is assumed to be a four-bayrigid frame structure and the loads are applied at the grid nodes ofthe pile cap. A combination of both gravity and wind loads isconsidered. A downward load of 24.6 MN due to the gravity load-ing of a 40-story building is applied to each grid node of the pilecap. Apart from the wind induced shear load of 0.8 MN on eachnode in the x-direction, the resultant downward loads due to grav-ity and wind induced moment are shown in Table 2. A finite-element analysis program, FB-Pier �BSI 2003�, is used to performanalyses on the pile groups taking into due consideration the non-linear behavior of the soil-structure interaction.

The GA parameters are set at a population size of 50, a cross-over rate of 0.5, and a mutation rate of 0.01. Using two bits torepresent the size of a pile and three bits for the thickness of thepile cap, the string length is 53. The optimization process is al-lowed to continue for 50 generations.

For this example, the end bearing pressure and the structuralcapacity of piles and the pile cap are categorized as stress con-straints. As any failure of a pile or the pile cap can cause a seriousconsequence, each pile is checked against violation in structural

Table 1. Summary of Soil Profile and Soil Properties

LayerMaterial

type

Level�m� Unit

weight�kN /m3Top Bottom

1 Sand 0 −39.9 18

2 Rock −39.9 −55.0 26

Table 2. Combined Gravity and Wind Induced Loads on Piles

Node numbera 1–5 6–10 11–15 16–20 21–25

Load on node �MN� 14.9 19.7 24.6 29.4 32.3a
Refer to Fig. 4 for node numbers.


capacity and end bearing pressure of rock. The maximum shearand moment in the pile cap are also examined. In other words,there are totally 52 stress constraints for the pile group of 25 piles.The displacement constraints include the total settlement, the lat-eral displacement, and the differential settlement. Only the maxi-mum values of these three displacements found out of all nodesare considered in the design optimization.

In order to properly reflect the relative importance of both thepile stress and displacement constraints, the weighting factors ofstress constraints k1 and displacement constraints k2 are testedbefore the optimization process. This is achieved by examiningthe stress and displacement penalty values for some randomlygenerated designs. For this example, the weighting factors k1 andk2 are chosen as 0.5 and 1.0, respectively.

Result by Simple GA

As shown in Fig. 5�a�, piles with larger diameters are placed atthe right-hand side of the pile group where the loads are larger.This arrangement can be due to two factors: larger wind-induceddownward loads at the right-hand side of the pile group, and thelarge lateral soil resistance in the front rows of piles due to groupeffect. It is believed the latter is not dominant because the appliedlateral wind load is relatively small as compared to the gravityload so that no lateral displacement constraint is found critical.The wind induced shear is found to have little influence on thepile location and sizes.

On the other hand, the algorithm suggests reducing the pilecap thickness to its minimum value of 1 m. By distributing con-struction material to the most favorable location where the loadsare large, the need for load redistribution through the pile cap canthen be lessened.

As the problem is so highly constrained that any removal ofpiles may lead to an infeasible design, no feasible design is founduntil the eighteenth generation �Fig. 5�b��. After the first feasibledesign is obtained, the algorithm converges to the final designwith a further reduction of 18% in the total volume of material.Besides, as reflected from the final configuration and design his-tory, the algorithm could not fine tune the 3 m diam pile on thefourth row of the group even after 30 generations.

Result by Hybrid GA

The result obtained by the hybrid GA is very much similar to theone obtained by pure GA; piles with larger diameters are placedat the right-hand side of the pile group �Fig. 6�a��. The only dif-ference is that the 3 m diam pile in the fourth row is reduced to a2.5 m diam pile and the 2.5 m diam pile in the fifth row is re-duced to a 2 m diam pile. The hybrid GA also obtained the firstfeasible solution at an earlier generation than the pure GA. The

Unconfinedcompressive

strength�MPa�

Frictionangle�deg�

Constant ofmodulus of

subgradereaction

�MN /m3�

Massmodulus�MPa�

— 35 20 —

25 — — 2,875

�

first feasible solution was found at the tenth generation �Fig.


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6�b��; and the final design was further reduced by 33% in the totalvolume of material, and the pile cap was reduced from3 m to 1 m in thickness by the GA operators.

In this design example, the local search operator managed toreduce pile sizes by exploiting useful design information from thetopological search. The design was fine tuned to the one closer tothe minimum point and the material consumption of piles wasfurther reduced. The final design obtained by the hybrid GA alsoappeared more reasonable and favorable in terms of cost. It isobserved that the penalized material volume remained high in thefirst few generations for both the simple GA and hybrid GA. Asthe design examples are highly constrained, many pile group to-pologies are deemed infeasible no matter what the pile sizes are.Therefore, the topological search, which relies on GA, took quitea few generations to reduce the number of designs with deadpenalty assigned.

The hybrid GA showed faster convergence. As the majordrawback of genetic algorithm is the large computational effort,the local search operator greatly enhances the performance of the

Fig. 5. Optimization resu

Fig. 6. Optimization resu



search algorithm. The sizing optimization mainly relies on theFSD operator, which can efficiently bring a design to its closestlocal minimum point. Therefore, much time is saved to locate theoptimum solution when compared with the simple GA. The firstfeasible solution was found at the tenth and the eighteenth gen-erations by the hybrid GA and the pure GA, respectively, indicat-ing a reduction of approximately 44% in the computation effortby the hybrid GA.

Practical Design Example

A pile group foundation originally designed for an office buildingin Beijing as shown in Fig. 7 is now considered to demonstratethe feasibility of the proposed optimization method for large-scalepractical foundation design. All structural loads acting on thebuildings are transferred through the columns and walls down tothe piles. Among all different load combinations, only the domi-

pile group by simple GA

pile group by hybrid GA

lts of

lts of


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nating gravity load alone was considered in this study.The original design layout of the pile group foundation is

shown in Fig. 7. All the piles are of the same diameter of 1.2 m.Piles are arranged in a triangular pattern with three diameter dis-tance between the centers of piles. Within the footprint of thebuilding, there are 350 piles under a pile cap of 64.8 m by64.8 m. Such an arrangement allows the maximum number ofpiles to be placed within the pile cap. According to the engineer,the major concern of this foundation design is to obtain sufficientcapacity for the tremendous amount of loading.

The geology of the site is modeled as seven layers of sand andclay �Table 3�. In this project, both the shaft resistance and the toeresistance are considered in foundation design. As a linear analy-sis is not sufficient to truly capture the behavior of pile group,nonlinear analyses were adopted in this study. The shaft and toeresistances are simulated by the nonlinear load transfer curves ofO’Neill and Reese �1999�; an uncorrected SPT-N of 50blows/0.3 m is used to determine the toe resistance. The lateralresistance in sand is simulated by the Reese et al. �1974� p-ycurve and the lateral resistance in clay is simulated by the O’Neilland Gazioglu �1984� p-y curve. The soil parameters are shown inTable 3.

Fig. 7. Graphic presentation of “zone” formulation

Table 3. Summary of Soil Profile and Soil Properties

LayerMaterial

type

Level�m�

Top Bottom

1 Sand −21 −32.4

2 Clay −32.4 −37.8

3 Sand −37.8 −44.1

4 Clay −44.1 −48.1

5 Sand −48.1 −55.7

6 Clay −55.7 −70.0

7 Sand −70.0 −82.0



Challenges

High-rise buildings generally require a higher demand for loadcarrying capacity of the foundation. This can be reflected fromthis building project with 350 piles in the original design. Al-though simple GA can be applied to any problem types once theyare properly formulated, the large computational effort often hin-ders its use in practical projects.

For a pile group with the scale comparable to this project, thecomputational time for a nonlinear analysis using FB-Pier isabout half an hour for a single load case, using a computer with aCPU of P4 3.0G and 512M RAM. That means only 48 analysescan be performed in one day.

For this design example, in which two bits are used for eachpile, the string length required for defining the entire pile group of350 piles is 700 bits. As the population size is about one to twotimes the length of the string, the population size should be about700 to 1,400 for this project using the previously described for-mulation. Such a formulation is obviously infeasible for such alarge scale foundation because at least 350 h are needed for eachgeneration for one load case. Therefore, a new formulation for theproblem is needed for this project, especially for the design vari-ables that govern the variation of pile groups in the search space.

Objective Function and Design Variables

For this building project, a wall-frame structural system is usedsuch that columns are located at the perimeter and a core wall iseccentrically located toward the south face of the building. It isanticipated that the eccentric core wall being located near thecenter of the building is subjected to a larger portion of the grav-ity loading. Piles should, thus, be more closely spaced around thecenter of the core wall. Two zones with different pile spacing arethen defined: one is at the center of the core wall with a closerpile spacing between piles �inner zone�; the other is around theperimeter of the building with a larger spacing �outer zone�.

With the consideration of maximizing the load carrying capac-ity of the pile group, it is determined that a minimum pile spacingequal to three times the pile diameter is used in the inner zone. Inthis design formulation, two design variables are used to definethe dimensions of the inner zone, a and b. A pile is located at thecenter of the pile group. Within the inner zone, the number ofcolumns of piles is defined by “2a+1” along the x-axis and thenumber of rows of piles is defined by “2b+1” along the y-axis�Fig. 7�. Outside the inner zone, a design variable n is used todefine the spacing between piles in the outer zone. Piles will be

niteight

/m3�

Undrainedshear

strength�kPa�

Frictionangle�deg�

Constant ofmodulus of

subgradereaction

�MN /m3�

4.3 — 37.82 40.7

4.3 56 — —

4.3 — 40.5 40.7

4.3 60 — —

4.3 — 39.81 40.7

4.3 64 — —

4.3 — 40.5 40.7

Uw

�kN

1

1

1

1

1

1

1


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arranged in nD spacing in the orthogonal direction within the
outer zone �Fig. 7�. The maximum number of gridlines for pilesand loading is limited to 30 by 30. For a practical reason ofmaintaining an even spacing of piles, the piles of the two zonesare temporarily assumed to have the same size during the GAoperations. Once the topological arrangement of piles is deter-mined by the genetic operators, the size of each pile can be re-sized using the full stressed design operator.

Design Constraints

Pile Spacing and Edge DistanceThe design constraints for this project were imposed with refer-ence to the Chinese code for design of building foundationGB50007 �MOC 2002�. According to GB50007 Clause 8.5.2.1,the minimum spacing between piles that primarily derive theirresistance from shaft friction shall be three diam; also fromClause 8.5.15, the minimum distance between edge piles to theedge of the cap shall be one diam.

Strength RequirementBased on recommended bearing resistance and side resistance ofeach layer of soil as well as SPT-N values provided by the engi-neer, the allowable capacities of piles of different diameters weredetermined with reference to the piling guidelines, as shown inTable 4. Soil parameters for analysis were back-calculated fromthe bearing resistance and side resistance using the formulas inthe FB-Pier manual; and the SPT-N values were used in FB-Pier.However, these soil parameters would give the ultimate capacityof pile in the analysis program, which was greater than the allow-able capacity adopted by the engineer. In this case, piles wouldreceive load up to their ultimate capacity during the analysis.When the design was assessed based on the constraints, it wouldbe infeasible due to a smaller allowable capacity. Therefore, thepile capacity constraint was introduced into the analysis program.Soil parameters were used to produce the curve defining thestress-strain relationship of piles in FB-Pier. Yet only the rela-tively linear portion of the curve was retained and the remainingpart was bounded at the design capacity.

For simplicity, the design of the pile cap was not included inthis study, and a rigid pile cap assumption was made in the com-puter model of the pile group. The strength checking of the pilecap could be done after the optimization of the pile group design.

Serviceability RequirementAccording to Table 5.3.4 of GB 50007 �MOC 2002�, for a high-rise building with height greater than 100 m, the allowable valueof overall rotation is 0.002, while the allowable averaged settle-ment for a high-rise building with uniform shape is 200 mm. As

Table 4. Pile Capacities Allowed in Design Project

Pile diameter �m� Axial capacity �kN�

2.0 22,287

1.8 19,688

1.7 18,419

1.5 15,943

1.2 12,384

1.0 10,114

there is no recommendation for the allowable lateral displacement



of the pile group, the constraint for lateral displacement is set at25 mm according to the general design practice suggested byPoulos et al. �2001�.

Genetic Algorithm Parameters

The two design variables a and b are allowed to be chosen fromone to four to define the dimensions of the inner zone. Therefore,two bits are needed to define each dimension. Eight choices areavailable for the design variable n defining the spacing betweenpiles in the outer zone, which is from 3.0 to 3.7 with an interval of0.1. Another two bits are needed to define the largest pile diam-eter in the group, of which the choices are 1.8, 1.7, 1.5, and1.2 m. A summary of design variables is as follows:

a = �1,2,3,4� = 22

b = �1,2,3,4� = 22

n = �3.0,3.1,3.2,3.3,3.4,3.5,3.6,3.7� = 23

D = �1.8,1.7,1.5,1.2� = 22

As four bits are needed for defining the dimensions of theinner zone, three bits for the spacing between piles in the outerzone, and two bits for the largest pile diameter, the string length lis 9. The population size is set at 10, with the crossover rate of 0.6and the mutation rate of 0.02. The rate of the FSD operator is setat 0.05. The optimization process is allowed to continue for 10generations to study the convergence history.

Optimization Results

The pile group suggested by the optimization algorithm for thepractical project is shown in Fig. 8. The pile group consisted of169 piles and all piles were evenly spaced with 3D spacing. Themaximum pile diameter was 1.5 m and about 37% of the pileswere reduced to 1.2 m by the fully stressed design operator.

The arrangement of the 1.5 m diam piles and 1.2 m diam pilesmatched the engineering intuition. As the central core wallsshifted a bit toward the south side of the building, less capacitywas required at the north side of the foundation. Thus, some piles

Fig. 8. Optimization result of practical project

could be reduced to 1.2 m in diam. The FSD operator acted very


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effectively to reduce the sizes of the piles based on their internalforces after each analysis. This could hardly be achieved by theGA operators alone, as experience indicates that GA is incapableof fine tuning individual pile sizes.

Rate of GA and FSD Operator

A brief study on the crossover rate with a fixed mutation rate andvarious FSD rates was attempted. The combination of crossoverrate of 0.2, 0.6, and 1.0, fixed mutation rate of 0.02, and the FSDrate of 0.05, 0.1, and 0.2 was studied in particular. Only a fewtimes of optimization on each setting were performed. Based onthe limited information, a medium crossover rate of 0.6 would besuggested. This is because a small crossover rate was found in-sufficient to provide the minimum solution so far obtained. How-ever, a very large crossover rate of 1.0 did not have stableperformance in the study, which might be due to the large distur-bance to the already formed good designs. Similar to the designexamples presented earlier, a small rate of FSD operator of 5%was sufficient to perform the local searching for the practicaldesign project. A large FSD rate was obviously unfavorable.

Practicality of OC-GA

Although the number of analyses needed in the optimization pro-cess can be reduced with the aid of a local search operator, thetime required for nonlinear analysis of a large-scale pile group isstill substantial in the viewpoint of optimization. An efficientanalysis method is of great importance to optimization of large-scale foundation projects. Nonetheless, the hybrid GA methodshowed good performance in the design project. Apart from thetopological searching by GA operators, the FSD operator success-fully resized every pile based on its internal forces. This couldhardly be achieved by genetic operators alone, especially withover a hundred piles.

Conclusions

An optimization methodology based on hybrid genetic algorithmswas developed for the topological and sizing design of pile groupfoundations. The optimization algorithm has been applied to pilegroups subjected to different loading conditions of gravity loadalone, and combination of gravity and wind-induced loads. Basedon the findings obtained in this study, the following conclusionscan be drawn:1. The optimization algorithm has the capability of distributing

construction material to the most favorable location in orderto minimize the pile cap thickness and, hence, the total vol-ume of the foundation;

2. The algorithm developed is capable of obtaining feasible de-signs even when the design problem is highly constrained;

3. While GA performs well in global search, it is weak at finetuning the design to a unique global solution. Oftentimes,only a close-to-optimal design could be achieved by GA;

4. The design constraints and penalty functions are problemdependent, which require engineering judgment. The weight-ing factors between the stress and displacement constraintsshould be carefully determined to truly reflect the relative
importance; and


5. As demonstrated by the design examples, the proposed hy-brid genetic algorithm has potential application to real lifedesign problems with different load combinations.

Acknowledgments

This research was partially supported by the Research GrantsCouncil of the Hong Kong Special Administrative Region, China�Project Nos. HKUST6126/03E and HKUST6302/04E�. The ad-vice and assistance provided through the practical example byPing Liu of Ove Arup and Partners Hong Kong Ltd. are alsogratefully acknowledged.

References

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Buildings Department. �2004�. Code of practice for foundations, Build-ings Dept., Hong Kong SAR.

Chan, C. M. �2001�. “Optimal lateral stiffness design of tall buildings ofmixed steel and concrete construction.” Struct. Des. Tall Build.,10�3�, 155–177.

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O’Neill, M. W., and Reese, L. C. �1999�. Drilled shafts: Constructionprocedures and design methods, U.S. Dept. of Transportation, FederalHighway Administration, Office of Implementation, McLean, Va.

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