degreeprojectin structural engineering and bridges

KTH ROYAL INSTITUTE OF TECHNOLOGY

DEGREE PROJECT IN STRUCTURAL ENGINEERING AND BRIDGESSECOND CYCLE, 30 CREDITSSTOCKHOLM, SWEDEN 2021

Optimization and Automatization of end-bearing pile groups

GABRIEL ANTONIO DEL POZO ALARCON

SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

Examiner

Gabriel Antonio del Pozo [email protected] Royal Institute of Technology

Author

Place for ProjectStockholm, Sweden

Supervisors

Raied KaroumiKTH Royal Institute of Technology

Raied Karoumi

KTH Royal Institute of Technology

Shaho Ruhani

SWECO

Abstract

This thesis consists of the development, trial an application of a computational tool that by the

use of the Genetic Algorithm optimization method provides the user with an alternative

approach from the traditional trial and error when faced with the challenge of designing end-

bearing pile groups.

The work presents the three main concepts needed for understanding the computational tool as

well as the structure of it. The scripting capabilities of a FEM software are exploited in order

to automatically generate several design alternatives that are then compared in quality in an

iterative way. The developed algorithm succeeds validation against pile group problems with

known solutions and is then applied to the analysis of three different real structures with

different levels of complexity.

The algorithm is suitable for optimizing problems with loading in three directions and several

load cases, within a time range consistent with an overnight run, when the density of the pile

group is not high. Another potential application is to automatize retaining wall resting on piles

type problems within a short amount of computational time.

Sammanfattning

Denna avhandling består av utveckling och prövning av ett beräkningsverktyg som med hjälp

av den genetiska algoritmoptimeringsmetoden ger användaren ett alternativt tillvägagångssätt

från den traditionella prövningen när den möter utmaningen att utforma bärande höggrupper.

Arbetet presenterar de tre huvudkoncepten som behövs för att förstå beräkningsverktyget och

dess struktur. Skriptfunktionerna för en FEM-programvara utnyttjas för att automatiskt

generera flera designalternativ som sedan jämförs i kvalitet på ett iterativt sätt. Den utvecklade

algoritmen lyckas validera mot höggruppsproblem med kända lösningar och tillämpas sedan

för analys av tre olika byggda strukturer med olika nivåer av komplexitet.

Algoritmen är lämplig för att optimera problem med belastning i tre riktningar och flera lastfall,

inom ett tidsintervall som överensstämmer med en löpning över natten, när täthetsgruppens

densitet inte är hög. En annan mycket intressant applikation är att automatisera kvarvarande

väggproblem inom en kort tid av beräkningstid.

Preface

This Master thesis has been produced during my scholarship period at KTH Royal Institute of

Technology, Stockholm, which was funded by the Swedish Institute. It has been written for the

Department of Civil and Architectural Engineering, the division of Structural Engineering and

Bridges.

It is of great importance to express my appreciation and gratitude for the opportunity to write

this thesis at SWECO. Specifically, a profound thanks to my excellent supervisor in the

company Shaho Ruhani, his constant guidance and support made this work possible. Big thanks

to my supervisor at KTH Raied Karoumi and Ph.D. candidate Elisa Khouri Chalouhi for their

always nice disposition and valuable help.

Finally, thanks to my friends for making this journey enjoyable and my dear family for being

my main drive and motivation in life.

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Contents

1. Introduction ..................................................................................................................................... 1

1.1 Background ............................................................................................................................. 1

1.2 Previous Studies ...................................................................................................................... 2

1.3 Aim and Scope ........................................................................................................................ 3

1.3.1 Aim ................................................................................................................................. 3

1.3.2 Scope ............................................................................................................................... 3

1.4 Theis Outline ........................................................................................................................... 3

2. Theory ............................................................................................................................................. 4

2.1 Pile Group Design ................................................................................................................... 4

2.2 Pile Group Modelling.............................................................................................................. 9

2.3 The Genetic Algorithm ......................................................................................................... 12

3. Optimization of end-bearing pile groups using the Genetic Algorithm ........................................ 16

3.1 The computational Tool ........................................................................................................ 16

3.2 Validation of the Algorithm .................................................................................................. 21

4. Case Studies .................................................................................................................................. 25

4.1 Case Study 1: Pedestrian Bridge Abutment .......................................................................... 25

4.2 Case Study 2: Pedestrian Bridge Pylon Abutment ................................................................ 28

4.3 Case Study 3: Pedestrian Retaining Wall.............................................................................. 30

5. Discussion and Conclusion ........................................................................................................... 32

5.1 Results Discussion ................................................................................................................ 32

5.2 Conclusions ........................................................................................................................... 32

5.3 Further Development ............................................................................................................ 33

References ............................................................................................................................................. 34

Appendix 1 ............................................................................................................................................ 36

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List of Figures Figure 1 Typical end bearing capped pile group. .................................................................................... 1

Figure 2 Design Considerations for pile groups from (Burland et al.) ................................................... 4

Figure 3 Deformation Control for a Piled Foundation ............................................................................ 5

Figure 4 Pile toe elevation ...................................................................................................................... 6

Figure 5 Pile head elevation .................................................................................................................... 7

Figure 6 Failure modes for pile groups (Burland et al.) .......................................................................... 7

Figure 7 Pile batter calculation ............................................................................................................... 8

Figure 8 FEM model of pile group & surrounding soil (Dezi et al., 2016) ............................................ 9

Figure 9 Characteristic variation of Shear Strength .............................................................................. 10

Figure 10 Defining properties of a single pile ...................................................................................... 10

Figure 11 Global forces and moments application ............................................................................... 11

Figure 12 Geometry-wise load application ........................................................................................... 11

Figure 13 Genetic Algorithm example from (Kramer, 2017) ............................................................... 13

Figure 14 Initial Matrix ......................................................................................................................... 13

Figure 15 Initial individuals’ fitness ..................................................................................................... 14

Figure 16 Crossover example ............................................................................................................... 14

Figure 17 Mutation example ................................................................................................................. 14

Figure 18 Population matrix for 3 piles and 8 individuals .................................................................... 15

Figure 19 Computational Tool Flow Diagram ...................................................................................... 16

Figure 20 User input ............................................................................................................................. 17

Figure 21 Load definition ..................................................................................................................... 17

Figure 22 Iteration of fitness function ................................................................................................... 18

Figure 23 Levels of Penalization .......................................................................................................... 21

Figure 24 Validation 1st trial ................................................................................................................ 22

Figure 25 Validation 2nd trial ............................................................................................................... 23

Figure 26 Validation 3rd trial................................................................................................................ 23

Figure 27 Solution space for double symmetry and vertical piles of 3rd trial ...................................... 24

Figure 28 Case Study 1: Bridge Abutment ........................................................................................... 25

Figure 29 Case Study 1: Fitness development ...................................................................................... 26

Figure 30 Case Study 1: Pile group distributions ................................................................................. 27

Figure 31 Case Study 2: Pedestrian suspension bridge pylon’s foundation ......................................... 28



Figure 34 Case Study 3: Retaining Wall ............................................................................................... 30



1

Chapter 1

1. Introduction 1.1 Background

Pile groups and their corresponding pile cap form a type of foundation utilized to transmit sets

of loads from a structure to the surrounding soil. Pile foundations are used when the magnitude

of these loads is too large for shallow foundations to work properly. Then, the loads are carried

to lower soil (or rock) strata at the level of the pile toes which have much better bearing

properties (Burland et al., 2012).

A typical end bearing capped pile group is shown in Figure 1.

Figure 1 Typical end bearing capped pile group.

Common practice for pile group design consists in manually determining the acting forces in

the piles and the corresponding deformed shape of the pile cap for a determinate pile group

geometry. The designer faces the challenge, with this manual approach, to propose an initial

pile configuration based on structural engineering knowledge and/or previous experience in

pile group design. This initial configuration, which is expected to work properly, is usually

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solved using a Finite Element Method approach (FEM onwards), and then compared to a set

of requirements on the behavior of the FEM model. If these requirements, usually stated by

norms and construction codes, are not met, the designer must choose between adjusting the

configuration or trying a rather different one. This process, regarded as a trial and error, could

substantially consume the designer’s time.

When considering a manual optimization procedure for a design that already fulfils the

minimum requirements for it to be feasible and norm compliant it becomes even more time

demanding. It could even end up in no improvements because of the low-robustness of manual

pile group solutions (Abedin and Ligai, 2018). Thus, it becomes interesting to explore the

possibilities to develop a different approach for end-bearing pile group design and optimization

such that takes advantage of modern computing possibilities.

1.2 Previous Studies The aim of designing optimal piled foundations is not a recent one, the oldest bibliographical

document found on the topic dates forty years back where the author presents a user guide for

a computer program named PILEOPT developed for the U.S. Army Engineer Division that

analyzes pile foundations and has as objective to minimize the cost while keeping pile loads

and cap displacements compliant with limitations (Hill, 1981).

The selection of the optimization objective is not limited to the cost of the pile foundation as

(Chow and Thevendran, 1987) explored the possibility of minimizing differential settlements

calculated by the method analysis in (Chow, 1986) and the transformation of a constrained

minimization into an unconstrained via Sequential Unconstrained Minimization Technique.

The optimized solutions of the analyzed problems have negligible differential settlements and

load differentials while the overall settlements of the pile groups are also controlled. Likewise,

the minimization of differential settlements of a piled raft foundation with three different load

conditions utilizing Recursive Quadratic Programing and simplifying the structural system by

idealizing the piles as coupled springs based on (Randolph and Wroth, 1979) yields pile

configurations where pile clustering that follows the loading conditions is allowed due to the

soil-raft interaction. (Kim et al., 2001).

An optimization procedure for steel piles under rigid slabs and based on optimality criteria with

Lagrange Multipliers satisfying Kuhn-Trucker Conditions that minimize the total weight of the

steel are applied to 2D problems (Truman and Hoback, 1992) and 3D problems with loads

acting only on two dimensions (Hoback and Truman, 1993). Where the influence of the initial

suggestion, needed for the procedure, on the behavior of the iterations and the limitation of the

maximum deformation constraint on the weight minimization are of interest. Moreover, as

expected by the authors, the last 3D problem does not show a large reduction of steel weight

because it is an existing structure with an extensive amount of engineering hours devoted to it.

The minimization of the total weight of steel of a 2D retaining wall pile foundation with a

program that uses optimality criteria with Lagrange Multipliers and the introduction of

weightless variables, which have an indirect effect on the objective function but not on the

weight, reach to a significant observation by the authors. They state that the program’s solution

is a local minimum and that their program can allow the designer to choose from a set of sub-

optimal possibilities (Hurd and Truman, 2006).

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Global search carried by a genetic algorithm and local search by the fully stressed section

criteria, a hybrid genetic algorithm, is used by (Chan et al., 2009) in order to minimize the total

concrete volume, including the pile cap. The importance and problem dependency of the design

constraints are highlighted.

1.3 Aim and Scope

1.3.1 Aim

The purpose of this thesis is to develop a computational tool suitable for optimizing the

structural behavior of capped pile groups and for helping structural engineers with the

automatization of the design and evaluation of them. The structural behavior is analyzed by a

FEM approach while the optimization and automatization are carried out utilizing a pre-

programmed optimization method.

1.3.2 Scope

The thesis project is focused on end-bearing steel piles with a tubular cross section. The

capacity of these piles is obtained from tables for RD-type piles, and it is governed by:

- Buckling of the element considering it surrounded by soil, section capacity and

amount of lost section due to rust.

- Geotechnical properties of the surrounding soil, strength properties of the rock at the

pile toe depth and the verification level of these geotechnical properties.

The computational tool is initialized in MATLAB®, where the user defined parameters of each

problem such as pile cap geometry and loads are introduced. The optimization method is the

Genetic Algorithm which works with three variables per pile:

- Coordinates of the pile heads in the pile cap (horizontal plane geometry).

- Direction angle of the pile (discretized by increments of 90 degrees).

The FEM analysis is carried out using ABAQUS scripting capabilities and it is limited to linear-

elastic and static considerations in Ultimate Limit State. The results extracted from it which are

then analyzed by the GA are:

- Deformation of the pile cap.

- Axial force in the piles.

1.4 Theis Outline This thesis has five chapters. The first one consists of an introduction to the topic and its

corresponding previous studies, also the aim and scope of the thesis are defined in chapter 1.

The theoretical background on pile group design, pile group modelling and the Genetic

Algorithm is presented in chapter 2. Chapter 3 is a thorough description and validation of the

computational tool which is then applied to the analysis of three different Case Studies in

chapter 4. The last chapter evaluates the results, draws conclusions from them and discusses

possible further studies.

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Chapter 2

2. Theory This chapter presents three theoretical concepts required to understand the structure and inner

functioning of the computational tool.

2.1 Pile Group Design Pile foundations are complex systems where several aspects, both from structural and

geotechnical engineering, must be considered for their correct design. Figure 2 extracted from

(Burland et al., 2012) shows some of the most important ones.

Figure 2 Design Considerations for pile groups from (Burland et al.)

The regulations to follow when designing a pile group are present in several sections within

the Eurocodes where section 7 of part 1 of (EN_1997-1:2005), Eurocode 7 on geotechnical

design, is devoted to pile foundations. Within section 2 of Eurocode 7, Basis of geotechnical

design, the actions on the structure can be found (in 2.4) as well as a reference to (EN_1992-

1-1:2005) (EN_1993-5:2007) and (EN_1994-1-1:2005) for concrete, steel and composite

section piles respectively. Annex A of Eurocode 7 provides with the partial factors for the

ultimate limit states. Furthermore, for the type of pile sections utilized in the present work,

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(EN_14199:2015) has extra considerations for micro-piles which are drilled piles with a

diameter not larger than 300 mm.

Section 2.4.7.1 of (EN_1997-1:2005) defines the limit states to be checked within the Ultimate

Limit State verification of foundations in general. Where, the relevant ones for this thesis are:

- EQU: Rigid body motion, and as consequence lack of equilibrium, of the

surrounding soil or structure. In this limit state the strength of the soil and structure

do not play a role in providing equilibrium.

- STR: Extreme deformation of the structure or failure of the structural elements. In

this limit state the strength of the structural materials and their sections is the main

provider of resistance.

- GEO: Excessive deformation or yield of the surrounding soil where the strength

properties if the soil material are of main importance.

To comply with the regulations and design piled foundations that do not fail these three limits

states several controls are carried out on a design:

- The first one consists of controlling the deformations of the structure. Figure 3 is a plan view

of a typical piled foundation, where the dotted lines represent the deformed shape of it.

Inequation (2.1) must be fulfilled for this control to be successful. Note that the rigid body

motion mentioned in the EQU limit state is also avoided if the inequality is valid.

Figure 3 Deformation Control for a Piled Foundation

𝑈𝑖 ≤ 𝐷𝑖 , 𝑖 = 𝑥, 𝑦, 𝑧 (2.1)

where:

𝐷𝑖 is the limitation on the deformation for axis 𝑖

𝑈𝑖are the absolute maximal deformations as shown in Figure 3

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- Just as the structure deformation must be kept under a limiting boundary, the forces acting on

the piles must also be within the range of the pile capacities. This range and verification as

defined in (2.2).

𝐹𝑅,𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑖𝑜𝑛 ≤ 𝑆𝐹𝑖 ≤ 𝐹𝑅,𝑡𝑒𝑛𝑠𝑖𝑜𝑛, 𝑖 ∈ [0, 𝑛𝑝𝑖𝑙𝑒𝑠] (2.2)

where:

𝑆𝐹𝑖 is the acting axial force on pile 𝑖

𝐹𝑅,𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑖𝑜𝑛 is the compressive capacity of the pile.

𝐹𝑅,𝑡𝑒𝑛𝑠𝑖𝑜𝑛 is the tensional capacity of the pile.

𝑛𝑝𝑖𝑙𝑒𝑠 is the number of piles.

It is important to note that a pile acting in tension, requires it to be anchored to the bedrock

(Figure 4) and a special detail in the connection to the pile cap (Figure 5).

Figure 4 Pile toe elevation

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Figure 5 Pile head elevation

- A minimum distance between pile heads is kept not only to minimize the risk for two piles to

be too close to each other in the space between the pile heads and the pile toes but also for there

to be the possibility to add extra piles due to eventual installation-related failure of a pile

(TDOK_2016:0203). Additionally, and corresponding to the GEO state, the pile head

separation is important to avoid overstressing the soil between the piles and to increase the

verification area of the block of soil around the piles and avoid block failure as shown in Figure

6.

Figure 6 Failure modes for pile groups (Burland et al.)

- A major contribution from traditional pile group design to this thesis, and a way to increase

the certainty that the pile group to verify in the EQU limit state, consists of setting the batter of

the piles to a fixed value. The criterion to determine the inclination value is governed by two

factors. The first one, follows the relationship between the global vertical and global horizontal

forces acting in the plane as shown in Figure 7. Then, this value is compared to the limitations

set by the machinery used to install the piles with discrete increments in the range of 10:1 5:1

4:1 with a lower bound of vertical piles and un upper bound of 3.5:1. The load case which has

a lower relation vertical over horizontal forces is the one that dominates the selection of the

pile batter as indicated in (2.3).

8

𝑆𝑖 = 𝑓 (𝐹𝑧

𝐿𝐶

𝐹𝑖𝐿𝐶) , 𝑖 = 𝑥, 𝑦 𝐿𝐶 = 1,2, … , 𝑛𝐿𝐶 𝑆𝑖 ∈ [∞, 10,5,4,3.5] (2.3)

where:

𝑆𝑖 is the inclination in plane 𝑖

𝐹𝑧𝐿𝐶 is the global force in the Z axis for load case 𝐿𝐶

𝐹𝑖𝐿𝐶 is the global force in the 𝑖 axis for load case 𝐿𝐶

𝑛𝐿𝐶 is the number of load cases

Figure 7 Pile batter calculation

9

2.2 Pile Group Modelling The axial forces of the piles and the deformation of the pile cap are onwards referred as the

structural behavior of the capped pile group. The common approach for solving pile groups,

and then discriminating if their structural behavior complies with the norms, is to build FEM

models.

The complexity of these FEM models depends largely on the definition on whether to model

the soil surrounding the piles or not. This strategy is of course possible by representing the soil

by solid and infinite elements. Specifically, in (Dezi et al., 2016) and as shown in Figure 8 two

types if solid elements and one type of infinite element are used to represent the soil.

Figure 8 FEM model of pile group & surrounding soil (Dezi et al., 2016)

Other option, where solid and infinite elements are avoided because they are computationally

expensive, is to model the soil-pile interaction by defining boundary elements that represent

the soil (Luamba and de Paiva, 2019), these boundary elements are coupled with the stiffness

element of the pile group and with the equivalent load vector. One of the most straightforward

techniques is to define a Winkler space where translational and rotational springs are attached

to the nodes of the beam elements that represent the piles. In these three techniques the soil is

represented by either boundary elements, volumetric and infinite elements, or springs. These

elements or springs require definitions within their formulation that are in direct relation with

soil properties, in the case of clay it is the shear strength. However, the shear strength, and any

soil property, varies with depth, as shown in Figure 9. Furthermore, these geotechnical

variables carry a large uncertainty around their values because of the heterogenous nature of

the soil. Models of this kind are in theory very precise. However, this increase in precision does

not imply an increase in accuracy.

10

Figure 9 Characteristic variation of Shear Strength

It is common practice to simplify the modelling of the foundation as if the soil surrounding the

piles would not interact with the structure because of the complications with modelling the

surrounding soil. Then, the axial forces obtained from the FEM model are compared with

values of maximum pile bearing force obtained from geotechnical calculations (where the pile

buckling is also considered). Moreover, the deformations and settlements are compared with

the limitations stated in the corresponding regulations and norms. Even though this

simplification can be deemed as significant; the common procedure of pile group design is not

simple since it is a three-dimensional problem with many load cases transmitted from the

structure. In addition, the complexity is increased in proportion with the number of piles

because each one of them implies the definition of at least three properties. These properties,

as shown in Figure 10, are the coordinates in the plane of the pile cap (X-Y plane in this thesis),

the angle of rotation in the same plane with respect to the X-axis.

Figure 10 Defining properties of a single pile

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When faced with the task to transmit the forces and moments from the structure to the pile

foundation, two different approaches can be taken. The first one, and most predominant in the

literature of pile group optimization as in (Abedin and Ligai, 2018) and (Bengtlars and

Väljamets, 2014), is to lump and apply them in a global manner. As shown in Figure 11 a single

set of three forces and three moments act in the gravity center of the plate while the application

point is coupled with the pile heads as if the pile cap were totally rigid.

Figure 11 Global forces and moments application

This modelling technique is not lacking the complications explained in section 2.1 because the

forces and moments set has several load cases. The approach taken in this thesis for the way of

applying the loads from the structure to the pile cap consists of taking the geometry of the

structure connected with it into consideration. The FEM software has the capability of applying

any number of point and line loads on the shell. An example of a case where the loads from the

superstructure are applied to the pile cap is shown in Figure 12. This way, and by the definition

of the pile cap as a thick shell element, a better approximation to the real behavior of the cap-

pile interaction is obtained.

Figure 12 Geometry-wise load application

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The intent to represent the cap-pile behavior in a more detailed way corresponds to the

guidelines present in Section 7.6.2.1 of (EN_1997-1:2005) where the stiffness of the pile cap

has direct effect on its ability to redistribute load. If the pile cap is completely rigid and because

it can redistribute load between piles, only if a significant quantity of piles fails together the

limit state would be reached, thus failure of only one pile can be disregarded. On the other

hand, if the pile cap is very flexible, failure of only one pile can be enough for the limit state

to occur. A real pile cap will be somewhere between these two extremes and a good

representation of the pile cap stiffness is necessary for the correct judgement of the limit states

when checking acting pile forces.

2.3 The Genetic Algorithm The Genetic Algorithm (GA onwards) in formal definition is a Meta-Heuristic optimization

method inspired by natural selection of the fittest species. It is an iterative process based on

natural evolution principles where the fittest individuals of a given population are preferred for

selection to form the next generation. Via combination of the selected individuals’ properties,

this next generation is formed. The concept of mutation is introduced to increase the diversity

of the generation of new individuals. Just as any optimization method, it has an objective

function (2.4), where the variables are bounded by constraining inequalities and equalities

(2.5).

𝑜𝑝𝑡𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝑓(𝑥) (2.4)

Subject to:

{ℎ𝑖(𝑥) ≥ 0, 𝑖 ∈ [1, … , 𝑚]

𝑘𝑖(𝑥) = 0, 𝑖 ∈ [1, … , 𝑝] (2.5)

where:

𝑥 are the optimization variables.

𝑧 is the objective function (fitness function for the GA)

ℎ𝑖 are the inequalities that constrain the optimization.

𝑚 is the number of inequalities.

𝑘𝑖 are the equalities that constrain the optimization.

𝑝 is the number of equalities.

This means that the GA is a smart search for the best solution, where a solution is composed

of a determined combination of the optimization variables that satisfy the problem constraints.

The terms solution and individual are interchangeable in the GA case, while the term fitness

refers to the numerical value obtained by each individual when evaluated in the objective

function, hence it is called fitness function. As it is an iterative process each loop run is also

named a generation. For this search to be smart, the GA relies on three operators which are

selection, crossover, and mutation. These operators come into play during each generation as

shown in Figure 13 extracted from (Kramer, 2017).

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Figure 13 Genetic Algorithm example from (Kramer, 2017)

The main interest of this thesis is the application of this optimization technique to the end-

bearing pile group design problem, therefore for more theory on the Genetic Algorithm refer

to (Kramer, 2017) and (Rothlauf, 2011). The selection of this technique is firstly based on the

capability it has for discontinuous solution spaces where it does not rely on the calculation of

gradients based on the differentiability of the objective function. Secondly, its stochastic nature

allows the user to trial several unbiased and very different configurations of pile groups.

Finally, because the concepts behind the algorithm, such as crossover and mutation, are

tractable and easy to understand.

The application of the GA to the problem of pile group design with the variables defined in

Figure 10 starts with forming a generation. For the example shown in Figure 14 where four

individuals and only one pile are considered the values of the position of the pile head are

generated randomly within the geometry of the pile cap, this is enforced by the constraining

inequalities (2.5).

Figure 14 Initial Matrix

The fitness of each individual in the initial population matrix, is calculated by evaluating the

fitness function (2.4) and associated to each row of the matrix, as shown in Figure 15.

14

Figure 15 Initial individuals’ fitness

Once each solution has a fitness associated the selection operator comes into play. This means

that a selection function is used where the individuals with better fitness values have a higher

probability of being selected for Crossover. In the example of Crossover shown in Figure 16

individuals 1 and 3 are selected and combined by a crossover function 𝑔 to form a new

individual which will have numerical values of X, Y, and direction between the ones from the

selected individuals.

Figure 16 Crossover example

This crossover process is repeated until a new matrix of individuals is produced, such as the

one shown in Figure 17a. Mutation is the final operator to which this matrix is subjected, where

random variation is some of the variables of some of the solutions is introduced.

Figure 17 Mutation example

Finally, the matrix shown in Figure 17b becomes the initial population matrix of the next

generation as the one shown in Figure 14 and the loop is repeated until the stopping criteria of

the algorithm is met, e.g. the fitness is good enough or the enhancement has not been

15

considerable after a defined number of iterations. The Elite Clones concept is used by the GA,

so the fitness of the next generations is at least as good as the last evaluated one. A small

number of the individuals with the best fitness of each generation is copied to the next

generation without crossover or mutation.

An example of an initial population matrix for a problem with 3 piles and 8 individuals is

shown in Figure 18.

Figure 18 Population matrix for 3 piles and 8 individuals

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Chapter 3

3. Optimization of end-bearing pile groups using

the Genetic Algorithm This chapter provides a complete description of the computational tool and the validation

process it was subjected to.

3.1 The computational Tool Figure 19 shows the flow diagram of the computational tool. It can be divided in three sections

which are (i) user input (ii) iteration of fitness function and (iii) finalization.

Figure 19 Computational Tool Flow Diagram

17

The user input section, which is done in MATLAB, can be closely seen in Figure 20b where

the problem geometry of the pile cap is defined. For the piles’ geometry, the section definition

refers to its diameter and thickness while the length corresponds to the vertical distance from

the pile cap mid-surface to the average depth of the bedrock. The bearing load of the piles refers

to the compression capacity of the section, one of the objectives of the algorithm is to avoid

piles acting in tension because of the reasons explained in section 2.1.

Figure 20 User input

The computational tool’s load definition, as Figure 20a shows, can manage any number of

point forces and moments as well as any number of line loads (forces and moments) for any

number of load cases. The definition of each load is done by the user in a matrix form following

the guide example of Figure 21 where point forces are defined.

Figure 21 Load definition

The deformation limits set by the user correspond to section 2.1 and must be norm compliant.

The separation of the pile heads, usually set to 4 times the diameter, is enforced by the

algorithm using non-linear constraints that MATLAB uses for this problem. This means that

18

no individual with pile heads closer to each other than the set limit has its fitness calculated in

the next section.

Referring to the GA parameters set by the user, the number of piles must be constant along the

generations because the optimization variables for this problem are directly proportional to the

number of piles and the GA requires the number of optimization variables to be constant. The

number of individuals as explained in section 2.3 dictates the number of rows of each

generations population matrix and has a direct impact on the diversity of the solutions but also

on the computational time. Therefore, correct judgement is necessary for the number of

individuals selection. The final GA parameter, the initial suggestion, is an optional one. Where

the difference between optimization of a given design is carried out by the computational tool

if this design is provided to the computational tool. A pure automatization of the design is done

if no suggestion is fed to the algorithm.

The iteration of the fitness function shown in more detail in Figure 22 consists of the fitness

evaluation of the different individuals throughout each generation. The user input, which will

be constant for every generation, along with the coordinates and directions of the piles for every

individual in the generation are exported to ABAQUS where the script (developed also by the

author) interprets the data and builds a model for every individual. Each model is solved, and

the results (cap deformations and pile forces for every load case) are sent back to MATLAB

where their fitness is evaluated.

Figure 22 Iteration of fitness function

The optimization process is a minimization of the fitness which as stated in (3.1) is composed

by the “raw” fitness (i.e., the differential settlement of the individual) and the penalization

terms, between parentheses.

𝑓𝑖𝑡𝑛𝑒𝑠𝑠 = 𝐷𝑑𝑖𝑓𝑓 + (𝑗𝑈 + 𝑗𝑅𝐹 + 𝑗𝑠𝑝𝑎𝑐𝑒) (3.1)

19

where:

Equation (3.2) selects the largest differential settlement produced by any of the load

cases.

𝐷𝑑𝑖𝑓𝑓 = max(𝐷𝑑𝑖𝑓𝑓𝐿𝐶 ) , 𝐿𝐶 = 1,2, … , 𝑛𝐿𝐶 (3.2)

where:

𝑛𝐿𝐶 is the number of Load Cases

and:

𝐷𝑑𝑖𝑓𝑓𝐿𝐶 = {

|(max(𝑈𝑧𝐿𝐶) − min(𝑈𝑧

𝐿𝐶))| , 𝑈𝑚𝑎𝑥𝐿𝐶 ≤ 𝐷𝑅𝐵𝑀

|𝑈𝑚𝑎𝑥𝐿𝐶 | , 𝑈𝑚𝑎𝑥

𝐿𝐶 > 𝐷𝑅𝐵𝑀

(3.3)

where:

𝑈𝑧𝐿𝐶 is the deformation of the pile cap in the Z-axis (vertical) for load case LC.

𝑈𝑚𝑎𝑥𝐿𝐶 is the absolute maximum deformation of the pile cap in any direction defined by

(3.4) for load case LC.

𝑈𝑚𝑎𝑥𝐿𝐶 = max(|max (𝑈𝑖

𝐿𝐶)|, |min(𝑈𝑖𝐿𝐶)|) , 𝑖 = 𝑥, 𝑦, 𝑧 (3.4)

𝐷𝑅𝐵𝑀 is the Rigid Body Motion limit (set by the user) which if surpassed, the

computational tool treats the load case as unstable, refer to section 2.1

𝑈𝑖𝐿𝐶 is the deformation of the pile cap along axis 𝑖 for load case LC

𝑖 is the axis along which the deformation takes place.

Equation (3.5) defines the first penalization term; it is related to the deformation of the pile cap

when a rigid body motion is not reached, and it selects the largest of any load case.

𝑗𝑈 = max ( ∑ 𝑗𝑈,𝑖𝐿𝐶

𝑖=𝑥,𝑦,𝑧

) + max(𝑗𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝐿𝐶 ) , 𝐿𝐶 = 1,2, … , 𝑛𝐿𝐶 (3.5)

where:

𝑗𝑈,𝑖𝐿𝐶 = {

0 , max(|max(𝑈𝑖𝐿𝐶)|, |min(𝑈𝑖

𝐿𝐶)|) ≤ 𝐷𝑖,𝑚𝑎𝑥

100 , max(|max(𝑈𝑖𝐿𝐶)|, |min(𝑈𝑖

𝐿𝐶)|) > 𝐷𝑖,𝑚𝑎𝑥

, 𝑖 = 𝑥, 𝑦, 𝑧 (3.6)

and:

𝑗𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝐿𝐶 = {

0 , ∆𝑈𝑥−𝑥𝐿𝐶 ≤ 𝐷𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛,𝑥−𝑥 ∩ ∆𝑈𝑦−𝑦

𝐿𝐶 ≤ 𝐷𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛,𝑦−𝑦

200 , ∆𝑈𝑥−𝑥𝐿𝐶 > 𝐷𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛,𝑥−𝑥 ∪ ∆𝑈𝑦−𝑦

𝐿𝐶 > 𝐷𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛,𝑦−𝑦

(3.7)

where:

𝐷𝑖,𝑚𝑎𝑥 is the deformation limit along axis 𝑖, refer to section 2.1

∆𝑈𝑥−𝑥𝐿𝐶 is the difference in deformation along the X-axis between the vertices of

any of the two edges (the algorithm selects the largest one) perpendicular to the

X-axis of the pile cap for load case LC.

20

∆𝑈𝑦−𝑦𝐿𝐶 is the difference in deformation along the Y-axis between the vertices of

any of the two edges (the algorithm selects the largest one) perpendicular to the

Y-axis of the pile cap for load case LC.

𝐷𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛,𝑥−𝑥 is the limit for the difference in deformation along the X-axis

between the vertices of any of the two edges perpendicular to the X-axis of the

pile cap.

𝐷𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛,𝑦−𝑦 is the limit for the difference in deformation along the Y-axis

between the vertices of any of the two edges perpendicular to the Y-axis of the

pile cap.

Equation (3.8) defines the second penalization term; it is related to the acting axial force in

each one of the piles for every load case.

𝑗𝑅𝐹 = max(𝑗𝑡𝑒𝑛𝑠𝑖𝑜𝑛𝐿𝐶 ) + max(𝑗𝑃𝑚𝑎𝑥

𝐿𝐶 ) , 𝐿𝐶 = 1,2, … , 𝑛𝐿𝐶 (3.8)

where:

𝑗𝑡𝑒𝑛𝑠𝑖𝑜𝑛𝐿𝐶 = ∑ 𝑗𝑡𝑒𝑛𝑠𝑖𝑜𝑛 ,𝑘

𝐿𝐶

𝑛𝑝𝑖𝑙𝑒𝑠

𝑘=1

, 𝑗𝑡𝑒𝑛𝑠𝑖𝑜𝑛 ,𝑘 𝐿𝐶 = {

10 , 𝑆𝐹1𝑘𝐿𝐶 < 0

0 , 𝑆𝐹1𝑘𝐿𝐶 ≥ 0

(3.9)

and:

𝑗𝑃𝑚𝑎𝑥

𝐿𝐶 = ∑ 𝑗𝑃𝑚𝑎𝑥 ,𝑘𝐿𝐶


𝑘=1

, 𝑗𝑃𝑚𝑎𝑥 ,𝑘𝐿𝐶 = {

20 ∙ 𝑎𝑏𝑠 (𝑆𝐹1𝑘

𝐿𝐶

𝑃𝑚𝑎𝑥) , 𝑆𝐹1𝑘

𝐿𝐶 > 𝑃𝑚𝑎𝑥

0 , 𝑆𝐹1𝑘𝐿𝐶 ≤ 𝑃𝑚𝑎𝑥

(3.10)

where:

𝑆𝐹1𝑘𝐿𝐶 is the axial force acting on pile 𝑘 for load case LC

𝑃𝑚𝑎𝑥 is the pile section compression capacity

𝑛𝑝𝑖𝑙𝑒𝑠 is the number of piles

Equation (3.11) defines the third penalization term; it avoids pile collision or too small pile

separation in the space between the pile heads and the pile toes planes. This is carried out by

controlling the distance between the center of the section of pairs of piles in five different

horizontal planes which are located at an even distance between each other and equally

distributed between the piles’ heads plane and piles’ toes plane.

𝑗𝑠𝑝𝑎𝑐𝑒 = ∑ ∑ 𝑗𝑠𝑝𝑎𝑐𝑒𝑚,𝑛

𝑛𝑝𝑖𝑙𝑒𝑠!

(𝑛𝑝𝑖𝑙𝑒𝑠−2)!∙2!

𝑛=1

𝑚=5

𝑚=1

(3.11)

where:

𝑗𝑠𝑝𝑎𝑐𝑒𝑚,𝑛= {

5 , 𝑠𝑚,𝑛 < 𝐷𝑚𝑖𝑛,𝑠𝑝𝑎𝑐𝑒

0 , 𝑠𝑚,𝑛 ≥ 𝐷𝑚𝑖𝑛,𝑠𝑝𝑎𝑐𝑒 (3.12)

𝐷𝑚𝑖𝑛,𝑠𝑝𝑎𝑐𝑒 is the minimum required separation between piles

𝑠𝑚,𝑛 is the separation between the pair of piles 𝑛 in verification plane 𝑚

21

𝑛 is the pile pair to be verified

𝑚 is the plane of verification between the pile cap and bedrock

Equations (3.3) (3.6) (3.7) (3.9) (3.10) and (3.12) define different penalization coefficients with

varying magnitudes. This difference in magnitude and value between them has the purpose of

guiding the search towards acceptable and feasible solutions. If each penalization coefficient

had the same value, an infeasible solution (e.g. rigid body motion) would have a very similar

fitness as a feasible but non-desirable solution (e.g. tension in some piles). Furthermore, in

order to obtain solutions that are useful from an engineering perspective, excessive deformation

of the pile cap is heavily penalized (also considering rotation). In that sense, the coefficients’

numerical values correspond to different levels of penalization shown in Figure 23.

Figure 23 Levels of Penalization

Having calculated the numerical values of the fitness, the GA operators (selection, crossover,

and mutation) as in section 2.3 produce a new generation matrix and the loop is repeated until

one of the stopping criteria of the GA are met. These criteria can be one of the following:

- Maximum number of generations reached. This is usually avoided, setting a very large

number for max generations, because reaching a fixed number of generations does not

necessarily mean that an optimum has been found.

- Fitness limit is reached. This could for practical applications be exploited when a differential

settlement below a limit is deemed good enough to finish the search.

- The fitness enhancement between a set number of generations is smaller than a fixed

tolerance. This criterion is mainly used in this thesis.

3.2 Validation of the Algorithm An internal validity problem arises when considering if the GA will yield correct solutions in

pile group design. To tackle it, the GA shall be evaluated against a set of known design cases

where the optimum is known and deductible from classic structural engineering theory. Once

the GA yields solutions equal (or acceptably close) to the theoretical best solutions of this cases,

22

internal validity is accepted. Then, external validity is tested by using the GA for real cases of

pile group design.

In that sense, the approach in this thesis was to gradually increase the complexity of the internal

validity trials. Three steps of analysis were used as following:

- 2D analysis of a pile cap with a centric column subjected to vertical and horizontal forces and

two inclined piles as shown in Figure 24a. The piles inclination is equal to the relation between

the vertical and horizontal forces. The problem has two variables which are x1 and x2. The

algorithm solution is shown in Figure 24b which is the correct one.

Figure 24 Validation 1st trial

- Figure 25a presents the 2nd validation trial, a 3D analysis of a square pile cap subjected only

to self-weight and supported by only one pile. The intuitive and theoretical solution, also found

by the GA after 12 generations is shown in Figure 25b. The solution space for this problem has

a size of 4805 possible combinations of the three variables (X Y and pile direction).

23

Figure 25 Validation 2nd trial

- The third trial of validation as presented in Figure 26 is the 3D analysis of a square pile cap

subjected only to self-weight and a vertical pressure supported by four piles with their positions

restricted to each one of the quadrants shown. For this case, the solution space grows to a size

of approximately 1.6 × 1012 possible combinations of the 12 optimization variables. In order

to be able to study the solution space, the portion of it for double symmetry around the two

horizontal axis (X and Y) and only vertical piles is plotted in Figure 27. For an approximate

formula to determine the size of the solution space for any problem applied to the algorithm

refer to Appendix 1.

Figure 26 Validation 3rd trial

24

Figure 27a and b show the complete solution space for double symmetry and only vertical piles,

where a clear valley can be seen starting from 0.3 towards the positive on both directions. When

only this valley is plotted, and as presented in Figure 27c and d, the global optimum can be

found at point (1.1 , 1.1). The solution given by the computational tool for this problem is not

double symmetric but can be averaged to the point (1.2 , 1.2). It is not the global optimum, but

it can be deemed as a good solution because the fitness is very close to the best possible one.

Given that the computational tool succeeds the three levels of validation it is deemed as

sufficient for applying it to three Case Studies to be introduced in the following chapter.

(a) solution space plan view

(b) solution space 3D view

(c) valley plan view

(c) valley 3D view

Figure 27 Solution space for double symmetry and vertical piles of 3rd trial

25

Chapter 4

4. Case Studies This chapter presents the three Case Studies of real structures to which the computational tool

was applied. For each case study at least three runs of the algorithm are done where two

optimizations are trialed and one automatization. For the first optimization the initial

suggestion is the actual design of the structure (designer’s solution) while the second

optimization has the user’s solution. Finally, the automatization is not given any initial

suggestion.

4.1 Case Study 1: Pedestrian Bridge Abutment The first case study consists of the abutment of a pedestrian bridge. An elevation of the

structure is shown in Figure 28a. As the 3D representation of the bridge abutment in Figure

28b shows, earth pressure and surcharge act on the front walls and the wing walls while loads

coming from the bearings are also present, thus the loads act in three directions. The designer’s

solution for the abutment is shown in Figure 28c where eight RD140/8 piles are considered

with four of them requiring anchoring. Three characteristic load cases, coupled with the loads

from the bearings, for retaining structures are considered.

(a) CS1: Elevation

(b) CS1: 3D view

(c) CS1: Designer Solution

Figure 28 Case Study 1: Bridge Abutment

The fitness evolution for the three initial conditions for the Case Study 1 is shown in Figure 29

while Figure 30 shows the plan view of the pile groups before and after their respective

optimizations and the final solution for the automatization. Figure 30a and b correspond to the

26

optimization of the designer’s solution while Figure 30c and d show the initial user’s input and

the optimization final individuals, respectively. Figure 30e presents the automatization final

solution.

Figure 29 Case Study 1: Fitness development

It is noteworthy that after around eight hours of iteration the automatization reaches a pile

distribution (Figure 30e) with a better fitness than the designer’s initial suggestion shown in

Figure 30a. Regarding the user’s optimization, it took only two hours for the computational

tool to reach a better fitness than the designer’s original. The main reason the algorithm can

find distributions with better fitness is because it reduces the number of piles acting in tension.

Furthermore, the computational time required for this problem, between ten to eight hours, is

in the same order of magnitude of an overnight run.

0 5 10 15

0

50

100

150

200

250

300

0 50 100 150 200

Computational Time (h)

Fitn

ess

Generations

Optimization:Designer

Automatization

Optimization:User

Figure 30c

Figure 30d

Figure 30e Figure 30a

Figure 30b

27

(a) Optimization: Designer Initial

(b) Optimization: Designer Final

(c) Optimization: User Initial

(d) Optimization: User Final

(e) Automatization: Final

Figure 30 Case Study 1: Pile group distributions

28

4.2 Case Study 2: Pedestrian Bridge Pylon Abutment The second case study consists of the foundation for the pylon of a suspension pedestrian

bridge. An elevation of the structure is shown in Figure 31a while the plan view is presented

in Figure 31b. The designer’s approach was to lump the forces from the superstructure,

surrounding soil, and self-weight into a single set of six forces and moments with 11 load cases

applied in the gravity center of the pile cap. The designer’s solution for the foundation is shown

in Figure 31c where 54 RD220/12.5 piles are considered.

(a) CS2: Elevation

(b) CS2: Plan


Figure 31 Case Study 2: Pedestrian suspension bridge pylon’s foundation


0 20 40 60 80 100

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 20 40 60 80 100 120 140 160


Fitn

ess

Generations


Automatization

Optimization:User 2

Optimization:User 1

Figure 33d

Figure 33b

Figure 33c

Figure 33a

29

The fitness evolution for the four algorithm runs for the Case Study 2 is shown in Figure 32

while Figure 33 shows the plan view of the final pile groups after their respective optimizations

and automatization. Figure 32a corresponds to the optimization of the designer’s solution while

Figure 32b and c show the final solution for the two optimizations where user’s input is not an

initial distribution but less piles. Figure 32d presents the automatization final solution.

(a) Optimization: Designer

(b) Optimization: User 1 Final

(c) Optimization: User 2 Final

(d) Automatization: Final


Note from the optimization trial for the user’s solution that the tool is not able to find a better

solution when a highly experienced engineer solution is suggested in the initialization.

Moreover, the algorithm has difficulties to reach a competitive fitness in the other

optimizations too. The main reason behind this is the density if the pile cap and as consequence

avoiding pile collision has been the main challenge for the algorithm. This case study allows

to note the limitation of the optimization technique but also shows the capability of the script

to process problems with several piles and more than ten load cases.

30

4.3 Case Study 3: Pedestrian Retaining Wall The third case study consists of the foundation for a pedestrian retaining wall. An elevation of

the structure is shown in Figure 34a while the 3D representation of the wall is presented in

Figure 34b. This is a typical case of a pseudo-3D problem because the loads, for which three

load cases are considered, only act on two directions. The designer’s solution for the foundation

is shown in Figure 34c where 10 RD170/10 piles are considered.

(a) CS3: Elevation

(b) CS3: 3D view


Figure 34 Case Study 3: Retaining Wall

The fitness evolution for the three algorithm runs for the Case Study 3 is shown in Figure 35b

while Figure 35a omits the automatization so the two optimizations can be better appreciated.

Figure 36a and b show the plan view of the designer’s pile group before and after optimization

while Figure 36c shows the final pile group for the user’s optimization and Figure 36d shows

the solution for the automatization. The user’s optimization could be considered closer to an

automatization because no initial suggestion is given to the computational tool but only the

instruction for the piles direction to be only along the X-axis (where the horizontal load acts).

(a) CS3: Fitness evolution only optimizations

(b) CS3: Fitness evolution


0 5 10 15

0

2

4

6

8

10

12

0 50 100 150


Fitn

ess

Generations

0 10 20 30

0

50

100

150

200

250

300

350

400

0 100 200 300


Generations


Automatization

Optimization:User

Figure 36a

Figure 36b

Figure 36c Figure 36d

31

(a) Optimization: Designer Initial

(b) Optimization: Designer Final

(c) Optimization: User Final

(d) Automatization: Final


For the result shown in Figure 36 the user took advantage of the type of problem, in literature

they are called pseeudo-3d cause the loads only act in 2 directions. Note from Figure 35b that

after just two hours of computational time, the solution has better fitness than the designers at

the same generation. Moreover, even the very first generation of the User’s optimization has a

better fitness than the designer’s original. This is because the algorithm solves 50 individuals

per generation, something that because of time constraints a designer cannot do.

32

Chapter 5

5. Discussion and Conclusion This chapter presents a discussion on the case studies and validation results, a general

conclusion on the thesis work and suggestions for future development of the computational

tool.

5.1 Results Discussion Pile group design is a time-consuming work specially when the traditional trial and error

procedure is used, this work shows an alternative design approach where modern

computational capabilities provide the option to optimize given solutions and or automatize the

procedure. The results from the validation not only allowed the internal validity problem to be

solved but also the analysis of the solution space for very basic and simple problems exposed

the rapidly increasing number of possible combinations as the number of piles and the size of

the pile cap grow. Considering the order of magnitude that the usual pile group design problem

solution space size it is logical to observe that the algorithm solutions to be local optima or

sub-optimal ones. Nevertheless, most of these solutions are sufficient for design purposes and

satisfy engineering requirements. They would need to be subjected to some kind of post process

where the user’s judgement is needed for making the solution more buildable.

When trialed with Case Studies with different levels of complexity and specific conditions

some more insight on the computational tool capabilities was obtained. From case studies one

and two it can be summed up that the algorithm is suitable for problems with loading in three

directions and several load cases. However, because of the stochastic nature of the optimization

technique it is important for the density of the pile cap to not be too high, so the algorithm has

liberty of trialing solutions without encountering pile collision in most of the individuals. These

two case studies also show that the initial guess has a great influence on the final optimization

product. From case study three, a retaining wall resting over piles (many projects have a large

volume of this type of structure), a useful industrial application of the developed tool was tested

because with minimum user’s manipulation the algorithm obtains a pile group with good

structural behavior in a small amount of computational time.

5.2 Conclusions Optimization of the structural behavior of end-bearing pile groups via a genetic algorithm has

been explored and achieved in this thesis, furthermore the automatization of the design process

shows promising results for retaining walls. Specific considerations are:

33

- The obtained solutions from the computational tool are not global optima.

- The optimization process is highly dependent on the quality of the initial guess.

- The procedure depends on the definition of the fitness function where the decision

to use an external FEM software to solve the structure has an important effect on

the computational time.

The trend in every sector is to turn into this kind of data driven solutions and by the time more

and more interest is focused on them. They provide the possibility to consulting companies,

given that more time is invested into research and development of them, to be part of the digital

transformation and keep their status as drivers of change. It comes with implications such as

the definition of prices, this is a tool programmed by a designer for designers, but the actual

human hours are reduced. An informed user on norms and design is needed for the results to

be correct and reliable must not be treated as a black box.

5.3 Further Development As one of the responsibilities of a pile group designer is to make the verification of the actual

installed (as-built by the contractor) pile group, some modifications can be done to a part of

the scripts so this verification can be done in an automatic way.

The effect on computational time from using an external FEM software to solve the pile group

can be diminished by the application of two computational enhancements of different

complexity. Firstly, a memory system that eliminates the redundancies in the fitness calculation

of pile groups that have been solved generations before. Secondly, the application of a smart

memory system that is trained by the iterations of the FEM solver, so the smart memory system

is eventually able to predict the fitness of the pile groups without calling the solver.

For structures as the third case study, where a good fitness is reached after a relative short time,

the stopping criteria of a fitness limit could be combined with another loop to be applied to the

flow diagram shown in Figure 19 where a range of number of piles are trialed by the

computational tool. Obtaining this way, a reduction in the material of the piles governed by the

structural behavior of the pile group.

The selection of the Genetic Algorithm as the optimization technique for pile groups is based

on the capabilities it has for discontinuous and integer mixed problems. However, an analysis

on several different Meta-Heuristics on the fitness function defined in this thesis could be

analyzed given that problems with known optima are used to make correct comparisons. On

the other hand, the fitness function definition is open for the structural behavior of the pile

group to be only part of the penalization terms and include a calculation of other considerations

such as an LCC analysis of the foundation.

34

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36

Appendix 1

Equation A1.1 can be used to calculate the number of possible combinations with good

precision for any kind of pile group optimization or automatization applied to the

computational tool developed in this thesis.

𝑁𝑠𝑜𝑙𝑢𝑡𝑜𝑛𝑠 ∝ (((𝐿𝑥

∆𝐿𝑥+ 1) ∙ (

𝐿𝑦

∆𝐿𝑦+ 1) −

𝜋𝐷ℎ𝑒𝑎𝑑𝑠2

∆𝐿𝑥 ∙ ∆𝐿𝑦∙ (𝑛𝑝𝑖𝑙𝑒𝑠 − 1)) ∙ 5)


(𝐴1.1)

where:

𝑁𝑠𝑜𝑙𝑢𝑡𝑜𝑛𝑠 is the size of the solution space.

𝐿𝑥 is the pile cap length in the X-direction.

∆𝐿𝑥 is the grid size in the X-direction.

𝐿𝑦 is the pile cap length in the Y-direction.

∆𝐿𝑦 is the grid size in the Y-direction.

𝐷ℎ𝑒𝑎𝑑𝑠 is the pile head minimum separation

𝑛𝑝𝑖𝑙𝑒𝑠 is the number of piles.

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