Optimization Of Pile Groups - DiVA Portal
DEGREE PROJECT, IN STRUCTURAL ENGINEERING AND BRIDGES , SECOND LEVEL STOCKHOLM, SWEDEN 2014 Optimization of Pile Groups A PRACTICAL STUDY USING GENETIC ALGORITHM AND DIRECT SEARCH WITH FOUR DIFFERENT OBJECTIVE FUNCTIONS ANN BENGTLARS & ERIK VÄLJAMETS KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT
Optimization of Pile Groups A practical study using Genetic Algorithm and Direct Search with four different objective functions ANN BENGTLARS ERIK VÄLJAMETS Master of Science Thesis Stockholm, Sweden 2014
TRITA-BKN. Master Thesis 409, 2014 ISSN 1103-4297 ISRN KTH/BKN/EX--409--SE Ann Bengtlars & Erik Väljamets 2014 Royal Institute of Technology (KTH) Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges KTH School of ABE SE-100 44 Stockholm SWEDEN
Abstract Piling is expensive but often necessary when building large structures, for example bridges. Some pile types, such as steel core piles, are very costly and it is therefore of great interest to keep the number piles in a pile group to a minimum. This thesis deals with optimization of pile groups with respect to placement, batter and angle of rotation in order to minimize the number of piles. A program has been developed, where two optimization algorithms named Genetic Algorithm and Direct Search, and four objective functions have been used. These have been tested and compared to find the most suitable for pile group optimization. Three real cases, two bridge supports and one culvert, have been studied, using the program. It has been difficult to draw any clear conclusions since the results have been ambiguous. This is probably because only three cases have been tested and the results are very problemdependent. The outcome depends, for example, on the starting guess and settings for the optimization. However, the results show that the Genetic Algorithm is somewhat more robust in its ability to remove piles than Direct Search and is therefore to prefer in pile group optimization. Keywords: Pile group optimization, objective functions, Genetic Algorithm, Direct Search, Pattern Search i
Sammanfattning Pålning är en dyr men ofta nödvändig åtgärd vid byggande av större konstruktioner såsom exempelvis broar. Vissa påltyper, såsom stålkärnepålar, är mycket kostsamma och det är därför av stort intresse att hålla ner antalet pålar i en pålgrupp. Denna rapport behandlar optimering av pålgrupper med avseende på placering, lutning och rotationsvinkel med syftet att minimera antalet pålar. Ett program har utvecklats, där två optimeringsalgoritmer, Genetic Algorithm och Direct Search, samt fyra målfunktioner har använts. Dessa har testats och jämförts för att hitta de bäst anpassade för pålgruppsoptimering. Tester med det färdigställda programmet har även utförts på tre olika verkliga fall, två brostöd och en kulvert. De erhållna resultaten har varit tvetydiga och några tydliga slutsatser har varit svåra att dra. Detta kan förklaras av att endast tre fall har studerats och att resultaten är mycket problemberoende. Resultaten är bland annat beroende av startgissningen och optimeringsinställningar. Dock visar resultaten att Genetic Algorithm är något mer robust i sin förmåga att ta bort pålar än Direct Search och är därför att föredra vid pålgruppsoptimering. Nyckelord: Pålgruppsoptimering, målfunktioner, Genetic Algorithm, Direct Search, Pattern Search iii
Preface This thesis has been written for the Department of Civil and Architectural Engineering, the division of Structural Engineering and Bridges at the Royal Institute of Technology, KTH. First we would like to thank our supervisors Christoffer Svedholm and Majid Solat Yavari for their support and advice. We would also like to thank our examiner Costin Pacoste who has been encouraging during the project. Our gratitude to ELU Konsult AB for the opportunity to carry out the project there, and for lending us material for our case studies. Thanks to all coworkers at ELU for inspiration and support. Finally we would like to give special thanks to Christoffer Svedholm for the great enthusiasm he has shown and for his help in developing the computer program. Stockholm, June 2014 Ann Bengtlars & Erik Väljamets v
Contents Abstract . i Sammanfattning . iii Preface . v List of Abbreviations . ix 1 Introduction. 1 1.1 Background . 1 1.2 Previous Studies . 2 1.3 Aim and Scope . 3 1.4 2 1.3.1 Aim . 3 1.3.2 Scope . 3 Outline of Thesis . 4 Theory . 5 2.1 Optimization . 5 2.2 Genetic Algorithm . 7 2.3 2.4 2.2.1 Overview . 7 2.2.2 Search method . 7 2.2.3 Genetic Algorithm in Matlab . 8 2.2.4 Genetic Algorithm for pile group optimization. 11 Direct Search . 11 2.3.1 Overview . 11 2.3.2 Direct search in Matlab . 12 2.3.3 Direct search for pile group optimization . 13 Pile Group Analysis . 14 2.4.1 Forces and deformations . 14 2.4.2 Resisting earth pressure . 16 2.4.3 Pile centre . 17 vii
3 Method . 19 3.1 4 5 3.1.1 General structure . 19 3.1.2 Variables . 20 3.1.3 Objective function . 21 3.1.4 Constraints . 25 A Simple Two-variable Optimization . 27 4.1 Method. 27 4.2 Results . 30 Case Studies . 35 5.1 General . 35 5.2 Case I - Intermediary Bridge Support A. 37 5.3 5.4 5.5 6 Pile Group Optimization Program . 19 5.2.1 Background . 37 5.2.2 Results . 40 Case II – Culvert . 48 5.3.1 Background . 48 5.3.2 Results . 50 Case III – Intermediary Bridge Support B . 53 5.4.1 Background . 53 5.4.2 Results . 54 Summary and Comparison . 58 Discussion and Conclusion . 63 6.1 Discussion . 63 6.2 Conclusion . 66 6.3 Future Research . 66 References . 69 A Number of Piles with Discrete Values . 71 B Load Cases – Support A . 72 C Pile Groups . 75 Case I . 76 Case II . 82 Case III. 86 viii
List of Abbreviations α weight factor λi Lagrange multiplier estimates, nonnegative and components of vector λ ρ the positive penalty parameter ϕj penalty parameter Aq transformation matrix ci the nonlinear inequality constraint ceqi the nonlinear equality constraint Cq transformation matrix DX horizontal deformation in x-direction EI pile bending stiffness f fitness value f1-6 forces and moments in pile fT maximum shear force fM maximum bending moment Fq force vector GA Genetic Algorithm i number of load combinations kd modulus of soil reaction Kq pile stiffness matrix L length of construction in y-direction ix
Le buckling length Nd,max,i maximum design normal force in piles in load combination i Nd,min,i minimum design normal force in piles in load combination i Pi ith pile PCdist,start sum of distances to pile centre in starting guess PCX x-coordinate of pile centre PCZ z-coordinate of pile centre Pq force vector at pile cap origin PS Pattern Search R external forces acting on pile cap origin Rd,c,i design compression capacity of piles in load combination i Rd,t,i design tension capacity of piles in load combination i U displacement vector at pile cap origin S global stiffness matrix of pile group si nonnegative shifts and components of the vector s VZ rotation around z-axis x1-6 displacements and rotations in pile xint,ij x-coordinate of intersection between line i and j Xq displacement vector zint,ij z-coordinate of intersection between line i and j x
INTRODUCTION. BACKGROUND Chapter 1 Introduction This chapter starts with a description of the background of the topic and presents previous studies made in the area of pile group optimization. The aims and scope of this thesis are then stated and followed by a brief outline of the report. 1.1 Background Pile groups are an essential part of many bridge constructions in Sweden. Piling is needed when the soil beneath the bridge supports is too weak to carry the loads, which is often the case. Designing a pile group is a difficult and sometimes very time-consuming task. One of the main reasons is that a pile group has a great number of input parameters and variables. There are geometric parameters such as the pile cap thickness, pile type and diameter, location, angle of rotation and batter (slope) of the piles and pile length. Other important parameters are the pile bearing capacity and external loads. All these parameters can be combined in an almost infinite number of ways. The larger the pile group is, the more complex the task becomes. A designer is always searching for an optimal solution, which usually means the most cost effective solution. Some pile types are very expensive, costing up to 300.000 SEK per pile. Therefore, finding a solution with as few piles as possible is often the most cost effective. It is difficult however for a designer to find an optimal solution when the problem is this complex. There are no strict rules or guidelines when designing a pile group, instead designers mostly rely on experience and engineering judgement to establish some of the parameters before starting the analysis. Today, most pile groups are designed by calculating the section forces and deformations in the piles for a specific pile group configuration and then slightly adjusting the configuration and recalculating, until the results are satisfactory. Satisfactory meaning, for example, that the pile group is able to carry the design loads and that there are as few piles as possible in the pile group. Even though several of the parameters have been set, the process of finding a satisfactory solution becomes very iterative. The iterative process and the large number of parameters makes pile group design an ideal target for computer optimization. One of the first attempts at pile group optimization using computers was made in 1981 by James Hill. Since then several more attempts focusing on 1
INTRODUCTION. PREVIOUS STUDIES different aspects have been made, such as those by Hoback and Truman and Chan et. al. (1992a) (2009). This thesis will examine optimization of pile groups. Different optimization algorithms will be used and compared to develop a computer program that performs the design and optimization of a pile group with respect to location, batter and angle of rotation. Today there are computer programs that calculate the forces and deformations in the piles but no programs that provide a practical design and optimization of pile groups such as the one developed in this thesis project. 1.2 Previous Studies As mentioned above, one of the earliest attempts at computerized pile group optimization was made by Hill (1981). In his report Hill presented a computer program that optimized the location and batter in order to reduce the cost of the pile foundation. The optimization was divided into several parts. First the pile cap was divided into a grid where piles were placed at all grid points and then the optimal batter was calculated. When the optimal batter had been obtained a search for the optimal spacing was carried out. This was done by a series of deletion passes where the least or most loaded piles were removed. When a set number of piles had been removed the spacing was increased. The process was repeated until the least number of piles possible was found. In the optimization Hill used the Nelder-Mead simplex method to find the optimal batter. The Nelder-Mead simplex method is a deterministic search method for unconstrained optimization. Hill transforms the constrained variables from the pile group optimization to unconstrained variables in order to use the Nelder-Mead method, which is said to be less time-consuming than methods for constrained optimization. Hoback and Truman presented two papers where they used the Optimality Criteria to optimize the weight of steel in pile foundations (1992a) (1992b). The Optimality Criterion is a numerical gradient-based optimization method, which requires that the problem is modelled mathematically. In their solution, Hoback and Truman varied the pile diameter and batter. Hurd and Truman published a similar research where they optimized the weight of steel for pile foundations varying batter, pile diameter and number of piles using a 3-D computer program (2006). The optimization was performed using the Optimality Criteria. No details are given explaining how the computer program works. The Genetic Algorithm has been used in different forms to optimize pile groups. A hybrid Genetic Algorithm was used to minimize the material volume of the foundation by mainly varying the pile diameter (Chan, et al., 2009). Liu et. al. used an Automatic Grouping Genetic Algorithm to minimize the cost (2012). They varied the pile diameter and the layout of the pile group. The piles were divided into different modules where all the piles had the same characteristics. Another example of pile optimization was performed by Kim et. al., who used recursive quadratic programming to minimize differential settlements in a piled raft foundation (2001). Hwang et. al. used a discrete Lagrange multiplier method to minimize the construction cost of a bridge foundation (2011). There are two examples of pile foundation optimization where only pile length is varied in order to minimize differential settlements of the pile cap. (Chow 2
INTRODUCTION. AIM AND SCOPE & Thevendran, 1987) (Leung, et al., 2010). Similar to Hill, Chow and Thevendran used a direct search method for unconstrained optimization, transforming their variables from constrained to unconstrained, however using a different method of transformation. Several articles have been published where different global optimization algorithms are used to optimize piled grillage foundations (Šešok, et al., 2010) (Belevičius & Šešok, 2008) (Belevičius, et al., 2011). They focus on the optimization methods Genetic Algorithm and Simulated Annealing. Belevičius et al. compared seven different algorithms and found that for grillages Simulated Annealing, Genetic Algorithm and the NEWUOA-algorithm were the best suited (2011). Several of the authors mentioned above have based their optimization programs on the current national design codes. This allows for a more complete design tool, but limits the use of the program to those specific codes. The program developed in this thesis project is based on basic structural mechanics and does not depend on design codes, which makes it a more general tool and will not need updating if the codes are changed. 1.3 Aim and Scope 1.3.1 Aim In this master thesis project a computer program will be developed that can design a pile group that is optimized with respect to certain parameters. These parameters are the coordinates in the horizontal plane, angle of rotation, batter and number of piles. Using this program, two main issues will be investigated. The first issue concerns the optimization algorithm. The two different optimization algorithms Genetic Algorithm and Direct Search will be compared to find out which one is most suitable for the task of optimizing a pile group. The second issue is the objective function of the algorithms. An objective function is the function that is optimized by the optimization algorithm and is the foundation for the optimization. Four different objective functions have been formulated and will be compared to see which is the most suitable for this problem. To conclude, this thesis will answer the following two questions. Which optimization algorithm in this thesis is the most suitable for optimizing pile groups? Which objective function gives the best results in pile group optimization? The answers to these questions will aid in the development of a program for practical optimization of pile groups, which is the final aim of the thesis. 1.3.2 Scope The thesis project will mainly focus on infrastructure foundations with end-bearing piles only. The optimization will be carried out in the programming language Matlab using two different 3
INTRODUCTION. OUTLINE OF THESIS algorithms, Genetic Algorithm and a type of Direct Search called Pattern Search. These algorithms are available in the Matlab optimization toolbox. An external program will be used to analyse the structural mechanics of the pile groups, i.e. no calculations on pile forces and deformations will be performed in the scope of the thesis project. The variables that will be used in the optimization are coordinates in the horizontal plane, angle, batter and number of piles. To improve the possibility of receiving good and usable results, some restrictions and limitations have been set up. Pile type, pile length, thickness of slab and economics in the form of prices of steel and concrete are not optimized. Including all of these parameters in the optimization would have made the problem too complex and too large to fit into the time frame of the thesis work. Even though the parameters pile type, pile length and the thickness of the slab are not optimized, they are still part of the optimization as constant parameters. The designer using the program will be able to define what type of pile or thickness of slab that is to be used in the current solution. Constraints are needed to ensure that the pile group obtained from the optimization works in practice. The following constraints are taken into account in the program; pile capacity, distance between piles, distance between piles and the edge of the slab, deformations and moment capacity. The optimization program will be tested on three different cases. To make the testing of the algorithms and objective functions as general as possible, these cases are different in type, size, number of piles, pile type and loading. 1.4 Outline of Thesis This thesis consists of six chapters. Chapter 1 introduces the topic and gives an overview of previous studies. The aims and scope of the thesis are also stated here. In chapter 2 a theoretical description of the optimization algorithms are presented. Here the Genetic Algorithm and Direct Search methods are explained in detail. The theory of pile group analysis is also described. Chapter 3 presents the structure of the optimization program. Chapter 4 presents a simple two-variable optimization in order to study the algorithms. Chapter 5 deals with the three case studies. Chapter 6 discusses the results and draws conclusions. Future research is also proposed in the final chapter. 4
THEORY. OPTIMIZATION Chapter 2 Theory Chapter 2 starts with an overview of optimization and optimization methods in general and motivates the choice of optimization algorithms used in the thesis. Further, the two algorithms chosen are presented and explained in detail. Finally, the theory of pile group analysis used for the program is described. 2.1 Optimization Optimization is the process of searching for the optimal value of a function. The optimal value can be a maximum or a minimum value, a local extreme point or a global extreme point. A function can have only one optimum or many, depending on the type of function. There are a large number of different search methods and different ways to categorize them. One way is to divide methods into local and global optimization methods, aiming at finding either local or global optima. In a book on the Genetic Algorithm, the author Goldberg mentions three main types of traditional search methods, namely: calculus-based, enumerative and random search methods (1989). The calculus-based methods are, as the name suggests, based on classic calculus. The idea of the enumerative search methods is to search the whole function space one point at a time. These methods are very time consuming and are not practical in the case of pile group optimization, since the number of variables and the function space is to large. The random search methods search the function space at total random and are, similar to the enumerative methods, too inefficient to work for larger problems. Apart from these, there are natural algorithms, such as the Genetic Algorithm, which is discussed further on. Search methods can also be divided into gradient- and non-gradient-based depending on whether they use the derivatives of the objective function. The gradient- or calculus-based methods are either direct or indirect (Goldberg, 1989). The indirect methods solve a system of equations based on the fact that the derivative is equal to zero at an extreme point. The direct method uses the concept of “hill climbing” moving in the direction of the steepest gradient to find an extreme point. The gradient-based methods are all local optimization methods, since they only find local optima. 5
THEORY. OPTIMIZATION The non-gradient-based methods are usually global optimization methods. They can be divided into deterministic and stochastic methods (Zabinsky, 1998). The stochastic methods use some form of randomness to aid the search while the deterministic methods use a predetermined procedure, which results in the same solution every time. Some search methods are capable of handling discrete values and others are restricted to continuous values. Figure 1 below describes one example of how Genetic Algorithm and Pattern Search can be categorized. Pattern Search can be either deterministic or stochastic depending on the poll method used. Gradient-based (local) Optimization algorithms Deterministic Non-gradientbased (usually global) Pattern Search Pattern Search Stochastic Genetic Algorithm Figure 1 – One example of categorization of algorithms. An optimization problem always needs an objective function, in Genetic Algorithm called the fitness function, stating what is to be optimized (Pedregal, 2004). It also needs a clear definition of constraints and boundaries that limit the variables and the objective function itself. For the problem of pile group optimization an algorithm is required that can handle non-linear constraints and a large number of variables. A non-gradient-based algorithm is preferable since the function space is non-smooth and the objective function cannot be modelled mathematically. It would be possible to use a gradient-based algorithm that approximates the derivatives, but this approximation can lead to errors such as loss of accuracy (Powell, 1998). The algorithm needs to be robust and efficient, since the problem includes many parameters and variables and uses a “black box” function. It is also favourable if the algorithm can handle discrete values, since all variables in a pile group are discrete in practice. Even though it is possible to calculate the coordinate of piles down to the last millimetre, the limited accuracy when constructing makes this impossible to carry out. To reduce the risk of error when constructing, it is also good to keep the coordinates as even as possible and to keep the number of different angles to a minimum. Table 1 shows the different algorithms available in the Matlab global optimization toolbox. Simulated Annealing is a powerful optimization algorithm, which would have been a good 6
THEORY. GENETIC ALGORITHM option if it was capable of handling nonlinear constraints. There are two algorithms that fulfil all the requirements above, namely the Genetic Algorithm and Pattern Search. That is why they were chosen for this project. Table 1 - The different algorithms available in Matlab optimization toolbox (The MathWorks, Inc., 2013). Smooth objective function Non-smooth objective function Linear constraints or bounds only Global Search, MultiStart Simulated Annealing All types of constraints Pattern Search, Genetic Algorithm Pattern Search, Genetic Algorithm 2.2 Genetic Algorithm 2.2.1 Overview One of the algorithms chosen for this thesis is the Genetic Algorithm (GA). The GA was developed by John Holland and his colleagues and students at the University of Michigan (Goldberg, 1989). The original aim was to create a search method that was more robust than traditional methods at the time. Inspiration was taken from nature and Darwin’s theory of evolution and survival of the fittest. It has since then been proved both theoretically and empirically to be a robust algorithm even in complex problems. In engineering problems one is often searching for a solution that is good enough within the available time-frame and not the best possible solution. This fits well with what Goldberg writes: The most important goal of optimization is improvement and It would be nice to be perfect: meanwhile, we can only strive to improve. This is one of the GA’s strong points. The
case. Designing a pile group is a difficult and sometimes very time-consuming task. One of the main reasons is that a pile group has a great number of input parameters and variables. There are geometric parameters such as the pile cap thickness, pile type and diameter, location, angle of rotation and batter (slope) of the piles and pile length.
To observe the design capacity, a test pile is constructed and estimated load is given upon the designed pile. There are three kinds of static pile load testing. 1. Compression pile load test. 2. Tension pile load test. 3. Lateral pile load test. A lobal Journal of Researches in Engineering V olume XVI Issue IV Ve rsion I 41 Year 201 E
PHC pile, with a diameter of 1000mm and a wall thickness of 130mm , is adopted for the wall pile frame structure of this project. The pipe pile prefabrication is carried out by dalian prefabrication factory. The parameters of the PHC prestressed concrete pipe pile are as follows: Table 1 parameters of PHC pile of wall pile frame structure .
sheet pile and ground vibrations during sheet pile vibratory driving. And second, to analyze a selected portion of the collected data with focus on the sheet pile - soil vibration transfer. Both aspects of the thesis work aim, more generally, to contribute to the understanding of ground vibration generation under vibratory sheet pile driving.
2.2 High-Strain Dynamic Pile Testing. 2.2.1 The contractor shall perform dynamic pile testing at the locations and frequency required in accordance with section 4.0 of this special provision. 2.2.2 Dynamic pile testing involves monitoring the response of a pile subjected to heavy impact applied by the pile hammer at the pile head.
The pile driving analyzer (PDA) was developed in the 1970's as a method to directly measure dynamic pile response during driving. As the name implies, it was developed to analyze pile driving and evaluate pile driveability, including the range of stresses imparted to the pile, hammer efficiency, etc.
to pile cap resistance to lateral loads. The focus of the literature review was directed towards experimental and analytical studies pertaining to the lateral resistance of pile caps, and the interaction of the pile cap with the pile group. There is a scarcity of published information available in the subject area of pile cap lateral resistance.
The nonlinear springs are defined using API p–y curves at regular depth . intervals, where p represents the lateral soil resistance per unit length of the pile and y is the lateral deflection of the pile (API, 2007). As it was discussed before response of a single pile is different from response of a pile in a pile group due to group effect. One of the most common methods of accounting for .
ANATOMI & FISIOLOGI SISTEM LIMFATIK DAN KONSEP IMUN Atika Dalili Akhmad, M. Sc., Apt . PENDAHULUAN 20 L cairan plasma difiltrasi keluar menuju bagian interstisial, 17 L direabsorpsi oleh pembuluh darah, BAGAIMANA 3 L SISANYA ? Sistem Limfatik sistem yang terdiri dari pembuluh, sel, dan organ yang membawa kelebihan cairan insterstisial ke dalam aliran darah dan filter patogen dari darah. FUNGSI .
