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SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

Parametric design and

optimization of pipe bridges

Automating the design process in early stage of design

ANDREAS GRANBERG JOEL WAHLSTEIN

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Andreas Granberg <agranb@kth.se>

Joel Wahlstein <joelwah@kth.se>

Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges KTH Royal Institute of Technology

Place for Project

Stockholm, Sweden

Examiner

Prof. Raied Karoumi Stockholm, Sweden

KTH Royal Institute of Technology

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pipe bridges

Automating the design process in early stage of design

Andreas GRANBERG Joel WAHLSTEIN

STOCKHOLM, June 2020

TRITA-ABE-MBT- 20198

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Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges Stockholm, Sweden

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Parametric design can be used for structural design. This approach has some clear advantages compared to the conventional point-based approach using different Computer Aided Design (CAD)-software, especially in early stage of design. Since the model is parametrically defined, alternate designs, that are within the scope of the parametric definition, can be explored with little effort from the user compared to the point-based models. In this way, optimization routines can be used to make more informed decisions about the design. Pipe bridges usually have a similar design that is suitable to be defined parametrically.

The aim of the thesis is to automate the modeling of pipe bridges in the early stages of design, to make an integrated analysis and to optimize the structure with regard to material cost and carbon dioxide equivalent-emissions as well as mass of the structure. Further, to investigate in what way these objectives are correlated.

This thesis improves an existing grasshopper script used to design pipe bridges and implement an automatic generation of a Bill of Quantity (BoQ).

The results of the thesis case study suggests that there is potential in using optimization with parametric design to minimize the cost of pipe bridges. With a good parametric design definition alternate designs can be explored with little effort from the user. This benefit to speed up the design process, and allowing the designer to work with adaptable design, could be reasons to turn to a parametric design method.

It should also be stressed that this thesis suggests a correlation between the cost of the structure and the carbon dioxide equivalent emission from the structure. Meaning that while minimizing emissions one could also be minimizing the cost.

Keywords

Parametric design, Optimization, Grasshopper, Wallacei, Pipe bridge, Master thesis, Genetic algorithm, Visual programming

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Parametrisk design kan användas för konstruktionsutformning. Denna metod har några tydliga fördelar i jämförelse med den konventionella punkt-baserade modelleringen som används i olika CAD-programvaror, särskilt i de tidiga skedena av projekteringen. Eftersom modellen är parametriskt definierad kan alternativa lösningar, inom ramen av den parametriska definitionen, utvärderas med lite arbete från användaren. Med hjälp av detta kan optimeringar användas för att ta informerade beslut kring konstruktionen. Rörbryggor har ofta en standardiserad utformning vilket gör dem till ett bra objekt att definiera parametriskt.

Examensarbetets målsättning är att automatisera modelleringen av rörbryggor i tidiga projekteringsskeden, att utföra en integrerad analys och optimering av konstruktionen med avseende på materialkostnad, koldioxidekvivalenter- utsläpp och massan hos konstruktionen. Vidare undersöktes det på vilket sätt dessa tre mål kan vara korrelerade. Examensarbetet vidareutvecklar ett befintligt Grasshopper-script för rörbryggor och implementerar en automatisk generering av en kostnadskalkyl.

Resultaten från arbetets fallstudie påvisar att det finns en potential att använda optimering med parametrisk design för att minimera kostnader hos rörbryggor.

Med en bra parametrisk definition kan alternativa utformningar utforskas med liten ansträngning hos användaren. Fördelen av en snabbare utformningsprocess, och att konstruktören kan arbeta med en anpassningsbar design, är en stor anledning att implementera parametrisk design. Arbetets resultat påvisar en korrelation mellan kostnaden och koldioxid-ekvivalenten för utsläpp hos konstruktionen. Detta betyder att kostnaden även minimeras genom att man minimerar utsläppen.

Nyckelord

Parametrisk design, Optimering, Grasshopper, Wallacei, Rörbrygga, Examensarbete, Genetisk algoritm, Visuel programmering

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This master thesis was carried out during spring 2020. The provider of the project is the company Sweco Structures AB.

To start off we would like to express our thanks to Sweco Structures for giving us the opportunity to write and conduct this thesis, as well as providing us with material.

We had the possibility to meet great engineers, and a lot of people showed interest in our work. A big thank you to Peter Neovius who helped us with the Bill of Quantity for costs and emissions. Furthermore, we would like to thank Sebastian Andersson for his help and knowledge about pipe bridges, Grzegorz Drapiñski for his interest and help regarding the analysis and Baptiste Woerli for taking his time to introduce us to the original script. We also want to thank Prof. Raid Karoumi at KTH for his willingness to discuss the thesis with us along the way and his continuous help with the report. Finally we want to thank our supervisor at Sweco, Samir El Mourabit, who have been very enthusiastic and supportive throughout the project. He also provided us with great insight and came up with ideas for both solutions and new perspectives on problems.

Stockholm, June 2020

Andreas Granberg & Joel Wahlstein

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FEM Finite Element Method

GH Grasshopper

PBD Point-Based Design PD Parametric Design BoQ Bill of Quantity

CO2e Carbon dioxide equivalent FE Finite Element

CAD Computer Aided Design

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1 Introduction

1

1.1 Aim and research question . . . 2

1.2 Demarcations. . . 3

1.3 Background . . . 3

1.3.1 Applications for parametric design based on previous studies . 4 1.3.2 Parametric design in civil engineering . . . 5

1.3.3 Pipe bridge . . . 6

1.3.4 Optimization . . . 6

1.4 Thesis overview . . . 6

2 Theory

7 2.1 Pipe bridges . . . 7

2.1.1 Conceptual design . . . 9

2.1.2 Truss types . . . 9

2.1.3 Cross-sections . . . 12

2.1.4 Manufacturing . . . 14

2.1.5 Transportation . . . 14

2.1.6 Assembly . . . 15

2.2 Optimization in structural design . . . 15

2.2.1 Optimization techniques . . . 17

2.2.2 Meta-heuristic optimization techniques . . . 18

2.2.3 Genetic algorithms . . . 18

2.3 Parametric design . . . 20

2.3.1 Rhinoceros . . . 21

2.3.2 Grasshopper . . . 21

2.3.3 Karamba3D . . . 23

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2.3.4 Galapagos and Wallecei . . . 24

3 Developing the Grasshopper definition

30 3.1 Previous work . . . 30

3.2 Modifications and improvements to the original script . . . 31

3.2.1 Modification of member groups . . . 31

3.2.2 Changing how intermediate support points are treated . . . 33

3.2.3 Bill of quantity . . . 34

3.2.4 Grasshopper and RFEM connection . . . 34

3.2.5 Foundations . . . 35

3.2.6 Broken geometry with specific directions . . . 36

3.2.7 Wider towers in corners . . . 36

3.2.8 Karamba3D . . . 38

3.3 Definition overview after modifications . . . 40

3.4 Optimizations used in the thesis . . . 42

3.4.1 Optimization . . . 42

3.4.2 Penalty functions. . . 43

4 Case-study

46 4.1 Geometry . . . 48

4.2 Loads . . . 51

4.2.1 Permanent loads . . . 51

4.2.2 Variable loads . . . 53

4.2.3 Omitted loads . . . 55

4.3 Load combinations . . . 55

4.3.1 Ultimate limit state . . . 56

4.3.2 Serviceability . . . 56

4.4 Settings for the optimization features . . . 56

4.5 Verification . . . 57

5 Results

60

6 Discussion and conclusion

68 6.1 Discussing the result from the case study . . . 68

6.1.1 Wallacei and Galapagos . . . 70

6.2 Discussion about parametric design approach . . . 72

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6.3 Conclusions . . . 73 6.4 Future work . . . 74

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2.1.1 A steel pipe bridge during construction to run pipes overhead. This particular pipe bridge was designed by Sweco to run pipes in several

levels. . . 8

2.1.2 A 3D-view of a normal form of design for a pipe-bridge. . . 9

2.1.3 Normal stress σ in case of tension (a) and bending (b) [6]. . . 10

2.1.4 Three common truss designs for bridges. The different colors indicate whether the members are in compression or tension. Red members are in compression and blue members are in tension. . . 11

2.1.5 Different parts of the truss pipe bridge. . . 12

2.1.6 Cross-sections of a typical truss pipe bridge structure. . . 13

2.1.7 Example of profile connections. . . 14

2.1.8 Oversized load in terms of length [50]. . . 15

2.2.1 Flowchart diagram that represent the Genetic algorithm. . . 20

2.3.1 Parametric design example. . . 21

2.3.2The process for drawing a line in Python and Grasshopper. . . 22

2.3.3 The pictures shows a definition of a simple truss geometry. The definition is created in Grasshopper and is previewed in Rhinoceros. The definition has been made by Karamba3D [7]. . . 23

2.3.4Galapagos component in Grasshopper. . . 27

2.3.5 Wallacei interface in Grasshopper. . . 28

2.3.6Flowchart of the basic NSGA-II procedure. . . 29

3.2.1 The subfigures show the different element IDs. In this work the horizontal beams over supports are divided into two different element IDs, namely (2) and (9). . . 32

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3.2.2A visualization of a tower and simple support foundation and excavation, foundation is gray and the excavation is red in the figure.

Parts of the excavation geometry is hidden for visual representation. . . 36

3.2.3 The difference of corners in the old and the new script, the top image is from the old script were the two pipe bridges shares the corner vertical. The bottom image is from the new script with a clear offset from each corner. . . 37

3.2.4The ”Loads”-component . . . 38

3.2.5 Example of a side mesh used for wind load. . . 39

3.2.6OptiCroSec component in Karamba3D. . . 40

3.3.1 Flowchart. . . 41

3.4.1 Four lists in Grasshopper presenting how the allowed placement is defined for each span. . . 45

4.0.1 The total length of the pipe bridge is approximately 178 meters and is a box truss bridge made of steel and designed by Sweco. The numbers indicate the support numbers. . . 46

4.0.2The pipe bridge shown from above. The numbers represent the support location. 4,5,7 are tower supports. The other is plane supports. . . 47

4.1.1 Plan and elevations of the pipe bridge. . . 49

4.1.2 Section of the pipe bridge supports. All lengths are theoretical lengths between system points in the computer model. . . 50

4.2.1 Pipe load. . . 52

4.2.2Equipment load. . . 53

4.2.3Snow load. . . 54

4.2.4Wind load acting in +y direction. . . 54

4.4.1 The location of the intermediate supports was allowed to move +/− 2 meters along the path of the pipe bridge from the case study location. . 57

4.5.1 Location of each support node with numbering. . . 58

4.5.2 Member location of the considered element 538 and the support node locations. . . 59

5.0.1 3D view in Rhinoceros of the case study (blue) and the optimization (red). 62 5.0.2YZ-plane view of the two designs. . . 62

5.0.3XZ-plane view of the two designs. . . 62

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5.0.4Penalty functions variation during an optimization. . . 64

5.0.5Fitness objective Cost. . . 65

5.0.6Fitness objective CO2. . . 66

5.0.7 Fitness objective Mass. . . 67

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Introduction

The building sector is competitive where many firms compete for the projects. It is of interest for the firms to find ways to make the work more efficient, for example by inventing new working methods to improve their productivity. In an early stage of design, a lot of time must be spent on making models and testing different design solutions. In many cases this must be done before the job has been assigned to the firm. This also applies when the project has been assigned, and sketches are sent back and forth between the company and the client. This is a common dialogue that aims to result in a satisfactory and feasible design. Therefore, it is of importance that this process in the early stage is both practical and time- and cost- effective [33]. Today, most companies use the same structural design process, which is a Point-Based Design (PBD) where the development of the design is based on a single decision in a step-by- step process [33]. This is a time-consuming process when changes needs to be made, meaning that the design often has to be remade from scratch. One possible way to make the structural design work in the early stage more efficient is by using a Parametric Design (PD) approach. Since the ”parameters” or ”variables” in a parametric design method easily can be adjusted, many different variants can be created to find the best solution [33]. This is something that the industrial department at Sweco has reflected upon, and has resulted in an interest on how this can be more effective - especially for pipe bridges.

A pipe bridge is a structure mainly used to guide pipes over ground from one position to another. These are often found in industrial and manufacturing facilities where the function of the structure is central, not the appearance. This means that the

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consultants who regularly take on projects of this type work according to designs that have worked before. The design therefore becomes a bit standardized. However, each assignment is different, meaning that there are no ready-made catalog solutions that can be put together into a finished product, and the modeling of the design needs to be done from scratch each time. This is a time-consuming process. Much can also change during the early stage of the design. It is rare that the original idea works, which means that many changes in the design has to be made. Since the structure often is built up with the same basic truss-structure, the implementation of PD for this type of structure would be truly beneficial since it gives the opportunity to automatize the modelling.

The total greenhouse gas emissions from the construction and the real estate sector in Sweden in 2017 were approximately 18 million tonnes of Carbon dioxide equivalent (CO2e), which accounted for about 30% of Sweden’s total greenhouse gas emissions.

Of the total emissions from the sector, construction represents around 50 % in 2017.

Compared to 2016, the emissions were increasing in both the real estate sector and construction, but most in the field of construction [5]. Structural engineers have a great responsibility in the design of a structure. The structural engineer’s expertise is to come up with a solution that holds the right dimensions and optimize when possible.

By not using more material than needed, it becomes cheaper and does not affect the environment as much. A PD approach to generate models of structures gives the opportunity to generate a lot of different variants with little effort that can be evaluated regarding the costs and the enviromental aspects.

The implementation of a PD approach can save time, have economical and environmental advantages, especially in early stages of preliminary design for structures with a similar design. This thesis therefore feels relevant to explore these possibilities.

1.1 Aim and research question

The aim is to automate the modeling and the structural analysis of pipe bridges in early stage of design. Further, to optimize the structure with regard to cost, CO2e emissions and mass of the structure, and to investigate how these objectives relate to each other for optimization.

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1.2 Demarcations

The Bill of Quantity (BoQ) that has been produced in this thesis include material costs and CO2e emissions of the material being used for the structure, including foundation.

The material costs are based on unit price times the quantity of a certain activity.

The unit price for the activities are based on the definition stated in AB04 [4], which includes the material costs, manufacturing in the workshop, transportation, assembly at the work site, labor and overhead costs for establishment, portacabin, machinery, staff management and contractor fees. The unit price does not include costs after final inspection, such as maintenance, repair and disposal. The unit price for the activities was provided by Sweco.

Further analytical and geometrical demarcations are listed below.

• Only one type of truss design is evaluated in this thesis, namely Pratt truss design.

• Detailing such as connections between parts of a long pipe bridge, welding, bolting and additional attachments are not included in the analysis.

• Not all loads and load combinations are considered for the case-study comparison. Due to the simplified nature of the geometry certain loads had to be omitted.

• The Finite Element (FE)-calculations are based on first order analysis.

Karamba3D allows for second order analysis, but for this preliminary design first order analysis is deemed enough.

• No FE-calculations are made for the substructure such as piles and foundations.

1.3 Background

In this section, background for parametric design and optimization will be described with regard to applications and previous studies. Further, a thesis overview will be described.

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1.3.1 Applications for parametric design based on previous studies

Nowadays, implementation of PD can be found in many fields, including the vehicle industry, the fashion industry, architecture and structural engineering. In the vehicle sector PD has been used to improve different systems in vehicles [19, 58, 60]. In the field of fashion, Wang et.al. [52] used PD for customizing shoes, and concluded that a PD approach makes this process easy and quick. In architecture, PD can be used to generate creative and complex geometries [12, 54]. Yu et.al. [57] found that the design creativity potentially is increasing when working in PD environments, compared to more conventional geometric modelling environments. In the field of structural engineering, PD can be used as support for the early stages of the design process, by easily and quickly providing several alternative solutions by adjusting some parameter values [29]. PD is an effective tool for reusing existing design solutions [26] and PD gives the opportunity to implement optimization routines of structures. Hasançebi et.al. [14] and Tejani et.al. [48] used PD as a tool to optimize the design of pin- jointed structures and truss structures, respectively. The most common optimization routines performed in the building sector is about determining the optimum cross- sectional dimensions for the members of a structure or the optimum geometry and/or optimum topology [1], something that Fleck Fagel Miguel et.al. [30] studied for truss structures. Kretov and Shataev [42] used optimization to determine the optimum topology of hot load-bearing structures. In another study Putra Gerry Liston et.al.

[32] optimized stiffening parts for ships with the aim to minimize the mass, and thus reduce CO2e emissions. The design variables included the number and type of stiffeners, stiffening distances and the thickness of the plate. By changing the variable values, an optimized ship could be evaluated. The focus of the research regarding optimization in the field of construction has mostly been on the development of algorithms and to evaluate which optimization techniques to use in different fields.

A collection of optimization techniques that are widely used are called meta-heuristic search techniques [15]. They provide good solutions at a reasonable time, even though an optimal solution cannot be guaranteed [45]. These are often metaphors of some natural or man-made process [45]. Some algorithms of this sort are genetic algorithms, simulated annealing, particle swarm optimization, harmony search and taboo search [15]. In research made by Lagaros [23] the environmental and economic effects of optimization, when used in construction, was investigated, demonstrated through real

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examples. Lagaros concludes that implementation of optimization in the construction industry would lead to major economic and environmental savings. He also identified problems within this area regarding optimization, mainly the skepticism from the practicing engineers to try new working methods. He believes that the state has an important role in implementing this working method, and real case studies are important to demonstrate its benefits. Rempling et.al. [33] compared three existing bridges, which were designed in a traditional point-based approach with an alternative design for each bridge developed with a PD approach. Since the parameters in a PD approach can be adjusted easily, many different variants of the bridges can be created.

The study showed a reduction of 20% - 60% in material cost and CO2e emissions for the evaluated bridges. Koch [21] mentioned in his research that the development of the PD script can be time-consuming and associated with high cost. Therefore many designers are reluctant to use PD, even though it can have time-saving effects once the script has been developed. Further, the authors conclude that visual programming can make modeling of parametric models more flexible, transparent and time-effective.

Grasshopper (GH) is a software for visual programming used for PD.

This thesis build upon and develop a previous master thesis performed during 2019 [2]. The original work presented a GH script that could build the geometry of a pipe bridge based on certain user defined parameters, and the only modelling the user had to perform was to draw a line representing the center-line of the bridge.

1.3.2 Parametric design in civil engineering

In the building sector, PD was first adopted by architects and is often associated with design of complex roofs and facades, which can be difficult to model in a conventional 3D CAD-modeling software. However, a PD approach can be used in order to obtain a lot of other purposes within this field as well. In repetitive work, parametric design can help automate the production of geometries that are essentially similar to one another.

For example, structures such as roof trusses and pipe bridges are suitable to automate with parametric design, if a script can be created in a smart way. Then you can adjust the input parameters to the dimensions you need for length, width, height and so on for the pipe bridge or roof truss you are looking to create. By defining the geometry on the basis of set parameters, it is easy and quick to make changes in appearance in the early stages, which would otherwise be a time-consuming work.

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1.3.3 Pipe bridge

A pipe bridge is a normal form of pipeline transportation over ground, which includes pipe bridge, anchoring system, pipe racks, pipe supporting, supporting structure etcetera. The structure usually sits in industry and manufacturing facilities and the pipes transport for example; gas, liquid and pulp. A common structure for the pipe bridges is a steel truss bridge, which is a strong and effective structure. The pipe bridge rests on hinged supports at both ends of the pipeline, and for long spans the bridge also rests on intermediate supports. A more in depth description of pipe bridges are presented in Chapter 2.1.

1.3.4 Optimization

In practice, engineers rely on experience, try out various design alternatives, and use computer programs for engineering calculations, sometimes together with for example cost models until a solution that can be considered satisfactory is obtained [28]. A more effective way is to implement an algorithm for solving structural optimization problems. By using PD it makes it possible to let a computer program try different combination of variables and evaluate the outcome to present you with a good solution.

1.4 Thesis overview

The thesis is organised as follows. In Chapter 2, the theories used in this thesis will be presented and explained. This includes a description of pipe bridges and their usage, while also presenting the optimization for the thesis and an introduction to parametric design and the used software. In Chapter 3, a review of what has been implemented into the GH definition and how the process of developing the script was performed. Chapter 4 describes a case study comparison of an optimized pipe bridge with an existing pipe bridge made with a conventional 3D CAD-modeling software.

The process of verifying the model produced by the script is also presented. The results of the case study comparison is presented in Chapter 5 together with results that compare the optimization software. Chapter 6 concludes the work and reflects on the implementation of parametric design for early stages in design for pipe bridges, and what conclusions that can be surmised from the case study.

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Theory

2.1 Pipe bridges

Pipes are often used to transport different mediums such as gas, liquid and manufacturing products with different temperature and pressure [19]. This applies, for example, to the distribution of water from the water treatment work to our homes, to ventilation ducts that have the task of transporting air from inside a building to outside or vice versa, to transport manufacturing products in an industrial park. These pipes can be located on the ground surface, below the ground surface or above the ground surface. When pipes are located above the ground surface and in an open place without the posibility to be attached to a wall or ceiling, bridges for these pipes are often used. These have the purpose to carry pipes over a certain distance [24]. Pipe bridges are often to be found in industry and manufacturing facilities. An example of the installation of a pipe bridge section can be seen in Figure 2.1.1. The reason for the common occurrence of pipe bridges in these operations is one; industry and manufacturing plants often need to transport different mediums from one place to another and two; it is often a relatively small area, also often occupied by other objects, which makes it difficult to place pipes at the ground surface. The piping via the pipe bridges usually consists of several pipes lying next to each other and at different levels. The pipes can vary in size and it is not uncommon with a diameter up to one meter.

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Figure 2.1.1: A steel pipe bridge during construction to run pipes overhead. This particular pipe bridge was designed by Sweco to run pipes in several levels.

Regarding the analysis of this type of structure, the guidance given by the building codes is minimal. Since pipe bridges are not considered to be buildings nor bridges, they are treated as “other structures” [3]. Lack of uniform industry standards for this topic makes the organizations adopt their own engineering standards [3]. The pipes are free to move longitudinally and transversally, but rails are present at each side of the pipes to limit the sideways movement of the pipes [3]. The loads considered are normally gravity load (dead load of structure and pipes), medium temperature, flow load and wind load [24]. The biggest loading that occur is liquid loading during test pressure of the pipes. This will cause strong dynamic loads of impact and could cause movements of the pipes. The longitudinal forces has to be taken care of by bracing of support structure, and should be considered in the design of the supporting structure. The structural design of pipe bridges varies widely depending on the plant operations and the associated plant standards. Failures of pipe support systems could potentially impact the health, welfare and safety of employees due to pipe breakage or leaks [3].

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2.1.1 Conceptual design

Pipe bridges do not have a specific design for all occasions. The design is adapted to what is required depending on the circumstances. For example, if the span is very long, a suspension bridge may be the only possible option. However, when several options exists regarding the type of structure to be used, the engineers may have their preferences about the design. For pipe bridges in general, appearance is not the primary factor, but the function of it. Pipe bridges for medium long to long lengths, with the possibility to have intermediate supports, are normally designed with parallel truss beams coupled with cross beams, as seen in Figure 2.1.2. The pipe bridges are usually built up of beams made in steel, and the joints are welded. The truss beams normally consists of square hollow sections (SHS- profiles).

Figure 2.1.2: A 3D-view of a normal form of design for a pipe-bridge.

2.1.2 Truss types

A truss is a structure made up of a collection of members connected to each other, usually in a triangular pattern. Further, two assumptions are made for a truss structure. One is that all joints in the structure can be represented by a pinned connection, meaning that members are free to rotate at the joints. The members of a truss are often rigidly connected using a gusset plate or by welding. But if the center line of all the members at a joint intersect at the same point, it is reasonable to assume that the joint behave as a pinned connection. This means that it only transfer axial forces and no bending moment. The second assumption is that loads only are applied to the joints of the truss. Trusses are used in many structures, often when high demands are placed on strength and stiffness and to withstand large spans.

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Examples of places where trusses are used are bridges, radio masts, tower cranes and roof structures. Trusses are generally made of two materials, steel and wood. The reason why trusses are so effective, from a structural point of view, is that the members are utilized very evenly, namely mostly under axial forces. In Figure 2.1.3 a square cross-section of a beam and its normal stress distribution is shown when loaded with axial force and bending moment. If only axial force is present in the cross section, the stress distribution is uniform, which means the whole cross-section can be evenly utilized. On the other hand, if a cross-section is subjected to bending, a linear stress- distribution with maximum stress occurring at the top and bottom of the cross-section.

The rest of the cross-section is not much utilized. At the horizontal line (z-axis) for example, the normal stress is zero. [17]

(a) Normal stress distribution σ in a beam cross section under tension.

(b) Normal stress distribution σ in a beam cross section under bending around the z-axis.

Figure 2.1.3: Normal stress σ in case of tension (a) and bending (b) [6].

There are many different truss designs, but a few truss designs are more popular and very common. The different designs carry loads in different ways. Three examples of common truss designs are shown in Figure 2.1.4, together with typical axial forces in the truss members under gravity load from the structure.

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Figure 2.1.4: Three common truss designs for bridges. The different colors indicate whether the members are in compression or tension. Red members are in compression and blue members are in tension.

In the Howe truss (top image), the vertical members are in tension, while the diagonal members are in compression upon the application of gravity load. The opposite applies to Pratt trusses (middle image), which has a geometry of a Howe truss if looking upside-down, where the vertical members are in compression and diagonal members in tension. Members in compression usually needs to be thicker then members in tension, to reduce the risk of buckling. The Pratt design is therefore more cost-effective than Howe truss design, since the longer, diagonal members, can be thinner. Warren truss (bottom image) uses equilateral triangles, meaning all members have the same length. It uses less members than Howe and Pratt trusses, meaning its more material efficient. The diagonals alternate between tension and compression, so it has some quite long members in compression [44].

The GH definition developed in this thesis will produce a Pratt truss bridge.

The different members are shown in Figure 2.1.5. The structure consists of two vertical parallel trusses, connected with crossbeams. At the bottom of the bridge diagonal bracing are supporting the structure from shear deformation and load acting

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horizontally to the side of the bridge.

Figure 2.1.5: Different parts of the truss pipe bridge.

2.1.3 Cross-sections

Figure 2.1.6 shows the typical cross-sections of the structural elements in a pipe bridge structure. Cold formed rectangular hollow sections (KKR)1 is preferred to hot rolled hollow sections (VKR) 2 since they are cheaper, even though some residual stresses reduce their resistance, which has to be considered in the design. Further, square profiles are preferred to rectangular profiles since they ensure a symmetry of resistance in both directions to resist load coming from different directions. HEA-profiles are normally used as an upper horizontal beam of the pipe bridge supports to ensure good support of the bridge. The columns in the supports can also have HEA-profiles, although KKR-profiles are more common.

1KKR stands for Kallformade KonstruktionsRör.

2VKR stands for Varmformade KonstruktionsRör.

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(a) KKR-profile [46]. (b) HEA-profile [47].

Figure 2.1.6: Cross-sections of a typical truss pipe bridge structure.

The material properties of the KKR-profiles is steel S355 J2H according to EC3 [46]

with fyk = 35.5kN /cm2. The geometric properties of KKR can be seen in standard EN 10219-2 [46]. The geometric properties of HEA can be seen in standard EN 10365:2017 [47]. The dimensions of the cross-sections vary along the pipe bridge, depending on the forces in the individual beams. Further, since it is not recommended to do welding in corners of the KKR-profiles, connecting profiles should have a dimension of four to six times the thickness smaller than the base profile, see Figure 2.1.7.

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Figure 2.1.7: Example of profile connections.

2.1.4 Manufacturing

Truss structures are well suited for being manufactured industrially in suitable transport units that can be assembled together at the construction site [13]. In the workshop, smaller parts are joined by welding, and preparation for the assembling of the truss beam units are prepared by hole punching and welding of connection details [51]. The time saved by the prefabrication of the elements means lower costs during erection of the structure and earlier commissioning [16]

2.1.5 Transportation

Transportation of maximum dimensions of loads are defined by the Swedish Transport Agency. Transportation of an oversized load can in some cases be carried out without dispensation. A load with a maximum width of 3.5 meters can be transported without dispensation if the conditions in the regulations given in TSFS 2010:141 [49] are followed. A load with a maximum length of 30 meters can be transported without dispensation if the conditions in the regulations given in TSFS 2010:142 [50] are followed. One of the conditions is that the load are not allowed to project in front of the transportation vehicle and more than 5.0 meters behind the center of the rear axis. The headroom for road bridges in Sweden are at least 4.5 meters. Based on the bed height of the trailers, which is 1.0 meter, the height of the load should not be more

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than 3.5 meters.

Figure 2.1.8: Oversized load in terms of length [50].

2.1.6 Assembly

At the construction site the pre-made sections of the pipe bridge are lifted into place, normally using mobile cranes, and the sections are joined together with bolts. One of the advantages with a bolted connection is that the assembly of the structure can be made quick and easy [51].

2.2 Optimization in structural design

Optimization has been of interest for a long time. For example, Euclid proved 300 BC that a square has the largest possible area of all rectangular shapes for a given total length of the sides [43]. Another example when optimization is being used in everyday life is when planning a weekend to Rome. When planning for visiting a set of attractions in a limited time frame, to walk in the most effective way, in terms of time and/or walk distance.

Optimization in structural design is a big research area, and a lot of research has been published on this topic. When designing a structure parametrically, optimization routines are possible for the practicing engineer. Determining the optimal topology, the optimal geometry and/or the optimal cross-sectional dimensions for the members of a structure, given that it should be able to carry a certain external load with minimum weight of the structure, are examples of what optimization in structural design can be about [15]. Most optimization problems can be mathematically stated as follows:

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[43]

FindX = (x1, x2, ..., xn)which minimizes fi(X) i = 1, 2, ..., no (2.1) subject to:

gj(X)≤ 0, j = 1, 2, ..., ng (2.2)

hk(X) = 0, k = 1, 2, ..., ne (2.3)

xlm ≤ xm ≤ xum, m = 1, 2, ..., ns (2.4)

Where X is the vector containing the design variables, fi(X)are the objective functions (or fitness functions), gj(X) are the inequality constraints, hk(X) are the equality constraints, and equation 2.4 are the side constraints. In equation 2.1, it is possible to formulate the objectives as a maximization problem. Also, the inequality constraint in equation 2.2 can be turned in the other way [55]. no, ng, ne and ns are the number of objective functions, number of inequality, equality and side constraints, respectively [43].

The design variables are the variables included in the optimization process, and describe shape, topology and geometry of the structure, or they define size or properties of structural elements [43]. The objective functions are decided by the user. In structural optimization, it can be to minimize construction cost, minimize life cycle cost, or to maximize stiffness [43]. If i = 1 in equation 2.1, there is only one objective function. The constraints in 2.2 and 2.3 are called design constraints that comes from building codes like Eurocode, where safety and serviceability requirements are stated [43]. The side constraints in 2.4 are non-behaviour constraints. These constraints restrict the acceptable range of potential solution of the problem, by limiting the number of the design variable xm. This is also called the search space. In the construction industry, type and size of available structural elements are available in a limited editions, and these are examples of non-behaviour constraints [43].

In the following sections of this chapter, different optimization techniques will first be explained, and than a deeper explanation will be made of one of these techniques, namely meta-heuristic optimization techniques. To end the chapter, the basic concept of the genetic algorithm will first be explained (which is one meta-heuristic optimization technique), and than a more detailed explanation on how the genetic

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algorithm work in the Galapagos component and the Wallacei component in GH (they both use the concept of Genetic algorithms but they are programmed in different ways).

2.2.1 Optimization techniques

The simplest optimization technique is the brute force technique where you iterate through every possible solution by trying every single combination of the design variables. This becomes very time-consuming rather quick. Consider a padlock that has four wheels where each wheel can take 10 digits. This rather simple lock allows for 104different solutions. To brute force this kind of problem you simply have to test all the solutions one at a time. For a person this could take quite some time but for a computer this can be done in a matter of seconds. But when the problems are more complicated this approach is not a very efficient method since the number of solutions quickly can become very numerous.

When high-speed computers were developed in the middle of the 20th century, it made it possible to do optimization on larger and more complex structures. Since then, a lot of different optimization techniques has been developed [43].

Optimization techniques can be divided into two categories, namely, function and parameter optimization techniques. In function optimization, an object under consideration is described by a number of unknown functions and in the end, the optimum form of these functions will be found and result in one optimum continuous function. In parameter optimization, the optimum values of design variables for a specific problem are obtained. Meta-heuristic methods are a subset of parameter optimization techniques [43].

You can classify optimization problems based on, for example: number of design variables (single variable/multi variable), number of objective functions (single objective, multi objective) and presence of constraints (unconstrained, constrained) [43]. There are a lot of numerical optimization techniques. This thesis focuses on one class (family) of numerical optimization techniques, called meta-heuristic optimization technique [43]. Optimization problems in structure design based on these techniques have proven to be robust and quite effective [15].

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2.2.2 Meta-heuristic optimization techniques

’Meta’ in ’Meta-heuristic’ means ’higher level’ and ’heuristic’ means ’to discover by trial and error’ [55]. Meta-heuristic methods are built on general ways to search for good solutions, but they do not guarantee to find the best solution [43]. The method works well if you are satisfied with obtaining good or almost optimal solutions [55].

Meta-heuristic search techniques are developed to find the optimized solution within an acceptable computer time [15]. There are a lot of different types of meta-heuristic techniques. Many of them are nature-inspired algorithms. Among these are genetic algorithms (GA), simulated annealing (SA), particle swarm optimizer and ant colony optimization [15], [55], [43]. Meta-heuristic algorithms are shaped by ideas inspired from nature, and the basic idea behind the algorithms is to mimic natural phenomena into a numerical algorithm, such as survival of the fittest and the cooling process of molten metals through annealing for genetic algorithms and simulated annealing, respectively [15]. There is no search algorithm that outperforms any other search algorithm in all cases, according to the ”No free lunch theorem” [53]. According to this theorem, the best performing algorithm depends on the problem to be solved. A lot of research has been done comparing different meta-heuristic search techniques for one specific problem, such as De Corte and Sörensen [8], Hasançebi et.al. [15], Yildiz [56], Deb and Gulati [9]. In the next section, one meta-heuristic search technique will be described, namely genetic algorithms. Genetic algorithm is a common technique to solve a wide range of optimization problems [55].

2.2.3 Genetic algorithms

Among the most well-known and common meta-heuristic search techniques are genetic algorithms, which is a population-based search method, which converge into a solution in an iterative process. It is inspired by the biological evolution based on Charles Darwin’s theory of natural selection in biological systems. The evolutionary process involves the genetic operators selection, crossover and mutation, which are implemented in the algorithm, to direct the population toward an optimum design [43, 55]. Since the development of the Genetic algorithm in the 1960s and 1970s by John Holland and his colleagues, several variants of this technique have been developed and are nowadays extensively used in structural optimization problems [43, 55]. The advantages with genetic algorithms is that it avoids being trapped in local optimum

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and works in many types of optimization problems [55]. Even if many variants of Genetic algorithms have been developed, the fundamentals are the same regarding the computational steps of this technique. They are outlined below and in the flowchart diagram in Figure 2.2.1 [15].

Step 1. Initial population: A predefined number of individuals (or genomes) are generated through a random approach. Each individual of the population is described by a string of values (or genes), equally many in number as the number of design variables. The individuals are assigned a random value (or gene) from each design variable [15]. The individuals are now distributed over the search space at random (think of a landscape) [43].

Step 2. Evaluation of population: Each individual is evaluated (or calculated) and is given a fitness score, based on the equation for the objective function [15]. Here, the objective function is a predefined function including the design variables. The values for the design variables (the individuals) works as input for the objective function, that calculates the value of the function (the fitness score), which indicates the merit of each individual [15].

Step 3. Selection: The selection process works in a randomized method. However, the individuals with high fitness scores are more likely to be selected and reproduced, compared to the least fit individuals that have a very small chance of being selected to reproduce [15].

Step 4. Crossover: The selected individuals create pairs of two. Each pair creates offspring by crossover, meaning that genetic information between the individuals are combined. There are a few ways in which this can be done, for example by taking the mean of mom and dads gene values [15].

Step 5. Mutation: When a child individual has been created, mutation is applied on the genes. The mutation operation is random for the population, as well as what gene to mutate [15]. Mutation is one factor that slows down the premature convergence, which is important to prevent getting stuck in local optimum [43].

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Step 6. Termination: The child population replaces the current population, and steps 2 - 6 are repeated until a specified number of generations is reached [15].

The Genetic algorithms can be summarized as in the flowchart diagram shown in Figure 2.2.1.

Figure 2.2.1: Flowchart diagram that represent the Genetic algorithm.

2.3 Parametric design

Parametric design is a method to create geometries based on a set of parameters and mathematical operations and rules in an algorithmic way. By changing the values of the parameters, the algorithm will produce a new version of the geometry. One way to use parametric design is with the visual programming tool GH, which is an add-on to Rhinoceros. The geometry is defined in GH and previewed in the Rhinoceros viewport.

This is also possible to do with a common programming language such as Python. The difference is that Python is text based and the other one is visual (Grasshopper). One strength of parametric design is how the model is structured. If a set of parameters define the curves that act as the base for the model, a change to the parameter will echo with the rest of the design and produce an updated model. A visual example of this can be seen in Figure 2.3.1.

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(a) Slider value for the height set to 10. (b) Slider value for the height set to 19.

Figure 2.3.1: Parametric design example.

2.3.1 Rhinoceros

Rhinoceros is a general 3D computer graphics and CAD software, specialized on geometries based on NURBS curves and surfaces. In Rhinoceros anything between simple 2D-lines and complex surfaces can be created. The area of application is big in Rhinoceros, not only for architects, but also for other industries such as furniture design, jewelry design, kitchen application and automotive sector [34]. The potential and functionality of Rhinoceros is greatly improved by the plug-in Grasshopper.

2.3.2 Grasshopper

Grasshopper is a programming editor incorporated in Rhinoceros. Grasshopper is used to design algorithms that represent a series of commands for how Rhinoceros should automate tasks [22]. But instead of programming textually, row by row, it is done graphically with functional blocks representing different operations into a sequence of actions in an algorithmic procedure. Figure 2.3.2 show the process for creating a line between two points in Python respectively in Grasshopper.

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(a) Python script.

(b) Grasshopper script.

Figure 2.3.2: The process for drawing a line in Python and Grasshopper.

A Grasshopper definition is a step by step procedure, meaning if you change data somewhere along the definition, Grasshopper will recompute the solution from that point and forward, and automatically the preview of the solution in Rhinoceros will update. There are two types of user objects in Grasshopper, parameters and components. Parameters store data and components perform actions that result in data. Data is used as inputs for the components, which results in new data that can be used by another components and so on.

Working with GH has a lot to do with working with tree structures that are implemented in the software. These tree structures can handle any kind of data that is produced in GH and allow the user to control the data flow through the definition.

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Figure 2.3.3: The pictures shows a definition of a simple truss geometry. The definition is created in Grasshopper and is previewed in Rhinoceros. The definition has been made by Karamba3D [7].

2.3.3 Karamba3D

Karamba3D is a Grasshopper add-on program for Finite Element Method (FEM) calculations. Once a geometry has been defined in Grasshopper with objects such as lines and surfaces, it can be converted into an analytical model using Karamba3D. The Karamba3D component can work with different elements such as beam-elements and shell-elements [31]. The set-up for a structural analysis using Karamba3D most times consists of the following steps, where all the steps are made using different Karamba3D components.

1. Define the model elements, in which GH-objects, such as line- and surface- objects, together with chosen material, and cross-sections is transformed into structural elements.

2. Assemble the model. A finite element model is then created from given entities like the beams/shells of the model from point 1 above, together with support conditions and by applying various types of loads and load cases.

3. Analyze model, in which the given model is calculated, for example the deformation and stresses of the model under external load.

4. Retrieve results and display results of the structural analysis, which lets you

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review the results in the Rhinoceros viewport, or by getting access to the results using Karamba3D components.

5. Disassembling of the structural model. This lets you dissect the structural model in more detail, for example extraction of cross-section types after a cross-section optimization, or in case you want to modify the existing model.

2.3.4 Galapagos and Wallecei

Once the parameterized geometric model has been turned into a FE model using Karamba3D it is possible to perform structural optimization algorithms on the finite element model. Two such optimization algorithms that can be used inside of Grasshopper is the genetic algorithm in Galapagos and Wallacei.

The genetic algorithm in Galapagos

In Grasshopper there is a component called Galapagos, which can be used for optimization routines. Galapagos implements two meta-heuristic solvers (one uses a genetic algorithm and one uses simulated annealing algorithm). In the following text, the genetic algorithm (called Evolutionary Solver in the Galapagos component) will be explained in a more detailed way, compared to the explanation of the basic Genetic algorithm in section 2.2.3.

Galapagos is a single objective optimization solver, meaning that it only con- sider one objective function. If you have several fitness values to optimize for, you have to combine these into one fitness function, with weighing factors for each fitness value to avoid one variable being stronger than the others [35]. A description of how the genetic algorithm works in Galapagos is presented below, and is based on a number of documents published by David Rutten, the creator of this algorithm, as well as the creator of the program Grasshopper.

In step 1, a number of individuals (or “genomes”) are created. Each individ- ual is defined by a number of values (or “genes”) that comes from the variables (or

“sliders”) included in the optimization algorithm. The variables are all set at random values. For example, an individual can be i = 3, j = 6, k = 16 if there are three variables, and where the values of ”i”, ”j” and ”k” are within the range of allowed values for each

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variable. Together, the individuals form the first population G[0]. Galapagos has a population multiplication factor for the first generation, called “Initial boost”, which multiplies the number of individuals with a factor bigger than 1. In this way more individuals are evaluated and reduces the risk of being trapped in local optimum, which could be the case if to few individuals are present [36, 41].

In step 2, the loop of the Genetic Algorithm starts. The initial population G[0] is passed into this step and becomes G[n].

In step 3, the solver calculates the score (or “fitness”) of the objective func- tion (or “fitness function”) that has been defined, for each and every individual and returns a single numeric value for each individual [36].

In step 4, the individuals are ranked based on their fitness, from most fit to the least fit. Since the individuals were created randomly and with only a few individuals, it cannot be assumed that the highest scoring individual has found the absolute best result for the optimization problem [36].

In step 5, the current population G[n] is now replaced by the next generation G[n+1]. In Galapagos, there are two ways in which this can happen. An individual can either “survive” the generation transition, or mate with another individual to produce offspring. The number of individuals for each generation is kept constant, except for the initial step where the possibility to multiply the population with a factor exists. In Galapagos, the procedure for creating a new generation is a combination of selection, breeding and mutation algorithms. They are explained in step 5a, 5b and 5c respectively.

Step 5a. This step is about selecting individuals from the current population G[n] for mating couples. The individuals can be picked isotropically (all individuals have the same opportunity to mate, regardless of their fitness), by exclusive selection (only the fittest K% are allowed to mate, everyone with the same chance to mate) and by biased selection (all individuals have a chance to mate, but fitter individuals have a higher chance of being picked to find a mate) [40].

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Step 5b. Individuals that have been selected to mate in step 5a need to find a partner. In Galapagos, a partner is chosen based on genetic distance (meaning how similar the gene values are to set individual). In Galapagos, the user decides whether the individual prefers very similar individuals, or very different individuals, or somewhere in between. This is what the setting ”inbreeding factor” in Galapagos is. None of the extremes are healthy. In genetic algorithms, high inbreeding result in incestuous couple. This is a risk since it rapidly decreases population diversity, which reduces the chances of finding a better solution and getting stuck in a solution which may not be as good as it could be. The other extreme is to exclude all individuals who are close to you. This results in zoophilic mating, which is also not preferable. If individuals mate, that are both fit but very different from each other, it is likely that the offspring will fall somewhere in between and become lame. It is preferable to balance in-breeding and out-breeding. In Galapagos, you specify an in-breeding factor (between -100% and + 100% , total out-breeding and total in-breeding, respectively).

50 - 75% usually gives a good result [39].

Step 5c. Now, when a mate has been found, offspring can be created. In this step it is determined how the genes of the parents are transmitted to the offspring. In Galapagos, this can be done in several ways. In Galapagos, all genomes have the same number of genes.

- Alternative 1: Crossover. The offspring inherits a random number of genes from the mother, and the remaining of the genes comes from the father. In Galapagos the genes are placed from left to right in the genome. If the genome consists of 5 genes and the first two come from the mother, then the last three comes from the father.

-Alternative 2: Interpolation. Brand new values for the genes of the offspring are generated by taking the mean value of the genes from mom and dad.

-Alternative 3: Preference blending. the genes are created in a similar way as in Alternative 2, but the blending is weighted with the relative fitness of both parents.

If the mother is fitter than the father for example, her gene values will be more prominent in the offspring [37, 39].

Step 5d. When the offspring has been created (in step 5a to 5c), there is a risk that the bio-diversity will decrease in the population. To counteract this, mutation is used and this is what this step is about – mutation of genes. In Galapagos, point

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mutations are exclusively used because it is not possible to increase or decrease the number of sliders, and therefore the number of genes are fixed. Mutation change gene values randomly, and the event is random [38].

Step 6. A new generation of individuals has been created G[n+1] that is no longer random. For each individual the fitness is computed, and hopefully this generation has approached a potentially good solution. To ensure that strong individuals do not disappear during the process, Galapagos has something called

”Maintain High Fitness”. This is set to ten percent. When offspring are created, the weakest ten percent of this generation are compared to the previous strongest ten percent. If the individuals of the previous generation are stronger than the current individuals, the previous individuals will take their place in the list.

Step 7. The loop starts over, from Step 2 to Step 6 (ignoring the poorest per- forming individuals to breed the best performing individuals). This continues until no progression has been made during a number of iterations, or until a specific fitness value has been reached, or until the maximum number of generations has been reached. Galapagos remembers all genomes being created and no genome will be created again that once existed. In Figure 2.3.4, the Galapagos component is shown.

The component needs two inputs, a fitness function to evaluate and the genomes, with values to play around with (that are the variables defining the fitness function).

The settings of the component is seen in the right hand side in Figure 2.3.4.

Figure 2.3.4: Galapagos component in Grasshopper.

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The genetic algorithm in Wallacei

Wallacei is a multi-objective optimization tool, meaning that the number of objective functions can be more than one. The solver considers several objective functions simultaneously to determine an optimum solution [43].

The basic interface for the Wallacei component in GH can be seen in Figure 2.3.5.

Compared to the Galapagos interface, Wallacei features more settings that gives the user more control over the optimization, graphs and plots to follow the optimization.

The solver also allows the user to save arbitrary data for each iteration, in Figure 2.3.5 the saved data is a list with the profiles used for each iteration. The possibility to run several objective functions simultaneously is a valuable property considering the optimization in this thesis is performed against both the total mass of steel in the structure but also the expected cost of the design as well as emissions.

Figure 2.3.5: Wallacei interface in Grasshopper.

Wallacei makes use of the multiobjective evolutionary algorithm nondominated sorting genetic algorithm II (NSGA-II) [27]. This specific algorithm was put forward in 2002 and aimed to lower the computational complexity of, at the time, current evolutionary algorithms using nondominated sorting from O(M N3) to O(M N2) (M is the number of objectives and N the population size) as the worst case scenario complexity, while also improving the lack off nonelitism and the need for specifying a specific sharing parameter. Elitism means that the algorithm keeps the best individuals from the parent, and child population for the next generation. The algorithm uses a crowding measure to give a rank to individual designs, it also sort

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the individuals with the nondominated scheme. These methods give a good spread of solutions and convergence close to the Pareto-optimal frontier. [10]

The algorithm is used in all manners of sectors ranging from optimization of stochastic computer networks to mixed-flow pump impeller to design of dry-type air-core reactor in electrical power distribution, and modifications to the core algorithm have been made to increase the performance further [18, 25, 59]. The basic algorithm of NSGA-II can be summarized as in the flowchart diagram shown in Figure 2.3.6.

Figure 2.3.6: Flowchart of the basic NSGA-II procedure.

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Developing the Grasshopper definition

3.1 Previous work

This thesis aims to further develop and build upon a previous master thesis performed during the spring of 2019. The original work presented a GH script that could build the geometry of a pipe bridge based on certain user defined parameters that defines the width, height, support location and height of the supports to mention some of the parameters, and the only modelling the user would have to perform was to draw a line representing the centerline of the bridge. The line is drawn in Rhinoceros as a poly-line with only perpendicular turns, points are placed at corners and on straight lines if an intermittent support is wanted. The parameters in the GH script range from fixation direction in towers to the height of supports. The script was made in two different versions, one that focused on the optimization of the individual elements utilizing the FEM tool Karamba3D in GH to compute utilization of elements and displacements.

The other version further developed the script so it would be able to handle different kinds of design of the bridge and the supports. Things like multiple horizontal and vertical bridges, but removing many of the analytical capabilities using Karamba3D [2].

This thesis improves the latter of the versions of the script to be able to keep the geometric improvements while also adding even further geometric freedom and other functions.

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The script has a live-link between Grasshopper and Tekla Structures, which means that the geometry is exported into a Tekla model automatically.

The script has been used by the department responsible for designing pipe bridges in Sweco Structures to evaluate the script in a real design scenario. The result showed that the script could be used to a large extent to model the pipe bridge, and it saved time in this stage. Some improvement areas with the script was recognized to make the script more efficient.

3.2 Modifications and improvements to the original script

3.2.1 Modification of member groups

The original definition grouped the different elements into group of members. This is important to be able to assign different properties, such as cross-sections and materials to different elements. The elements in the previous work were divided into 9 group of members. The member groups have been modified from being a collection of 9 members to 10 members, as visualized in Figure 3.2.1 and seen in Table 3.2.1. The horizontal beam over supports has been modified to be two different types of horizontal beam over support. One, named loaded horizontal beam, is underneath where the bridge is supported on the tower supports. The second is called not loaded horizontal beam which is located on the tower supports at the upper tower edges not affected by the load of the pipe bridge. This was made to be able to have the freedom of having more local control of the members making up the bridge structure.

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(0) Diagonals (1) Support Bracing (2) Not loaded Horizontal beams

(3) Diagonal Bracing between Chords

(4) Columns in Supports (5) End Span Verticals

(6) Chords (7) Cross-beams (8) Verticals

(9) Loaded Horizontal Beams

Figure 3.2.1: The subfigures show the different element IDs. In this work the horizontal beams over supports are divided into two different element IDs, namely (2) and (9).

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Table 3.2.1: Member group names and element ID.

Element ID Group name

0 Diagonals

1 Support Bracing

2 Not loaded Horizontal Beams 3 Diagonal Bracing between Chords 4 Columns in Supports

5 End Span Verticals

6 Chords

7 Cross-beams

8 Verticals

9 Loaded Horizontal Beams

3.2.2 Changing how intermediate support points are treated

The support location in the previous definition was fixed to the location the user had defined when drawing the center line of the pipe bridge. This by specifying the points along the polyline in Rhinoceros. To be able to use the support location as a parameter for optimization the creation of these nodes was changed. The input curve was changed to a polyline that only contained the points at the start, the end and the perpendicular corners. A parameter that controlled the number of supports in each span was created, ranging from 0 to 3. To configure the placement of these points along the curve another parameter was implemented. Numerical sliders with a range of 0 to 1 with two decimals was used to specify the parametric placement of each point along the curve.

To control the placement of each support, each span was divided into segments according to the number of supports the sliders indicate. If the gene controlling the number of supports in one span shows 1 for that particular span, a single point will be inserted along the curve of that span. The second gene controlling the placement is then able to move this point from 0 to 1 with 0 meaning that the division is at the start of the span and 1 being at the end of the span. These two placements will result in the created point overlapping with the start and end point of the span, and such duplicate points are purged to avoid duplicate geometry. If the number of supports is 2 for a specific span that span length is divided in the middle and 1 point is created in each half of the span. These points can move freely in their half of the span, but if both of the points are placed in the middle of the span, one of the points will be purged to once again avoid duplicates. This is to avoid duplicate solutions that would appear with for example a placement of 0.25 and 0.75 compared to 0.75 and 0.25.

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3.2.3 Bill of quantity

By implementing a calculation, as well as a method of printing a BoQ of selected elements required for the chosen design of the pipe bridge, the designer can make even further informed decisions about the design. It also gives the designer an extra tool in their communication with the client and an easy way to show the client the economic effect of decisions. A more comprehensible BoQ makes it possible to translate some of the quantities to emissions, like CO2e equivalents. This makes it possible to optimize the design with regards to the cost of the design and to emission level. Certain steel profiles may have better structural qualities, but the cost of the profile or the emission might be higher than other similar profiles. Regarding the emission levels at this stage, only benchmark values of CO2e equivalents for individual elements in the BoQ supplied by Sweco and the effect that emissions will have on the price have been implemented.

Grasshopper is a very good environment for this kind of computations since the script can both read data from excel- or csv-files that contains any kind of data regarding the price or the CO2e equivalents, and with proper use of indexes and names this data can be connected to the constructed geometry. By constructing a template for the BoQ, Grasshopper can write the computed data directly to the template with the press of a button. Much like how changes in Rhino and Grasshopper can be seen directly by the user, the written values will change live according to changes made in the GH design.

The template will be made with space for the user to add other elements that the script does not take into account. These own additions will not be effected by any change made to the GH design since GH only overwrites the specific excel-cells and not the whole document.

As mentioned in section 1.2 the whole life costing of the bridge is not included. Ongoing activities such as maintaining and repairing for bridges are something that is very expensive [20]. This thesis does not include any costs after final inspection of the bridge.

3.2.4 Grasshopper and RFEM connection

A popular and powerful program for finite element analysis is RFEM. It was of interest to see how the data exchange between Grasshopper and RFEM worked, which Dlubal

References

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Närmare 90 procent av de statliga medlen (intäkter och utgifter) för näringslivets klimatomställning går till generella styrmedel, det vill säga styrmedel som påverkar

Den förbättrade tillgängligheten berör framför allt boende i områden med en mycket hög eller hög tillgänglighet till tätorter, men även antalet personer med längre än