MScThesis

Computational Mechanics(Int. Master’s Program)

Technische Universitat Munchen

Master’s Thesis

Linear and Nonlinear Instability Analysis of the WindTurbine Tower using Model-Driven Software

Development

Author: Milad Kiaee1st supervisor: Dr. -Ing. Roland Wuchner2nd supervisor: Dipl. -Ing. Claudio HillmannThesis handed in on: January 15, 2013

I hereby declare that this thesis is entirely the result of my own work except where other-wise indicated. I have only used the resources given in the list of references.

February 4, 2013 Milad Kiaee

Acknowledgments

This thesis would not have been possible without the guidance and the help of severalindividuals who in one way or another contributed and extended their valuable assistancein the preparation and completion of this study.

First and foremost, my utmost gratitude to Dr. Wuchner , deputy of the structural anal-ysis chair at Technical University of Munich, and encouragement I will never forget. Dr.Wuchner has been my inspiration as I hurdle all the obstacles in the completion this re-search work.

Besides my supervisor at the university, I would like to thank Claudio Hillmann, my su-pervisor at Fraunhofer IWES, for the insights in Java he has shared and all the staff of theFraunhofer IWES especially Paul Poisson and Michael Strobel for all the help they offered.

Last but not the least, my family: my parents and my sisters for supporting me spirituallythroughout my life.

February 4, 2013 Milad Kiaee

v

Abstract

The idea of hierarchical and tool independent models is discussed in this thesis. UsingEngineering Design Data (EDD) concept developed at Fraunhofer IWES models are de-fined in object oriented and parametric manner. Main models introduced within the scopeof this thesis are two towers named TowerSegmentedConical and TowerElastic whichare detailed and reduced models respectively. These two models are implemented us-ing Eclipse Modeling Framework and extended in Java. This provides the user with thepossibility to generate the code required for the simulation directly from pre-modelingstage before entering pre-processor phase of a tool-specific software. The main challengeat this level besides software development is to distinguish the appropriate parametersand sub-components required for the analyses. This requires both deep knowledge inphysical behavior of the model and software skills. Furthermore in order to distinguishthe needed level of detail developer should be aware of the analysis type which is going tobe performed on the model. The chain of detail parametric hierarchy starts from the mostdetailed tower (TowerSegmentedConical) and the information is passed until the coars-est tower (BeamTowerFE) through the so called Transformation classes. This coarse towerthen enters the overall wind turbine simulation to get back the load results in time seriesfor final assessment in the detailed model.

Furthermore in this thesis the goal is to perform suitable Instability Analysis on thetower models and to discover which parameters have the major and minor effects on thesimulation results. Difficulty to distinguish Geometric Imperfection and over predictingload factors in linear stability analysis usually forces designers to perform the nonlinearanalysis. Elaboration and time cost of such analyses usually push users back from imple-menting such analysis when it does not seem to make much difference in results. Insidethis thesis Geometric Nonlinear Instability Analysis (GNIA) is performed in order toverify the effect of the door and other geometric imperfections on the buckling load fac-tor results. Implementing the door and joints for the TowerSegmentedConical model isshown to have impact on buckling load factors. The extra simulation time by adding thesedetailed features is a huge drawback. To distinguish the qualitative effect that these detailshave on the results is one of other goals of this thesis. Having models in different levels ofdetail gives the opportunity to the user or further designer to choose the appropriate levelof detail which fits ones desired solutions and resources.

Finally models representing National Renewable Energy Laboratory (NREL) 5MW windturbine tower in different levels of detail are compared by performing linear and nonlinearstability analysis. Following this approach the effect of reduction or ignoring parametersin different types of buckling analysis is distinguished.

vii

viii

Contents

Acknowledgements v

Abstract vii

Outline of the Thesis xiii

I. Introduction and Theory 1

1. Introduction 31.1. Fraunhofer IWES and The OneWind Project . . . . . . . . . . . . . . . . . . . 31.2. Engineering Design Data at Fraunhofer . . . . . . . . . . . . . . . . . . . . . 31.3. Project OneWindTower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4. Procedure and goal inside the scope of this Thesis . . . . . . . . . . . . . . . 6

2. Free-Standing steel tubular towers 132.1. Buckling Strength of Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Buckling Theory 173.1. Euler buckling for long columns . . . . . . . . . . . . . . . . . . . . . . . . . 173.2. Buckling of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3. Effective Parameters on Buckling of Symmetric Shells . . . . . . . . . . . . . 19

3.3.1. Geometric Imperfection and Residual Stresses . . . . . . . . . . . . . 203.3.2. Ring Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.3. Non-linearity and Boundary conditions . . . . . . . . . . . . . . . . . 20

3.4. Specific to the Tubular Steel Tower . . . . . . . . . . . . . . . . . . . . . . . . 233.5. EuroCode-3 A rough approximation for buckling [16] . . . . . . . . . . . . . 233.6. Solution Techniques for Buckling Analysis . . . . . . . . . . . . . . . . . . . 25

3.6.1. Block Lanczos Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6.2. Linear Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . 253.6.3. Nonlinear Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4. Eclipse Modeling Framework and OneWindSDK 314.1. Eclipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2. Eclipse Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3. Meta-Model and Actual Model . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4. Making models using EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5. Xtext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6. XDoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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Contents

4.7. Advantages of using EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.8. Engineering Design Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.8.1. Creation of an EDD using Plug-ins . . . . . . . . . . . . . . . . . . . . 334.9. JUnit test plug-ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.10. OneWind and OneWindSDK . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.10.1. Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.10.2. Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.10.3. Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

II. Modeling 37

5. The Two Towers 395.1. Philosophy of detailed and coarse parametric models . . . . . . . . . . . . . 395.2. A Discussion on the selection of parameters . . . . . . . . . . . . . . . . . . . 405.3. TowerSegmentedConical Model Plug-in . . . . . . . . . . . . . . . . . . . . . 40

5.3.1. TowerSegmentedConical Class . . . . . . . . . . . . . . . . . . . . . . 445.3.2. TowerSegmentedConicalSegment Class . . . . . . . . . . . . . . . . . 445.3.3. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3.4. TowerSegmentedConicalToTowerElasticTransformation Class . . . . 465.3.5. Generation of the ANSYS inputfile . . . . . . . . . . . . . . . . . . . . 475.3.6. Benefits and Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4. TowerElastic Model Plug-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4.1. Segments and Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4.2. EulerBucklingLoad Method . . . . . . . . . . . . . . . . . . . . . . . . 565.4.3. HomogenBeamInfo Method . . . . . . . . . . . . . . . . . . . . . . . . 565.4.4. Representation of the Door in the TowerElastic Model . . . . . . . . . 565.4.5. Analytic Buckling Assessment based on Eurocode-3 . . . . . . . . . . 575.4.6. Transformation to the Beam Tower Model . . . . . . . . . . . . . . . . 575.4.7. Benefits and Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5. Fixed Tower Model - the Simplified Wind Turbine . . . . . . . . . . . . . . . 58

III. Implementation, Results and Conclusion 61

6. Implementation and Results 636.1. Buckling Analysis in Different Level of Detail . . . . . . . . . . . . . . . . . . 64

6.1.1. Euler Column Representation . . . . . . . . . . . . . . . . . . . . . . . 646.1.2. TowerElastic buckling assessment based on EUROCODE 3 . . . . . . 656.1.3. ANSYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.1.4. TowerSegmentedConical, Shell Model in Different Levels of Detail . 666.1.5. Linear Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 696.1.6. Nonlinear Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . 77

6.2. Two Tower Models, General Discussion on Results . . . . . . . . . . . . . . . 906.2.1. TowerSegmentedConical Model . . . . . . . . . . . . . . . . . . . . . 906.2.2. TowerElastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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Contents

7. Conclusion 93

Appendix 97

Bibliography 97

xi

Contents

Outline of the Thesis

Part I: Introduction and Theory

CHAPTER 1: INTRODUCTION

This chapter presents an overview of the thesis and its purpose in a general approach.

CHAPTER 2: FREE-STANDING STEEL TUBULAR TOWERS

An introduction to wind turbine towers and buckling phenomena of this component ofthe wind turbine.

CHAPTER 3: BUCKLING THEORY

Provides the detailed discussion on the buckling theory and different methods of bucklinganalyses from euler column theory to the much more sophisticated nonlinear approach forthe shell buckling analysis.

CHAPTER 4: ECLIPSE MODELING FRAMEWORK AND ONEWINDSDKA short introduction of Eclipse IDE and Eclipse Modeling Framework (EMF) as the maintool for modeling inside this thesis.

Part II: Modeling

CHAPTER 5: THE TWO TOWERS

The modeling procedure, purposes and discussion on the goal of implementation of twospecific tower models (TowerSegmentedConical and TowerElastic) are main part of thischapter.

Part III: Implementation, Results and Conclusion

CHAPTER 6: IMPLEMENTATION AND RESULTS

This chapter is the most important part of the thesis. Buckling analysis is performed ontowers with different level of detail (column, beam, shell with different components). Fur-thermore two types of buckling analyses (linear and nonlinear) are considered for capablemodels.CHAPTER 7: CONCLUSION

Conclusion based on buckling simulation results. This chapter talks about the accomplish-ment of goals of this thesis which are implementation and comparison of the differenttypes of buckling analysis on towers with different levels of detail and effects of these de-tails on buckling load handed by the respective simulation type.

xiii

Part I.

Introduction and Theory

1

1. Introduction

1.1. Fraunhofer IWES and The OneWind Project

Fraunhofer IWES is a branch of Fraunhofer Society for the advancement of applied re-search, which deals with the wind energy systems and more specifically wind turbines.

Wind turbines are designed using complex simulation tools. The Fraunhofer Institutefor Wind Energy and Energy System (IWES) in Bremerhaven, Germany has experiences,in loads simulation and consulting in system design as well as the development of its ownsoftware family called OneWind.

The aim of OneWind is to develop a software system that allows to combine all thecomponents of a wind farm in a numerical model with different levels of detail.

The scientists and engineers of system simulation and evaluation at Fraunhofer IWESdeal with the theoretical prediction of the behavior of the entire wind turbine, consideringall loads acting on it. These include material procurement, site conditions, mechanical andtechnical processes and predictable environmental impacts. All these factors determinethe complex and highly dynamic nature of a wind Turbine. Experiments on the simulationmodel provide early, important insight that can be incorporated into the design of systemcomponents. In parallel, simulation tools are continuously developed and optimized forthe growing demands in the market.

In development of the OneWind software family insights from years of cross-systemresearch at Fraunhofer IWES such as, control, structural, aerospace and mechanical engi-neering, computer science and physics are implemented. In addition, results of numerousresearch projects, such as the Offshore Code Comparison Collaboration(OC) projects of theInternational Energy Agency (IEA), are implemented directly in software development atIWES. As a result it made a more accurate load calculation possible, which leads to the de-velopment of more reliable and less expensive equipment. The OneWind family follows acomplete range of simulation tools that cover all successive steps in the development of amodern wind turbine. OneWind is a tool set which deals with all models, at different levelof detail, in a consistent manner throughout the design process.

1.2. Engineering Design Data at Fraunhofer

The concept of Engineering Design Data (EDD) has been developed by Fraunhofer IWES inorder to create a consistent physical model of the wind turbine containing different levelsof detail. As a consequence, parametric models for physical components are created. EDDrepresents a parametric model which describes the geometry and the physical behavior ofa component in an object oriented manner.

In order to define any conceptual object there must be a sufficient number of indepen-dent parameters which are capable of building the space of the object as a part of the

3

1. Introduction

underlying world.

For instance in Fig 1.1 one can see the black vector object can be instantiated either byusing the two purple or the two red vectors, hence both of them could represent the objectspace.

Figure 1.1.: Object Space

Different modeling theories and analyses for the description of the same physical com-ponent require different sets of parameters. This forces the EDD at different levels of detailto contain different parameter sets. An EDD can be converted then to make a respectivelyreduced EDD by applying a transformation on parameters of a more detailed EDD.

4

1.2. Engineering Design Data at Fraunhofer

Figure 1.2.: Transformation of EDD

As It is depicted in Fig 1.2, defining a reduced EDD requires mapping and reduction ofthe parameter set.

Defining the EDD only by sets of parameters makes the model independent of the sim-ulation tool since it does not know in advance about any simulation which will be per-formed on it in the future. This can be translated as data encapsulation at different levelsof modeling and simulation with the same philosophy as that of the object oriented pro-gramming concept.

The Generator has the responsibility to generate input files related to the specific sim-ulation software. This should not be confused with the model generator that is a part ofan EMF meta-model (will be introduced in following chapters), which generates the baseclasses for the EDD model.

Each specific type of simulation in every kind of software requires information withrespect to the geometric and material characteristics. Simulation of the model can be per-formed independent from the parametrization, hence the only difference that the genera-tor should consider is in translating the model input for different tools from the parametricinformation which the EDD provides. For example transformation of parameters into dif-ferent units set of parameters for the different tools, which describe the same physicalmodel.

There are many advantages to define models using the EDD concept. As an examplein this way the user defines the model by means of inputting parameter values for thedetailed EDD and then transformer creates reduced models and the generator generatesinput for the specific simulation tool. In the whole sequence the user does not need tohave any special knowledge about the simulation software syntax. Furthermore reducedmodels do not need to be explicitly redefined after each time user input parameters for the

5

1. Introduction

detailed model are changed, as the procedure will be executed automatically. This showsitself more clearly when the iterative design process is implemented, which itself requiresmany transformations and generations.

The EDD concept is defined and implemented using the Eclipse Modeling Framework(EMF) and the generator uses Xtext syntax definitions to create the appropriate input filefor specific simulation tool (ANSYS in this thesis) 1.

1.3. Project OneWindTower

The main approach of the OneWindTower project is to provide a detailed hierarchy of theparametric tower models from the most detailed EDD to the coarsest one. The informationwhich has been set in the detailed EDD gets categorized and more simple as the model getsreduced.

Different simulation types need only a specific amount of information in order to obtainthe desired level of accuracy. For instance the buckling analysis requires more geometricinformation than the Eigen-frequency analysis.

All tower models in fact intend to represent the same reference tower but in differentlevels of detail. Reduction of the EDD is done by means of the transformation classes. Thesuitable EDD which contains the set of parameters format for the Modelica overall turbinesimulation 2 enters the simulation in order to obtain load results. The load result thenis transformed back to the detailed model which contains desired amount of parametricinformation for the user and is used for further detailed simulation and assessment.

OneWindTower is enhanced with a product launcher implemented in eclipse which is agraphical user interface. By using this product launcher it is possible to follow the wholeprocedure from the modeling to the analysis phase in a user friendly manner.

The OneWindTower project is intended to be merged into the OneWind of the Fraun-hofer IWES.

1.4. Procedure and goal inside the scope of this Thesis

In this thesis the buckling analysis of the wind turbine tower is accomplished by using theEDD concept of Fraunhofer IWES. The wind turbine in addition to the tower consists ofseveral parts and in order to create an accurate tower simulation one must take couplingeffects of the other components also into account.

From the theory [7] it is proved that the buckling phenomena is sensitive to geometricand material imperfections as well as to the type of boundary condition. This requires ahigh level of detail in modeling of the structure. Within this thesis suitable models fordifferent types of buckling (local and global) and with different levels of accuracy of thebuckling load factor and the effect of inclusion of various detailed parameters are created.

Two basis models are introduced within the period of this thesis. The detailed modelis named TowerSegmentedConical and the name of the reduced model is TowerElastic.

1EDD and EMF will be explained in more details in following chapters.2Modelica overall simulation gather all components of the wind turbine in order to calculate the load in time

series

6


TowerSegmentedConical is the main basis for the buckling analysis in ANSYS which itselfcan represent various levels of detail by removing some parameters from it.

In Fig 1.3, the procedure, which is meant to be followed in the OneWindTower project,is depicted.

7

1. Introduction

Figure 1.3.: Thesis framework

8


The procedure of the design process can be explained shortly as the following algorith-mic work flow:

1. A user defined TowerSegmentedConical object is instantiated representing the de-tailed model.

2. The transformation is performed on the TowerSegmentedConical and creates theTowerElastic as the reduced model.

3. *Another transformation should be performed on TowerElastic in order to create theBeamTowerFE.3

4. *BeamTowerFE should generate the input for the overall simulation of the wind tur-bine in Modelica.

5. *After the simulation load results are known in time series.

6. *Load results should be transformed back to the TowerSegmentedConical.

7. TowerSegmentedConical generates the input file for the ANSYS. This input file iscombined with the create and solve commands in order to perform the bucklinganalysis of the detailed tower.

8. Linear and nonlinear buckling simulations are performed in ANSYS.

9. *The buckling load should be transformed back to the TowerSegmentedConical inorder to generate the assessment documents based on the GL handbook 4 for thewind turbine tower.

The relation between different models of towers can be seen in Fig 1.4. The main concernfor the detailed analysis is to investigate the effect of the inclusion of parameters on theresults. In Fig 1.5 one could see tower in different levels of detail.

3star sign (*) means that it is outside of the scope of this thesis4Standard format for documentation of wind turbines

9

1. Introduction

Figure 1.4.: Project OneWindTower Models [1]

10


Figure 1.5.: Tower model, Level of details is increasing from left to right

11

1. Introduction

12

2. Free-Standing steel tubular towers

The tower is an important component of the wind turbine. This importance could be seenin different aspects. The cost of manufacturing and installing the tower can rise up to 20percent 1 of the overall turbine cost . Increasing the tower height is beneficial for energycapture but it makes installation and maintenance of other components such as rotor bladeand nacelle more difficult, hence more expensive. On the other hand increasing the towerheight makes usage of larger diameter rotor blades possible which yields more power.An optimization process could realize the appropriate tower height with various desiredtechnical cost constraints.

The cost of construction increases with more slope as the height of the tower increases.Additionally the site conditions may have a great influence on design. The difference be-tween wind loads behavior at inland and off-shore sites play and important role in select-ing the optimum tower height with respect to highest potential of the wind. These factorsalong with requirements from the costumer and the manufacturer can make selecting asuitable height a challenging problem.

The most common tower type currently in use is the free-standing steel tube or monopole.Mastery of the vibration behavior has made it easier to use this type so that steel tubulartowers with very low design stiffness can be implemented. It has thus become possible tolower the structural mass, and thus the costs of the towers, considerably.

Figure 2.1.: Different wind turbine tower types [2]

2.1. Buckling Strength of Towers

One crucial criterion, which plays an important role especially in thin-walled steel tubulartowers, with low natural bending frequency, is the resistance to local buckling of the tube

1This share is getting more for offshore wind turbines

13


wall. In Fig 2.2 one can see an example of catastrophic buckling of the tubular steel windturbine tower. Turbine number 10 of the 11 turbine Searsburg wind facility in Searsburg ,Vermont in the USA collapsed on October 16, 2008 due to extreme wind conditions. In thisfailure one of the turbine blades hit the base causing the tower to buckle and the nacelleand rotor assembly to crash to the ground. In modern tubular steel towers, optimizationof tower weight by reducing wall thickness, makes the buckling margin an important op-timization constraint.

Figure 2.2.: Buckling Failure in Tower, Searsburg VT turbine number 10 [3]

When the ratio of the tower height to diameter gets large, the wall thickness which isrequired for more buckling resistance could be determined by the stiffness requirement.

Tower stiffness is characterized by several natural frequencies but only first two bendingand the first torsion natural frequencies have practical values. In most towers the firstnatural torsion frequency is much higher than the first natural bending frequency. Thetorsion frequency of free-standing steel tubular tower is approximately three times higherif their diameter/wall thickness ratio lies within normal limits. Hence it is sufficient to usethe first bending natural frequency as a measure of ”stiff tower” design.

A stiff tower design, that is one has a 1st bending mode sufficiently higher than the low-est expected excitation frequency, is simple and one of the safest solutions with regard tothe vibration behavior, however using this approach for design will require large amountof mass to be added to the tower structure.

In wind turbines with tower heights of more than 80 m, a stiff tower design can, there-fore, no longer be implemented in practice. For economic reasons, the stiffness shouldbe kept as low as technically feasible. For simple tower geometries, for example a cylin-drical steel tube, analytical approaches were developed which permit the required wallthickness to be calculated by using relatively simple formula, on the basis of the said loadcases with a given height, tower head mass and the chosen stiffness of the wind turbine.These models are mainly suited for demonstrating the influence of the dimensioning param-eters, thus helping to understand their significance with regard to the tower optimization.

14

2.1. Buckling Strength of Towers

In reality, the calculated masses are often lower. Manufacturers increasingly tend to fa-vor more complicated designs such as wall thickness varying in stages with diameter, orweight-optimized tapered shapes to minimize the tower mass and thus the costs.

Mentioned all above above about buckling and the stiff design of the tower, still thewind turbine tower is mainly considered to act like a simple cantilever beam in these ap-proaches. However, in reality tower section are formed from a thin-walled cylindrical orconical shells (cans) and therefore, several issues arise during the analysis, such as the localbuckling of the conical shell structure or the effect of added geometric imperfection into thetower model which must be thoroughly examined. This will be explained in more detailin other chapters of the thesis and is the foundation of buckling modeling and simulationinside this work.

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16

3. Buckling Theory

There are two major phenomenons leading to the sudden failure of a mechanical compo-nent: material failure and structural instability, which is often called buckling. Materialproperties are distinguished by experimental tests. The geometry of such test is standard-ized. Thus, geometry is not specifically mentioned in defining material properties, suchas yield stress. Geometry enters the game only when the stresses are calculated by theanalytic or numerical methods. Predicting material failure can be accomplished using thelinear finite element analysis. This is done by solving a linear algebraic system of equationsfor displacements corresponding to applied loads

Kδ = F (3.1)

Stresses determined with FEA are compared to the design stress within the component.Inside regions where these allowable are exceeded, it is assumed that material failure mayoccur. The load at which the buckling happens depends on the stiffness of the component,not the strength of the material. Two phenomenons are governed by different differentialequations. Buckling failure is primarily characterized by a loss of structural stiffness and isnot modeled by the usual linear finite element analysis, but by a finite element eigenvalueand eigenvector solution 1. The buckling calculation gives a multiplier that scales the mag-nitude of the load (up or down) to that required to cause buckling. Slender componentsunder compression stress are in susceptible buckling. Structures can show two types ofinstability. One state is when structure tends to acquire new out of current space state 2

but the change is still smooth. This point is called Bifurcation Point.The other type of instability is when structure which is under smooth loading snaps

through after a specific point. This point is called the Limit Point. The limit point can causethe structure to fail suddenly hence it can be catastrophic. Both of these points can becategorized as critical points of a structure under compression.

3.1. Euler buckling for long columns

When a long slender member subjected to a compression force moves lateral to the direc-tion of that force, it is at risk of the so called Euler column buckling. The force, F, whichis necessary to cause such a buckling motion varies by a factor of four depending only onhow the two ends are restrained. Therefore, buckling analyses are much more sensitive tothe component boundary conditions than a simple stress analysis.

1In the linear case2Current space is the non-buckling load-displacement behavior e.g. the compression in the global buckling

case

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3. Buckling Theory

Figure 3.1.: Euler buckling modes for long columns [4]

The topic of buckling is still seem unclear because keywords such as stiffness, long andslender are not consistently defined on all texts. Most of these concepts are developedhistorically from 1D studies. For a material, stiffness refers to either its elastic modulus, E,or to its shear modulus

G =E

(2 + 2v)(3.2)

where ν is the Poisson ratio. Slender is a geometric property which is quantified by theradius of gyration. The radius of gyration, r, describes the pattern in which the area of across section is distributed. If the area is concentrated far from the centroidal axis it willhave a greater value of r and a greater resistance to buckling. The section buckles in thedirection of the smallest value for radius of gyration. The radius of gyration, r, is definedas:

r =

ÊI

A(3.3)

Where I and A are the area moment of inertia, and area of the cross section. For a circleof radius R, one could obtain r = R/2. For a rectangle of large length R and width b onecan obtain rmax = 0.29R and rmin = 0.29b . Length is a geometric concept that is quantifiedby the dimensional slenderness ratio L/r or simply λ , where L corresponds to the axial

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3.2. Buckling of Shells

length of the component. The slenderness ratio is defined to be long when it obeys theinequality

λ > πK2

Ê2E

σy(3.4)

Where K is a constant that depends on the restraints of the two ends of the column (seeFig 3.1). A long slenderness ratio is typically in the range of more than 120. The aboveequation is the dividing point between long (Euler) columns and intermediate (empirical)columns. The critical compression stress which causes the buckling always decreases asthe slenderness ratio increases. Euler long column buckling is quite sensitive to the endrestraints. The critical buckling force is

FEuler =π2EA

(KL/r)2(3.5)

3.2. Buckling of Shells

Buckling in shells can happen in both local and global types. The global buckling of a longtubular shell (tower case) can be roughly interpreted as beam buckling. Theories showsthat the post critical behavior for a bar is neutral, and for plates is stable but in shellswith non-negative curvature, buckling tends to go toward instability [6]. In other wordscolumns loose their stiffness against the load after the critical point. Stiffness in plates alsoreduces but never goes to zero as the load increases. In shells stiffness becomes negativeafter the critical load, which means the structure can not resist under current load andcollapses. Another significant issue with shells is that the critical load of an imperfect shellis dramatically lower than that of the ideal, perfect structure, and this makes the geometrya decisive factor in design against buckling.

3.3. Effective Parameters on Buckling of Symmetric Shells

The buckling strength of an isotropic axially compressed thin shell depends on its [7]

• Geometry (radius r, thickness t, length L and taper angle β)

• Elastic modulus E

• Yield stress σY

• Amplitudes and forms of imperfections in the geometry

• Ends boundary conditions

• Pattern and direction of the loading

Ring stiffeners and changes of the shell thickness could be considered as boundaries be-tween segments and can be analyzed independently. Experiences show that strengths ofshells measured in real physical tests are far below the ideal theoretical strength, which re-quires the use of safety factors when a theoretical approach is implemented for the design.

19

3. Buckling Theory

Furthermore, adding stiffeners to structures seems to be ineffective for axially compressedshells. Although this may result in the structure having a higher load factor for buckling,it is expected to weaken the post buckling path significantly due to the added geometricimperfection and will cause the buckling to be even more unpredictable and catastrophic.

Consequently buckling is usually a sudden and unpredictable phenomena which leadsto a rapid loss of the load-carrying capacity and stability, which demands a careful andaccurate treatment of parameters as stress reach the critical region.

3.3.1. Geometric Imperfection and Residual Stresses

The most important factor which cause the difference between the theory and experimentsof the shell buckling is the geometric imperfection. Imperfection sensitivity is being inves-tigated in a large scale by various scientists. For instance [8] explains the buckling behaviorof cylindrical shells with many imperfection types under various forms of uniform axialload. When geometric imperfections exist, residual stress in the structure could be com-paratively large. As a type of the geometric imperfection, residual stresses can also causedifferences between experimental and theoretical results. Finally, thickness changes cancause also geometric imperfections in thin shells. The effect of thickness variation of theshell on buckling behavior is outside of scope of this thesis.

3.3.2. Ring Stiffeners

Ring stiffeners can be used in shell structures in order to strengthen the structure againstbuckling. At shell bolted joints, where the singularity and sudden variation of geometricproperties exist, these stiffeners can act as a support for the structure. The ring itself isusually created from the same materials and has more thickness but it can buckle itself.Two approaches can be implemented in order to model the ring stiffeners in finite elementmethod based on [8]:

1. Another shell segment

2. Separated ring

The first approach can be inaccurate due to the shell theory to capture all behaviors at thejunction which needs special care. The disability of this theory is proved when buckling isdominated by nonlinear strains caused by in-plane displacements [9].

Inside the scope of this thesis the second approach is followed by considering the towerjoint as the ring stiffeners modeled by brick elements in ANSYS which is the standardelement to model flanges of the wind turbine tower.

3.3.3. Non-linearity and Boundary conditions

Linear buckling theory assumes that the state of the stress before buckling is perfectly uni-form and consists of membrane stresses alone. This means that the shell is free to expandunder only the Poisson effect. In the real structure the bottom boundary condition is notusually free, resulting in bending, which reduces the buckling load. Stresses increase non-linearly as the load increases (see Fig 3.2). The effect of different boundary conditions on

20

3.3. Effective Parameters on Buckling of Symmetric Shells

the buckling strength of shells was explored extensively, with attention to boundary condi-tions that affect displacements and membrane stresses during the buckling process. Undermembrane state (linear case) stresses, the greatest sensitivity to boundary conditions occurif the shell ends are free to displace in the circumferential direction during the buckling,halving of the buckling stress. Other boundary condition changes have a much smallereffect. [10]

Real shells (tower base) are restrained against displacements at their ends by boundaryconditions that may take a variety of forms. When one end of a shell structure is rota-tionally restrained, local bending happens in the boundary region, which leads itself tothe local loss of strength against buckling, hence differences from the ideal linear classicbuckling theory. Different type of displacements at the bottom of the shell could be seenin Fig 3.3.

Since wind turbine tower shells are usually long, boundary condition type are expectedto have minor effect on the buckling load3.

3This could be seen by considering the tower as a long column, which sees all types of applicable boundaryconditions, simply as clamped at the bottom.

21

3. Buckling Theory

Figure 3.2.: Pressure-Displacement graph for buckling of the shell [7]

Figure 3.3.: Possible displacements at the bottom [7]

22

3.4. Specific to the Tubular Steel Tower

3.4. Specific to the Tubular Steel Tower

Steel towers could be ideally considered as tubular conical shells, with flanged boltedjoints as ring represent stiffeners and the door at the bottom and probable thickness vari-ations which can cause significant geometric imperfections. In the buckling aspect andspecifically the geometric imperfection there are three fabrication classes in production:

• Class A: Excellent

• Class B: Good

• Class C: Normal

For example towers which are created by REPOWER are categorized as class B [14], Henceimperfect towers are common in production which makes buckling phenomena seriousdesign consideration.

3.5. EuroCode-3 A rough approximation for buckling [16]

Following the industry standard EuroCode-3 spread sheet can be created in order to makean approximation of buckling strength by using the ideal longitudinal buckling load inconical shape tower shells.

By defining the table of parameters the ideal longitudinal limit value for the bucklingstress can be calculate by

σx,Rcr =0.605CxE

rxe/t(3.6)

where

rxe =r

cos(β)(3.7)

and

Cx = 1.0 if 1.7 ≤ ω ≤ rxe2t

(3.8)

Cx = Cx,N if ω >rxe2t

(3.9)

In which

ω =L√rxet

(3.10)

and Cx,N is the greater of 0.6 and

Cx,N = 1 +0.2

Cxb(1− 2ω

t

rxe) (3.11)

The value derived by this calculation is used in order to get a rough estimate of bucklingstrength or load factor in TowerElastic model, which is explained in next chapters.

23

3. Buckling Theory

Figure 3.4.: Table of Parameters of Eurocode3 for Buckling [1]

24

3.6. Solution Techniques for Buckling Analysis


For finding the buckling load factor(s) different methods could be used depending on themodel and analysis type. Within scope of this thesis these methods are ANSYS built-insolving algorithms, which are briefly introduced here. [20]

3.6.1. Block Lanczos Method

In Eigen-buckling approach, considered to act linear 4. The buckling load factors in thisapproach could be found by solving the following Eigenvalue problem.

(K + λmKF )δm = 0 (3.12)

Where λm is the buckling load factor for the mth mode, KF is the additional geometricstiffness due to the stresses caused by the loading, F, and δm is the associated bucklingdisplacement shape for the mth mode. Block Lanczos eigenvalue extraction method is asolver which ANSYS provides in order to find eigenvalues of large symmetric matrices.The Block Lanczos method employs automated shift strategy combined with Sturm se-quence check, which checks to ensure that requested eigen-frequencies beyond the shiftedfrequency are found without missing any modes. The use of the Block Lanczos method forlarge systems can cost a large amount of computer memory. This problem can be solved byusing the so called PSG Lanczos method, which does not use automated shift and Sturmsequence check in order to find eigenvalues.

3.6.2. Linear Perturbation Analysis

Linear Perturbation method is a feature ANSYS provides and can be employed in orderto solve buckling eigenvalue problem. This method has an iterative procedure which isdescribed as follows:

1. Perform Static analysis

2. Restart the analysis from the load point of interest in order to generate the tangentstiffness matrix KT .

3. Define the perturbation load.

4. Solve the new state of the problem.

5. The buckling load will be handed by calculating

FBuckling = FRestart + λFPerturbed (3.13)

where λ is the Eigenvalue.

Linear perturbation analysis can be used simply as a linear eigenvalue buckling solvermethod when FRestart is set to 0. Although there will be no difference in this point between

4Linear response of the displacement with respect to the applied load.

25

3. Buckling Theory

directly using the Eigen-buckling approach and the linear perturbation approach, the per-turbation analysis is still functional when the stiffness matrix is indefinite which cause thelinear eigenvalue buckling analysis using the Block Lanczos method to fail. The Stiffnessmatrix could be indefinite when the applied load is large enough to trigger the singularpoint in the nonlinear solution process.

3.6.3. Nonlinear Approach

Most shell structures show nonlinear behavior of the displacement with respect to theapplied load after some level of loading reached. Some shell structures, specially imperfectshells, show extreme nonlinearity even before the buckling happens. This is due to thenonlinear change in the stiffness matrix with load history, which is caused by the highlocal stress intensity and displacements.

Hence two main reasons exist which make nonlinear analysis an important touchstone:

1. Nonlinear analysis is more accurate in comparison with the linear analysis. Dueto considering the load-displacement history no theoretical prediction happens inadvance in order to figure out the buckling load.

2. It is the only way to capture the post critical behavior of the structure.

ANSYS has the ability to perform the nonlinear analysis. Nonlinear buckling analysis isusually performed in order to check the effect of imperfections and mainly the differenceit can cause 5 and in order to check the post-buckling behavior of the structure.

There are several methods for following the nonlinear path. Three main methods are:

1. Load Control

2. Displacement Control

3. Arc-Length Control

Figure 3.5.: line search methods (a) load control (b) displacement control (c) arc length con-trol [5]

Inside this thesis displacement control is the method which is used for the nonlinear linesearching.

Some issues are common to be considered when incremental load factors is being usedfor investigating the stability analysis by the Finite Element Method solver programs.These can be categorized as Element Distortion and Nonlinear Stabilization.

5Between linear and nonlinear approaches

26


Element Distortion Problem

When the value of the load increases rapidly and the mesh is not refined enough at largegradient regions, the solution is at risk of facing element distortion, which is also promptedin the ANSYS console as ”Error In Element Formulation”. This happens when at leastone eigen-value and consequently the determinant of the Jacobean of the stiffness matrixbecomes negative. Geometrically it means that a the node sequence in element is violated.

Figure 3.6.: Element distortion by moving the corner node [5]

From finite element analysis for every solution to converge two criteria are needed:

• Consistency

• Stability

Element distortion is one problem which violates the stability requirement in the so-lution procedure, which can prevent the convergence of the solution. Fig 3.7 shows thisproblem as a prompt in the console

Figure 3.7.: Element distortion prompt in ANSYS

Fig 3.8 shows the effect of element distortion on the total behavior of the sample towermodel in post buckling.

27

3. Buckling Theory

Figure 3.8.: Element distorted at high gradient region

The load sub-steps close to the buckling region where the stiffness reduces significantlymust be chosen small enough to capture the behavior of the structure correctly before avery large deflection happens due to local changing sign of the stiffness matrix slope 6.

Nonlinear Stabilization

With ANSYS, one can apply three techniques to solve instability problems:

• Nonlinear stabilization

It is a method for dealing with local instabilities as well as global instability. One canuse it together with nearly any other nonlinear solution technique, such as line searchand automatic time stepping. The only exception is using the arc-length method.

• Arc-length method

This method can investigate the global instability. The special power of this methodis that it has the ability to simulate the negative slope portion of the load-displacementcurve.

• Running the static problem representing the dynamic analysis

In this method one uses a dynamic effect to prevent divergence. This method is diffi-cult to use because the analysis type and system changes, so the user must input therequired parameters gradually. ANSYS in this matter recommends trying nonlinearstabilization or the arc-length method first. [20]

The instability of the structure could be global or local. Arc-Length method can notdistinguish the local instability of the structure. This is the main reason that inside thisthesis for the buckling of shell structure of the tower, incremental static analysis by usingthe energy stabilizer is used.

The energy dissipation ratio is the ratio of work done by stabilization forces to the ele-ment potential energy.

6Mesh refinement is another remedy

28


The energy dissipation ratio should be between 0 and 1 [20]. As the value of the energyratio gets bigger the stabilization force also gets larger which means the system becomesmore and more stiff.

The specified value should be large enough to avoid the divergence of the nonlinearsolution, but small enough to represent the actual stiffness of the system. The ideal valuedepends on the specific problem, the time of the load step, and the number of substeps.[13]

One can use the nonlinear stabilization to achieve the convergence in an analysis ofunstable nonlinear problem7. Although it can be accomplished by activating the nonlinearstabilization at the start of the solution, it causes the result to be more desirable if it isactivated in a multiframe restart 8.[20]

Since it is not possible to predict when a system will become unstable before an analysisperformed, it is logical to run the nonlinear analysis while saving restart files. If the anal-ysis fails to converge because of the instability, there is a possibility to restart the analysiswith the stabilization activated from some substep prior to the last converged substep. 9

During the nonlinear analysis within this thesis the nonlinear energy stabilizer is usedin order to prevent divergence of the solution.

7Nonlinear buckling analysis in this thesis8Saving and monitoring all steps of the nonlinear activity9ANSYS needs one substep to prepare the data for the stabilization.

29

3. Buckling Theory

30

4. Eclipse Modeling Framework andOneWindSDK

4.1. Eclipse

Eclipse is an integrated development environment for many programming languages andmainly Java. Eclipse first announcement goes back to 2001. The first version with an actualcode-name was Callisto which has been published on 2006 and from that time it has annualsimultaneous release. On June 2011 Indigo (the 8th version of Eclipse) has been published[12]. Indigo is the version of eclipse which is used inside this thesis and OneWindSDKof Fraunhofer IWES. Eclipse is an open source software which means the source code andcertain other rights which are reserved for copyright holders are provided under an Eclipselicense (LGPL) that permits users to study, change, improve and at times also to distributethe software. The Eclipse IDE is extendable by using components called plug-ins. SeveralOpen Source projects and companies have extended the Eclipse IDE. The Extendability ofthe Eclipse made the creation of the OneWind project at Fraunhofer IWES possible.

4.2. Eclipse Modeling Framework

The Eclipse Modeling Framework (EMF) is used in order to create the domain model. EMFhas a distinction between the meta-model and the actual model(see below). It provides aplugable framework to store the model information and uses XMI (XML meta-data In-terchange) 1 to persist 2 the model definition. Once the EMF meta-model is specified thecorresponding Java classes are generated from it. EMF provides the possibility that thegenerated code can be safely extended manually.

4.3. Meta-Model and Actual Model

Inside the Eclipse Modeling Framework the meta-model describes the structure of themodel and the model itself (actual model) is the instance of this meta-model. Meta-modelsare divided into

• ecore

• genmodel

ecore holds the information required for the definition of the model and allows to definedifferent elements:

1It is a standard for exchanging meta-data information via Extensible Markup Language (XML).2A persistent object is an object that has been assigned a storage location in federated database.

31

4. Eclipse Modeling Framework and OneWindSDK

• EClass: represents a class, with zero or more attributes and references.

• EAttribute: represents an attribute which has a name and a type.

• EReference: represents one end of an association between two classes. It has flag toindicate if it represented a containment and a reference class to which it points.

• EDataType: represents the type of an attribute, e.g. int

The genmodel generates the code with respect to the defined model and creates the basisfor the further development. It contains additional information for the code generation,e.g. the path and file information. The genmodel contains also the control parameter howthe coding should be generated. [11]

4.4. Making models using EMF

As Eclipse is a plugin 3 based framework the files described above, which define the EMFmodel are also to be stored in a plugin. To construct them one can use references tothe existing classes of the registered packages, classes of the plug-ins which exist at theworkspace or by using existing eclipse java libraries as imported attributes.

4.5. Xtext

Xtext is a textual framework for developing programming languages and domain-specificlanguages. Unlike standard parser (text analyzer) generators, Xtext not only generates aparser, but also a class model for the Abstract Syntax Tree. [12] Xtext is being developed inthe Eclipse Project as part of the Eclipse Modeling Framework Project. Xtext has been usedat Fraunhofer IWES in order to create a connection point between the parametric modeland the target simulation tool in the generator shape. Inside this thesis it is used in orderto create the ANSYS Parametric Design Language (APDL) input file.

4.6. XDoc

XDoc is a language created at Fraunhofer IWES which is based on Xtext to create the doc-umentation of the OneWind Software and the contained models. Different types of docu-mentation is generated using XDoc, such as pdf Handbook, html help pages on the server,online help and context sensitive help in the software.

4.7. Advantages of using EMF

With EMF one make the domain model explicit which helps to provide clear visibility ofthe model. EMF also provides change notification functionality to the model in case ofmodel changes. EMF will generate interfaces and factory to create objects, therefore ithelps to keep the application clean from the individual implementation classes. Another

3See 4.8.1

32

4.8. Engineering Design Data

advantages is that one can regenerate the Java code from the model at any point of time.[11]

4.8. Engineering Design Data

Engineering Design Data (EDD) has been developed and used in Fraunhofer IWES in or-der to model different types of components and sub-components of wind turbines such astower, rotor blade etc in different levels of detail. The EDD is purely parametric and onlycontains physical properties(geometry and material). An EDD model holds all the infor-mation necessary to describe the geometery and the physical behavior of a component in aspecific level of detail, without any simulation tool specific information. Inside this thesisEDD is used in order to create tower models. The procedure of modeling using the EDDconcept is discussed in more details in the following chapter.

4.8.1. Creation of an EDD using Plug-ins

A plug-in is a software component that adds specific abilities to a larger software appli-cation. If supported, plug-ins enable customizing the functionality of an application. InFig 4.1 the TowerSegmentedConical 4 plug-in is depicted as an example. An EDD Plug-inshould contain several parts:

• Generated Source Package

• Extended Source Package

• Meta-Model

• Doc

• META-INF

4TowerSegmentedConical 5.3 is the most detailed tower model inside this thesis

33


Figure 4.1.: Example of plugin in eclipse

Generated and Extended Source

Generated Source is generated by the Genmodel using the information provided by theEcore. This folder contains itself three packages

• PluginName

• PluginName.impl

• PluginName.util

The package PluginName contains interfaces which are going to be implemented by theabstract class of PluginName.impl. These abstract classes themselves will be extendedmanually by the developer inside the extended source packages.

34

4.9. JUnit test plug-ins

Meta-Model

The Model folder contains the ecore model with the definition of the metamodel and thegenmodel to generate the source codes which appears in the generated source packages.

Doc

This folder contains ”documents.xdoc” file which can be used for the user interface or thedocumentations related to the plugin.

META-INF

The META-INF folder contains the MANIFEST.MF which has the responsibility of linkingthe plug-in to the surrounding world. For example inside the dependencies one can addplug-ins which are required for the current plug-in. Eclipse procedure to describe the metainformation of a plugin, used in the packed jar for this plugin.

4.9. JUnit test plug-ins

A unit test is a piece of code written by a developer that executes a specific functionality inthe code under test. Unit tests ensure that code is working as intended and validate thatthis is still the case after code changes. Typically unit tests are created in their own projector their own source folder to avoid that the normal code and the test code is mixed [11].Furthermore unit test can provide a basis to instantiate the template objects which takessome user (developer) defined default parameters. This enables to work with the plug-inbefore the whole framework gets functional. Each Plug-in inside the OneWind is testedusing JUnit test and by adding ” test” at the end of the plug-in name.

4.10. OneWind and OneWindSDK

OneWindSDK is a software that contains all components of OneWind product family. TheSDK is designed in particular based on the fact that to give users the possibility to createtheir own product from respective OneWind products. The following are provided toenhance OneWind:

• EDD

• Generators

• Transformators

OneWindSDK has been built based on Eclipse.

4.10.1. Transformation

The transformation is mapping the information from detailed model to the reduced modelwhile keeping as much information as required. This is done by using Transformationclasses.

35


4.10.2. Generator

The key concept of OneWind 5 is separation of model specifics and tool specifics. Simula-tion tool specific information 6 is not stored in the EDD. The class which has the responsi-bility for translating the model specific to tool specific is the Generator.

4.10.3. Assessment

Assessment classes have the responsibility to get back the result from the simulation andgenerate the assessment tables (or documents).

5Introduced in 1.16Information which is only meaningful for a specific simulation software

36

Part II.

Modeling

37

5. The Two Towers

Two towers are introduced in the scope of this thesis. The TowerSegmentedConical andthe TowerElastic represent detailed and coarse parametric models respectively.

5.1. Philosophy of detailed and coarse parametric models

The idea of Finite Element Method in mathematics and engineering has been shaped onthe basis of that all physical models could be defined in a discretized approach by usingfinite elements which have similar general parameters. This made defining sophisticatedgeometries and physical properties, in a consistent and classified way, possible.

By using such specific approach it is hard (if it is not impossible) for the model to remainindependent of the physical theory and solution method behind a specific simulation ap-proach. It means the model which has been created with the purpose to be used inside aFE solver can not be used inside a symbolic solver. This requires to implement a more gen-eral parametric model from which the specialized models for different solving techniquescan be generated.

Parametric modeling has use for different purposes:

• Simulation tool independent model definition

• Having the possibility to perform multi-analysis on different simulation tools.

• Having hierarchy of models referencing the same physical component at differentlevel of detail suitable for different type of analyses and other goals of the user

• Consistency of models at different level of detail

Since purely parametric models only define some general information about geomet-ric and physical properties of the model they can be captured and translated to the tar-get simulation tool by using a suitable Generator (see Eclipse Modeling Framework andOneWindSDK chapter). Therefore the user does not have to deal with the difficulties aris-ing with the use of different simulation software. It also creates a start point before theusual time and energy consuming and mistake raising implementations.

The need for the reduced tower model shows itself in many cases. Two of these casesare:

• Investigation of other components of the wind turbine in detail like the substructurefor example. Wind turbine systems are usually simulated as a whole and it mayrequire the tower 1 to be analyzed in less detail because of the software developmentconvenience and more importantly due to the computational cost.

1It may show less property gradient or less effect on the interest focus area

39

5. The Two Towers

• The analysis type which is going to be performed on the model needs a specificamount of detail about geometric and physical properties. In this case defining themodel only by using coarse parameters suffices for the specific simulation purposeand the information must be extracted from the detailed model.

The transformation 2 is being used in order to create the coarse tower model from themore detailed one. Obviously the new model itself also represents a parametric modelhence it is still tool independent and can be used in various types of simulations.

In order to become simulate-able in the target simulation software, the input file shouldbe generated by obtaining the required information from the parametric model. This isdone by the Generator 3.

5.2. A Discussion on the selection of parameters

As it is mentioned in the introduction chapter there is a wide variety of possibilities tochoose parameters which define the tower. During the accomplishment of this thesis itoften happened that one set of parameters for the definition of the tower model had to bemodified in order to maintain the desired software and modeling requirements.

Choosing parameters depends on the physical component which is going to be modeledand the simulation type which is being performed as well as the desired accuracy.

For the reduced model represented by the TowerElastic the main goal is not the anal-ysis itself but passing the information from the detailed model represented by the Tow-erSegmentedConical to a model (Beam Finite Element Model) which is simulate-able inthe overall turbine simulation. This is needed to get back load results from the totallycoupled wind turbine simulation which considers all components and their interactions.Hence the reduced TowerElastic model is a gateway between the TowerSegmentedConicaland BeamTowerFE models.

5.3. TowerSegmentedConical Model Plug-in

The TowerSegmentedConical model is the most detailed model inside the OneWindTowerproject 4 and also inside this thesis. This model is meant to provide the input for theANSYS linear and nonlinear buckling simulation.

Specifically for the TowerSegmentedConical model, the reference instance model 5 isthe ”NREL 5MW” wind turbine tower. 6 It is designed for linear and nonlinear bucklinganalyses. The first guess for the set of parameters here is that the detailed geometry ofthe tower is needed in order to capture the accurate buckling load factor. The next stepof analysis is the nonlinear buckling analysis capturing geometric imperfections and thepost-buckling behavior. This requires detailed material and geometric properties for the

2See 4.10.13See 4.10.24See 1.35It is an instance of the tower metamodel6The ”NREL 5MW reference wind turbine is a artificial representative utility-scale multi-megawatt turbine.

40


whole tower. Hence one of the most important questions might be asked at this point is:”Which parameters effects the buckling behavior?”

In order to answer this question, the buckling simulation of the reference NREL 5MWwith and without the detailed parametric components 7 seems to be necessary. Basedon the simulation results, the buckling theory and finally by considering the Eurocode-3standard as the reference for buckling of shells, several parameters distinguished to play arole for the buckling behavior of the tower:

• Shell Thickness t

• Shell Meridian Curve or Taper Angle if Linear β

• Boundary Conditions specially at the bottom and at the top (similar to the Eulerbuckling modes but here the model is 3D)

• Geometric Imperfections represented by the door and joints

• Shell Slenderness λ

• Sectional Areas A

• Material Properties

Hence the TowerSegmentedConical model as the detailed model for buckling simula-tion should provide enough information about the mentioned parameters. The TowerSeg-mentedConical plug-in contains 8:

• TowerSegmentedConical extends Tower

• TowerSegmentedConicalSegment extends ModelElement

• TowerJoint extends ModelElement

• TowerDoor extends ModelElement

• TowerInternalComponent extends ModelElement

• TowerCoating extends ModelElement

• TowerSegmentedConicalAssessment9 extends ComponentAssessment

• TowerSegmentedConicalAnsysGenerator extends AbstractAnsysGenerator

7Tower can contain different detailed components such as door and flanges.8Still there are more components inside this model which are not fully implemented due to the limit of the

time. e.g. TowerLift or TowerCable9It is not implemented due to being outside of the scope of the thesis

41

5. The Two Towers

Fig 5.1 shows the UML diagram of the model.

Figure 5.1.: TowerSegmnetedConical model UML diagram

42


Fig 5.2 shows the ecore-model of the TowerSegmentedConical model in more detail.

Figure 5.2.: Ecore of TowerSegmentedConical Model

43

5. The Two Towers

5.3.1. TowerSegmentedConical Class

This class basically holds the information which is required to define the tower and itincludes:

• Segment

• Coating

• height Method

By using segments, tower geometry and material is created. TowerCoating defines thecoating of the tower structure. The height method has the responsibility to calculate theheight of the tower up to the specified segment number (integer) which it takes as an input.

5.3.2. TowerSegmentedConicalSegment Class

As it is mentioned before the TowerSegmentedConical model is divided into segmentswhich construct the whole tower. Each of these segments contains

• Body Material

• Top Joint

• isDoorSegment

• Door

• Taper Angle

• Thickness Taper Angle 10

The necessity of the mentioned parameters could be realized more clearly by the type ofanalysis which is performed and properties which are contained in the reference physicalmodel.

For instance in this thesis NREL 5MW is used as the reference which has a linear changeof wall thickness. It requires the definition of a thickness taper angle. The body materialhas the type of IsotropicMaterial which is the class for defining isotropic materials. Thedoor and joints show themselves as possible decisive parameters for a buckling analysis.The door seems to add an important geometric imperfection to the model. The effect onthe buckling load factor caused by adding the door to the detailed model is investigatedin the implementation part of this thesis considering both linear and nonlinear analyses.At this point the doubt rises about if the implementation of the door is necessary to beconsidered for the buckling analysis. This is due to the tower slender structure causes thecolumn global buckling state to dominate the overall buckling behavior. Furthermore, Byimplementing joints inside the tower model they show themselves as ring stiffeners in theoverall tower buckling analysis.

The isDoorSegment parameter inside each segment object has the boolean type and en-sures to instantiate the Door only for the door segment.

10Thickness taper angle defines the angle which the thickness changes by.

44


One segment of the TowerSegmentedConical model is depicted in Fig 5.3.

Figure 5.3.: TowerSegmentedConical Segment [1]

45

5. The Two Towers

5.3.3. Boundary Conditions

From the Euler column buckling theory it is understood that the top and bottom boundarycondition affect the buckling load. The bottom boundary condition is also expected toaffect the tower shell buckling as it is discussed in detail in the buckling theory chapter.

The boundary conditions are not implemented explicitly as an object inside the Tow-erSegmentedConical but it is an additional condition of the tower when it is simulatedfor buckling. For instance, the FixedTowerBuckling contains the TowerSegmentedConi-cal model which is clamped at the bottom and refers to the word ”Fixed”. Other typesof boundary conditions are also investigated inside the implementation chapter. This isdue to the reuse of the tower model in the overall wind turbine system with substructureand nacelle connections, which make up the boundary conditions in the case of the overallsystem simulation.

5.3.4. TowerSegmentedConicalToTowerElasticTransformation Class

The transformation is mapping the information of the detailed model to the reduced modelwhile keeping as much information as required. In the transformation from the TowerSeg-mentedConical to the TowerElastic the parametric information of the detailed componentsand the geometrical information is reduced and reformed in a number of cylindrical com-ponents named TowerElasticElement. The tower part containing the door needs more ofthese cylindrical components.

In Fig 5.4 one can see a rough outlook of this transformation.

Figure 5.4.: TowerSegmnetedConical model transformation

Furthermore the effects of the coating and the internal components such as the Tower-Ladder are considered in the reduced model as distributed masses at the TowerElasticEle-ment level. Finally flanges and the door as the most detailed features of the TowerSeg-mentedConical are represented in the TowerElastic model by elements with special cross

46


section types 11.

5.3.5. Generation of the ANSYS inputfile

The class TowerSegmentedConicalAnsysGenerator takes a TowerSegmentedConical ob-ject and creates input files for ANSYS based on the parameters inside the model. It extendsthe base class Generator 12 and is integrated in the OneWind Framework.

The buckling simulation of the TowerSegmentedConical is accomplished by ANSYS.The Required scripts for the buckling simulation are 13:

• Create Script (TowerCreate APDL.inp)

• Linear Solve Script (SolveLinearBuckling.inp)

• Nonlinear Solve Script (SolveNonlinearBuckling.inp)

• Input File (TowerParameters.ini)

The create script remains unchanged as the tower parameters change and it adapts tothe new tower by using the respective variables. The solve command does not depend ontower parameter values instead it follows the type of the simulation which is performedon the model. Hence as long as the simulation type is not changed this command filealso remains unchanged. The main concern for the generator is the .ini file. This inputfile is generated by the TowerSegmentedConicalAnsysGenerator class depending on theparameters of the tower object. Ideally the whole procedure should be executed in thebatch format in cmd 14. This allows the ANSYS simulation to be executed from insideOneWind and used in an automated design or the optimization loop. An overview of thisimplementation could be seen in Fig 5.8.

Tower Model Parameters for ANSYS

The file TowerParameters.ini contains the following parts:

• Tower main parameters:The parameters of the tower such as height, number of sections etc.

• Tower door parameters:Door shape parameters, will be ignored if Door = 0

• Tower Joint parameters:Joint Parameters, has two forms detailed and simple, joint will be simple ifDetailed = 0

11RingArc12The other example for generators in OneWind is the ModelicaGenerator which deals with generating the

input file for the Modelica.13 See 5.3.514OneWind was not yet ready to support ANSYS experiments consequently, automated procedure abandoned

in this thesis

47

5. The Two Towers

Figure 5.5.: Tower main parameters [1]

Figure 5.6.: Tower door parameters [1]

48


Figure 5.7.: Tower joint parameters [1]

Create Script

The create script with the name TowerCreate APDL create the model geometry withthe specified parameters needed from the tower parameters input file. The mesh has theability to be refined at specific areas. This is required several times during simulationwhich is explained in more detail in 6.1.6.

As one can observe only specific information from the TowerSegmentedConical needs tobe generated for the TowerCreate APDL.inp. In the ideal case the whole scripts includingthe Create and Solve should also be generated automatically which is outside of the timeand scope of this thesis.

49

5. The Two Towers

Linear Solve Script

The linear solve script is an automated procedure of solving eigen-buckling (linear buck-ling) problem. It has also the ability to make use of the linear perturbation method to dealwith the indefinite stiffness matrix for the special cases which are explained in the Buck-ling Theory chapter. 3.6.2 This script has a straight forward routine and extracts a specifiednumber of buckling modes for the tower structure.

Nonlinear Solve Script

The nonlinear solve script actually contains the nonlinear static simulation in ANSYS. Itrequires manually following of the .mntr file 15 of the multi-frame restart file to realizethe buckling load. Furthermore it requires special techniques to secure the convergence ofthe solution and stabilize the structure in the post buckling phase to capture the realisticbehavior in that domain. Due to extreme caution and awareness required for the manualobservation of the procedure, this script seems to be a severe task (if is possible) to begenerated automatically in advance without having precise knowledge about the bucklingof the structure. This happens since even the smallest changes in the structure geometriccharacteristic makes a huge difference in the buckling load and consequent techniqueswhich are required in this phase. The procedure of this simulation type is explained inmore detail in 6.1.6.

The more detailed generation process of the ANSYS input file for the OneWIndSDK canbe seen in Fig 5.8.

15ANSYS activity monitor file

50


Figure 5.8.: Simulation in ANSYS procedurec [1]

5.3.6. Benefits and Drawbacks

Some benefits of using the TowerSegmentedConical model can be categorized as:

1. One significant advantage of this model is using the integration of detailed geometricparameters. For instance, the inclusion of the door can cause effect on the bucklingload and this model considers the door as a detailed feature.

2. Another benefit of this detailed parametric model is the possibility of using the op-timization and the design procedure by performing the FOR loops in commandprompt on different files generated by different values of parameters and calling thesimulation software 16 afterwards. This will be implemented generically inside theOneWind framework in the future and this model and simulation can then be useddirectly.

The drawbacks for such a model can be classified as:

1. One of unavoidable drawbacks of this model is being specific due to being specific.This means that not many generic types have been defined but almost every compo-nent has its own specific name and detailed information. For instance the TowerDoorclass has all geometric properties defined specifically by names 17 and this makes de-riving the new more detailed type of the door to require creating the completely new

16ANSYS in this thesis17They have the Length type

51

5. The Two Towers

class and following the same approach. This causes further practical derivation fromthis model to be elaborate if it is possible.

2. The detailed model of the tower requires lots of computational cost even for linearbuckling analysis in ANSYS depending on how much detailed parameters are beingimplemented inside. This can be observed by triggering the door ON and OFF to seethe difference in computational cost. As one can conclude the remedy could be toreduce the computational cost by removing some of the parameters especially whenthe model is inside an iterative optimization or design loop.

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5.4. TowerElastic Model Plug-in


The TowerElastic model is the reduced model in comparison with the TowerSegmented-Conical model and carries the information needed for the BeamTowerFE for the totallycoupled simulation of the wind turbine. This model is meant to create a gateway betweenthe detailed parametric model and the beam finite element parametric model. Implemen-tation of such a model should ideally map information provided by the detailed model inmore general and classified way. For instance only by using a special type of cross sectionnamed RingArc the tower door geometry is modeled in a discretized way.

The TowerElastic plug-in contains several classes as follows:

• TowerElastic extends Tower

• TowerElasticSegment extends ModelElement

• TowerElasticElement extends ModelElement

• TowerElasticInternalAssessment extends ComponentAssessment

• RingArc extends Ring extends CrossSection

The UML diagram of the model can be seen in Fig 5.9:

Figure 5.9.: TowerElastic model UML Diagram

By the use of towerElements inside each towerSegment of the TowerElastic model alldetailed parametric information which are coming from the detailed model can be rep-resented in a consistent way. Internal parts and the coating of the detailed tower modelcan be represented here as optional distributed and point masses as explained in the Tow-erSegmentedConical transformation to the TowerElastic model.

Fig 5.10 shows the TowerElastic ecore-model.

53

5. The Two Towers

Figure 5.10.: TowerElastic Ecore Model

54


5.4.1. Segments and Elements

The TowerElastic model is divided into a number of physical segments. Each segmentcaptures the information about the constant material within it and also represents a basisfor the joint element. Elements are defined by using only geometric information whichis implemented as element top cross section. They can also represent different physicalcomponents of the tower such as:

• Joint

• Tower Door

• Discretized segment body

Fig 5.11 shows how the TowerElastic is divided into segments and elements.

Figure 5.11.: Tower Elastic Segments and Elements

As it is obvious it is just matter of assumption to consider the element tapered in cross-section, radius and thickness as they are not included in TowerElastic metamodel attributesexplicitly. The tapering can be derived by some methods of the TowerElasic model whichare discussed in the following sections . The relation between segments and elements canbe seen in Fig 5.12 in more detail.

55

5. The Two Towers

Figure 5.12.: TowerElastic [1]

5.4.2. EulerBucklingLoad Method

This method of the TowerElastic model takes two input arguments: (1) Number of modesto be extracted (N) and (2) Column effective length factor (K) which depends on the Eulertop and bottom boundary conditions and then calculates and returns the Euler columnbuckling loads. This is the roughest possible approximation for the buckling load of thetower.

5.4.3. HomogenBeamInfo Method

The TowerElastic model provides the possibility to build towers with varying cross section(radius and thickness) and transform it to the more reduced model BeamTowerFE. Forthis purpose the HomogenBeamInfo method is defined inside each element and has theability to return a list of HomogenBeamProperties objects which carry the informationneeded for the beam finite element model. Radius and thickness within each elementis assumed to change linearly and the value in between is calculated simply by using alinear interpolation of TopCrossSection of the current and the previous towerelement. Forfurther details on how this method is used in the transformation into the TowerBeamFEmodel please refer to [18].

5.4.4. Representation of the Door in the TowerElastic Model

The detailed TowerSegmentedConical model has the ability to consider the door. In theTowerElastic model this effect is considered by using the special cross-section type calledRingArc which is extended from the Ring class. This cross-section considers additionaloptions in order to represent the entrance, casing and radial position of the door. In Fig5.13 one can see how this special cross-section looks like.

56


Figure 5.13.: Tower Elastic RingArc

For the lower part of the door there is no entrance region but still there exist the casingfor this element cross section. This is instantiated by setting width = 0 in that specific crosssection. For the non door area the cross section will call the super class methods which issimply the Ring.

5.4.5. Analytic Buckling Assessment based on Eurocode-3

This assessment at the TowerElastic level seems to be a simple approach as this model isnot being simulated by any external certified simulation software. This assessment deliv-ers a quick and rough validation for the TowerElastic model and can be used to check theeffect of the reduction of information on the buckling load factors. The core methods forthis assessment are formulas offered by the Eurocode-3 standard [16].

5.4.6. Transformation to the Beam Tower Model

As it is mentioned before the tower top loads ideally must come from the overall simu-lation in which effects of other parts of wind turbine such as nacelle, blade, . . . are con-sidered. Each simulation demands a specific type of model. The BeamTowerFE is thename of the model for the overall wind turbine simulation. BeamTowerFE is a finite ele-ment description of the tower and contains Euler-Bernouli beam elements. Obtaining theBeamTowerFE object representing the same reference tower which the TowerElastic modelpoints to requires the transformation 18. 19

5.4.7. Benefits and Drawbacks

The advantages of using the TowerElastic model are:

• By using Segments and Elements, the transformation to the BeamTowerFE is moreconvenient. In this way the HomogenBeamInfo method of the TowerElastic is calledand creates beam elements for the BeamTowerFE object.

18From the TowerElastic to the BeamTowerFE19For more information regarding this part please read more in [18]

57

5. The Two Towers

• The Use of segments makes it possible to distinguish the place of the flange and italso carries the information which is the same in the elements of each segment. Forinstance the material is the same among all the elements of the same segment. Thisproperty is shifted from the element to the segment level.

• In comparison with the detailed TowerSegmentedConical, the TowerElastic is moreclassified. For example, TowerElasticElements are used to define the whole geometryof the tower .

• Finally and more relevant to the scope of this thesis the tower elastic assessmentcan present a rough overview of the buckling load factor when the time costing andelaborate simulation of the more detailed TowerSegmentedConical is not desired.

The Drawbacks and features which are not covered inside this model are:

• Since the main purpose for creation of this model is to be the model in between 20 fortransformation from the detailed model to the finite element model, no generator foran external simulation is implemented.

• Some information is lost in the transformation from the detailed model. This is aninevitable drawback which is the cost of information reduction.

• Elements should know their parent segments and segments also should know theirparent tower. The reason is to be able to access the information of parents at childrenlevel. For instance finding the next element inside a method of the TowerElasticEle-ment requires to go one level up to the segment level and find the next element. Thisis similar to what is called element connectivity table in finite element language butfar more simple. This requirement is covered by using opposite references. Usingopposite reference to connect elements, segments and the tower in both ways is anobstacle against readability and efficiency of the program and obviously a drawback.

5.5. Fixed Tower Model - the Simplified Wind Turbine

The simple model 21 for wind turbine which can be modeled for the linear and nonlinearbuckling analysis is the FixedTowerBuckling model. The FixedTowerBuckling takes onereference object from the TowerSegmentedConical class and also uses generator of thisclass. With additional mass at the top representing the dead mass of nacelle, blade andother components of the wind turbine and also clamped boundary conditions at the bot-tom, FixedTowerBuckling represents a wind turbine. The extra mass at the top is supposedto have additional effects on the buckling analysis of the model because of the pre-stressof the structure caused by the gravity. In Fig 5.14 the simple UML diagram of such modelis depicted.

20with respect to level of detail21Other parts of the wind turbine are represented here in the least amount of detail

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5.5. Fixed Tower Model - the Simplified Wind Turbine

Figure 5.14.: Tower Fixed Buckling UML diagram

Furthermore clamped bottom boundary conditions is defined inside the FixedTower-Buckling and allows it to represent a simple inland wind turbine 22. The bottom boundarycondition is implemented by the name itself which is fixed means the tower is fixed at thebottom (clamped).

The FixedTowerBucklingAnsysGenerator class generates the ANSYS input file for thebuckling simulation. The FixedTowerBuckling is defined to be the basis for the simulationsince boundary conditions and the top mass are considered in this model 23.

22Different types of boundary conditions can be used for off-shore based or soft-land based wind turbines23These two conditions allow it to represent a wind turbine in contrast with the more general isolated Tow-

erSegmentedConical tower model

59

5. The Two Towers

60

Part III.

Implementation Results andConclusion

61

6. Implementation and Results

The main purpose of this chapter is to determine the effect of detailed parameters on thebuckling of the tower. This is accomplished by simulating models having a range of detailsvarying from the column to the shell and also by considering components such as the doorand bolted joint flanges. The other goal of this chapter is to determine how the nonlinearanalysis of buckling gives different results. The analysis overview could be seen in Fig. 6.1

Figure 6.1.: Analysis overview

Models of the tower in different levels of detail are simulated and results are investigatedin this part. These models are:

• Euler column tower model

• Simplified beam tower model

• TowerElastic using Internal Buckling Assessment Class

• FixedTowerBuckling without middle joints or a door

• FixedTowerBuckling with middle joints but without the door

• FixedTowerBuckling without middle joints but with the door

• FixedTowerBuckling with middle joints and with the door

• TowerSegmentedConical considering the pin joint boundary condition at the bottom.

All tower models represent the NREL 5MW wind turbine tower. The TowerElastic internalassessment is performed in accordance with the Eurocode-3 standard. The beam model isdirectly modeled and simulated inside ANSYS. For TowerSegmentedConical the top massis included using the FixedTowerBuckling Model. In order to validate the correctnessand show the need for some geometric imperfections both linear and nonlinear bucklinganalyses are performed in ANSYS.

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6.1. Buckling Analysis in Different Level of Detail

6.1.1. Euler Column Representation

In order to find out the coarsest approximation for the buckling of the tower Euler columnbuckling is the first step. In Fig 6.2 two types of boundary conditions for the column areshown.

Figure 6.2.: Two Euler column with different boundary conditions (left) Free to rotate atthe top (right) Rotationally clamped at the top

The Euler column buckling theory predicts the buckling load by the following formula:

FEuler =π2EA

(KL/r)2(6.1)

and second moment of area can be calculated approximately as

I = πr3t (6.2)

The buckling load for the first mode when the top boundary condition is free to rotateis obtained by setting K = 2. For the column representing the NREL with the constantthickness and cross section diameter results are as follows:

FtopCS = 32, 271, 890 NFmeanCS = 95, 918, 015 NFaverage = 102, 327, 054 NFbottomCS = 172, 382, 218 N

(6.3)

For the case in which the bottom and the top boundary conditions are clamped withoutconsidering the top mass effects, K is equal to 1. This case gives a lower and upper bound

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and also a mean value calculated by assuming an average value for the cross section di-ameters.

FtopCS = 129, 087, 562 NFmeanCS = 383, 672, 063 NFaverage = 409, 308, 218 NFbottomCS = 689, 528, 875 N

(6.4)

6.1.2. TowerElastic buckling assessment based on EUROCODE 3

TowerElastic internal assessment is developed in Java by implementing the formulas of-fered by Eurocode-3. By using the transformation on the TowerSegmentedConical and bycreating the TowerElastic representation of the NREL 5MW buckling results are shown inFig 6.3

Figure 6.3.: TowerElastic Built-in assessment based on Eurocode-3

6.1.3. ANSYS

ANSYS is a simulation software which offers various types of numerical modeling andsimulations. ANSYS has been used for many years in order to examine models beforephysical experiment and commercial use of the product happens. It is a black-box 1 soft-ware which implements different types of built-in solvers in order to overcome adventchallenges in technology. Shell representation of towers inside this thesis are simulatedusing ANSYS in order to figure out buckling loads in linear and nonlinear cases.

NREL 5MW as Reference Model

Models in this thesis represent the reference National Renewable Energy Laboratory (NREL)5MW wind turbine tower. In Fig 6.4 one can see properties for this specific tower.

1The source code can not be seen or altered by the user in contrast with the open-source software such asOpenFOAM

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Figure 6.4.: NREL 5MW tower information [1]

6.1.4. TowerSegmentedConical, Shell Model in Different Levels of Detail

Having a stand-alone tower as the simulation input model does not seem to be an accurateway of analysis. However for static loads such as the tower top mass, the gravity onthe tower body and some user defined loads at the top of the tower this gives the roughapproximation of actual loads. It also provides a basis for the investigation of the bucklingand the post buckling behavior of the tower in different conditions. Tower linear bucklinganalysis is performed by using the eigen-buckling analysis. Furthermore the nonlinearapproach is followed by using nonlinear incremental static analysis. The element used foreach segment body is the SHELL181 which allows both the membrane and the bendingstate analyses of the shell model. Volumes of joints and the door are meshed with brickelements. Frictional contact elements are implemented for the friction section betweenflanges in the detailed joints. [1]

Tower middle joints

TowerSegmentedConical segments are connected by means of joints. These joints are de-fined in two forms one simple and one detailed inside the ecore of this model. The jointbetween two segments of the tower is simple if all bolt(i) = 0 and detailed if there ex-ist at least one bolt(i) 6= 0. In Fig 6.5 and Fig 6.6 two types of tower joints are depicted.Joints in middle have much weight and thickness in comparison with the segment bodyshell. This fact makes them act as ring stiffeners in the tower buckling analysis. The imple-

66


mented joint inside this thesis is the simple joint model. The further implementation couldbe the detailed model for the shell by considering detailed bolts and the frictional contactbetween the flanges. However implementing this detailed type of the flange is time con-suming and due to lack of computational resources outside of the scope of this thesis. InFig 6.6 one example of such joint considering the bolts and other detailed components isdepicted.

Figure 6.5.: Simple Joint [1]

Figure 6.6.: Detailed Joint [1]

Tower Door

Most wind turbine towers have a door in the base segment. As mentioned before, the doorcould effect the buckling analysis by adding the geometric imperfection to the model. Thetower door is created inside the model by setting Door = 1 inside the ANSYS script. Thedoor can be removed from the model simply by setting Door = 0 inside the script forANSYS and re-creating the mesh. The first drawback of implementing the door inside themodel shows itself in the pre-processor phase when the mapped meshing is not possibleanymore due to the complex geometry. This is the first impact which implementation ofthe door has on the computational cost.

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Figure 6.7.: Tower door at bottom segment [1]

Spider beams

Point loads at the top of the tower are applied using the Spider Beams and Spider Node atthe place where beams join together. The weight of the beam is small and the stiffnessis very large in comparison with other elements and so it forms effectively a rigid regionat the top which transmits the applied load to other elements. By implementing the topjoint and spider beams at the top of the tower one could assure that the shape will remaincircular at the top . This may be a realistic assumption if connection of the top joint tothe nacelle does not allow a significant amount of local deformations 2. The bucklingsimulation without using spider beams and node at the top which is less realistic andmore elaborate is also performed during this thesis however it turned out it has a verysmall effect on the buckling load despite the deformation shape at the top.

2If no local buckling deformation happens at the top

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Figure 6.8.: Spider beams and node at top segment [1]

6.1.5. Linear Buckling Analysis

Linear buckling analysis (also known as the eigenvalue buckling analysis) is one of themost simple case of buckling analysis and the most simple one in ANSYS. ANSYS usesBlock-Lanczos (mentioned in buckling theory chapter) eigen-solver to find buckling modeshapes and respective eigenvalues which are buckling load factors respectively. In thisapproach the material is considered to behave linearly and the load history and also geo-metric imperfections are expected to have minor effect on the buckling load factor. In orderto perform the eigenvalue buckling analysis in ANSYS a single unit compression load isapplied on the structure and then after a static analysis which cause the pre-stress to betaken into account 3 load factors are determined by performing the Eigen-Buckling solvecommand. These load factors are multiplied by the current load in order to get bucklingloads of the structure under the specific load type. Of course only the lowest buckling loadvalue is of interest to the tower designer.

3Pre-stress is caused by the tower head mass and the gravity.

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Simplified beam tower buckling simulation

Next approach to get a more clear idea about the effect of the level of detail on bucklingload factors is to perform the linear buckling simulation of beam representation of thetower model in ANSYS. Results in Table 6.1 are first three buckling load factors of thetower. As it is trivial there is no local buckling in the beam model which causes the furtherbuckling mode loads to be relatively large. This happens since inside the tower which isrepresented by the shell model the local buckling happens between global buckling loads.By restraining the rotational degree of freedom at the top of the beam 4 the linear buckling

units are in N Beam buckling load Euler buckling loads1st Mode 0.11398E+09 0.94127E+082nd Mode 0.78910E+09 0.376511E+093rd Mode 0.19947E+10 0.84715E+09

Table 6.1.: Linear buckling for the beams, bottom clamped top free

simulation is performed once more and it leads to Table 6.2 which contains load factorsof the new analysis. Results show that even the simple constant taper angle in the beam

units are in N Beam Buckling Loads Euler buckling loads1st Mode 0.35414E+09 0.376516E+092nd Mode 0.13208E+10 0.15060E+103rd Mode 0.27159E+10 0.33886E+10

Table 6.2.: Linear buckling for the beams, bottom and top clamped

model can effect the buckling load factors which are coming from the simple Euler formu-lation. However boundary conditions still cause the same qualitative effect on the bucklingload.

Linear buckling of the NREL without the door and middle joints

This model is a simple conical shape of the NREL 5MW tower with only one flange at thetop, at the place where the load is applied. Simple linear isotropic material is used for thismodel with the NREL specified properties. This model does not possess any geometricimperfection and the flanges in the middle and the door being considered. Fig 6.9 showsthe buckling pattern of such section under the compression load at the top. The colorcontour plots represent amount of displacements which has taken place, but only in arelative sense 5. In Table 6.3 first three buckling load factors are shown

4This boundary conditions does not happen for wind turbine tower in practice. The purpose here is toinvestigate the buckling analysis of the structure itself.

5The actual magnitude has no meaning.

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units are in N Buckling load factors1st Mode 0.11422E+092nd Mode 0.53929E+093rd Mode 0.57583E+09

Table 6.3.: Linear buckling of the NREL tower without the door and middle joints

Figure 6.9.: First four buckling mode of the tower without flanges with the door

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Linear buckling of the NREL tower with middle joints

Following the similar instruction for the simulation of the NREL 5MW tower the linearbuckling analysis is performed in ANSYS and this time by considering flanges in the mid-dle of the tower body. In Fig 6.11 first four buckling modes with corresponding nodal(relative) displacements of the tower structure are shown. In Table 6.4 first three bucklingload factors are shown One interesting phenomena in the linear buckling analysis of the


Table 6.4.: Linear buckling of the NREL tower without the door and with middle joints

tower shows itself at 5th buckling mode. As it is shown in Fig 6.10 the flange prevents thewave to propagate. This is shown comparatively in the tower with and without middleflanges.

Figure 6.10.: The middle flange in the left tower prevents buckling to propagate

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Figure 6.11.: First four buckling mode of the tower with the flange without the door73


Linear buckling of the NREL tower with the door

The next model is created in order to perform the linear (eigenvalue) buckling simulationwith considering the door in the tower. Fig 6.12 shows the third buckling mode in whichthe door region is clearly buckled.

Figure 6.12.: Linear Buckling analysis

The overall tower buckling mode shapes are similar to the case without middle jointsflanges except for the local difference at the door region. In Table 6.5 the first three bucklingload factors are shown.


Table 6.5.: Linear buckling of NREL tower with door and middle joints

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Linear buckling of the NREL tower with pin-joint boundary condition at the bottom

As explained in the theory chapter bottom boundary conditions can play an importantrole in the buckling of the structure. This effect can be seen in systems from the simpleEuler column buckling to the sophisticated shell model of the tower. In order to investi-gate the effect of the bottom boundary condition another linear analysis is performed byusing the pin-joint at the bottom. This pin-joint at the bottom allows nodal rotations butdisplacements are restrained. The model which is investigated here does not contain anymiddle flanges in between but contains the door in the base segment. In order to considerthe worst case, the bending force acts in the direction of the door.

Figure 6.13.: Bottom pin boundary condition

Table 6.6 shows results for buckling load factors under both compression and bendingloads. As one can observe from the results the different boundary conditions has some

units are in N Bend/Clamped Bend/Pinned Comp/Clamped Comp/Pinned1st 0.94019E+07 0.95754E+07 0.11400E+09 0.11404E+092nd 0.13063E+08 0.13304E+08 0.51237E+09 0.51237E+093rd 0.13386E+08 0.13633E+08 0.53914E+09 0.53914E+09

Table 6.6.: Linear buckling for the NREL5MW tower under compression bottom pinnedand clamped

influence for the bending load case but nearly no influence for the compression load case.The reason for the reduction of the buckling load for the bending case is the added stress

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to the structure in boundary regions. Although it is not expected that the linear bucklinganalysis would be affected by these concentrated stresses at the boundary, due to changingthe initial stiffness matrix (added degrees of freedom) there is a small influence. More exactsolution is expected from the nonlinear analysis, which has the ability of taking stressesduring applying the load into account. Generally the tower is a slender structure due to itslarge height and therefore both bottom boundary conditions behave simply as clamped.

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6.1.6. Nonlinear Buckling Analysis

All buckling analysis so far were performed linearly. Buckling load factor results from thecompression load cases show no significant differences between tower models with andwithout geometric imperfections.

Performing the nonlinear buckling analysis 6 is difficult for many reasons. Some of themcan be categorized as:

• Especially for the detailed model of the tower the simulation is time consuming 7

• It requires the manual detection of the buckling load and the displacement .

• Convergence problems may occur.

• Post-buckling behavior is physically unstable.

However in order to investigate the effect of the geometric imperfection on the bucklingload factor this approach seems to be necessary. The tower already contains the geometricimperfection caused by the door at the bottom hence there is no need for defining the extraartificial imperfection or perturbation load which may cause the problem to be changedcompletely [20].

Beam representation of NREL: Nonlinear approach

In this analysis the beam model of the NREL tower is investigated by the nonlinear buck-ling analysis using ANSYS. The compression load acting at the top is 2e8 N and is morethan the over-predicted value of 1.14e8 N 8. Furthermore a lateral load of 1.0e4 N is appliedin order to trigger the buckling. By following the monitor file in Fig 6.14

Figure 6.14.: Minitor of nonlinear solution

6Here the nonlinear solver is ANSYS7only for the first attempt simulation takes 4 hours in a PC with 4GB RAM and Core i5 3GHz CPU8This load predicted before in the linear analysis

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The value for the buckling load is found to be nearly the same as the predicted valuefrom the linear analysis. The post-buckling behavior of the tower can be also seen in Fig6.15.

Figure 6.15.: Nonlinear buckling of the beam model

The displacement-force graph can be seen in Fig 6.16.

Figure 6.16.: beam nonlinear graph

The problem with this nonlinear analysis is that without adding or with a later load,or by adding only a lateral load, the perfectly straight beam (both geometric and elementshapes) can not buckle, which in turn leads the nonlinear analysis to fail. On the other

78


hand adding a large lateral load the deformation is not the axial load buckling since thereis a large amount of bending caused by the lateral load which makes the structure bucklebelow the expected axial buckling load. In the next approach instead of adding the lateralload small fraction (0.0005) of the first 10 linear buckling modes are summed to create acurved beam and then the nonlinear analysis is performed again with the same 2e8 N ofthe compression load.

Figure 6.17.: beam nonlinear graph with added imperfection

The result now shows more buckling strength of the tower but the post critical behavioris more abrupt.

NREL with considering the door: Nonlinear approach

The procedure requires some initial explanation. In this analysis the load value of 1.5E8N (more than the over-predicted linear analysis) is applied at the top of the tower and theinitial nonlinear static analysis is performed on the model. In the first attempt to solve theproblem the initial load acting on the structure is set to 1/50 of the total defined load andthe minimum number of equilibrium iterations is set to 10. Further in use options are thatpredictor is OFF to prevent the additional divergence of the solution and the time steppingis set to be automatic. The solution in this case is terminated at the load step number 28

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without reaching the convergence. In Fig 6.18 one can observe the monitor of the firstattempt of nonlinear solution.

Figure 6.18.: Minitor of nonlinear analysis

Results from the TowerSegmentedNonlinear.mtr file show the sudden increase of thedisplacement at the load-step number 25 corresponds to the load factor of 0.74696 and thisaccording to the ANSYS HELP shows the possibility of snap through behavior or simply

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buckling of the structure. So far it is clear that the buckling happens between the loadfactors of 0.71433 and 0.74696 (between 107.15E6 and 112.04E6 N of load) which is close tothe range of the predicted linear eigen-buckling analysis but as expected from the theorysmaller. Still, investigation of the behavior of the structure in the post-buckling regionis possible. Running the simulation again from the multiframe restart file and from thespecified problematic load step number 24 and applying the nonlinear energy stabilizerof 0.0002 and reduced initial load step factor of 1/10000 allows the analysis to go furtherand terminate at the load step number 45. With increasing the stabilizer to the large valueof 0.0009 9 and decreasing the load initial factor to the 1/20000 the solution terminates atthe load step number 56 and at the similar area of the maximum displacement of around-5.2m at the tip of the tower. These results may be explained by one of the following:

1. The structure goes to a new unstable configuration (high displacement gradient)which requires mesh refinement in the large displacement regions in order to pre-vent element distortion. As explained in previous chapters this effect is common forshell elements in the post-buckling domain.

2. The load step is too large, such that the singular state is exceeded before the conver-gence occurs.

To capture more of the post critical region several nonlinear simulation are performed onthe model with the same loads and with different constant values of nonlinear energy sta-bilizer. Also several mesh refinements are also performed at different areas where elementdistortion seems possible. The load step sizes is decreased in order to capture the buckledshape and the load behavior in the post-critical domain as much as possible before theelement distortion or other divergence problem happens.

Fig 6.19 shows local buckling which happens in the shell structure in the post criticaldomain.

9ANSYS prompts in console that this value seems to be too large in comparison with the reaction force

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Figure 6.19.: Post buckling shape of the Tower

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In the first glance the nodal displacement figure shows the global buckling of the toweris in the door direction. This shows itself by following same color regions of the displace-ment gradient and also shows that the trajectory lines are not straight anymore, as theywere for the linear buckling analysis. These qualitative expressions are the obvious dif-ference between the linear and nonlinear analysis on models with the same level of detailand the same loading at the top. The displacement at the top and the reaction force at thebottom graph can be seen in Fig 6.26

Figure 6.20.: Nonlinear buckling

In this graph one could see that the geometric imperfection caused by the door at thebottom has barely effected the qualitative path although it defines the buckling direction.From existing experiments on the idealized and imperfect cylinder it is expected that theimperfect shell will change path more smoothly than the perfect shell. In the Fig 6.21 onecould observe the von Mises and shear stress at the door region in the post buckling state

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Figure 6.21.: door region von mises and shear stresses

Here the geometric imperfection manifest as stress concentrations. In Fig 6.22 stressesinside the door frame is shown. High stress gradient in this area cause it to be one of thecritical regions

Figure 6.22.: door von mises and shear stresses

NREL tower with considering the door and middle joints: Nonlinear approach

By adding flanges to the tower the new nonlinear analysis is performed on the tower. Themonitor of the nonlinear analysis is shown in Fig 6.23

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Figure 6.23.: Monitor of the nonlinear analysis

The result shows that the buckling happens before the load factor of 0.75232 whichequals to the load of 112.848E6 N which is (as expected) less than the value which is pre-dicted by the linear analysis. In Fig 6.24 one could see the stress density and the shearstress distribution inside flanges in this model. As it is clear the stress gradient in flangesis critical

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Figure 6.24.: Stress intensity inside flanges

As it is mentioned before ring stiffeners 10 are also in danger of buckling. Fig 6.25 showsthe second flange close to the buckling domain. (left) This picture shows that during thebuckling of the tower the flange just moves without any significant deformation but it canalso buckle if the stress gets large enough inside the region. (right) Shows buckling of theflanges long after buckling happened.

10Here flanges could be assumed as ring stiffeners

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Figure 6.25.: Second flange deformation in buckling and post buckling phases

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NREL experimenting the global generated geometric imperfection: Nonlinearapproach

In order to investigate how much a large global geometric imperfection can reduce thebuckling load of the structure and affect the load-displacement path, an analysis whichconsiders an added geometric imperfection is performed. The load of 2e8 N is added atthe top of the tower. In this approach the geometric imperfection is added to the structureby summing up the first 10 modes of the linear buckling of the structure scaled by 0.1. Byfollowing the same approach of the monitor tracking one can see the load which causesbuckling in Fig 6.26.


The monitor shows the buckling of the structure happens between the sub-steps 13 and14 which corresponds to the load of between 0.258 and 0.303 of the applied 2e8 N load

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which is inside the boundary of 51.6E6 and 60.6E6. This is much less than the value of thenonlinear analysis without adding the artificial global geometric imperfection. Fig 6.27now shows the effect of the imperfection, which (as expected) leads the structure to bucklewith a significantly lower load factor.


As results show this approach reduces the buckling load factor significantly due to theinitial imperfection of the structure.

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6.2. Two Tower Models, General Discussion on Results

Inside the scope of this thesis two main models for the tower are introduced. These modelsrepresent the tower in different levels of detail. In this section results from simulations andcalculations of buckling are compared in order to distinguish advantages and drawbacksof using these models for different types of buckling analyses.

6.2.1. TowerSegmentedConical Model

In the following tables buckling load factors for models of different levels of detail of theNREL 5MW reference tower and different analysis types, with different load types actingon the tower are shown. For the linear buckling analysis with a unit compression loadat the spider node at the top of the tower, the first three buckling load factor results areshown in the Table 6.7. For the linear buckling analysis with an added bending load atthe spider node at the top of the tower in direction of the door (critical loading case), thefirst three buckling load factors are shown in the Table 6.8. Furthermore Results for upperbound of buckling loads from nonlinear analysis are shown in the Table 6.9

units in N (Lin-ear Axial)

w/o Door w/oJoints

with Door w/oJoints

w/o Door withJoints

with Door withJoints

1st 0.11422E+09 0.11400E+09 0.11430E+09 0.11402E+092nd 0.53929E+09 0.51237E+09 0.61837E+09 0.51428E+093rd 0.57583E+09 0.54001E+09 0.64739E+09 0.61845E+09

Table 6.7.: Linear buckling for the NREL5MW tower under compression

units in N (Lin-ear Bend)

w/o Door w/oJoints

with Door w/oJoints

w/o Door withJoints


1st 0.12883E+08 0.94019E+07 0.12734E+08 0.94163E+072nd 0.13067E+08 0.13063E+08 0.12913E+08 0.13111E+083rd 0.13305E+08 0.13386E+08 0.13144E+08 0.13428E+08

Table 6.8.: Linear buckling for the NREL5MW tower under bending

units in N (Ax-ial)

w/o Door w/oJoints

with Door w/oJoints


Added GlobalImperfection

Buckling LoadFactors

0.11303+09 0.11204E+09 0.11285E+09 0.0606E+09

Table 6.9.: Nonlinear buckling for the NREL5MW tower under compression

By comparing results, it is clear that effect of the door is significant for the tower with thebending moment at the top. This effect shows itself even in the linear case where imple-menting the door reduces the buckling strength by 3500 KN. The effect of implementingflanges in the middle in the case of bending does not have a significant impact on buckling

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6.2. Two Tower Models, General Discussion on Results

loads, although it increases it by 150 KN. On the other hand the effect of the door is not asgreat as it is for the bending load case in the linear analysis. The middle flange again adds780 KN to the buckling strength. By performing the nonlinear analysis for the tower underthe compression load the door shows much more impact on the reduction of the bucklingload. In this case the door reduces the buckling strength by 3000 KN. On the other hand,middle flanges add 800 KN to the buckling load. Finally by generating artificial globalimperfection the buckling load reduced dramatically. This shows the small scale of the ef-fect caused by the geometric imperfection made by implementing the door. Usage of twotypes of boundary conditions (clamped and pinned) are examined in the linear analysisand it is shown that it alters the buckling load only in the bending case and only by a smallamount. This is due to implementation of a relatively slender tower, which make the effectof boundary conditions at the bottom have a minor effect on buckling behavior. 11

6.2.2. TowerElastic Model

TowerElastic model is the same NREL 5MW tower which is transformed from the Tow-erSegmentedConical model. Although the main use of this model is to transform the in-formation to the BeamTowerFE model, the internal assessment class of this model uses EU-ROCODE 3 description of the ideal buckling load factor. The buckling load factor resultsfrom this assessment class show good agreement with the result which comes from thesimulation of the more detailed tower shell model. However the TowerElastic model is un-able to consider the door and flanges and other detailed geometric detail effects. Consid-ering the main purpose of this model which is transformation of information, one shouldbe aware of the approximation that happens in calculation of idealized buckling load fac-tors resulting from this model. Nevertheless by ignoring geometric imperfections such asthe door and flanges results are quite close to the result of the shell buckling analysis inANSYS.

11They show themselves as clamped boundary conditions

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7. Conclusion

The idea of modeling detail hierarchy of the wind turbine tower is investigated inside thisthesis with the focus on the buckling analysis. Linear and nonlinear buckling analyses areperformed on tower models with different levels of detail.

Two approaches are followed within the scope of this thesis. Finding the effect of im-plementation of models in different levels of detail and analysis types on buckling loadresults and the post critical behavior of the wind turbine tower model. This although isbeing discussed in general way about shell structures under axial load in many references1 but inside this thesis these effects are investigated specifically for the tower model andall simulations are accomplished based on this.

It is a significant task to use the suitable model for a specific real world tower and desiredsimulation goals. Inside this thesis different time consuming and elaborated modeling ap-proaches are compared with their impact on simulation results. This is accomplished inorder to provide the possibility for the user to distinguish if these extra efforts are neces-sary for the desired specified accuracy.

In this thesis it is shown that by playing with the set of parameters which describes themodel it is possible to alter amount of details 2 inside the model without considerablychanging simulation results.

There are many possible set of parameters which are suitable for a specific type of anal-ysis. Geometric imperfections which show themselves in 3D models play an importantrole when the buckling is expected to occur. By implementing the door and middle jointseffects of geometric imperfection on the buckling load are examined.

Since wind towers are usually classified as a very long shell 3 structure, the door is notexpected to have a decisive impact as the geometric imperfection on the buckling loadfactor. This fact is being proved inside this thesis by comparing results with and withoutimplementing the door inside the model. Furthermore the nonlinear analysis is performedon the model and it turned out even in this analysis the door and bottom realistic boundaryconditions 4 do not make a big difference. This also is compared by adding more geometricimperfection to the tower model which caused the critical load to reduce significantly.Furthermore , It is shown that adding the door to the model does not change the nonlinearbehavior of the structure and the kink still remains in displacement-force graph inside thepost-critical domain (similar to the perfect geometry).

Nonlinear buckling analysis is implemented for two reasons. Firstly in order to fig-ure out if usage of such analysis alter results which come from the over-predicting linearanalysis. Secondly in order to investigate and highlight the computational cost and othermanual elaboration it can causes in comparison with the amount of changes in results.

1For instance see [7]2Obviously the goal is to reduce the level of detail and get the nearly same results3Slender4Which can be implemented in the real tower structure

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7. Conclusion

It is also concluded that clamped boundary conditions for medium and long length ofshells do not significantly alter the buckling load, which contrasts strongly with their effectin the short shell buckling 5.

Finally additional linear and nonlinear analyses are performed by applying bendingload at the top of the tower. Buckling of the tower under bending could only be investi-gated by using the 3D model and by using shell elements in the finite element analysis.This is because it does not create any longitudinal global buckling inside the model. Im-plementation of the door in this case showed more effect on the reduction of the bucklingload.

As the overall conclusion this thesis showed the quantitative analysis of the bucklingand gave a through description of the buckling modeling and simulation of the wind tur-bine tower.

Suggestions for future works

Although the TowerSegmentedConical model and its defined parameters show variety ofsimulation capabilities there still remain further features and possibilities which could beadded and examined 6. These features could be:

• Performing further simulation types on the model such as fatigue and strength anal-ysis. These analyses can be also performed and examined in the future. Speciallythe more sophisticated time-dependent dynamic buckling analysis which is morerealistic can be complement of this work.

• Defining plasticity inside the model by changing the material type from isotropic tobe plastic could be also another interesting suggestion. This type of buckling analysisis called Geometric and Material Nonlinear Instability Analysis (GMNIA) whichis the most sophisticated type of buckling analysis for the wind energy structures inindustry [14]. This could itself add variety of detail sub-hierarchy 7 to the detailedmodel and could be implemented from the most simple linear material to the mostcomplicated nonlinear plastic material which expected to have impact on the buck-ling nonlinear analysis and definitely on the computational cost.

• Many other nonlinear treatment of the structure are open for further investigation.For instance following the negative slope of the displacement-force curve is only pos-sible by implementing the arc-length method. This method has its own drawbacksas it is mentioned in theory chapter.

5These types are dominated by the local shell buckling6These were outside of the scope and time of this thesis7Detail hierarchy inside TowerSegmentedConical model

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Appendix

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Bibliography

[1] P. Poisson ” OneWindTower Project” , Fraunhofer IWES, Bremerhaven, Germany, 2012.

[2] F. Miceli ”WTG Technology” , May 31, 2012 .

[3] ”Industrial Wind Action” , Release 2008.

[4] G. W. Herbert ”Steel Construction Manual”, 8th edition, 2nd revised printing, AmericanInstitute of Steel Construction, 1987

[5] C. Fellipa ”Nonlinear Finite Element Method” ,University of Colorado , Colorado, UnitedStates, 2009.

[6] K. Bletzinger ”Theory of Shells” ,Technical University of Munich , Chair of StructuralAnalysis, Munich, Germany, 2007

[7] J. G. Teng, J. M. Rotter ”Buckling of Thin Metal Shells” , ASME Appl Mech ISBN 0-203-30160-9, April 2004.

[8] J. G. Teng ”Buckling of thin shells: Recent advances and trends” , ASME Appl Mech Revvol 49, no 4, April 1996.

[9] J. G. Teng ”Shell restraint to ring buckling at cone-cylinder intersections” , journal of Engi-neering Structures, July 1996

[10] J. O. Almroth ”Influence of edge condition on the stability of axially compressed cylindricalshells” , AIAA J, 4(1), 134-140, 1966

[11] L. Vogel ”Eclipse Modeling Framework” , vogella GmbH, 2012

[12] ”eclipse.org” , Oracle, 2012

[13] P. Wrigger ”Nonlinear Finite Element Methods” , Springer, 2008

[14] C. Baniotopoulos ”Effects of geomterical imperfection to the resistance of the tower” ,AUTH, FCTUC, April, 2009.

[15] G. J. Simitses ”Buckling and postbuckling of imperfect cylindrical shells: A rewiew” , ApplMech Rev vol 39, no 10, Oct 1986.

[16] ”EUROCODE-3 Stahlbau” Band 2, Beuth Verlag GmbH , 2012.

[17] M. Strach ”Modeling Offshore Wind Turbine Substructures Using Engineer Design Data aNewly Developed Parametric Approach” Fraunhofer IWES 2011.

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Bibliography

[18] C. Dimitropoulos ”Modeling of the wind turbine tower using engineering design data andmodal analysis” ,Master Thesis, Fraunhofer IWES 2012.

[19] ANSYS Element Reference (2009). Element Reference. ANSYS, Inc., ANSYS Release14.0.

[20] APDL Guide (2009). ANSYS Parametric Design Language Guide. ANSYS, Inc., Re-lease 14.0.

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