Structural reliability analysis and robust design of offshore wind turbine support structures

TESI DE MÀSTER Màster Màster en Enginyeria de Camins, Canals i Ports Títol Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Autor Marc Garcia Lladó Tutor Prof. Dr. Albert de la Fuente Antequera Co-tutors Prof. Dr. Michael Muskulus Lars Einar Stieng Intensificació Especialitat en Enginyeria de l’Aigua Data 3 de Juliol de 2015 i NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF CIVIL AND TRANSPORT ENGINEERING Report Title: Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Date: 29/06/2015 Number of pages (incl. appendices): 184 Master Thesis X Project Work Name: Garcia Lladó, Marc Professor in charge/supervisor: Prof. Dr. Michael Muskulus Lars Einar Stieng Abstract: The purpose of this master thesis is to develop and evaluate a novel approach for performing structural reliability analysis and robust design of offshore wind turbine support structures. Offshore wind turbines are structures especially prone to uncertainties due to its exposure to environmental conditions. Therefore, the models that take into account probabilistic descriptions of variability are more natural approaches. In this sense, a structural reliability analysis and a robust design are two complementary approaches that allow for incorporating uncertainty and randomness in the design process. To achieve the objective of this study, it has been developed a simplified model of a monopile structure that allows for doing the necessary calculations efficiently. The simplified model is implemented as a time-domain simulation in MATLAB and allows to solve the transient dynamics of a beam. In parallel, the rotor loads are obtained from rotor simulations in FEDEM Windpower software. Uncertainty is studied in the following system parameters: aerodynamic damping, soil stiffness and cross section of the support structure. Simple probability distributions are used in order to describe the uncertainty on these parameters. The basis of the research is a probability analysis based on several approaches. First, it is raised an individual analysis for each system parameter and wind speed. Secondly, the way used to add the aerodynamic damping in the simplified model has given rise to two different analyses. The effect of the soil stiffness and the structure cross section is investigated through a correlated and uncorrelated sampling of both variables. Finally, a combined probability analysis is performed in order to observe the effect of analysing the three system parameters at the same time. Additionally, the reference offshore wind turbine is also analysed in order to enable evaluate the effects of the other analyses. The probabilistic analysis is performed with a variable number of samples and the displacements at the tower top and the bending moments at the mud-line of the structure are obtained as a result. These outputs are used to evaluate the equivalent fatigue damage of the support structure. The wind turbine considered is the 5-MW NREL, with the reference support structure defined in the OC3 project. The results show that the response of the proposed simplified model is significantly different depending on the system parameter. The soil stiffness and the aerodynamic damping have shown a weak effect on the response of the structure. The reason could be an underestimation of the uncertainties associated to these parameters. Conversely, the structure cross section has shown a greater impact, probably due to dynamic amplifications linked to resonance problems. The uncorrelated analysis of the soil stiffness and the structure cross section has shown the greater effects on the response of the structure, given that the sampling approach of this case leads to a greater scattering of the system parameters and, consequently, of the results. Keywords: 1. Offshore Wind Turbine 2. Structural Reliability Analysis, Robust Design 3. Simplified Model, MATLAB 4. Probability Analysis, Uncertainty __________________________________ ii iii En record de la Roser Parés Blasco. Gràcies, Iaia. Descansa en Pau. iv v Preface This thesis, Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures, has been written as a Master Thesis of Marc Garcia Lladó. The final work is the result of a 2-years studies of Master in Civil Engineering at the Universitat Politècnica de Catalunya (UPC). The thesis has been written in Trondheim, in the group of Marine Civil Engineering for the Department of Civil and Transport Engineering at the Norwegian University of Science and Technology (NTNU). The main aim of the work is to develop and evaluate a novel approach for performing structural reliability analysis and robust design of offshore wind turbine support structures. Acknowledgment I would like to express my gratitude to those who helped me accomplish this study. I would like first to thank to my supervisors in charge: Michael Muskulus and Lars Einar Stieng, for their useful comments and remarks throughout all the meetings that we have had. I would like also to thank to my supervisor at the UPC: Albert de la Fuente Antequera, who has been always willing to help me, in spite of knowing that I was in Norway doing the thesis. Secondly, I would like to express my sincerest gratitude and appreciation to my family for their unconditional love and support during this year. Special thanks to my grandmother, Roser Parés Blasco, and Manolo Rodríguez Pazos, to who would have loved to see this work finished. Last but not least, I would like to thank to all my friends, especially to Patricia Hernández, Manon Pelzer and Judit Tomás for your support and advice. Department of Civil and Transport Engineering, NTNU June 2015, Trondheim Marc Garcia Lladó vi vii Abstract The purpose of this master thesis is to develop and evaluate a novel approach for performing structural reliability analysis and robust design of offshore wind turbine support structures. Offshore wind turbines are structures especially prone to uncertainties due to its exposure to environmental conditions. Therefore, the models that take into account probabilistic descriptions of variability are more natural approaches. In this sense, a structural reliability analysis and a robust design are two complementary approaches that allow for incorporating uncertainty and randomness in the design process. To achieve the objective of this study, it has been developed a simplified model of a monopile structure that allows for doing the necessary calculations efficiently. The simplified model is implemented as a time-domain simulation in MATLAB and allows to solve the transient dynamics of a beam. In parallel, the rotor loads are obtained from rotor simulations in FEDEM Windpower software. Uncertainty is studied in the following system parameters: aerodynamic damping, soil stiffness and cross section of the support structure. Simple probability distributions are used in order to describe the uncertainty on these parameters. The basis of the research is a probability analysis based on several approaches. First, it is raised an individual analysis for each system parameter and wind speed. Secondly, the way used to add the aerodynamic damping in the simplified model has given rise to two different analyses. The effect of the soil stiffness and the structure cross section is investigated through a correlated and uncorrelated sampling of both variables. Finally, a combined probability analysis is performed in order to observe the effect of analysing the three system parameters at the same time. Additionally, the reference offshore wind turbine is also analysed in order to enable evaluate the effects of the other analyses. The probabilistic analysis is performed with a variable number of samples and the displacements at the tower top and the bending moments at the mud-line of the structure are obtained as a result. These outputs are used to evaluate the equivalent fatigue damage of the support structure. The wind turbine considered is the 5-MW NREL, with the reference support structure defined in the OC3 project. The results show that the response of the proposed simplified model is significantly different depending on the system parameter. The soil stiffness and the aerodynamic damping have shown a weak effect on the response of the structure. The reason could be an underestimation of the uncertainties associated to these parameters. Conversely, the structure cross section has shown a greater impact, probably due to dynamic amplifications linked to resonance problems. The uncorrelated analysis of the soil stiffness and the structure cross section has shown the greater effects on the response of the structure, given that the sampling approach of this case leads to a greater scattering of the system parameters and, consequently, of the results. xiv Effect of the Soil Stiffness ........................................................................................ 104 Effect of the Structure Geometry ............................................................................ 105 Combined Analysis................................................................................................... 105 Correlated and Uncorrelated Analysis .................................................................... 106 Equivalent Fatigue Damage Analysis ....................................................................... 106 Conclusions and Recommendations ..................................................................................... 109 Recommendation for Further Work ........................................................................ 110 References ............................................................................................................................. 113 Appendix A ............................................................................................................................. 117 A.1. Stiffness Matrix ........................................................................................................ 118 A.2. Mass Matrix ............................................................................................................. 119 Appendix B ............................................................................................................................. 121 B.1. Main MATLAB script ................................................................................................ 122 B.2. MATLAB script used to obtain the geometry of the structure ................................ 137 B.3. MATLAB script used to obtain the soil stiffness ...................................................... 139 B.4. MATLAB script used to perform the FE analysis ..................................................... 141 B.4. MATLAB script used to obtain the fatigue damage ................................................ 149 Appendix C ............................................................................................................................. 151 xv List of Tables Table 2.1: Properties of the 5MW NREL wind turbine (Jonkman et al., 2009) .......................... 7 Table 2.2: Geometrical dimensions of the support structure. ................................................... 8 Table 4.1: Number of elements for each part of the support structure discretized. .............. 38 Table 4.2: Wind speeds selected to perform the simulation of this research. ........................ 44 Table 4.3: Mean and standard deviation of the aerodynamic damping for each wind speed. .................................................................................................................................................. 46 Table 4.4: Angle of internal friction for each range of depth (Fischer, 2010). ........................ 47 Table 4.5: Analysis cases when the aerodynamic damping is distributed in the entire structure. .................................................................................................................................................. 52 Table 4.6: Analysis cases when the aerodynamic damping is added to the finite element located in the top of the structure. .......................................................................................... 53 Table 4.7: Additional analysis for the study cases of the degree of damping and the geometry of the structure. ....................................................................................................................... 54 Table 4.8: Analysis cases for the reference support structure of the offshore wind turbine. 54 Table 5.1: Natural frequency of the first mode obtained in FEDEM Windpower software and the FEM in MATLAB. ................................................................................................................. 57 Table 5.2: Mean, standard deviation and variation coefficient value of the displacements in the y and x axes on the top of the structure for the reference case. ...................................... 58 Table 5.3: Mean, standard deviation and variation coefficient value of the bending moments in the y and z axes on the mud-line of the structure for the reference case. ......................... 60 Table 5.4: Mean value of the equivalent fatigue damage of displacements and bending moments for the reference case. ............................................................................................. 60 Table 5.5: Mean, variation coefficient and variation with respect to the reference case of the displacements in the y and x axes for the study case of the damping. ................................... 62 Table 5.6: Mean, variation coefficient and variation with respect to the reference case of the bending moments in the y and z axes for the study case of the damping. ............................. 63 Table 5.7: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the study case of the damping. .................................................................................................................................................. 64 Table 5.8: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the study case of the damping. ................................................................................................................................... 66 xvi Table 5.9: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the additional study case of the damping. ............................................................................................................................ 66 Table 5.10: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the additional study case of the damping. ........................................................................................................................ 70 Table 5.11: Mean, variation coefficient and variation with respect to the reference case of the displacements in the y and x axes for the study case of the soil stiffness............................... 71 Table 5.12: Mean, variation coefficient and variation with respect to the reference case of the bending moments in the y and z axes for the study case of the damping. ............................. 72 Table 5.13: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the study case of the soil stiffness. ................................................................................................................................... 74 Table 5.14: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the study case of the soil stiffness. ................................................................................................................................... 76 Table 5.15: Mean, standard deviation, variation coefficient and variation with respect to the reference case of the natural frequency of the structure for the study case of the soil stiffness. .................................................................................................................................................. 76 Table 5.16: Mean, variation coefficient and variation with respect to the reference case of the displacements in the y and x axes for the study case of the cross section of the structure. .. 80 Table 5.17: Mean, variation coefficient and variation with respect to the reference case of the bending moments in the y and z axes for the study case of the cross section of the structure. .................................................................................................................................................. 81 Table 5.18: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the study case of the cross section of the structure. ........................................................................................................... 83 Table 5.19: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the study case of the cross section of the structure. ........................................................................................................... 85 Table 5.20: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the additional study case of the cross section of the structure. ........................................................................................... 87 Table 5.21: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the additional study case of the cross section of the structure. ....................................................................................... 87 xvii Table 5.22: Mean, standard deviation, variation coefficient and variation with respect to the reference case of the natural frequency for the additional study case of the cross section of the structure. ............................................................................................................................ 90 Table 5.23: Mean, variation coefficient and variation with respect to the reference case of the displacements in the y and x axes for the combined study case. ............................................ 92 Table 5.24: Mean, variation coefficient and variation with respect to the reference case of the bending moments in the y and z axes for the combined study case. ...................................... 94 Table 5.25: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the combined study case. . 96 Table 5.26: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the combined study case. .................................................................................................................................................. 98 Table 5.27: Mean, standard deviation, variation coefficient and variation with respect to the reference case of the natural frequency of the structure for the combined study case. ....... 98 Table 0.1: Standard deviation for the study cases with the aerodynamic damping distributed in the entire structure. ........................................................................................................... 152 Table 0.2: Standard deviation of additional analyses for the study cases of the degree of damping and the geometry of the structure with the aerodynamic damping added to the finite element located in the top of the structure. ......................................................................... 152 Table 0.3: Standard deviation for the study cases with the aerodynamic damping added to the finite element located in the top of the structure. ................................................................ 153 xviii xix List of Figures Figure 1.1: Wind farm in the Baltic Sea, neat Zingst, Germany. ................................................ 3 Figure 2.1: Representation of the different parts of an offshore wind turbine. Some of the terminology to be used in the rest of the thesis. ....................................................................... 6 Figure 2.2: Dimensions of the offshore wind turbine. ............................................................... 8 Figure 3.1: Discretization a bridge with finite elements (Oñate, 2009). .................................. 12 Figure 3.2: Examples of free vibration decays for four different wind speed (Schafhirt, 2014). .................................................................................................................................................. 16 Figure 3.3: Proportional damping scheme. .............................................................................. 18 Figure 3.4: Some typical foundation concepts: Gravity-based foundation, monopile foundation, caisson foundation, multiple foundation, multi-caisson foundation and jacket foundation, from left to right respectively (Kallehave et al., 2014) ........................................ 20 Figure 3.5: Transfer of horizontal loads and moments in monopile structures and a schematic representation of the pile deformation. .................................................................................. 21 Figure 3.6: Representation of the time domain record of measured mud-line bending stress variation (top) and the frequency domain spectrum of the same time trace (bottom). ........ 22 Figure 3.7: Superposition of the time dependent and mean wind velocities (Van der Tempel, 2006). ........................................................................................................................................ 23 Figure 3.8: Illustration of the typical excitation ranges of a modern offshore wind turbine and the location of the 1st and 2nd mode natural frequencies. ...................................................... 24 Figure 3.9: SN-curves for steel in seawater with catholic protection (DNV, 2012). ................ 25 Figure 3.10: Some normal density functions with different mean and standard deviation. .. 28 Figure 3.11: Some log-normal density functions with different mean and standard deviation. .................................................................................................................................................. 28 Figure 3.12: Concept of robust design. .................................................................................... 30 Figure 4.1: Beam segment of a space frame showing displacements and rotations at the nodal coordinates. .............................................................................................................................. 34 Figure 4.2: Schematic representation of the support structure and the finite element model, with lateral springs under the mud-line where they represent the soil stiffness. .................. 39 Figure 4.3: Illustration of time-varying acceleration. ............................................................... 40 Figure 4.4: Histogram of the probability of occurrence of different wind speeds. ................. 44 xx Figure 4.5: Rotor loads from rotor simulations in FEDEM Windpower software. The time-series loads correspond to the X, Y and Z axis, from top to bottom respectively. ............................ 45 Figure 4.6: Variation of the total damping with wind speed. .................................................. 46 Figure 5.1: Distribution of amplitudes of the displacements in the y axis for the reference case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). .................................................................................................................................................. 59 Figure 5.2: Distribution of amplitudes of the bending moments in the z axis for the reference case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). .................................................................................................................................. 61 Figure 5.3: Boxplot of the amplitudes of the displacements obtained in the rainflow counting for the study case of the damping. Each graph correspond to the displacements in the y axis (top) and in the x axis (bottom). .............................................................................................. 62 Figure 5.4: Boxplot of the amplitudes of the bending moments obtained in the rainflow counting for the study case of the damping. Each graph correspond to the displacements in the y axis (top) and in the z axis (bottom). .............................................................................. 63 Figure 5.5: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the study case of the damping. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ........................................................................... 65 Figure 5.6: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the study case of the damping. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). .................................................................... 67 Figure 5.7: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the additional study case of the damping. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ................................................................. 68 Figure 5.8: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the additional study case of the damping. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ..................................................... 69 Figure 5.9: Boxplot of the amplitudes of the displacements obtained in the rainflow counting for the study case of the soil stiffness. Each graph correspond to the displacements in the y axis (top) and in the x axis (bottom). ....................................................................................... 72 Figure 5.10: Boxplot of the amplitudes of the bending moments obtained in the rainflow counting for the study case of the soil stiffness. Each graph correspond to the displacements in the y axis (top) and in the z axis (bottom). ........................................................................... 73 Figure 5.11: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the study case of the soil stiffness. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). .................................................................... 75 xxi Figure 5.12: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the study case of the soil stiffness. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ................................................................. 77 Figure 5.13: Distribution of natural frequencies for the study case of the soil stiffness. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 78 Figure 5.14: Boxplot of the amplitudes of the displacements obtained in the rainflow counting for the study case of the cross section of the structure. Each graph correspond to the displacements in the y axis (top) and in the x axis (bottom). .................................................. 81 Figure 5.15: Boxplot of the amplitudes of the bending moments obtained in the rainflow counting for the study case of the cross section of the structure. Each graph correspond to the displacements in the y axis (top) and in the z axis (bottom). .................................................. 82 Figure 5.16: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ............................................ 84 Figure 5.17: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ..................................... 86 Figure 5.18: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the additional study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). .................. 88 Figure 5.19: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the additional study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). .................. 89 Figure 5.20: Distribution of natural frequencies for the study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ........................................................................................................................... 91 Figure 5.21: Boxplot of the amplitudes of the displacements obtained in the rainflow counting for the combined study case. Each graph correspond to the displacements in the y axis (top) and in the x axis (bottom). ....................................................................................................... 93 Figure 5.22: Boxplot of the amplitudes of the bending moments obtained in the rainflow counting for the combined study case. Each graph correspond to the displacements in the y axis (top) and in the z axis (bottom)......................................................................................... 95 Figure 5.23: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the combined study case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ..................................................................................... 97 Figure 5.24: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the combined study case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ........................................................................... 99 xxii Figure 5.25: Distribution of natural frequencies for the combined study case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). ................ 100 Figure 6.1: Coefficient of variation of each study case analysis. The curves relate to the different wind speeds and the damping approaches. ........................................................... 107 xxiii Nomenclature Abbreviations DOF Degree of freedom FEM Finite Element Method FORM First Order Reliability Method MSL Mean-sea level MPP Most Probable Point NREL National Renewable Energy Laboratory SORM Second Order Reliability Method Roman Symbols 𝑎(𝑒) Nodal displacements 𝑏 Diameter of the pile 𝐵 Constitutive matrix 𝑐 Chord of turbine blade, viscous damping parameter, elemental damping matrix 𝐶 Global damping matrix 𝐶1,𝐶2,𝐶3 Coefficients of the API p-y method 𝑐𝑑𝑎𝑚𝑝𝑖𝑛𝑔 Damping coefficient, aerodynamic damping 𝐶𝐿𝛼 Lift coefficient 𝑐𝑣,𝐶𝑉 Coefficient of variation 𝐷 Accumulated damage, Miner sum 𝑑𝑉𝑑 Change in wind speed perpendicular to rotor plane 𝑓 Frequency 𝑓1𝑃 Rotational frequency of the rotor 𝑓3𝑃 Blade-passing frequency 𝐹 Load projection into each DOF 𝐹𝑉 Prescribed body forces 4 Chapter 1 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Structure of the Report This research has been divided in 7 main chapters. The first chapter corresponds to this section, which represents the introduction of the thesis. The rest of the thesis is structured as follows.  In Chapter 2 basic information about wind energy is presented, and the scope and limitations to the objectives of the thesis are given. Here, the dimensions and concepts used for the offshore wind turbine in the simulations, are described as well.  Chapter 3 presents relevant background information needed for a complete understanding of the thesis. It including the finite element method, damping and its effects, environmental conditions and probabilistic analysis.  In Chapter 4 the methods and models used to perform the structural reliability analysis and robust design are described, as well as how the relevant aspects such as damping effects or soil interaction have been treated.  The results of the simulations are presented in Chapter 5  Chapter 6 presents the discussion of the results obtained in the previous chapter in terms of the objectives of the thesis.  In Chapter 7, conclusions regarding the obtained results and suggestions for further work are given. 5 CHAPTER 2 Scope and Limitations System Definitions Terminology For a complete understanding of the rest of this study, the basic terminology that will be used is introduced. Such terminology refers to different topics for offshore wind turbines, with special emphasis on the components of the support structure (DNV, 2011). The terms referring the different parts of the support structure are defined in Figure 2.1.  Blades: The flat panels on a wind turbine that are connected to a center shaft that coverts the push of the wind into a circular motion in a wind turbine. Most commercial turbines have three blades.  Cut-in speed: The wind speed at which the turbine blades begin to rotate and produce electricity, typically around 4.47 m/s.  Cut-out speed: The wind speed at which some wind turbines automatically stop the blades from turning and pitches out of the wind to avoid damage to the turbine, usually around 24.59 to 29.06 m/s.  Foundation: The base structural and/or geotechnical component of the offshore wind turbine, from the transition piece down to the seabed. 6 Chapter 2 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 2.1: Representation of the different parts of an offshore wind turbine. Some of the terminology to be used in the rest of the thesis.  Gearbox: The gears connect the low-speed shaft to the high-speed shaft and increase the rotational speed of the shaft to the speed required by the generator.  Generator: Device that produces electricity from mechanical energy, in this case from the rotating turbine shaft.  Nacelle: The structure at the top of the wind turbine tower just behind the wind blades. It houses the key components of the wind turbine, including the rotor, gearbox and generator.  Pitch: The angle between the edge of the blade and the plane of the blades rotation. Blades are turned, or pitched, out of the wind to control the rotor speed. Support structure Tower Substructure Foundation pile Mean-sea level (MSL) Mud-line Transition piece Chapter 2 7 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures  Rated wind speed: The wind speed at which the turbine is producing power at its rated capacity. The rated wind speed generally corresponds to the point at which the turbine can perform most efficiently.  Rotor hub: The center of a turbine rotor, which holds the blades in place and attaches to the shaft. The rotor refers to both the turbine blades and the hub.  Shaft: The rotating part in the center of a wind turbine that transfers power. A highspeed shaft drives the generator. A low-speed shaft is turned by a rotor at about 0.5 to 1 Hz.  Substructure: Section of the support structure composed by the transition piece and the foundation, from the tower down to the seabed.  Support structure: The base structure that supports and elevates the wind turbine rotor and nacelle.  Tower: The top structural component of the support structure, from the transition nacelle down to the transition piece.  Transition piece: The structural component of the support structure that connects the tower and the foundation. It is composed by a platform, ladder and boat landing.  Yaw: The rotation of a horizontal-axis wind turbine around its tower or vertical axis. Dimensions and Model Specifications The offshore wind turbine of this thesis with the U.S. Department of Energy’s National Renewable Energy Laboratory (NREL) 5-MW turbine, as defined by Jonkman et al. (2009). This reference wind turbine will be used for the simulation because it provides standardized and accurate data for the properties of a realistically sized offshore wind turbine. The properties of this wind turbine model are shown in Table 2.1. Table 2.1: Properties of the 5MW NREL wind turbine (Jonkman et al., 2009) Wind Turbine Properties Rating 5 MW Rotor Orientation, Configuration Upwind, 3 blades Control Variable speed, Collective Pitch Drivetrain High speed, Multiple-stage Gearbox Rotor, Hub Diameter 125.8 m, 3.0 m Cut-in, Rated, Cut-out Wind Speed 3 m/s, 11.4, m/s, 25 m/s Cut-in, Rated Rotor Speed 0.115 Hz, 0.202 Hz Rated speed 80.0 m/s Overhand, Shaft Tilt, Precone 5 m, 5o, 2.5o Rotor Mass 110 000 kg Nacelle Mass 240 000 kg The support structure of the wind turbine is the offshore reference support structure defined by Jonkman et al. (2009) in the OC3 project, and it is constituted by a tower, substructure and foundation. The tower has different geometric properties along its height, with an outer base diameter of 6 m and a base thickness of 0.027 m, and an outer top diameter of 3.87 m and a top thickness of 0.019 m. Both diameter and thickness are assumed to be linearly tapered from base to top. The tower is connected to the substructure, which is constituted by a monopile with a constant diameter of 6 m and a constant thickness of 0.060 m. 8 Chapter 2 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya The tower base begins at an elevation of 10 m above the mean-sea level until the tower top at an elevation of 87.60 m. The substructure extends from the tower base down to the base foundation, which is at 40 m below the mud-line, as can be shown in Figure 2.2. The properties of this support structure model are shown in Table 2.2. Table 2.2: Geometrical dimensions of the support structure. Support structure dimensions Total height of the support structure 147.6 m Height of the submerged monopile 60 m Height of the foundation 40 Height of the grouted part of monopile 70 m Outer diameter of tower at bottom 6 m Inner diameter of tower at bottom 5.946 m Outer diameter of tower at top 3.87 m Inner diameter of tower at top 3.832 m Outer diameter of substructure 6 m Inner diameter of substructure 5.88 m Figure 2.2: Dimensions of the offshore wind turbine. 10 m 77.60 m 40 m 20 m Mean-sea level (MSL) Mud-line 147.6 m Chapter 2 9 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures The material used for the support structure is steel with a Young’s modulus of 210 GPa and a shear modulus of 80.8 GPa. Taking into account the effect of paint, bolts, welds and flanges in the thickness, the mass density of steel is increased from 7850 kg/m3 to 8500 kg/m3. The geographical location of the offshore wind turbine is a relevant factor to determine some environmental parameters, such as the soil properties. The North Sea is the location chosen due to the numerous investigations carried out in that location and used in the background of this thesis. Limitations The model used to develop and evaluate a novel approach for performing structural reliability analysis and robust design of offshore wind turbine support structures is based on a simplified model of a monopile structure. In other words, the complex behaviour of an offshore wind turbine is here simplified, making impossible to obtain the real response of the structure. Although that might be true, it should not be forgotten that the general methods used to simulate the response of a structure are, with more or less accuracy, approximations of this response. That said, the simplified model has been implemented by using the finite element method and the support structure has been simplified by using beam elements to represent the structure. Additionally, the analysis of the structure is not integrated, since the contribution of the rotor has been obtained from an external software. In conclusion, the different simplifications taken to define the simplified model have implied a limitation in this research due to the reduction in the real response of the structure that they suppose. The time has been another limiting factor for several reasons. One of the objectives to implement a simplified model is to do the necessary calculations efficiently. Nevertheless, the finite element method requires choice between accuracy and computational cost, or in other words, time. In a practical sense this means that, by using the finite element method, it is necessary to select a determinate number of elements which provide enough accuracy in a reasonable time. Obviously, this is achieved with sufficient time, but the format of this research, a master’s thesis, does not allow it, so it is required to reduce the accuracy of the model to save time. On the other hand, the probability analysis raises a similar problem with regard to the time. This analysis is performed for a determinate number of samples, and this number of samples has to be large enough to ensure reliable results that give robustness to the probability analysis. The challenge, therefore, is similar to the one discussed in the previous paragraph, it is necessary to select a determinate number of samples which reduces the variance of the probability analysis, while maintaining a reasonable computational time. Furthermore, each sample implies a simulation of the simplified model, so the number of samples should also be taken into consideration when the time needed to perform each simulation is selected. To conclude, the number of samples required in the probability analysis is an important limitation in this thesis due to the significant amount time that will demand. 10 Chapter 2 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya 11 CHAPTER 3 Background and Literature Review In this chapter, relevant background theory for the scope of this thesis will be presented. The information provided will give the necessary background for a good understanding of the objectives that will be investigated, the results obtained and the subsequent discussion. Finite Element Method The main objective of an engineer is always to analyse reality to extract the most relevant information and create a calculation model that allows study. All calculation models are based on a number of assumptions that simplify the object of the study without departing unduly from reality. Until recently, these calculation models were limited by the number of variables and elements that could be included because there were not the necessary tools for calculation. Now, with the advent of computers, the scope of these models is broadened. The main drawback of models with discrete elements is the limitation in the representation of reality. The Finite Element Method (FEM) is a calculation tool that can represent an entire continuum, by grouping parts with similar properties and characteristics in elements of variable size. This increase of the number of elements and the variability of their size allow the use of differential equations associated to the problem studied. In this way, the loss of information when building the calculation model is reduced. For this reason it is possible to 12 Chapter 3 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya say that the finite element method has been a breakthrough in the world of engineering, and particularly structural engineering. The use of differential equations allows studying one-dimensional, two-dimensional and three-dimensional problems, as well their evolution in time. In particular, for the calculation of structures, different types of models can be used as, for instance, bar problems, 2D elasticity, plates and shells, solids of revolution, 3D models, fatigue or problems with heat fluxes. The only requirement is a precise knowledge of the constitutive equations and evolution over time. Finite Element In the FEM, the continuum is discretized in finite elements, which can be visualized as small portions of this continuum. The word finite distinguishes such a portion from the infinitesimal elements of differential calculus. The geometry of the continuum is considered to be formed by the assembly of a collection of non-overlapping domains with simple geometry termed finite elements. It is usually said that a mesh of finite elements “discretizes” the continuum (Figure 3.1). Since the exact analytical variation of such parameters is complex and generally unknown, the FEM only provides an approximation to the exact solution. Deriving the General Equation of Motion The principle of virtual work states that: “A structure is in equilibrium under a set of external loads if after imposing to the structure arbitrary (virtual) displacements compatible with the boundary conditions, the work performed by the external loads on the virtual displacements equals the work performed by the actual stresses on the strains induced by the virtual displacements” (Oñate, 2009). Figure 3.1: Discretization a bridge with finite elements (Oñate, 2009). As shown by Cook et al. (2002), the principle of virtual work can be used to derive a general form of the equation of motion for a finite element discretized structural system. A virtual Chapter 3 13 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures displacement, 𝛿𝑢, is called to any imagined small motion that satisfies essential boundary conditions and displacement continuity between elements. As shown in equation below, the equilibrium of work can be expressed for a single element of volume 𝑉and surface area 𝑆. ∫{𝛿𝑢}𝑇𝐹𝑉𝑑𝑉 𝑉+∫{𝛿𝑢}𝑇𝐹𝑆𝑑𝑆 𝑆+∑{𝛿𝑢}𝑖𝑇𝑝𝑖 𝑛 𝑖=1 =∫[{𝛿𝑢}𝑇𝜌𝑢󰇘+{𝛿𝑢}𝑇𝑐𝑢󰇗+{𝛿𝜀}𝑇𝜎]𝑑𝑉 𝑉 ( 3.1 ) Where, 𝐹𝑉: prescribed body forces 𝐹𝑆: prescribed surface tractions 𝑝𝑖: prescribed concentrated loads acting in the 𝑛 corresponding virtual displacements {𝛿𝑢}𝑖 𝜌: mass density 𝑐: a viscous damping parameter. {𝛿𝜀}: strain associated with the virtual displacement {𝛿𝑢} The finite element discretization of the continuous structure leads to the relationships below, where the generalized coordinates 𝑞 are functions of time and the shape function matrix 𝑁 depends on spatial position. 𝑢=𝑁𝑞 ( 3.2 ) 𝑢󰇗=𝑁𝑞󰇗 ( 3.3 ) 𝑢󰇘=𝑁𝑞󰇘 ( 3.4 ) 𝜖=𝑑 𝑑𝑥𝑁𝑞=𝐵𝑞 ( 3.5 ) Substituting these relationships into the principle of virtual work, and assuming that the concentrated loads 𝑝𝑖 act directly in nodes, would now provide, {𝛿𝑞}𝑇(∫𝜌𝑁𝑇𝑁𝑑𝑉𝑞󰇘 𝑉+∫𝑐𝑁𝑇𝑁𝑑𝑉𝑞󰇗 𝑉+∫𝐵𝑇𝜎𝑑𝑉 𝑉) −{𝛿𝑞}𝑇(∫𝑁𝑇𝐹𝑉𝑑𝑉 𝑉+∫𝑁𝑇𝐹𝑆𝑑𝑆 𝑆+∑𝑝𝑖 𝑛 𝑖=1 ) =0 ( 3.6 ) 20 Chapter 3 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 3.4: Some typical foundation concepts: Gravity-based foundation, monopile foundation, caisson foundation, multiple foundation, multi-caisson foundation and jacket foundation, from left to right respectively (Kallehave et al., 2014) API p-y Method The American Petroleum Institute (API) p-y method models soil-pile resistance using a series of non-linear springs along the length of the pile, where the deflection of a certain soil spring at position x below the mud-line is denoted by y. The soil-pile resistance p for sands based upon Winkler Foundation theory are defined below. 𝑝=𝐴𝑝𝑢tanh(𝑘𝑥 𝐴𝑝𝑢𝑦) ( 3.24 ) Where, 𝐴={(3−0.8𝑥𝑏)≥0.9 𝑓𝑜𝑟 𝑠𝑡𝑎𝑡𝑖𝑐 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 0.9 𝑓𝑜𝑟 𝑐𝑦𝑐𝑙𝑖𝑐 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 ( 3.25 ) 𝑝𝑢=𝑚𝑖𝑛{(𝐶1𝑥+𝐶2𝑏)𝛾′𝑥 𝐶3𝑏𝛾′𝑥 ( 3.26 ) 𝐶1, 𝐶2, and 𝐶3 are coefficients, 𝑘 is the modulus of subgrade reaction determined as a function of 𝜑′ using correlations provided by API, 𝑏 is the diameter of the pile and 𝛾′ is the submerged unit weight of the soil (API, 2005). For large diameters there is a combination of soil stiffness overestimation at large depths and a very large bending stiffness typical of large diameters. As a result, there may lead to overestimation of pile-soil stiffness using the API p-y method for large-diameter piles. Although other models exists, this method is proven to be one of the most suitable for offshore wind design. Chapter 3 21 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Foundation Modelling Vertical loads, horizontal loads and moments must be transferred directly to horizontal soil reactions. The pile is free to rotate and translate because it is not fixed at the top. In the type of substructure selected, the pile must be long enough to mobilize enough soil over its length to transfer all loads and prevent displacements of the tip of the pile. Figure 3.5: Transfer of horizontal loads and moments in monopile structures and a schematic representation of the pile deformation. The soil reaction loads can be modelled using a set of soil springs with non-linear properties described in standards. This non-linear spring model can be created using the finite element method by coupling the non-linear spring model to the stiffness matrix of the structure. Although other models exists, this model is proven to be one of the most suitable for offshore wind design. Environmental Conditions Stochastic Processes Offshore wind turbines are subjected to irregular external loads as they are excited by wind and waves. The generation of these loads is due to stochastic processes, so it varies with time and cannot easily be reproduced or predicted in advance. In order to analyse the data from these two variables, it is necessary to assume that both parameters can be modelled as stationary processes over a certain time period. The wind process is assumed stationary for a period of 10 minutes, even if it is a quite fluctuating process. Wave requires a longer time period to assume it is stationary, three hours. This difference between both processes could be a problem when performing analysis, but in this study only the wind is used as an external load, so this problem will not be treated. 22 Chapter 3 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Time Domain The response of offshore wind turbines varies in time because load also vary with time, so it is possible to analyse this response in the time domain. The analysis time series in the time domain contains information regarding mean, maximum and minimum values, standard deviation, strange peaks or slow variations of the process. This information shall be sufficient to perform properly the different parts of this study, but it is interesting to explain that the time series can be transformed into the frequency domain to make the data more accessible, as can be seen in the Figure 3.6. Figure 3.6: Representation of the time domain record of measured mud-line bending stress variation (top) and the frequency domain spectrum of the same time trace (bottom). Wind The wind conditions acting on a structure in the main wind direction are characterized by the mean wind speed and the fluctuating wind velocity. The total wind speed as a function of height and time is commonly described as the sum of a mean wind velocity as a function of height, and a turbulent component dependent on height and time. 𝑉𝑡𝑜𝑡(𝑧,𝑡)=𝑉(𝑧)+𝑣(𝑧,𝑡) ( 3.27 ) Frequency [Hz] Spectral density [(N/m2)2*s] Stress [N/m2] Time [s] 1st natural frequency Rotor frequency Wave Slowly varying wind speed Chapter 3 23 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 3.7: Superposition of the time dependent and mean wind velocities (Van der Tempel, 2006). The mean velocity could be described by the Normal Wind Profile (NWP). 𝑉(𝑧)=𝑉(𝑧ℎ𝑢𝑏)∙( 𝑧 𝑧ℎ𝑢𝑏)𝛼 ( 3.28 ) The exponent 𝛼 is dependent on the surface roughness and has a recommended value of 0.12 for sites located offshore, a low value compared to onshore sites, where 𝛼 has typically a value of 0.2 due to higher surface roughness (Böker, 2009). Just as was started earlier, over a period of ten minutes the wind conditions are assumed to be stationary. Hence, the mean wind speed and standard deviation due to turbulence are defined over this period of time and there are different statistical options to describe these parameters. DNV (2010a) recommends the Kaimal turbulence spectrum for representation of the wind speed spectral density. 𝑆𝑢(𝑓)=𝜎𝑈24𝐿𝑘 𝑈10 (1+6𝑓∙𝐿𝑘 𝑈10) ( 3.29 ) Natural Frequencies and Dynamic Response The resonance of a structure is an important factor to keep in mind due to the negative effect of this phenomena on the fatigue life. The response depends closely on the natural frequency of the first mode and the dynamic interaction with the external loads. On the other hand, the 24 Chapter 3 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya natural frequency of offshore wind turbines cannot be within the range of frequencies corresponding to the rotational frequency of the rotor, 𝑓1𝑃, and the blade-passing frequency, 𝑓3𝑃. These frequencies are produced by rotor imbalances and aerodynamic impulse loads when the blades pass the tower. The natural frequency of the first mode needs to be between the 𝑓1𝑃 and 𝑓3𝑃 frequency ranges because the peak frequency of linear wave excitation is usually situated below the rotational frequency of the rotor (Kallehave, 2014), as can be seen in the following Figure 3.8. Figure 3.8: Illustration of the typical excitation ranges of a modern offshore wind turbine and the location of the 1st and 2nd mode natural frequencies. The dynamic response of the structure, therefore, can be described as the dynamic amplification due to loads with a frequency close to the natural frequency of the structure. The natural frequency of the first mode would resonate with the waves at low wind speeds and be exposed to high spectral energy of waves during higher wind speeds. During operation of the turbine, the response of the first natural frequency due to loading acting orthogonal to the rotor plane will be low due to a rotor in operation provides a high degree of aerodynamic damping. Consequently, if the aerodynamic damping does not contribute during stand-still periods, the fatigue loading can become very severe. However, this consideration will not be taken into account in the study. Fatigue Loading Fatigue is a phenomenon associated with variable loading or more precisely to cyclic stressing or straining of a material. Thus fatigue loading is primarily the type of loading which causes variations in the applied stress or strain on a component, so any variable loading is basically a fatigue loading. Frequency [Hz] Power Spectral Density Wind Spectrum Waves Spectrum f1P f 3P 1st mode natural frequency 2nd mode natural frequency 0.1 0.2 0.3 0 0.4 0.5 Chapter 3 25 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures In structural perspective, fatigue loading produces fatigue damage, which is an important parameter to consider for the design of offshore wind turbines. Environmental loading, that is to say, loads form wind and waves are responsible for the main contributions to fatigue loading. The mean value of these loads and the deviation of these loads from their mean value are of significance in a fatigue perspective. 3.6.1. Evaluation of Fatigue Damage Fatigue damage may be calculated under the assumption of linear cumulative damage, which is done by the Palmgren-Miner rule. This method consists of the sum of the damage from each stress range in the stress history. 𝐷=∑𝑛𝑖 𝑁𝑖 𝑘 𝑖=1 ( 3.30 ) Where, 𝑛𝑖: Number of cycles for each stress range i. 𝑁𝑖: Constant amplitude endurance for the given stress range. Failure due to fatigue will occur if the Palmgren-Miner sum is above 1 for the given stress history (Berge, 2006). In order to obtain the total number of stress cycles before failure, an SN curve is used for this propose. log(𝑁)=log(𝑎)−𝑚∙log(∆𝜎(𝑡 𝑡𝑟𝑒𝑓)𝑘) ( 3.31 ) Figure 3.9: SN-curves for steel in seawater with catholic protection (DNV, 2012). 26 Chapter 3 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya The mathematical formulation of the S-N curve is expressed by the equation above, where log(𝑎) is the intercept of the log(𝑁) axis, 𝑚 is the negative inverse slope of the S-N curve, ∆𝜎 is the stress range, 𝑡 and 𝑡𝑟𝑒𝑓 are the ratio between the thickness in the expected direction of crack propagation and the reference thickness, and finally 𝑘 is the thickness exponent on fatigue strength. The number of cycles to be counted from the stress time history can be obtained using several methods such as crossing counting, peak counting, simple counting or rainflow counting. However, rainflow counting is generally considered as the most suitable of these method due to the fact that this counting procedure, for wide-banded loading, produces the same stressstrain loops as a material undergoing the same loading history (Almar-Næss, 1985). Probabilistic Analysis Models of Uncertainty The modeling of uncertainty is an important point in the formulation of structural problems of offshore wind turbines. There are several mathematical models of uncertainty when dealing with structural design problems, such as the probabilistic model. Stochastic randomness is the most common model for uncertainties in structural engineering (Doltsinis, 1999). In this analysis, there is uncertainty associated with the value of some parameters. On the one hand, uncertainty on the load side is based on the stochastic variation of the environmental forces; and, on the other hand, there is uncertainty on some system parameters as the degree of damping or the soil stiffness. A probability distribution of a random variable is a mathematical function that assigns the probability of occurrence to each measurable subset of the possible outcomes of a random experiment. The probability distribution is defined over the set of events and each event is the range of values of the random variable. The probability distributions are related to the frequency distribution. In fact, the probability distribution can be imagined as a theoretical frequency distribution. Theoretical frequency distribution is a probability distribution which describes how the results should change. Since these distributions deal with the expectation that something happens, these models are useful to make inferences and decisions regarding uncertainty (Badii et al., 2007). The probability density function and cumulative distribution function are used to define the occurrence properties of uncertain quantities which are random in nature. The statistical description of a random variable 𝑋 can be completely described by a cumulative density function 𝐹𝑋(𝑥) or probability density function 𝑓𝑋(𝑥), as given by, 𝐹𝑋(𝑥)=𝑃(𝑋≤𝑥)=∫ 𝑓𝑋(𝑥)𝑑𝑥 𝑥 −∞ ( 3.32 ) Associated with the probability distribution are some parameters used to describe how the variables are distributed, the statistical moments. The most common statistical moments are the first and second moment, known as mean value 𝜇(𝑋), also referred to as expected value Chapter 3 27 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures and denoted by 𝐸(𝑋), and variance denoted by 𝑉𝑎𝑟(𝑋) or 𝜎2(𝑋), respectively. These two statistical parameters are defined below. 𝑀𝑒𝑎𝑛: 𝜇(𝑋)=𝐸(𝑋)=∫ 𝑥𝑑𝐹𝑋(𝑥)= ∞ −∞ ∫ 𝑥𝑓𝑋(𝑥) ∞ −∞ 𝑑𝑥 ( 3.33 ) 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒: 𝜎2(𝑋)=∫ (𝑥−𝜇(𝑋))2 ∞ −∞ 𝑑𝐹𝑋(𝑥) =∫ (𝑥−𝜇(𝑋))2 ∞ −∞ 𝑓𝑋(𝑥)𝑑𝑥 ( 3.34 ) Log-normal, Weibull and uniform are the probability distributions most commonly used in structural engineering. Additionally, the coefficient of variation (CV) will be defined since it a standardized measure of the dispersion of a probability distribution and it will be useful in the data analysis of the results. 𝑐𝑣=𝜎𝜇 ( 3.35 ) 3.7.1.1. Normal and Log-Normal Distributions The normal or Gaussian distribution is a very common continuous probability distribution because many natural phenomena are often approximated by this distribution. The density function of this probability distribution is defined below. 𝑓(𝑥;𝜇,𝜎)=1 𝜎√2𝜋𝑒−(𝑥−𝜇)2 2𝜎2=1𝜎𝜑(𝑥−𝜇 𝜎) ( 3.36 ) Where, 𝜑(𝑥)=1 √2𝜋𝑒−12𝑥2 ( 3.37 ) The density function is defined by 𝜑(𝑥), and the graph of this density function has a bell shape and is symmetric about its mean. The term Gaussian bell curve is used to refer to the curve observed in the graph, as it can be seen in the Figure 3.10. The log-normal distribution is another continuous probability distribution of a random variable whose logarithm is normally distributed. In order to model a variable as log-normal, it is required that the variable is the multiplicative product of many independent random variables, each of which is positive. The central limit theorem in the log domain justifies this criteria. The density function of this probability distribution is defined below. 28 Chapter 3 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 3.10: Some normal density functions with different mean and standard deviation. 𝑓(𝑥;𝜇,𝜎)=1 𝑥𝜎√2𝜋𝑒−(𝐼𝑛(𝑥)−𝜇)2 2𝜎2 ( 3.38 ) In the Figure 3.11 it can be seen the shape of the curve obtained with the log-normal distribution function defined above. Figure 3.11: Some log-normal density functions with different mean and standard deviation. Chapter 3 29 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Reliability Analysis In structural reliability theory, uncertainties in, for instance, system parameters and in environmental loads are treated in a rational way. Structural engineering with a deterministic approach design is based on specified minimum material properties and specified load intensities. Furthermore, stresses and deflections follow a certain procedure of calculation that is often prescribed in deterministic based codes. However, it has been recognized that this deterministic thinking involves a high degree of uncertainty. The uncertainties mentioned combined with several similar forms of uncertainty result in an uncontrolled risk. Moreover, deterministic design leads to very expensive and conservative designs because it is required a too high safety level due to not handling uncertainties properly. Therefore, the deterministic approach poses a serious issue because a real measure of the safety or reliability of the structure is not obtained. In modern structural reliability theory it is manifestly recognized that some risk of structural failure must be accepted. A probabilistic model is a suitable approach to obtain some measure of this risk, that is to say, the probability of failure. This approach allows to design a structure with an acceptable level of failure probability during the lifetime of the structure. The basic structural reliability problem is going to be considered to illustrate this theory. Consider now only one load effect 𝑆 resisted by one resistance 𝑅. Each parameter is described by a known probability density function, 𝑓𝑆 and 𝑓𝑅 respectively, and both 𝑅 and 𝑆 are expressed in the same units. The safety of a structural element will be the unique consideration taken. The structural element will be considered to reach failure if the resistance R is less than the stress resultant S acting on the structural element. The probability of failure of the structural element 𝑝𝑓 can be stated with the following equation (Melchers, 1999): 𝑝𝑓=𝑃(𝑔(𝑅,𝑆)≤0) ( 3.39 ) Where 𝑔 is termed the limit state or performance function and the probability of failure is identical with the probability of limit state violation. The Equation 3.39 can be adapted to a probabilistic model to obtain the probability of failure for the number of samples considered in the analysis. The statistical description of the failure of the performance functions 𝑔𝑖 (𝑖=1,2,…,𝑛) requires a reliability analysis. Prior to the reliability analysis, the statistical characteristics of the random quantities are first defined using the tools explained in the previous section, that is, by suitable probability distributions. Then the probability of failure is evaluated by numerically stable and affordable procedures. Numerically stable and affordable procedures are then used to evaluate the probability of failure. There are several methods developed with the aim of obtaining the probability integration in the structural reliability analysis (Rackwitz, 2001). In the direct Monte Carlo simulation or Importance Sampling method, the probability of failure is derived from the test data of a large amount of samples. Conversely, in the First Order Reliability Method (FORM), the Second Order Reliability Method (SORM) or the Advanced Mean Value method, an additional nonlinear constrained optimization procedure is required for locating the Design Point or Most Probable Point of failure (MPP) and thus the reliability based design optimization 36 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya 𝐾(𝑒)= [ 𝐸𝐴 𝐿 012𝐸𝐼𝑧 𝐿3𝛽𝑦 0 0 12𝐸𝐼𝑦 𝐿3𝛽𝑧 𝑆𝑌𝑀 0 0 0 𝐺𝐽 𝐿 0 0 −6𝐸𝐼𝑦 𝐿2𝛽𝑧0𝛾𝑧𝐸𝐼𝑦 𝐿 06𝐸𝐼𝑧 𝐿2𝛽𝑦0 0 0 𝛾𝑦𝐸𝐼𝑧 𝐿 −𝐸𝐴 𝐿0 0 0 0 0 𝐸𝐴 𝐿 012𝐸𝐼𝑧 𝐿3𝛽𝑦0 0 0−6𝐸𝐼𝑧 𝐿2𝛽𝑦012𝐸𝐼𝑧 𝐿3𝛽𝑦 0 0 −12𝐸𝐼𝑦 𝐿3𝛽𝑧06𝐸𝐼𝑦 𝐿2𝛽𝑧0 0 0 12𝐸𝐼𝑦 𝐿3𝛽𝑧 0 0 0 −𝐺𝐽 𝐿0 0 0 0 0 𝐺𝐽 𝐿 0 0 −6𝐸𝐼𝑦 𝐿2𝛽𝑧0𝜂𝑧𝐸𝐼𝑦 𝐿0006𝐸𝐼𝑦 𝐿2𝛽𝑧0 𝛾𝑧𝐸𝐼𝑦 𝐿 06𝐸𝐼𝑧 𝐿2𝛽𝑦0 0 0 𝜂𝑦𝐸𝐼𝑧 𝐿0 −6𝐸𝐼𝑧 𝐿2𝛽𝑦0 0 0 𝛾𝑦𝐸𝐼𝑧 𝐿 ] ( 4.9 ) The different parameters that compose the local stiffness matrix of the Equation 4.9 are detailed in the Appendix A. 4.1.1.2. Element Mass Matrix The way to obtain the element mass matrix is analogous to the procedure followed in order to obtain the element stiffness matrix. In this case, however, the analysis is based on the external work of the inertial forces and, after modifications, the following relationship is obtained: 𝑊𝑒𝑥𝑡=−∫ [𝛿𝑎(𝑒)]𝑇𝐵𝑇𝜌𝐵𝑎󰇘(𝑒) 𝑑𝑥 𝑙(𝑒)=−[𝛿𝑎(𝑒)]𝑇(∫ 𝐵𝑇𝜌𝐵 𝑑𝑥 𝑙(𝑒))𝑎󰇘(𝑒) ( 4.10 ) The local element mass matrix is then expressed as: 𝑀(𝑒)=∫ 𝐵𝑇𝜌𝐵 𝑑𝑥 𝑙(𝑒) ( 4.11 ) The shear deformation effects are introduced in the element mass matrix using the parameters Φ𝑦 and Φ𝑧 defined above. Additionally, the rotatory inertia has been taken into account introducing the radius of gyration. Chapter 4 37 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures 𝑟𝑦=√𝐼𝑦 𝐴 ( 4.12 ) 𝑟𝑧=√𝐼𝑧 𝐴 ( 4.13 ) Finally, after some calculations the local mass matrix for the element is obtained as can be seen bellow. 𝑀𝑒𝑙= [ 1/3 0 𝑎𝑧 0 0 𝑎𝑦 𝑆𝑌𝑀 0 0 0 𝐽 3𝐴 0 0 −𝑐𝑦0𝑒𝑦 0𝑐𝑧0 0 0 𝑒𝑧 1/6 0 0 0 0 0 1/3 0 𝑏𝑧0 0 0𝑑𝑧0𝑎𝑧 0 0 𝑏𝑦0−𝑑𝑦0 0 0 𝑎𝑦 0 0 0 𝐽 6𝐴 0 0 0 0 0 𝐽 3𝐴 0 0 𝑑𝑦0𝑓𝑦0 0 0 𝑐𝑦0 𝑒𝑦 0 −𝑑𝑧0 0 0 𝑓𝑧0 −𝑐𝑧0 0 0 𝑒𝑧 ] ( 4.14 ) As with the local stiffness matrix, the different parameters that compose the local mass matrix of the Equation 4.14 are detailed in the Appendix A. 4.1.1.3. Element Damping Matrix The damping matrix have been calculated using the Rayleigh proportional damping as a linear combination of the mass and stiffness matrices. This method to get the damping can be applied to a singular element of the finite element model, obtaining the element damping matrix as a linear combination of the element mass and element stiffness matrices. 𝐶(𝑒)=𝛼𝑀(𝑒)+𝛽𝐾(𝑒) ( 4.15 ) As can be seen in the equation above, the element damping matrix depends on the mass coefficient (𝛼) and stiffness coefficient (𝛽). This two parameters depends on the different damping contributions taken into account in this study, and their values will be described on the following chapters. Discretization The discretization of the support structure is the modelling process of the body of the structure consisting in the equivalent division of itself in a system conformed by smaller 38 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya bodies, which are called finite elements. These are interconnected through common points or nodes, which form surfaces that behave as independent control volumes. The finite element analysis does not attempt to solve the problem as if it were a single piece, but rather the body is subdivided into a finite number of elements which in turn give individual results that finally merge to create a single solution. The support structure defined in Scope and Limitations has been discretized taking into account the main objectives of this study, as it can be seen in Figure 4.2. The accuracy of the FEM depends on the number of elements used on the analysis, where a large number of elements implies a large accuracy, but at the same time a large computational cost. For this reason, the structure have been treated differently depending on the importance of each part of the structure. 4.1.2.1. Number of Elements The tower has different geometric properties along its height, so it is necessary a minimum number of elements to observe the contribution of this variance on the results of the finite element model. Furthermore, a segment of the foundation, from the mud-line to the bottom of the monopile, has an important function since it introduces the contribution of the soil stiffness in the model. The soil is modeled as a series of linear lateral springs, which represents the stiffness of the soil at different sections. In order to form a unified stiffness matrix of both soil and structure, these sectional soil stiffness matrices are added to the structural stiffness matrix of the support structure. The distribution of selected elements for each section of the support structure can be seen in Table 4.1. Table 4.1: Number of elements for each part of the support structure discretized. Structural section Number of elements Tower 11 Bottom tower to mud-line 3 Below mud-line 40 4.1.2.2. Rotor Mass The mass matrix defined above is based on the properties of the support structure of the offshore wind turbine. Nevertheless, rotor and nacelle have a significant mass that must be taken into account. To achieve that, the rotor and nacelle mass has been added to the FE model in the same way as the soil stiffness, that is to say, adding the sectional mass matrix of the turbine to the structural mass matrix in the tower top element of the structure. Boundary and Load Conditions The boundary conditions of the structure can be divided into two different groups, the ones regarding to the lower part of the foundation pile and the linear lateral springs added to the foundation to simulate the contribution of the stiffness of the soil. Chapter 4 39 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures The lowest part of the foundation pile has been designed as a three-dimensional mobile articulate support, so at this point the movements are allowed in the two directions y and z, and the rotations are allowed in the three directions x, y and z. The only restriction has been imposed to the movements in the x directions, where the displacements are not allowed. Figure 4.2: Schematic representation of the support structure and the finite element model, with lateral springs under the mud-line where they represent the soil stiffness. On the other hand, the contribution of the soil stiffness in the model has been introduced to the model adding linear lateral springs from the mud-line to the bottom of the monopile. The sectional soil stiffness matrices introduced by these springs have been added to the structural Soil stiffness springs k1 k2 kn-1 kn t1 t2 t3 ti tn-1 tn n1 n2 n3 ni nn nn-1 Mud-line Tower top Tower bottom Monopile bottom 40 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya stiffness matrix of the support structure in the directions y and z, and on alternate nodes, so these contributions are added to every two nodes. The load conditions of the structure are marked by the influence of the wind on the wind turbine, situated on the top of the tower. The loads, therefore, are introduced in the top of the structure in the three directions x, y and z. The kind of loads and the way used to obtain them will be explained in the following chapters. Newmark Method In Background and Literature Review, it has seen that it is required to solve a system of coupled second-order ordinary differential equations in time in order to find the response of the structure when it is subject to a dynamic load. There are different methods capable of solving this problems, they are primarily the implicit and explicit schemes of step-by-step integration. The implicit methods present the major advantage that the solution is not artificially amplified whatever the time increment selected for the integration. For this reason and for its widespread use, the Newmark method is chosen to solve the system of coupled second-order ordinary differential equations (Canet, 2013). Let consider the variation of the acceleration 𝑢󰇘(𝑡) between the different times 𝑡𝑖 and 𝑡𝑖+1= 𝑡𝑖+∆𝑡. The variable change 𝜏=𝑡−𝑡𝑖 is introduced in order to obtain that 𝜏 is null when 𝑡= 𝑡𝑖, and 𝜏 is equal to ∆𝑡 when 𝑡=𝑡𝑖+1. The acceleration vector at the time 𝜏≤∆𝑡 will be expressed as follows, 𝑢󰇘(𝜏)=𝑢󰇘𝑖+𝑓(𝜏)(𝑢󰇘𝑖+1−𝑢󰇘𝑖) ( 4.16 ) Figure 4.3: Illustration of time-varying acceleration. 𝑢󰇘(𝑡) 𝑢󰇘𝑖+1 𝑢󰇘𝑖 𝑡 𝑡𝑖 𝑡𝑖+𝑡𝑖+1 𝜏 𝑢󰇘𝑖(𝜏)=𝑢󰇘𝑖+𝑢󰇘𝑖+1 2 Chapter 4 41 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Where the function 𝑓(𝜏) is equal to zero when 𝜏=0 and is equal to one when 𝜏=∆𝑡. The previous equation implies that the variation’s law of accelerations at the times [𝑡𝑖,𝑡𝑖+1] is the same for all the degrees of freedom. Therefore, the velocity 𝑢󰇗(𝜏) can be obtained integrating the acceleration expressed before. 𝑢󰇗(𝜏)=𝑢󰇗𝑖+∫𝑢󰇘(𝜏) 𝑑𝜏 𝜏 0=𝑢󰇗𝑖+∫𝑢󰇘𝑖 𝑑𝜏 𝜏 0+∫𝑓(𝜏)(𝑢󰇘𝑖+1−𝑢󰇘𝑖) 𝑑𝜏 𝜏 0 ( 4.17 ) 𝑢󰇗(𝜏)=𝑢󰇗𝑖+𝑢󰇘𝑖𝜏+(𝑢󰇘𝑖+1−𝑢󰇘𝑖)∫𝑓(𝜏) 𝑑𝜏 𝜏 0 ( 4.18 ) This expression can be rewritten by substituting the following relations in the equation. 𝑔(𝜏)=∫𝑓(𝜏) 𝑑𝜏 𝜏 0 ( 4.19 ) ∆𝑡𝛾=∫ 𝑓(𝜏) 𝑑𝜏 ∆𝑡 0 ( 4.20 ) Obtaining, 𝑢󰇗(𝜏)=𝑢󰇗𝑖+𝑢󰇘𝑖𝜏+(𝑢󰇘𝑖+1−𝑢󰇘𝑖)𝑔(𝜏) ( 4.21 ) And for the case of 𝜏=∆𝑡, 𝑢󰇗𝑖+1=𝑢󰇗𝑖+𝑢󰇘𝑖∆𝑡+(𝑢󰇘𝑖+1−𝑢󰇘𝑖)𝛾∆𝑡 =𝑢󰇗𝑖+[(1−𝛾)𝑢󰇘𝑖+𝛾𝑢󰇘𝑖+1]∆𝑡 ( 4.22 ) Using the same idea to obtain the equation of velocity, the displacements can be calculated from the integration of the velocity. 𝑢(𝜏)=𝑢𝑖+𝑢󰇗𝑖𝜏+𝑢󰇘𝑖𝜏2 2+(𝑢󰇘𝑖+1−𝑢󰇘𝑖)∫ 𝑔(𝜏)𝑑𝜏 ∆𝑡 0 ( 4.23 ) Introducing the following relation, ∆𝑡2𝛽=∫ 𝑔(𝜏) 𝑑𝜏 ∆𝑡 0 ( 4.24 ) 42 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya And for the case of 𝜏=∆𝑡, it is finally obtained, 𝑢𝑖+1=𝑢𝑖+𝑢󰇗𝑖∆𝑡+[(12−𝛽)𝑢󰇘𝑖+𝛽𝑢󰇘𝑖+1]∆𝑡2 ( 4.25 ) The Equations 4.22 and 4.23 are the difference equations of Newmark. These equations, together with the differential equation of the motion, allow to obtain the displacements, velocities and accelerations at the time 𝑡𝑖+1 depending on the values of 𝑡𝑖. As can be observed, the Newmark’s method depends on the value of the parameters 𝛼 and 𝛽, and they require an accurate choice because the stability of the method depends on the values selected. In this case, the set of choices correspond to the Average Acceleration Method, where the values of both parameters are as follows, 𝛾=12 ; 𝛽=14 As earlier pointed out, the equation of motion of a structure with several degrees of freedom for the time 𝑡=𝑡𝑖+1 can be expressed as, 𝑀𝑢󰇘𝑖+1+𝐶𝑢󰇗𝑖+1+𝐾𝑢𝑖+1=𝐹𝑖+1 ( 4.26 ) With M, C and K as the mass, damping and stiffness matrices, respectively; and F as the vector of externally applied loads. The Equations 4.22 and 4.25 are directly introduced into equation of motion, which leads to a set of algebraic equations which can be linear or non-linear depending on the type of problem. With 𝑢𝑖+1 as the resulting unknowns, the equations can be written as, 𝑢󰇘𝑖+1=1 𝛽∆𝑡2[𝑢𝑖+1−𝑢𝑖−𝑢󰇗𝑖∆𝑡]+(1 2𝛽−1)𝑢󰇘𝑖 ( 4.27 ) 𝑢󰇗𝑖+1=𝛾 𝛽∆𝑡2(𝑢𝑖+1−𝑢𝑖)+(1−𝛾𝛽)𝑢󰇗𝑖+(1−𝛾 2𝛽)∆𝑡𝑢󰇘𝑖 ( 4.28 ) Finally, substituting these equations into the equation of motion is obtained a system of equations as follows, 𝐾𝑢𝑖+1=𝑃𝑖+1 ( 4.29 ) Where, 𝐾=𝐾+ 1 𝛽∆𝑡2𝑀+ 𝛾 𝛽∆𝑡𝐶 ( 4.30 ) Chapter 4 43 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures 𝑃𝑖+1=𝐹𝑖+1+𝑀[ 1 𝛽∆𝑡2𝑢𝑖+1 𝛽∆𝑡𝑢󰇗𝑖+(1 2𝛽−1)𝑢󰇘𝑖] +𝐶[𝛾 𝛽∆𝑡𝑢𝑖+(𝛾𝛽−1)𝑢󰇗𝑖+(𝛾 2𝛽−1)∆𝑡𝑢󰇘𝑖] ( 4.31 ) The displacements are obtained at the time 𝑡𝑖+1 solving the system of equations in Equation 4.29. Substituting this value in the Equations 4.27 and 4.22, the accelerations and velocities are finally obtained. 4.1.4.1. Algorithm Newmark’s method can be synthesized with the following scheme. The algorithm is divided into two parts. First, basics parameters are obtained, which will allow to perform the second part of the algorithm. Secondly, the intended outputs, in this case displacements, velocities and accelerations, are obtained at each time step. Initialization: - Initial conditions 𝑢0, 𝑢󰇗0 and 𝑢󰇘0. - Choice of ∆𝑡, 𝛾 and 𝛽. - Assembly of K, M, and C. - Computation of the effective stiffness matrix, 𝐾. - Cholesky factorization of 𝐾. At each time step: - Computation of the effective loading 𝑃𝑖+1. - Solution of the system 𝐾𝑢𝑖+1=𝑃𝑖+1, obtaining the value of 𝑢𝑖+1. - Calculation of the velocities and accelerations, 𝑢󰇗𝑖+1 and 𝑢󰇘𝑖+1, at time 𝑡𝑖+1. Rotor Loading The wind acting on the wind turbine rotor produces loads which are transferred to the support structure of the offshore wind turbine. The wind speeds, therefore, are an important parameter since they determine the magnitude of the loads acting on the support structure. The histogram in Figure 4.4 below gives to probability of occurrence for each wind speed for a wind turbine generator situated at a height of 100 meters. As can be seen from the histogram presented in Figure 4.4, the wind speeds between 3 m/s and 15 m/s have the largest probability of occurrence. The time needed to perform the probability analysis requires a limited number of wind speeds to perform the analysis, so it has been decided to analyse three wind speeds: 8 m/s, 14 m/s and 18 m/s. The design standards of offshore wind turbines specify a higher number of wind speeds to perform the structural analysis of these structures. Nevertheless, the aim of this thesis is to study a specific problem of the support structure of offshore wind turbines, so a simplified model has been considered to perform the analysis and, consequently, the number of wind speeds analysed has been reduced. 44 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 4.4: Histogram of the probability of occurrence of different wind speeds. As has been mentioned, the wind acting on the wind turbine generator produces rotor loads which are transferred to the support structure, so it is not possible to introduce the wind speeds in the finite element model as a load. Consequently, in order to obtain the rotor loads, it has performed rotor simulation in FEDEM Windpower software. Table 4.2: Wind speeds selected to perform the simulation of this research. Case Wind Speed [m/s] 1 8 2 14 3 18 The rotor of the wind turbine described in Scope and Limitations has been modelled using this software and subsequently the rotor simulations for the three wind speeds have been performed. Additionally, the simulations have been done taking into account that FEDEM Windpower allows to perform the simulation with turbulent wind, which allows to perform a more realistic analysis. In the Figure 4.5 below it can be seen the loads in the top of the support structure from the rotor simulations performed in FEDEM Windpower for a wind speed of 8 m/s. The fluctuation of the results are caused by the turbulent component of the wind. Furthermore, the loads in the X-axis, so the axis aligned with the support structure shaft, have a mean value corresponding to the total weight of the rotor, approximately equivalent to 350 000 kg. 0 2 4 6 8 10 12 14 16 18 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 Probability [%] Wind speed [m/s] Chapter 4 45 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 4.5: Rotor loads from rotor simulations in FEDEM Windpower software. The timeseries loads correspond to the X, Y and Z axis, from top to bottom respectively. Damping Estimation The damping contributions that are relevant for offshore wind purposes are caused by aerodynamic, soil and structural effects. In this study, however, only aerodynamic and structural damping are going to be considered to determine the global damping of the structure. The structural damping, as noted before, is the damping produced by the mechanical properties of the structure. The structural damping had been taken from the OC3 project, where a structural damping ratio of 1% is specified for the monopile support structure (Jonkman et al,. 2007). The aerodynamic damping of the offshore wind turbine is obtained from the study of aerodynamic damping under constant and turbulent wind realized in Schafhirt (2014). The aerodynamic damping is predicted by performing logarithmic decay tests in FEDEM Windpower. The simulations are performed for the entire offshore wind turbine under turbulent wind with a turbulence intensity of 10% and for different wind speeds between 3 m/s and 24 m/s. 52 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Table 4.5: Analysis cases when the aerodynamic damping is distributed in the entire structure. Case Damping Variable Correlation Wind speed [m/s] DAD8 Distributed damping (DD) Aerodynamic damping - 8 DAD14 14 DAD18 18 DS8C Soil Correlated 8 DS14C 14 DS18C 18 DS8N No correlated 8 DS14N 14 DS18N 18 DA8C Area Correlated 8 DA14C 14 DA18C 18 DA8N No correlated 8 DA14N 14 DA18N 18 DC8CC Combined Correlated - Correlated 8 DC14CC 14 DC18CC 18 DC8C Correlated – No correlated 8 DC14C 14 DC18C 18 DC8NC No correlated - Correlated 8 DC14NC 14 DC18NC 18 DC8N No correlatedNo correlated 8 DC14N 14 DC18N 18 The different cases presented above have been analysed probabilistically performing 200 simulations. This number of samples is enough to ensure reliable results that give robustness to the probabilistic analysis. Furthermore, the computation cost, measured in time, to perform 200 simulations is approximately of 20 minutes, so this number of simulations is appropriate to the total number of cases of the probabilistic analysis of this study. As has been seen in relation to the wind speeds, the design standards of offshore wind turbines specify a higher number of simulations to perform the structural analysis of these structures. Nevertheless, the aim of this thesis is to study a specific problematic, without needing to perform a complete analysis of the support structure of an offshore wind turbine. Furthermore, the computational effort required to carry out the number of simulations specified in these standards cannot be assumed in this research. Chapter 4 53 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Table 4.6: Analysis cases when the aerodynamic damping is added to the finite element located in the top of the structure. Case Damping Variable Correlation Wind speed [m/s] TAD8 Top tower damping (TD) Aerodynamic damping - 8 TAD14 14 TAD18 18 TS8C Soil Correlated 8 TS14C 14 TS18C 18 TS8N No correlated 8 TS14N 14 TS18N 18 TA8C Area Correlated 8 TA14C 14 TA18C 18 TA8N No correlated 8 TA14N 14 TA18N 18 TC8CC Combined Correlated - Correlated 8 TC14CC 14 TC18CC 18 TC8CN Correlated – No correlated 8 TC14CN 14 TC18CN 18 TC8NC No correlated - Correlated 8 TC14NC 14 TC18NC 18 TC8NN No correlatedNo correlated 8 TC14NN 14 TC18NN 18 Additionally, when the degree of damping and the cross section of the structure are analysed probabilistically, the results obtained for the fatigue damage seem to show that the number of samples is not enough, as will be seen later. Therefore, the analyses of these cases have been repeated by performing 1000 simulations instead of 200. With this consideration is not intended to explain the results obtained in the following chapter, but rather that the total number of cases is clear due to their importance in the methodology of this thesis. All simulations have been performed with a simulation time of 720 seconds and a time step of 0.025 seconds. The two first minutes of each simulation have been deleted from the results order to avoid errors in the outputs caused by distortions in the first instants of simulation. By this way, the results presented in the following chapter correspond to the 10 last minutes (600 seconds) of simulation. 54 Chapter 4 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Table 4.7: Additional analysis for the study cases of the degree of damping and the geometry of the structure. Case Damping Variable Correlation Wind speed [m/s] TAD8_2 Top tower damping (TD) Aerodynamic damping - 8 TAD14_2 14 TAD18_2 18 TA8C_2 Area Correlated 8 TA14C_2 14 TA18C_2 18 TA8N_2 No correlated 8 TA14N_2 14 TA18N_2 18 Finally, a similar probabilistic analysis has been made for the reference support structure of the offshore wind turbine. The results of this analysis have enabled to validate the structure modelled in the finite element model and, at the same time, have been used to understand the response of the structure from the different analysis explained above by comparing the outcomes of both analysis. Table 4.8: Analysis cases for the reference support structure of the offshore wind turbine. Case Damping Wind speed [m/s] DRef8 Distributed damping (DD) 8 DRef14 14 DRef18 18 TRef8 Top tower damping (TD) 8 TRef14 14 TRef18 18 55 CHAPTER 5 Results This section presents representative results obtained in this study. Simulations have been performed to investigate the effect of damping, soil stiffness and geometry of the structure on the response of an offshore wind turbine analysed probabilistically. The MATLAB scripts used to perform the calculations in this section can be found in Appendix B. The outputs analysed are the displacements on the top of the support structure and the bending moments in the mud-line. The displacements are analysed in the direction of the y and x axes, corresponding to the axes where the main load of the wind is applied and the axis of the support structure, respectively; and the bending moments are analysed in the y and z axes. The displacements in the direction of the y axis and the bending moments in the direction of the z axis are prioritized with respect to the following analysis due to their higher influence on the response of the structure. On the other hand, the fatigue damage from the displacements and the bending moments of the support structure represents the most relevant output of this thesis and for this reason a detailed analysis of the obtained results have been performed. Additionally, the dynamic response of the support structure is linked to the variables analysed probabilistically and, in turn, the variation of these variables is reflected in the global mass 56 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya and stiffness matrix of the FEM. Therefore, the natural frequencies of the first mode have been analysed in order to have a better understanding of the results. Obviously, all the results of these outputs cannot be presented here, so it is necessary to select some representative values or statistical parameters. To this end, the analysis of the results is partially done by analyzing the mean, standard deviation, variation coefficient and the difference, in percentage, between the mean value of the outcomes and the results of the reference case. In some cases the standard deviation is not presented since it is not possible to present all the parameters and it is also considered that the variation coefficient is a better way to express the scattering of the results. Anyway, all standard deviation values con be consulted on Appendix C. With the intention of getting a more flexible analysis of the data, the name of certain parameters are simplified and the terminology introduced in the Table 4.5, Table 4.6, Table 4.7 and Table 4.8 is used to refer to the different cases of the analysis. Consequently, it will be used terminology as DD (or DStr) to refer to the case with the aerodynamic damping is distributed in the entire structure, DT (or DTop) to refer to the case with the aerodynamic damping located in the top of the support structure, or the numbers 8, 14 and 18 to refer to the different wind speeds. Finally, the probabilistic analysis performed requires special tools to analyse the data obtained. With that purpose in mind, histograms and boxplots are a convenient way of graphically depicting groups of numerical data. As will be seen later, the boxplots are represented with a box, where the central mark is the median, the edges of the box are the 25th and 75th percentiles and the whiskers extend to the most extreme data points not considered outliers. Chapter 5 57 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Reference Case The reference structure has been used to validate the FE model and to analyse the damping approach. The validation of the model will be performed by comparing the natural frequency obtained in the proposed model with the one obtained in a model built in FEDEM Windpower software. The analysis of the damping approach will allow the study of the main differences between the alternatives proposed and thereby simplify the others analysis of the thesis. Validation of the Model The structure has been completely defined in terms of geometry, materials and boundary conditions using MATLAB software. The validation of the structure modelled is required to ensure that the results obtained are correct. For that propose, the reference structure has been modelled in FEDEM Windpower software in order to compare the results of this model with the results obtained in the finite element model presented in this study. The validation has been based on the natural frequency of the structure. Although others outputs could be used to validate the model, the natural frequency has been selected since it gives relevant information about the response of the structure. This is due to the fact that the natural frequency of the structure is obtained by solving a problem of eigenfrequencies where the main inputs are the mass and stiffness matrix of the structure. These two parameters contain a great deal of information about the structure so, in conclusion, it is appropriate to study the natural frequency of the structure in order to validate the FE model. Furthermore, some time series of displacements have been analysed in FEDEM and it has been verified that they were mostly similar with the ones of the simplified FE model. Nevertheless, these results are not shown here. Table 5.1: Natural frequency of the first mode obtained in FEDEM Windpower software and the FEM in MATLAB. Model Natural frequency FEDEM Windpower 0.2771 FEM in MATLAB 0.2871 The natural frequency of the first mode in FEDEM Windpower is slightly lower than the same frequency obtained in the simplified finite element model, with a percentage variance of 3.6%. This difference could be caused by the fact that the proposed approach assumes some simplifications/assumptions respect to the model used in FEDEM Windpower. For instance, the way used to model the rotor is different in each model. The model in FEDEM Windpower proposes a complete modelling of the rotor, where each blade is modelled using beams. Nevertheless, in the proposed approach the rotor is not modelled, instead the loads engendered by the rotor are obtained from rotor simulations in FEDEM Windpower and the mass of the rotor is added to the global mass matrix of the structure. Moreover, the number of elements is lower in the case of the simplified FE model, resulting in a reduced accuracy in the results. Anyway, this difference in the value of the natural 58 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya frequency of the first node is small enough to claim that the proposed model is valid for the analysis to be carried out in the research. Damping Approach In the model, the aerodynamic damping may be distributed in the entire structure, i.e. the damping is added in each element damping matrix, or may be added to one finite element only, which in this case is the element that represents the wind turbine and is located in the top of the support structure. A summary of the results is presented below. This analysis is based on the response of the reference structure to the two alternatives presented in order to consider the aerodynamic damping. The results consist of are displacements, bending moments and fatigue damage of the structure. 5.1.2.1. Displacements The displacements in the y and x axes for each case of the reference support structure are listed in Table 5.2 and Figure 5.1. The displacements for the case with the aerodynamic damping distributed in the entire structure have mean values lower than those obtained for the case with the aerodynamic damping located in the top of the structure. In both cases, the mean value and the standard deviation of the displacements increase when the wind speed is higher. Table 5.2: Mean, standard deviation and variation coefficient value of the displacements in the y and x axes on the top of the structure for the reference case. Case Top displacements Y Top displacements X μ [m] σ [m] CV [%] μ [m] σ [m] CV [%] DRef8 3,98E-02 3,75E-02 94,26 1,29E-05 1,50E-05 116,72 DRef14 7,29E-02 8,52E-02 117,00 4,02E-05 3,79E-05 94,27 DRef18 9,55E-02 8,22E-02 86,12 5,73E-05 5,05E-05 88,29 TRef8 7,11E-02 4,76E-02 66,88 1,42E-05 1,53E-05 108,14 TRef14 1,38E-01 1,12E-01 81,69 3,58E-05 3,96E-05 110,61 TRef18 3,10E-01 1,53E-01 49,30 4,66E-05 5,38E-05 115,46 5.1.2.1. Bending Moments Values obtained for bending moments in the y and z axes for each case of the reference support structure are given in Table 5.3 and Figure 5.2. For the case of the aerodynamic damping distributed in the entire structure, the bending moments have mean values lower than the ones obtained for the case with the aerodynamic damping located in the top of the structure. Nevertheless, the distribution of amplitudes observed in the histograms show that the scattering of the bending moments is different since the most part of the amplitudes are in a reduced range of values. Chapter 5 59 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 5.1: Distribution of amplitudes of the displacements in the y axis for the reference case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 2,0E+03 4,0E+03 6,0E+03 8,0E+03 1,0E+04 1,2E+04 1,4E+04 1,6E+04 0,00 0,03 0,06 0,09 0,12 0,15 0,18 0,22 0,25 0,28 0,31 0,34 0,37 0,40 0,43 Frequency [-] Amplitudes [m] Wind speed of 8 m/s Y displ. for DStr. Y displ. for DTop 0,0E+00 2,0E+03 4,0E+03 6,0E+03 8,0E+03 1,0E+04 1,2E+04 1,4E+04 1,6E+04 0,00 0,04 0,07 0,11 0,15 0,19 0,22 0,26 0,30 0,34 0,37 0,41 0,45 0,49 0,52 Frequency [-] Amplitudes [m] Wind speed of 14 m/s Y displ. for DStr. Y displ. for DTop 0,0E+00 2,0E+03 4,0E+03 6,0E+03 8,0E+03 1,0E+04 1,2E+04 0,00 0,05 0,10 0,16 0,21 0,26 0,31 0,37 0,42 0,47 0,52 0,58 0,63 0,68 0,73 Frequency [-] Amplitudes [m] Wind speed of 18 m/s Y displ. for DStr. Y displ. for DTop 60 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Table 5.3: Mean, standard deviation and variation coefficient value of the bending moments in the y and z axes on the mud-line of the structure for the reference case. Case Mud-line bending moment Y Mud-line bending moment Z μ [Nm] σ [Nm] CV [%] μ [Nm] σ [Nm] CV [%] DRef8 8,50E+05 8,31E+05 97,77 3,94E+06 4,70E+06 119,25 DRef14 1,38E+06 1,44E+06 104,33 6,77E+06 9,92E+06 146,44 DRef18 2,19E+06 2,43E+06 110,97 1,01E+07 1,07E+07 105,90 TRef8 9,65E+05 1,18E+06 122,41 4,74E+06 6,23E+06 131,49 TRef14 2,31E+06 2,87E+06 124,55 7,64E+06 1,22E+07 159,29 TRef18 4,56E+06 5,32E+06 116,57 2,60E+07 2,53E+07 97,47 5.1.2.1. Fatigue Damage As seen from Table 5.4 the equivalent fatigue damage of the reference structure for the different outputs generally increases when the wind speed is higher. Here also, the fatigue damage is higher for the case with the aerodynamic damping located in the top of the structure. This behaviour is reasonable because it has been observed that with this damping approach the displacements and the bending moments are higher, which in turn results in higher damage in the structure. Table 5.4: Mean value of the equivalent fatigue damage of displacements and bending moments for the reference case. Case Top displacement Y [m] Top displacement X [m] Mud-line bending moment Y [Nm] Mud-line bending moment Z [Nm] DRef8 2,73E-03 1,31E-06 5,30E+04 3,68E+05 DRef14 4,49E-03 3,03E-06 1,01E+05 5,99E+05 DRef18 4,33E-03 4,00E-06 1,58E+05 5,86E+05 TRef8 3,29E-03 1,53E-06 7,36E+04 4,60E+05 TRef14 5,89E-03 3,63E-06 1,80E+05 8,12E+05 TRef18 9,86E-03 4,85E-06 3,14E+05 1,38E+06 5.1.2.2. Summary The mean values of the results obtained for the case with the aerodynamic damping distributed in the entire structure are lower than those obtained for the case with the aerodynamic damping located in the top of the structure. This means that the fact of locating the aerodynamic damping in the top of the structure results in considerable increases of the displacements, bending moments of and, in turn, fatigue damage in the support structure. As already mentioned in the previous chapter, the fact of locating the aerodynamic damping in the top of the structure is an approach more realistic and conservative of the dynamic response of the structure and it correspond to the approach more frequently in the modelling of offshore wind turbines. For this reason, the results presented in the following subchapters are focused in the case with the aerodynamic damping located in the top of the structure. Chapter 5 61 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures The results presented in this subchapter are used in the following sections to compare the results obtained in the different cases with the dynamic response of the reference support structure of this research. Figure 5.2: Distribution of amplitudes of the bending moments in the z axis for the reference case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 1,0E+04 2,0E+04 3,0E+04 4,0E+04 5,0E+04 6,0E+04 7,0E+04 8,0E+04 9,0E+04 1,0E+05 0,E+00 6,E+06 1,E+07 2,E+07 2,E+07 3,E+07 4,E+07 4,E+07 5,E+07 6,E+07 Frequency [-] Amplitudes [N.m] Wind speed of 8 m/s Z Bending M. for DStr. Z Bending M. for DTop 0,0E+00 1,0E+04 2,0E+04 3,0E+04 4,0E+04 5,0E+04 6,0E+04 7,0E+04 8,0E+04 9,0E+04 1,0E+05 0,E+00 8,E+06 2,E+07 2,E+07 3,E+07 4,E+07 5,E+07 5,E+07 6,E+07 7,E+07 Frequency [-] Amplitudes [N.m] Wind speed of 14 m/s Z Bending M. for DStr. Z Bending M. for DTop 0,0E+00 5,0E+03 1,0E+04 1,5E+04 2,0E+04 2,5E+04 3,0E+04 3,5E+04 0,E+00 1,E+07 2,E+07 3,E+07 4,E+07 5,E+07 6,E+07 8,E+07 9,E+07 1,E+08 Frequency [-] Amplitudes [N.m] Wind speed of 18 m/s Z Bending M. for DStr. Z Bending M. for DTop 68 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 5.7: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the additional study case of the damping. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+01 1,0E+02 1,5E+02 2,0E+02 2,5E+02 3,0E+02 3,288E-03 3,289E-03 3,289E-03 3,290E-03 3,291E-03 3,291E-03 3,292E-03 3,292E-03 3,293E-03 Frequency [-] Amplitudes [m] Wind speed of 8 m/s Y displ. for Aero. D. Case Reference damage 0,0E+00 5,0E+01 1,0E+02 1,5E+02 2,0E+02 2,5E+02 3,0E+02 3,5E+02 5,881E-03 5,882E-03 5,882E-03 5,883E-03 5,884E-03 5,885E-03 5,886E-03 5,886E-03 5,887E-03 Frequency [-] Amplitudes [m] Wind speed of 14 m/s Y displ. for Aero. D. Case Reference damage 0,0E+00 1,0E+01 2,0E+01 3,0E+01 4,0E+01 5,0E+01 6,0E+01 7,0E+01 8,0E+01 9,0E+01 9,839E-03 9,843E-03 9,847E-03 9,851E-03 9,856E-03 9,860E-03 9,864E-03 9,869E-03 9,873E-03 Frequency [-] Amplitudes [m] Wind speed of 18 m/s Y displ. for Aero. D. Case Reference damage Chapter 5 69 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 5.8: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the additional study case of the damping. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+01 1,0E+02 1,5E+02 2,0E+02 2,5E+02 3,0E+02 4,593E+05 4,595E+05 4,596E+05 4,597E+05 4,598E+05 4,599E+05 4,600E+05 4,601E+05 4,602E+05 Frequency [-] Amplitudes [Nm] Wind speed of 8 m/s Z Bending M. for Aero. D. Case Reference damage 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,4E+02 1,6E+02 1,8E+02 8,105E+05 8,110E+05 8,115E+05 8,120E+05 8,124E+05 8,129E+05 8,134E+05 8,139E+05 8,144E+05 Frequency [-] Amplitudes [Nm] Wind speed of 14 m/s Z Bending M. for Aero. D. Case Reference damage 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,377E+06 1,378E+06 1,379E+06 1,380E+06 1,380E+06 1,381E+06 1,382E+06 1,383E+06 1,384E+06 Frequency [-] Amplitudes [Nm] Wind speed of 18 m/s Z Bending M. for Aero. D. Case Reference damage 70 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Table 5.10: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the additional study case of the damping. Case Damage -Mud-line bending moment Y Damage -Mud-line bending moment Z μ [Nm] CV [%] Diff. with ref. case [%] μ [Nm] CV [%] Diff. with ref. case [%] TAD8_2 7,36E+04 4,43E-02 1,13E-03 4,60E+05 4,02E-02 6,99E-03 TAD14_2 1,80E+05 4,55E-02 6,29E-03 8,13E+05 1,33E-01 9,09E-02 TAD18_2 3,14E+05 1,14E-01 9,89E-03 1,38E+06 9,88E-02 7,39E-03 Summary It can clearly be observed from the results presented that the displacements and the bending moments are comparable to which has been observed in the reference case, particularly in the TD case. In this sense, the fact of analysing the aerodynamic damping in a probabilistic manner does not translate into significant results of these two variables. As regards fatigue damage, here too the mean values of the equivalent fatigue damage are equivalent to which has been observed in the reference case. Nevertheless, the distribution of the amplitudes shows that the peak frequencies of fatigue damage are obtained for higher values than the average values of the reference case, which implies greater damage in the structure. Furthermore, it has been seen some histograms with irregular distributions that suggested an inadequate number of samples. For this reason, some simulations have been repeated by increasing the number of samples to 1000. However, the results obtained have been equivalents and it has not been possible to extract additional conclusions. Chapter 5 71 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Case of Study: Soil Stiffness Displacements and Bending Moments Figure 5.9 and Figure 5.10 present the results obtained in the study case of the soil stiffness. These figures show the displacements and bending moments with boxplots. Additionally, the Table 5.11 and Table 5.12 present the mean values, variation coefficients and the percentage variances between the mean values of the different cases analysed in this section and the reference cases. Table 5.11: Mean, variation coefficient and variation with respect to the reference case of the displacements in the y and x axes for the study case of the soil stiffness. Case Top displacements Y Top displacements X μ [m] CV [%] Diff. with ref. case [%] μ [m] CV [%] Diff. with ref. case [%] DS8C 3,99E-02 94,01 0,112 1,29E-05 116,72 0 DS14C 7,23E-02 117,64 -0,764 4,02E-05 94,27 0 DS18C 9,56E-02 86,01 0,145 5,73E-05 88,29 0 DS8N 3,99E-02 93,99 0,119 1,29E-05 116,72 0 DS14N 7,25E-02 117,45 -0,466 4,02E-05 94,27 0 DS18N 9,54E-02 86,11 -0,035 5,73E-05 88,29 0 TS8C 7,20E-02 66,15 1,253 1,42E-05 108,14 0 TS14C 1,38E-01 82,04 0,460 3,58E-05 110,61 0 TS18C 3,09E-01 49,89 -0,363 4,66E-05 115,46 0 TS8N 7,20E-02 66,20 1,192 1,42E-05 108,14 0 TS14N 1,37E-01 82,20 -0,100 3,58E-05 110,61 0 TS18N 3,09E-01 49,92 -0,416 4,66E-05 115,46 0 The displacements in the y and x axes and the bending moments in the y and z axes obtained in this section are presented below. In all cases, the mean value and standard deviation of the displacements increase when the wind speed is higher. The standard deviation has different behaviour for each axis and for each damping approach, as has been observed in the results of the previous section. On the one hand, the variation coefficient of the displacements of the y axis has an irregular behaviour and it is observed that the maximum and minimum values are obtained for a wind speed of 14 m/s and 18 m/s, respectively. The variation coefficient of the displacements of the x axis for the DD case is reduced when the wind speed is higher, whereas for the TD case it is increased. The behaviour of the bending moments is analogous to which has been observed in the displacements. The mean value and standard deviation of the bending moments in both axes increase when the wind speed is higher. The standard deviation also has an irregular behaviour as well as the variation coefficient, making difficult to achieve a clear conclusion. Looking at the differences between the correlated and non-correlated case, it can be seen that there is almost not variations in the results obtained. There are small differences between the two cases but it is not possible to observe any general trend. 72 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 5.9: Boxplot of the amplitudes of the displacements obtained in the rainflow counting for the study case of the soil stiffness. Each graph correspond to the displacements in the y axis (top) and in the x axis (bottom). Table 5.12: Mean, variation coefficient and variation with respect to the reference case of the bending moments in the y and z axes for the study case of the damping. Case Mud-line bending moment Y Mud-line bending moment Z μ [Nm] CV [%] Diff. with ref. case [%] μ [Nm] CV [%] Diff. with ref. case [%] DS8C 8,49E+05 97,89 -0,187 3,94E+06 119,18 0,058 DS14C 1,38E+06 104,71 -0,378 6,73E+06 146,90 -0,604 DS18C 2,18E+06 111,10 -0,180 1,01E+07 105,58 0,509 DS8N 8,49E+05 97,89 -0,144 3,94E+06 119,18 0,073 DS14N 1,38E+06 104,58 -0,278 6,74E+06 146,83 -0,464 DS18N 2,18E+06 111,18 -0,280 1,01E+07 105,64 0,410 TS8C 9,72E+05 122,92 0,683 4,80E+06 130,37 1,279 TS14C 2,37E+06 123,37 2,688 7,79E+06 158,38 1,960 TS18C 4,47E+06 117,29 -1,991 2,55E+07 99,24 -2,064 TS8N 9,68E+05 122,86 0,294 4,80E+06 130,54 1,212 TS14N 2,36E+06 123,73 2,348 7,73E+06 158,66 1,232 TS18N 4,49E+06 117,10 -1,617 2,55E+07 99,08 -1,985 Chapter 5 73 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 5.10: Boxplot of the amplitudes of the bending moments obtained in the rainflow counting for the study case of the soil stiffness. Each graph correspond to the displacements in the y axis (top) and in the z axis (bottom). On the other hand, by comparing the results obtained in these cases with the ones of the reference case, one can see that the differences are small and the mean values are located around the mean values of the reference case. It is worth stressing that the mean values of the displacements in x axis are the same as for the reference case, observing, therefore, a null variation percentage between both cases. Conversely, the widest variations are to be found in the bending moments of the TD case, with variations around 1.5%. Fatigue Damage 5.3.2.1. Displacements From the fatigue damage results presented in Table 5.13 and Figure 5.11, one can see that, in general, the fatigue damage equivalent displacement increases when the wind speed is higher, as well as its standard deviation. It is interesting to observe that the standard deviation of the displacements of the y axis for the TD and uncorrelated cases has a different behaviour, with a maximum value for a wind speed of 14 m/s. Although the variation coefficient has different values for each case and it is hard to achieve an overall conclusion, the variation coefficient is an order of magnitude higher in the TD case, which results in a higher scattering of the results and, in turn, in a higher fatigue damage of the structure. Additionally, the displacements of the x axis has really small values of the variation coefficient. 74 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya The correlated and uncorrelated cases show analogous results. The differences between these two alternative are very small, making difficult to know which case the greatest fatigue damage occurs. Looking at the differences with the reference case, the mean values of the fatigue damage obtained here are approximately the same as the ones obtained in the reference case, with a deviation of less than 1% with respect to the mean values of the reference case. From the histograms presented in Figure 5.11, one can see that the peak frequency of the equivalent fatigue damage is close to the mean value of the reference case for the three wind speeds. The distribution of the amplitudes has a certain degree of symmetry around an average value corresponding, approximately, to the mean value of the reference case. Table 5.13: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the study case of the soil stiffness. Case Damage - Top displacements Y Damage - Top displacements X μ [m] CV [%] Diff. with ref. case [%] μ [m] CV [%] Diff. with ref. case [%] DS8C 2,73E-03 1,79E-01 6,43E-02 1,31E-06 1,13E-13 7,75E-13 DS14C 4,49E-03 4,81E-01 -1,23E-01 3,03E-06 8,40E-14 0 DS18C 4,33E-03 5,24E-01 -1,96E-02 4,00E-06 2,76E-13 -2,33E-13 DS8N 2,73E-03 1,57E-01 4,81E-02 1,31E-06 1,13E-13 7,75E-13 DS14N 4,49E-03 3,48E-01 -7,44E-02 3,03E-06 8,40E-14 0 DS18N 4,33E-03 4,53E-01 -1,22E-02 4,00E-06 2,76E-13 -2,33E-13 TS8C 3,28E-03 1,38 -2,97E-01 1,53E-06 3,46E-13 0 TS14C 5,93E-03 3,56 6,62E-01 3,63E-06 3,51E-13 -2,69E-13 TS18C 9,86E-03 1,28 -2,63E-02 4,85E-06 3,67E-13 2,09E-13 TS8N 3,29E-03 1,25 -1,85E-01 1,53E-06 3,46E-13 0 TS14N 5,91E-03 2,50 3,56E-01 3,63E-06 3,51E-13 -2,69E-13 TS18N 9,86E-03 1,32 -1,54E-02 4,85E-06 3,67E-13 2,09E-13 5.3.2.1. Bending Moments As seen from Table 5.14 and Figure 5.12, the fatigue damage equivalent bending moments generally increase when the wind speed is higher. The scattering of these results is difficult to analyse because there is not a clear trend that explains the distribution of the values of the standard deviation and the variation coefficient. Here too, the variation coefficient is an order of magnitude higher in the TD case, with the resulting consequences that involves. As in the case of the displacements, the correlated and uncorrelated cases show analogous results. The differences between this two cases are very small, making difficult to observe if there is more fatigue damage in the correlated or uncorrelated case. Regarding the differences with the reference case, the mean values of the equivalent fatigue damage obtained here are similar to those obtained in the reference case, with a maximum deviation of 1.5% with respect to the mean values of the reference case. The other Chapter 5 75 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures considerations made in the analysis of the fatigue damage equivalent displacement regarding the distribution of the results are analogous in this case. Figure 5.11: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the study case of the soil stiffness. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 3,153E-03 3,186E-03 3,220E-03 3,253E-03 3,286E-03 3,319E-03 3,352E-03 3,385E-03 Frequency [-] Amplitudes [m] Wind speed of 8 m/s Y displ. for Soil CC Y displ. for Soil NCC Reference damage 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 5,608E-03 5,761E-03 5,914E-03 6,068E-03 6,221E-03 6,374E-03 6,527E-03 6,680E-03 Frequency [-] Amplitudes [m] Wind speed of 14 m/s Y displ. for Soil CC Y displ. for Soil NCC Reference damage 0,0E+00 1,0E+01 2,0E+01 3,0E+01 4,0E+01 5,0E+01 6,0E+01 7,0E+01 9,201E-03 9,359E-03 9,517E-03 9,676E-03 9,834E-03 9,992E-03 1,015E-02 1,031E-02 Frequency [-] Amplitudes [m] Wind speed of 18 m/s Y displ. for Soil CC Y displ. for Soil NCC Reference damage 76 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Table 5.14: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the study case of the soil stiffness. Case Damage - Mud-line bending moment Y Damage - Mud-line bending moment Z μ [Nm] CV [%] Diff. with ref. case [%] μ [Nm] CV [%] Diff. with ref. case [%] DS8C 5,30E+04 2,93E-01 2,03E-02 3,68E+05 3,36E-01 1,24E-02 DS14C 1,01E+05 1,08E-01 1,33E-01 5,99E+05 2,25E-01 -1,46E-03 DS18C 1,58E+05 2,12E-01 -8,13E-02 5,86E+05 1,03E-01 -1,33E-02 DS8N 5,30E+04 2,56E-01 4,64E-02 3,68E+05 3,04E-01 4,19E-03 DS14N 1,01E+05 1,03E-01 1,20E-01 5,99E+05 2,03E-01 1,50E-03 DS18N 1,57E+05 1,61E-01 -9,56E-02 5,86E+05 7,69E-02 -3,19E-02 TS8C 7,41E+04 2,93 6,93E-01 4,60E+05 1,68 -6,63E-04 TS14C 1,82E+05 1,74 1,19 8,19E+05 3,63 8,59E-01 TS18C 3,10E+05 2,30 -1,52 1,38E+06 2,11 -1,35E-01 TS8N 7,39E+04 2,65 3,94E-01 4,60E+05 1,52 1,19E-01 TS14N 1,82E+05 1,44 1,23 8,16E+05 2,53 5,34E-01 TS18N 3,10E+05 2,27 -1,38 1,38E+06 2,17 -2,06E-01 Natural Frequencies The variations introduced in this study case results in different value of the soil stiffness. This, in turn, is translated in different natural frequencies for each simulation due to the dependence of this parameter on the stiffness matrix of the structure. As seen from Table 5.15 and Figure 5.13, the natural frequencies of the different cases analysed in this section have approximately the same average value. It is interesting to note that the mean values are very similar to those obtained in the reference case, with a maximum variation of the 0.0192%. Furthermore, these variations with the reference case do not seem to indicate a clear trend, since in some cases the average natural frequency is above the mean value of the reference case and in other cases the opposite is observed. Table 5.15: Mean, standard deviation, variation coefficient and variation with respect to the reference case of the natural frequency of the structure for the study case of the soil stiffness. Case μ [Hz] σ [Hz] CV [%] Diff. with ref. case [%] TS8C 0,287102 7,32E-04 0,2548 -0,0162 TS14C 0,287097 8,03E-04 0,2799 -0,0180 TS18C 0,287203 6,81E-04 0,2370 0,0192 TS8N 0,287162 6,73E-04 0,2345 0,0048 TS14N 0,287125 6,08E-04 0,2116 -0,0082 TS18N 0,287159 6,76E-04 0,2353 0,0035 Chapter 5 77 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 5.12: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the study case of the soil stiffness. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 4,434E+05 4,483E+05 4,532E+05 4,581E+05 4,631E+05 4,680E+05 4,729E+05 4,778E+05 Frequency [-] Amplitudes [Nm] Wind speed of 8 m/s Z Bending M. for Soil CC Z Bending M. for Soil NCC Reference damage 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 7,766E+05 7,978E+05 8,190E+05 8,402E+05 8,614E+05 8,826E+05 9,038E+05 9,250E+05 Frequency [-] Amplitudes [Nm] Wind speed of 14 m/s Z Bending M. for Soil CC Z Bending M. for Soil NCC Reference damage 0,0E+00 1,0E+01 2,0E+01 3,0E+01 4,0E+01 5,0E+01 6,0E+01 1,245E+06 1,274E+06 1,304E+06 1,334E+06 1,364E+06 1,393E+06 1,423E+06 1,453E+06 Frequency [-] Amplitudes [Nm] Wind speed of 18 m/s Z Bending M. for Soil CC Z Bending M. for Soil NCC Reference damage 84 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 5.16: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 2,615E-03 2,980E-03 3,344E-03 3,708E-03 4,073E-03 4,437E-03 4,801E-03 5,166E-03 Frequency [-] Amplitudes [m] Wind speed of 8 m/s Y displ. for Area CC Y displ. for Area NCC Reference damage 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 3,726E-03 5,067E-03 6,408E-03 7,750E-03 9,091E-03 1,043E-02 1,177E-02 1,311E-02 Frequency [-] Amplitudes [m] Wind speed of 14 m/s Y displ. for Area CC Y displ. for Area NCC Reference damage 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 3,971E-03 6,622E-03 9,272E-03 1,192E-02 1,457E-02 1,722E-02 1,987E-02 2,253E-02 Frequency [-] Amplitudes [m] Wind speed of 18 m/s Y displ. for Area CC Y displ. for Area NCC Reference damage Chapter 5 85 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Looking at the differences with the reference case, the behaviour of the equivalent fatigue damage obtained here do not follow a regular pattern and in some cases the average values are above the mean values of the reference case and in other cases the opposite trend is observed. As has been noted in the displacements, the distributions of amplitudes are irregular and it is difficult to analyse the histograms. Table 5.19: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the study case of the cross section of the structure. Case Damage - Mud-line bending moment Y Damage - Mud-line bending moment Z μ [Nm] CV [%] Diff. with ref. case [%] μ [Nm] CV [%] Diff. with ref. case [%] DA8C 5,55E+04 15,83 4,841 3,73E+05 9,53 1,382 DA14C 1,00E+05 4,16 -0,432 5,75E+05 6,21 -4,069 DA18C 1,61E+05 1,51 1,946 5,90E+05 4,02 0,797 DA8N 5,00E+04 6,87 -5,610 3,41E+05 8,36 -7,150 DA14N 9,97E+04 3,30 -1,220 5,67E+05 7,79 -5,378 DA18N 1,64E+05 1,71 3,805 6,10E+05 5,31 4,119 TA8C 8,23E+04 16,43 11,767 4,85E+05 12,75 5,386 TA14C 1,71E+05 12,37 -4,660 9,00E+05 9,57 10,809 TA18C 3,12E+05 13,94 -0,733 1,27E+06 15,08 -7,925 TA8N 7,75E+04 8,95 5,271 4,36E+05 11,40 -5,154 TA14N 1,67E+05 11,09 -6,898 9,65E+05 9,69 18,823 TA18N 3,42E+05 11,38 8,895 1,38E+06 15,84 -0,278 Additional Fatigue Damage Analysis The magnitude of the effects observed in the equivalent fatigue damage of this study case and the irregular behaviour of the distributions of amplitudes observed in the histograms have motivated to repeat some simulations by increasing the number of samples. The reason behind is to validate the previous results and deepen into the analysis. From the fatigue damage results presented in Table 5.20 and Table 5.21 and Figure 5.18 and Figure 5.19, one can see that the results obtained are similar to those obtained for 200 simulations. By increasing the number of simulation until 1000 samples, the histograms show a better distribution of amplitudes of displacements and bending moments. The distribution of amplitudes of the displacements shown in Figure 5.18 and Figure 5.19 indicates that the uncorrelated case has a more even distribution compared with the distribution observed in the correlated case. Looking at other aspects, the comments made about the mean values, standard deviations and variation coefficients are analogous to which has been indicated previously, as well as the differences with the reference case. 86 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 5.17: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 4,0E+01 4,5E+01 5,0E+01 3,010E+05 3,848E+05 4,686E+05 5,524E+05 6,363E+05 7,201E+05 8,039E+05 8,877E+05 Frequency [-] Amplitudes [Nm] Wind speed of 8 m/s Z Bending M. for Area CC Z Bending M. for Area NCC Reference damage 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 7,475E+05 8,138E+05 8,802E+05 9,465E+05 1,013E+06 1,079E+06 1,145E+06 1,212E+06 Frequency [-] Amplitudes [Nm] Wind speed of 14 m/s Z Bending M. for Area CC Z Bending M. for Area NCC Reference damage 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 9,955E+05 1,132E+06 1,268E+06 1,405E+06 1,541E+06 1,678E+06 1,814E+06 1,951E+06 Frequency [-] Amplitudes [Nm] Wind speed of 18 m/s Z Bending M. for Area CC Z Bending M. for Area NCC Reference damage Chapter 5 87 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Table 5.20: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent displacement in the y and x axes for the additional study case of the cross section of the structure. Case Damage -Top displacements Y Damage -Top displacements X μ [m] CV [%] Diff. with ref. case [%] μ [m] CV [%] Diff. with ref. case [%] TA8C_2 3,51E-03 11,37 6,510 1,62E-06 21,27 5,362 TA14C_2 6,78E-03 28,75 15,165 3,77E-06 21,41 4,001 TA18C_2 9,64E-03 32,37 -2,266 5,06E-06 20,64 4,263 TA8N_2 3,70E-03 10,44 12,455 2,19E-06 22,85 43,158 TA14N_2 8,35E-03 18,83 41,832 5,18E-06 21,89 42,943 TA18N_2 1,18E-02 23,83 19,761 6,87E-06 22,68 41,590 Table 5.21: Mean, variation coefficient and variation with respect to the reference case of the fatigue damage equivalent bending moment in the y and z axes for the additional study case of the cross section of the structure. Case Damage - Mud-line bending moment Y Damage - Mud-line bending moment Z μ [Nm] CV [%] Diff. with ref. case [%] μ [Nm] CV [%] Diff. with ref. case [%] TA8C_2 8,40E+04 20,28 14,021 4,91E+05 18,10 6,687 TA14C_2 1,72E+05 12,36 -4,372 8,95E+05 9,39 10,271 TA18C_2 3,14E+05 12,98 -0,107 1,27E+06 15,60 -8,093 TA8N_2 7,71E+04 8,54 4,752 4,33E+05 12,21 -5,813 TA14N_2 1,65E+05 10,62 -8,127 9,73E+05 8,91 19,843 TA18N_2 3,44E+05 12,48 9,441 1,38E+06 15,33 0,147 Natural Frequencies The variations introduced in this study case results in different values of the cross section of the support structure. This, in turn, is translated in different natural frequencies for each simulation due to the dependence of this parameter on the stiffness matrix of the structure. The results presented in Table 5.22 and Figure 5.20 correspond to the natural frequencies obtained for the analysis performed with 1000 samples. The variations in the cross section of the structure are not affected by a variation of the wind speed, which is why the mean values of the natural frequencies have a similar value for the correlated and uncorrelated case. The natural frequencies of the correlated case are on average a 0.6% lower than the natural frequency of the reference case. On the other hand, the natural frequencies of the uncorrelated case are on average a 7.5% lower than the natural frequency of the reference case. 88 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 5.18: Distribution of amplitudes of the equivalent fatigue damage displacement in the y axis for the additional study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 2,358E-03 2,740E-03 3,122E-03 3,503E-03 3,885E-03 4,267E-03 4,649E-03 5,031E-03 5,412E-03 Frequency [-] Amplitudes [m] Wind speed of 8 m/s Y displ. for Area CC Y displ. for Area NCC Reference damage 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,4E+02 3,163E-03 4,518E-03 5,873E-03 7,228E-03 8,583E-03 9,938E-03 1,129E-02 1,265E-02 1,400E-02 Frequency [-] Amplitudes [m] Wind speed of 14 m/s Y displ. for Area CC Y displ. for Area NCC Reference damage 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,4E+02 1,6E+02 4,158E-03 7,798E-03 1,144E-02 1,508E-02 1,872E-02 2,236E-02 2,600E-02 2,964E-02 3,328E-02 Frequency [-] Amplitudes [m] Wind speed of 18 m/s Y displ. for Area CC Y displ. for Area NCC Reference damage Chapter 5 89 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 5.19: Distribution of amplitudes of the equivalent fatigue damage bending moment in the z axis for the additional study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+01 1,0E+02 1,5E+02 2,0E+02 2,5E+02 2,751E+05 4,305E+05 5,858E+05 7,412E+05 8,965E+05 1,052E+06 1,207E+06 1,363E+06 1,518E+06 Frequency [-] Amplitudes [Nm] Wind speed of 8 m/s Z Bending M. for Area CC Z Bending M. for Area NCC Reference damage 0,0E+00 1,0E+01 2,0E+01 3,0E+01 4,0E+01 5,0E+01 6,0E+01 7,0E+01 8,0E+01 9,0E+01 7,389E+05 8,003E+05 8,618E+05 9,232E+05 9,846E+05 1,046E+06 1,108E+06 1,169E+06 1,230E+06 Frequency [-] Amplitudes [Nm] Wind speed of 14 m/s Z Bending M. for Area CC Z Bending M. for Area NCC Reference damage 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 9,787E+05 1,158E+06 1,338E+06 1,517E+06 1,697E+06 1,876E+06 2,056E+06 2,236E+06 2,415E+06 Frequency [-] Amplitudes [Nm] Wind speed of 18 m/s Z Bending M. for Area CC Z Bending M. for Area NCC Reference damage 90 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya It must be emphasized that the distribution of amplitudes of the histograms shown in Figure 5.20 has approximately the same shape for the correlated and uncorrelated cases. This behaviour is reasonable due to the fact that the natural frequencies are not affected by a variation of the wind speed. This can be clearly observed because of the large number of samples, in contrast to the results obtained in the case study of the soil stiffness. Table 5.22: Mean, standard deviation, variation coefficient and variation with respect to the reference case of the natural frequency for the additional study case of the cross section of the structure. Case μ [Hz] σ [Hz] CV [%] Diff. with ref. case [%] TA8C_2 0,284662 2,36E-02 8,2791 -0,8661 TA14C_2 0,285966 2,41E-02 8,4357 -0,4118 TA18C_2 0,285406 2,36E-02 8,2748 -0,6068 TA8N_2 0,265242 1,52E-02 5,7365 -7,6291 TA14N_2 0,265898 1,49E-02 5,5979 -7,4006 TA18N_2 0,265887 1,51E-02 5,6721 -7,4044 Summary The study case of the geometry of the structure has a notable impact on the dynamic response of the structure as has been seen in the results obtained in this section. First, the displacements and the bending moments are higher than those obtained in the reference case, both on the mean values and the scattering of the results. This difference is particularly large in the uncorrelated case. Looking at the differences between the correlated and noncorrelated case, the results of the correlated case are lower than the ones obtained in the uncorrelated case. As regards fatigue damage, on average the mean values of the equivalent fatigue damage are higher than those obtained in the reference case. This is especially evident in the case of the displacements, where it is reached a maximum variation of 47%. Comparing the correlated with the uncorrelated case, the fatigue damage is higher in the uncorrelated case. It is worth stressing that, in both cases, the distribution of amplitudes is shifted to the right in reference to the mean value of the reference case, translating to higher fatigue damage. Furthermore, the irregular distributions observed in the histograms have suggested that the number of samples used in the analysis could be inadequate and, for this reason, some simulations have been repeated by increasing the number of samples to 1000. Overall, the results obtained are analogous to those have been obtained with 200 samples, but it can be observed a more regular and natural distribution of amplitudes. The natural frequencies of the structure indicate a clear difference between the correlated and uncorrelated case. The results show that the natural frequencies of the uncorrelated case are a 7% lower than the ones obtained in the correlated case. In both cases, the natural frequencies are higher than those obtained in the reference case. Furthermore, the additional simulations with 1000 samples enable to indicate that the natural frequencies are not affected by a variation of the wind speed and it can clearly be observed the bell-shape of the histograms that are used to represent the distribution of natural frequencies. Chapter 5 91 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Figure 5.20: Distribution of natural frequencies for the study case of the cross section of the structure. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,4E+02 1,6E+02 1,8E+02 2,043E-01 2,280E-01 2,517E-01 2,754E-01 2,992E-01 3,229E-01 3,466E-01 3,703E-01 Frequency [-] Natural frequencies [Hz] Wind speed of 8 m/s Nat. Freq. for Area CC Nat. Freq. for Area NCC 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,4E+02 1,6E+02 1,8E+02 1,967E-01 2,201E-01 2,434E-01 2,667E-01 2,900E-01 3,133E-01 3,366E-01 3,599E-01 Frequency [-] Natural frequencies [Hz] Wind speed of 14 m/s Nat. Freq. for Area CC Nat. Freq. for Area NCC 0,0E+00 2,0E+01 4,0E+01 6,0E+01 8,0E+01 1,0E+02 1,2E+02 1,4E+02 1,6E+02 2,017E-01 2,238E-01 2,460E-01 2,681E-01 2,902E-01 3,124E-01 3,345E-01 3,567E-01 Frequency [-] Natural frequencies [Hz] Wind speed of 18 m/s Nat. Freq. for Area CC Nat. Freq. for Area NCC 92 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Case of Study: Combined Analysis The results obtained in this study case are a combination of the performances observed in the previous sections. This means that many of the behaviours noticed in the other cases are the same here and, therefore, they will not be mentioned. Displacements and Bending Moments Figure 5.21 and Figure 5.22 present the results obtained in the study case of the combined analysis. These figures show with boxplots the displacements and bending moments. Additionally, the Table 5.23 and Table 5.24 present the mean values, variation coefficients and the percentage variances between the mean values of the different cases analysed in this section and the reference cases. Table 5.23: Mean, variation coefficient and variation with respect to the reference case of the displacements in the y and x axes for the combined study case. Case Top displacements Y Top displacements X μ [m] CV [%] Diff. with ref. case [%] μ [m] CV [%] Diff. with ref. case [%] DC8CC 4,69E-02 91,68 17,74 1,38E-05 119,09 6,75 DC14CC 8,13E-02 114,91 11,63 4,03E-05 101,98 0,17 DC18CC 1,10E-01 96,91 14,98 5,75E-05 97,85 0,42 DC8C 4,81E-02 92,43 20,72 1,38E-05 118,05 7,29 DC14C 8,04E-02 115,46 10,37 4,06E-05 101,18 1,01 DC18C 1,14E-01 92,53 19,21 5,67E-05 97,99 -0,88 DC8NC 4,63E-02 101,26 16,22 2,00E-05 116,63 55,37 DC14NC 1,02E-01 107,70 40,07 5,73E-05 101,51 42,64 DC18NC 1,46E-01 85,34 52,89 8,00E-05 98,72 39,78 DC8NN 4,52E-02 103,74 13,48 1,87E-05 115,45 44,99 DC14NN 1,05E-01 106,37 44,59 5,77E-05 104,09 43,46 DC18NN 1,43E-01 86,34 49,26 7,96E-05 97,18 38,99 TC8CC 7,74E-02 73,89 6,15 1,50E-05 112,62 6,78 TC14CC 1,77E-01 80,09 28,87 3,74E-05 115,63 4,40 TC18CC 2,73E-01 72,21 -6,65 4,89E-05 119,61 9,24 TC8CN 7,61E-02 73,73 6,14 1,49E-05 111,28 5,06 TC14CN 1,78E-01 77,73 29,25 3,72E-05 114,31 6,09 TC18CN 2,78E-01 67,70 -12,80 4,86E-05 118,45 3,79 TC8NC 7,31E-02 82,44 4,30 2,14E-05 107,32 54,09 TC14NC 2,29E-01 69,20 71,10 5,31E-05 110,20 52,89 TC18NC 3,44E-01 62,96 10,40 6,97E-05 113,29 60,53 TC8NN 7,44E-02 83,25 2,46 2,22E-05 107,24 52,35 TC14NN 2,38E-01 68,58 73,17 5,50E-05 109,94 53,80 TC18NN 3,46E-01 62,43 15,81 7,22E-05 112,83 54,97 In the study case of the combined analysis, one can clearly see which variables analysed probabilistically have a large effect on the dynamic response of the structure. Both displacements and bending moments have shown the geometry of the structure as the variable with the large effects on the results. The uncorrelated cases of this variable show the Chapter 5 93 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures higher mean values of displacements and bending moments. The effects of the other two variables, degree of damping and soil stiffness, have analogous responses to which have been observed in the previous sections. Therefore, the effects observed are small and irregular, making difficult to achieve a clear conclusion of their trend. Looking at the differences with the reference case, in general the cases analysed here have mean values of displacements and bending moments higher than those obtained in the reference case. Taking into account only the effect of the geometry of the structure, these differences are as expected. Additionally, it is reasonable to think that the superposition of the different effects of each variable results in higher differences with regard to the reference case. In this sense, it has been obtained the higher difference with respect to the reference case, with a variation around 73% in the displacements of the y axis. Figure 5.21: Boxplot of the amplitudes of the displacements obtained in the rainflow counting for the combined study case. Each graph correspond to the displacements in the y axis (top) and in the x axis (bottom). Fatigue Damage From the fatigue damage results presented in Table 5.25 and Table 5.26, and Figure 5.23 and Figure 5.24, one can see that, here too, the main effects on the equivalent fatigue damage of displacements and bending moments are caused by the geometry of the structure, as a probabilistic variable. The histograms of the Figure 5.23 and Figure 5.24 show a clear trend for the cases in which the geometry of the structure is treated as correlated, and a different trend when it is treated as uncorrelated. As noted in the study case of the geometry of the structure, the higher equivalent fatigue damage is obtained for the uncorrelated case of the geometry 100 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Figure 5.25: Distribution of natural frequencies for the combined study case. Each histogram correspond to a wind speed: 8 m/s (top), 14 m/s (centre) and 18 m/s (bottom). 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 2,146E-01 2,355E-01 2,565E-01 2,775E-01 2,985E-01 3,194E-01 3,404E-01 3,614E-01 Frequency [-] Natural frequencies [Hz] Wind speed of 8 m/s Nat. Freq. for Comb. C-C Nat. Freq. for Comb. C-NC Nat. Freq. for Comb. NC-C Nat. Freq. for Comb. NC-NC 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 4,0E+01 2,146E-01 2,368E-01 2,591E-01 2,814E-01 3,037E-01 3,259E-01 3,482E-01 3,705E-01 Frequency [-] Natural frequencies [Hz] Wind speed of 14 m/s Nat. Freq. for Comb. C-C Nat. Freq. for Comb. C-NC Nat. Freq. for Comb. NC-C Nat. Freq. for Comb. NC-NC 0,0E+00 5,0E+00 1,0E+01 1,5E+01 2,0E+01 2,5E+01 3,0E+01 3,5E+01 2,146E-01 2,355E-01 2,565E-01 2,775E-01 2,985E-01 3,194E-01 3,404E-01 3,614E-01 Frequency [-] Natural frequencies [Hz] Wind speed of 18 m/s Nat. Freq. for Comb. C-C Nat. Freq. for Comb. C-NC Nat. Freq. for Comb. NC-C Nat. Freq. for Comb. NC-NC Chapter 5 101 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures To conclude, the behaviour noticed in the displacements, bending moments and fatigue damage is also observed in the natural frequencies of the structure. The larger effects occur in the uncorrelated case of the geometry of the structure analysed probabilistically. On the other hand, the largest differences with respect to the reference case are mainly due to the sum of contributions that supposes analyse probabilistically all the variables jointly. 102 Chapter 5 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya 103 CHAPTER 6 Discussion Damping of the Offshore Wind Turbine The degree of damping has been divided in two contributions, structural and aerodynamic damping. One of the lines of research has been the study of the way used to add the aerodynamic damping in the FE model, by using two different approaches. On the one hand, the aerodynamic damping has been distributed in the entire structure, that is, it has been added at each element damping matrix of the structure. On the other hand, it has been considered that this type of damping could only affect one finite element, which represents the wind turbine and is located in the top of the support structure. The results obtained have shown that the fact of locating the aerodynamic damping in the top of the structure has resulted in a considerable increase of the displacements, bending moments and, in turn, fatigue damage. In other words, the dynamic response of the structure has been more critical in the case of the aerodynamic damping located in the top of the structure than in the case of this damping distributed in the entire structure. This behaviour is due to the overestimation of damping when the aerodynamic damping is distributed in the entire structure, which results in lower displacements and bending moments because the dynamic response of the structure is compensated by a high degree of damping. 104 Chapter 6 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Therefore, the fact of locating the aerodynamic damping in the top of the structure is an approach more realistic and conservative of the dynamic response of the structure. Effect of the Aerodynamic Damping The aerodynamic damping has been analysed probabilistically in order to study the effect of uncertainties associated to this system parameter. The study of Schafhirt (2014) has been used as a basis to obtaining the mean value of the aerodynamic damping corresponding to each wind speed and its standard deviation. The simulations have shown that the aerodynamic damping has had a reduced impact on the response of the structure as it can be deduced from the small differences observed with respect to the reference case. This behaviour was not expected given that it is supposed a great importance of the aerodynamic damping in the dynamic response of the structure. The reason of this behaviour could be linked to the values of the standard deviation used in the analysis, which have a coefficient of variation between 2 and 7 per cent. These values are obtained from the variability of the results observed in the logarithmic decay tests in FEDEM Windpower and the degree of uncertainty taken into account in these tests can be insufficient for the purpose of this research. It is worth pointing out that the highest dispersion of the fatigue damage has been observed in the case of the aerodynamic damping distributed in the entire structure. This behaviour means here a great fatigue damage of the structure. However, it is contrary to the considerations explained in the previous section. This is caused by the high scattering associated to the fact of analysing all the elements of the FE model probabilistically, instead of only one specific element. Effect of the Soil Stiffness The effect of uncertainties have also been studied by analysing probabilistically the stiffness of the soil. In this case, the analysis is based on the study of Carswell et al. (2014) and the value of the soil stiffness depends on the angle of internal friction, which varies with the depth. The probability analysis of the soil stiffness has shown a weak impact on the results obtained from the simulations, as illustrated by the low differences with respect to the reference case have been obtained in the analysis of this system parameter. As in the case of the aerodynamic damping, this lesser influence was not expected because of the importance of the foundation on the dynamic response of an offshore wind turbine. The degree of uncertainty associated to the soil stiffness can be the answer to these results, since it has been used the coefficient of variation described in Carswell et al. (2014), with a value of 5%. This value of variation coefficient is conservative because it does not include all the uncertainties connected to the real behaviour of the soil stiffness. A more appropriate value of the variation coefficient could be around 20%, which takes into account a larger number of uncertainties associated with this system parameter. Chapter 6 105 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures Effect of the Structure Geometry The geometry of the structure has been studied by analysing probabilistically the cross section of the offshore wind turbine support structure described in this study. This analysis enable optimization and development of a cost effective design in the future and, to that end, it has been used a coefficient of variation of 5%. The results obtained from the simulations have shown that the cross section of the support structure is the system parameter with the larger impact on the dynamic response of the structure. This effect is observed in the high differences with respect to the references case, where the variations concerning the mean values of the fatigue damage equivalent displacement reach 47%. It is interesting to note that the mean values of the fatigue damage are generally higher than for the reference case, translating to greater damage in the structure. To correctly understand this behaviour it is advisable to observe how the natural frequencies of the structure vary with the probability analysis of the cross section of the support structure, because differences in eigenfrequencies are often correlated with, but not cause of, differences in fatigue damage. In this sense, it may be seen that the natural frequencies are lower than those obtained in the reference case, and this diminution coincides with an increase of the fatigue damage. Furthermore, the lower the natural frequency, the higher the fatigue damage. The increase of the fatigue damage with a reduction of the natural frequency of the structure can be caused by dynamic resonance. These types of problem occur when a structure is excited by external forces with a frequency close to the natural frequency of the structure. In this case, the reduction observed on the natural frequency implies that this frequency is nearer to the rotational frequency of the rotor, 𝑓1𝑃, which results in higher fatigue damage due to a dynamic amplification of the response of the structure. Combined Analysis The effect of uncertainties have also been studied by analysing probabilistically all the system parameters at the same time. To that end, the different system parameters have been analysed following the same criteria applied in their individual analyses. The combined analysis has shown that the results are a combination of the effects observed in the individual analysis corresponding to each system parameter. This has enabled to compare the magnitude of the effects of each variable in the dynamic response of the structure. In this sense, the cross section of the structure has been the system parameter with the highest effects on the combined analysis. The effect of other system parameters may also be observed, but their magnitude is lower and, in some, cases is difficult to detect. The magnitude and the reason of these effects are analogous to which has been described in the different analyses of the system parameters. It is interesting to highlight that the combined analysis of the system parameters is a simple superposition of the effects produced by each variable. In other words, this combined analysis does not produce an amplification or reduction of the response of the structure due to the fact that all the variables are analysed probabilistically at the same time. 106 Chapter 6 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya Correlated and Uncorrelated Analysis This approach raises that the system parameters with different values along the height of the support structure can be analysed probabilistically in correlation with a reference value or without correlation. Consequently, this alternative analysis has been applied to the soil stiffness and the cross section of the structure, since both system parameters have different values along the height of the support structure. On the one hand, the results obtained from the simulations concerning the soil stiffness have shown that the differences between the correlated and uncorrelated cases are not significant and there is no clear trend that explains the results observed. This makes it impossible to find a comprehensive explanation of how the correlation analysis of this variable affects the dynamic response of the structure. On the other hand, the results derived from the analysis of the cross section of the structure show large differences between the correlated and uncorrelated cases. In this case, it is observed a notable trend towards higher equivalent fatigue damage in the uncorrelated analysis. In order to understand these results, it is interesting to focus once again on the behaviour of the natural frequency of the structure. As has been seen previously, in the uncorrelated case the natural frequencies are, on average, 7.5% lower than in the reference case. There is a considerable difference by comparison with the variation observed in the correlated case, which is around 0.5%. The large variation in the uncorrelated case is reasonable if one considers the following aspects. For the correlated case, the dimensions of the elements in the FE model come from just one sample, whose mean value is at the reference value and, therefore, it is expected that the natural frequencies are distributed around the reference frequency. However, in the uncorrelated case, each element is sampled individually. Taking into account how this affects the mass matrix of the structure, it is expected to obtained, on average, the same total mass of the structure, but not each individual mass of the FE model. This difference in the distribution or shape of the sampling may explain the variations observed between the correlated and uncorrelated cases. Equivalent Fatigue Damage Analysis The investigation of the fatigue damage has been done by obtaining the equivalent fatigue damage of the support structure as it is described in Blasques et al. (2013). This way to analyse the fatigue damage does not allow to calculate the actual probability of failure, which is the basis of a structural reliability analysis. Instead, it has been obtained general or qualitative statements about which parameters produce higher or lower fatigue damage based on the distribution of the results. First, the probability analysis corresponding to each study case has shown that the differences with respect to the reference case depend highly on the system parameter analysed. Anyway, these differences are similar to the ones exposed in previous sections, so it is not necessary to go into detail. That said, the study case of the aerodynamic damping shows small variations with respect to the reference case, but large differences between the two damping approaches. The lowest differences are observed in the probability analysis of the soil Chapter 6 107 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures stiffness, where the maximum variation with respect to the reference case does not reach the 2%. In contrast, the greatest differences are observed in the probability analysis of the cross section of the structure, where the maximum variation with respect to the reference case reaches the 46%. Obviously, these greatest differences are obtained in the probability analysis of the cross section of the structure if it is not considered the combined case, where the superposition of the effects of each variable increases the equivalent fatigue damage and, therefore, the differences with respect to the reference case. Figure 6.1: Coefficient of variation of each study case analysis. The curves relate to the different wind speeds and the damping approaches. Regarding the scattering of the equivalent fatigue damage results, Figure 6.1 gives the variation coefficients obtained in the analysis corresponding to each study case in this research. This figure shows that the magnitude of the effects in the variation coefficient is similar to which has been observed in previous analyses. In other words, the study case of the cross section of the support structure has the greatest impact on the scattering of the equivalent fatigue damage. As previously stated, for the study case of the aerodynamic damping, the highest variation coefficient is observed in the case of the aerodynamic damping distributed in the entire structure, which is contrary to the general trend of the other cases. The reason can be comparable to the one presented in order to explain the differences between the correlated and uncorrelated cases in the study case of the cross section of the structure. That is to say, distributing the aerodynamic damping in all the elements of simplified FE model instead of only one element results in considerable increase of the scattering of the equivalent fatigue damage. On the other hand, the variation coefficients of the study case of the soil stiffness show a lower impact of this system parameter on the results in comparison to the other variables. 0 5 10 15 20 25 30 35 TAD TS-C TS-N TA-C TA-N TC-CC TC-CN TC-NC TC-NN CV [%] Cases DD_8 DD_14 DD_18 TD_8 TD_14 TD_18 108 Chapter 6 Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya This behaviour, however, has been observed in all analyses and shows the general trend of the effects of this variable on the dynamic response of the structure. To conclude, in the study case of the cross section of the structure the highest values of the variation coefficient are obtained in the correlated analysis. This observation is worth pointing out because the highest mean values of the equivalent fatigue damage have been obtained in the uncorrelated case. Nevertheless, the highest scattering has been obtained in the correlated case. 109 CHAPTER 7 Conclusions and Recommendations In this thesis an investigation to develop and evaluate a novel approach for performing structural reliability analysis and robust design of offshore wind turbine support structures is performed. The effect of uncertainty associated to some system parameters is investigated by performing a probabilistic analysis in terms of the degree of uncertainty corresponding to each system parameter. The effect of these uncertainties is also investigated in terms of their contribution to the fatigue damage of the offshore wind turbine. A simplified model of a support structure is developed in order to allow for doing the necessary calculations of the probability analysis efficiently. This model is implemented as a time-domain simulation in MATLAB by using the finite element method, and allows to solve the transient dynamics of a beam. The simplified FE model is based on the analysis of beam elements with 12 degrees of freedom, 3 translations and 3 rotations per node. The equation of motion is solved by using a direct numerical time integration method, the Newmark method, and the rotor loads are obtained from rotor simulations with an external software, FEDEM Windpower. The model represents the reference offshore wind turbine described in the OC3 project. The degree of uncertainty associated to the aerodynamic damping and the soil stiffness is investigated on the basis of the studies performed in other researches. On the other hand, the geometry of the support structure is also investigated through a probability analysis. In this 116 References Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya 117 Appendix A In this appendix are defined the parameters that comprise the local stiffness and mass matrices of the element defined in Methodology. 118 Appendix A Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya A.1. Stiffness Matrix In this appendix, the different parameters that compose the local stiffness matrix are defined. In order to facilitate the understanding, the local stiffness matrix is again defined and then the different parameters are defined. 𝐾(𝑒)= [ 𝐸𝐴 𝐿 012𝐸𝐼𝑧 𝐿3𝛽𝑦 0 0 12𝐸𝐼𝑦 𝐿3𝛽𝑧 𝑆𝑌𝑀 0 0 0 𝐺𝐽 𝐿 0 0 −6𝐸𝐼𝑦 𝐿2𝛽𝑧0𝛾𝑧𝐸𝐼𝑦 𝐿 06𝐸𝐼𝑧 𝐿2𝛽𝑦0 0 0 𝛾𝑦𝐸𝐼𝑧 𝐿 −𝐸𝐴 𝐿0 0 0 0 0 𝐸𝐴 𝐿 012𝐸𝐼𝑧 𝐿3𝛽𝑦0 0 0−6𝐸𝐼𝑧 𝐿2𝛽𝑦012𝐸𝐼𝑧 𝐿3𝛽𝑦 0 0 −12𝐸𝐼𝑦 𝐿3𝛽𝑧06𝐸𝐼𝑦 𝐿2𝛽𝑧0 0 0 12𝐸𝐼𝑦 𝐿3𝛽𝑧 0 0 0 −𝐺𝐽 𝐿0 0 0 0 0 𝐺𝐽 𝐿 0 0 −6𝐸𝐼𝑦 𝐿2𝛽𝑧0𝜂𝑧𝐸𝐼𝑦 𝐿0006𝐸𝐼𝑦 𝐿2𝛽𝑧0 𝛾𝑧𝐸𝐼𝑦 𝐿 06𝐸𝐼𝑧 𝐿2𝛽𝑦0 0 0 𝜂𝑦𝐸𝐼𝑧 𝐿0 −6𝐸𝐼𝑧 𝐿2𝛽𝑦0 0 0 𝛾𝑦𝐸𝐼𝑧 𝐿 ] ( A.1 ) Where, 𝛽𝑦=(1+Φ𝑦) ( A.2 ) 𝛽𝑧=(1+Φ𝑧) ( A.3 ) 𝛾𝑦=(4+Φ𝑦) (1+Φ𝑦) ( A.4 ) 𝛾𝑧=(4+Φ𝑧) (1+Φ𝑧) ( A.5 ) 𝜂𝑦=(2−Φ𝑦) (1+Φ𝑦) ( A.6 ) 𝜂𝑧=(2−Φ𝑧) (1+Φ𝑧) ( A.7 ) Appendix A 119 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures A.2. Mass Matrix In this appendix, the different parameters that compose the local mass matrix are defined. In order to facilitate the understanding, the local mass matrix is again defined and then the different parameters are defined. 𝑀𝑒𝑙= [ 1/3 0 𝑎𝑧 0 0 𝑎𝑦 𝑆𝑌𝑀 0 0 0 𝐽 3𝐴 0 0 −𝑐𝑦0𝑒𝑦 0𝑐𝑧0 0 0 𝑒𝑧 1/6 0 0 0 0 0 1/3 0 𝑏𝑧0 0 0𝑑𝑧0𝑎𝑧 0 0 𝑏𝑦0−𝑑𝑦0 0 0 𝑎𝑦 0 0 0 𝐽 6𝐴 0 0 0 0 0 𝐽 3𝐴 0 0 𝑑𝑦0𝑓𝑦0 0 0 𝑐𝑦0 𝑒𝑦 0 −𝑑𝑧0 0 0 𝑓𝑧0 −𝑐𝑧0 0 0 𝑒𝑧 ] ( A.8 ) Where, 𝑎𝑦=13 35+7 10Φ𝑧+13Φ𝑧2+65(𝑟𝑦 𝐿)2 (1+Φ𝑧)2 ( A.9 ) 𝑎𝑧=13 35+7 10Φ𝑦+13Φ𝑦2+65(𝑟𝑧 𝐿)2 (1+Φ𝑦)2 ( A.10 ) 𝑏𝑦=9 70+3 10Φ𝑧+16Φ𝑧2+65(𝑟𝑦 𝐿)2 (1+Φ𝑧)2 ( A.11 ) 𝑏𝑧=9 70+3 10Φ𝑦+16Φ𝑦2+65(𝑟𝑧 𝐿)2 (1+Φ𝑦)2 ( A.12 ) 𝑐𝑦=11 210+11 120Φ𝑧+1 24Φ𝑧2+(1 10+Φ𝑧 2)(𝑟𝑦 𝐿)2 (1+Φ𝑧)2𝐿 ( A.13 ) 𝑐𝑧=11 210+11 120Φ𝑦+1 24Φ𝑦2+(1 10+Φ𝑦 2)(𝑟𝑧 𝐿)2 (1+Φ𝑦)2𝐿 ( A.14 ) 120 Appendix A Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya 𝑑𝑦=13 420+3 40Φ𝑧+1 24Φ𝑧2+(1 10+Φ𝑧 2)(𝑟𝑦 𝐿)2 (1+Φ𝑧)2𝐿 ( A.15 ) 𝑑𝑧=13 420+3 40Φ𝑦+1 24Φ𝑦2+(1 10+Φ𝑦 2)(𝑟𝑧 𝐿)2 (1+Φ𝑦)2𝐿 ( A.16 ) 𝑒𝑦=1 105+1 60Φ𝑧+1 120Φ𝑧2+(2 15+Φ𝑧 6+Φ𝑧2 3)(𝑟𝑦 𝐿)2 (1+Φ𝑧)2𝐿2 ( A.17 ) 𝑒𝑧=1 105+1 60Φ𝑦+1 120Φ𝑦2+(2 15+Φ𝑦 6+Φ𝑦2 3)(𝑟𝑧 𝐿)2 (1+Φ𝑦)2𝐿2 ( A.18 ) 𝑓𝑦=− 1 140+1 60Φ𝑧+1 120Φ𝑧2+(1 30+Φ𝑧 6+Φ𝑧2 6)(𝑟𝑦 𝐿)2 (1+Φ𝑧)2𝐿2 ( A.19 ) 𝑓𝑧=− 1 140+1 60Φ𝑦+1 120Φ𝑦2+(1 30+Φ𝑦 6+Φ𝑦2 6)(𝑟𝑧 𝐿)2 (1+Φ𝑦)2𝐿2 ( A.20 ) 121 Appendix B In this appendix are presented the main MATLAB scripts used to build the simplified model of an offshore wind turbine support structure. The simplified model is composed of five parts: the main script that connects the different scripts and perform the probability analysis, the script used to obtain the geometry of the structure, the script used to obtain the soil stiffness, the script used to perform the FE analysis and, finally, the script used to obtain the fatigue damage of the support structure. 122 Appendix B Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya B.1. Main MATLAB script %% PROBABILISTIC ANALYSIS clear all; close all; clc disp(' PROBABILISTIC ANALYSIS ') %% Input Data: global Damping WindSpeed ProbAnalysis numberSamples % Damping: DStr (Structure), DTop (Top support structure) Damping = 'DTop'; fprintf('\nDamping: %s.\n',Damping) % Wind Speeds: 8 m/s, 14 m/s, 18 m/s WindSpeed = '18 m/s'; fprintf('- Wind Speed: %s.\n',WindSpeed) % Prob. Analysis: Area, Soil, Aerodynamic Damping, Combined ProbAnalysis = 'Combined'; fprintf('- Variable Analysis: %s.\n',ProbAnalysis) % Study case: Correlated case, Non-Correlated case Case = 'Correlated'; fprintf('- Case: %s.\n',Case) % Study case: Correlated case, Non-Correlated case CombinedAreaCase = 'Non-Correlated'; fprintf('- Combined Area Case: %s.\n',CombinedAreaCase) % Study case: Correlated case, Non-Correlated case CombinedSoilCase = 'Non-Correlated'; fprintf('- Combined Soil Case: %s.\n\n',CombinedSoilCase) % Number of samples numberSamples = 200; tic; % Start clock ttim = 0; % Initialize time counter % Units: m; N; kg; Pa global E G poisson rho Ls E = 210e9; % E: modulus of elasticity G = 80.8e9; % Shear modulus (G = E/2/(1+poisson) poisson = 0.30; % poisson: Poisson's ratio rho = 8500; % rho: specific weight of steel Ltp = 77.6; % Ltp: Tower-Top Height Above Ltb Ltb = 10; % Ltb: Tower-Base Height Above MSL Lw = 20; % Lw: Water Depth (From MSL) Ls = 40; % Ls: length of the foundation L = Ltp+Ltb+Lw+Ls; % L: length total %% Generation of coordinates and connectivities Appendix B 123 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures global numberElementsTowerTop numberElementsTowerBase ... numberElementsFoundation numberElementsTowerTop=11; numberElementsTowerBase=3; numberElementsFoundation=40; numberElements=numberElementsTowerTop+numberElementsTowerBase+... numberElementsFoundation; nodeCoordinatesBeam=[linspace(0,Ls,numberElementsFoundation+1)';Ls+... (linspace((Ltb+Lw)/numberElementsTowerBase,(Ltb+Lw),... numberElementsTowerBase))'; Ls+Ltb+Lw+(linspace(Ltp/numberElementsTowerTop,Ltp,... numberElementsTowerTop))']; nodeCoordinates=[nodeCoordinatesBeam zeros(numberElements+1,1) ... zeros(numberElements+1,1)]; numberNodes=size(nodeCoordinates,1); elementNodes=zeros(numberElements,2); bendingCoord=zeros(numberElements,1); for i=1:numberElements; elementNodes(i,1)=i; elementNodes(i,2)=i+1; bendingCoord(i) =(nodeCoordinates(elementNodes(i,1)) + ... nodeCoordinates(elementNodes(i,2)))/2; end global NElement GDofPN GDof GDofPE NElement=2; % NElement: number of nodes per element GDofPN=6; % GDofPN: degrees of freedom per node GDof=GDofPN*numberNodes; % GDof: global number of degrees of freedom GDofPE=GDofPN*NElement; % GDofPE: degrees of freedom per element ttim = Timing('Time needed to set initial values',ttim); %Reporting time %% Boundary conditions % prescribed DOF prescribed_Dof=(1:1)'; % active DOF active_Dof=setdiff((1:GDof)',prescribed_Dof); %% Loads addpath('Loads'); switch WindSpeed case '8 m/s' data1 = dlmread('FxWs8.txt'); data2 = dlmread('FyWs8.txt'); data3 = dlmread('FzWs8.txt'); 124 Appendix B Norges Teknisk-Naturvitenskapelige Universitet – Universitat Politècnica de Catalunya case '14 m/s' data1 = dlmread('FxWs14.txt'); data2 = dlmread('FyWs14.txt'); data3 = dlmread('FzWs14.txt'); case '18 m/s' data1 = dlmread('FxWs18.txt'); data2 = dlmread('FyWs18.txt'); data3 = dlmread('FzWs18.txt'); end global t dt load1 load2 load3 t=data1(:,1); dt=data1(2,1)-data1(1,1); load1=data3(:,2); % Load in the x-axis load2=data1(:,2); % Load in the y-axis load3=data2(:,2); % Load in the z-axis ttim = Timing('Time needed to read the input file',ttim); %Reporting time %% Probability distribution % Area analysis PDArea = zeros(numberElements,numberSamples); Di = zeros(numberElements,1); Ai = zeros(numberElements,1); % Soil analysis phi = [38;35;38;38;42;42.5]; PDSoil = zeros(length(phi),numberSamples); % Aerodynamic analysis PDAeroDy = zeros(numberElements,numberSamples); switch Damping case 'DStr' switch ProbAnalysis % PDF Area case 'Area' switch Case case 'Non-Correlated' for i=1:numberElements [A,~,~,~,Dint] = Beam_Geometry (elementNodes(i,1),... numberElements); mean1=A; sd1=0.05; Appendix B 125 Structural Reliability Analysis and Robust Design of Offshore Wind Turbine Support Structures mu = log((mean1^2)/sqrt(sd1+mean1^2)); sigma = sqrt(log(sd1/(mean1^2)+1)); X = lognrnd(mu,sigma,1,numberSamples); PDArea(i,:) = X; Di(i)=Dint; end case 'Correlated' for i=1:numberElements [A,~,~,~,Dint] = Beam_Geometry (elementNodes(i,1),... numberElements); Ai(i)=A; Di(i)=Dint; end mean1=Ai(1); sd1=0.05; mu = log((mean1^2)/sqrt(sd1+mean1^2)); sigma = sqrt(log(sd1/(mean1^2)+1)); X = lognrnd(mu,sigma,1,numberSamples); PDArea(1,:) = X; for i=2:numberElements mean2 = Ai(i); sd2 = 0.05; X=mean2*(((PDArea(1,:)-mean1)/mean1)*(sd2/sd1)+1); PDArea(i,:) = X; end end % PDF Soil case 'Soil' switch Case case 'Non-Correlated' for i=1:length(phi) mean1 = phi(i); sd1 = phi(i)*0.05; mu = log((mean1^2)/sqrt(sd1+mean1^2)); sigma = sqrt(log(sd1/(mean1^2)+1)); X = lognrnd(mu,sigma,1,numberSamples); PDSoil(i,:) = X;