STOCKHOLM SWEDEN 2016,
Extreme loading and fatigue analysis of a wave energy device
EGIL GUSTAFSSON
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
This is a master thesis in Maritime Engineering (course code SD271X). The thesis have been done in cooperation with CorPower Ocean, to whom I am grateful for letting me on the team and helping me with various questions through out the time spent at their office. I am especially grateful to Oscar Hellaeus who have answered many of my questions but also challenged me with more. Gunnar Steinn Ásgeirsson for helping me with questions regarding the hydrodynamic model developed at CorPower Ocean. Also many thanks to Anders Rosén at the Royal Institute of Technology (KTH) and Prof. Harry B. Bingham at the Technical University of Denmark (DTU) who have helped me understanding and exploring various aspects of the hydrodynamic and solid mechanic theory that have been used through out the thesis. Fi- nally I would also like too acknowledge Pär Johannesson and Thomas Svensson at SP Technical research institute of Sweden for discussions in various fatigue methods and uncertainties that these might bring.
Stockholm, June 2016 Egil Gustafsson
Wave energy is one of the possible solutions for meeting the future energy demand in a clean and sus- tainable way. Extracting large amounts of energy, a wave energy device would be subjected to extreme and fatigue loads from the waves. Designing such a device, a trade off needs to be done between making a device that is strong enough to withstand the loads and on the same time not too heavy making it inefficient and too costly. Having good estimations of extreme and fatigue loads are therefore critical when designing an efficient wave energy device. This thesis has aimed to create a tool that can be used between the already existing hydrodynamic and solid mechanic models available at CorPower Oceean.
The goal has been that the tool shall extract the extreme and fatigue loads from the hydrodynamic model and format them in a way so that they can be used in the solid mechanical model.
Four different tools have been created and compared for calculating fatigue using amplitude and spectral methods, where the amplitude methods also are able to estimate extreme loads. The fatigue tools have been evaluated against each other in a simple example showing that the estimated accumulated fatigue damage can be decreased by using several variables. An application of the tools has been done on a critical sub system of the wave energy device developed by CorPower Ocean. Where in this application critical points against extreme loading and fatigue have been localized. A new design has been suggested based on the strength analysis from the first one. Increasing the number of variables and using the tools developed in this thesis can significantly improve the fatigue damage estimations of the system. What fatigue method to use depends on the details for each case.
Keywords: Renewable energy, wave power, wave energy device, wave energy converter, wave scatter, offshore structures, design load case, hydro dynamics, solid mechanics, extreme loading, fatigue, am- plitude methods, spectral methods, Rainflow Cycles, Rainflow Matrix, accumulated damage, statistical analysis, Rainflow Projection, finite element analysis.
Vågenergi är en av flera potentiella lösningar för att möta ett framtida energibehov på ett rent och håll- bart sätt. Genom att extrahera stora mängder energi kommer ett vågkraftverk vara utsatt för extrem- och utmattningslaster. I konstruktionen av ett vågkraftverk behövs en avvägning göras mellan att göra ett vågkraftverk som är starkt nog att stå emot alla vågbelastningar samtidigt som det inte får vara för tungt vilket skulle göra det ineffektivt och alltför dyrt. Att ha en god uppfattning av extrem och utmat- ningslaster är därför kritiskt när man vill konstruera ett effektivt vågkraftverk. Det här examensarbetet har haft som syfte att skapa ett verktyg som kan användas mellan de redan existerande hydrodynamiska och strukturmekaniska modellerna som finns hos CorPower Ocean. Målet har varit att verktyget ska extrahera extrem och utmattningslaster från den hydrodynamiska modellen och formatera dem på ett sätt att de kan användas för hållfasthetsberäkningar i den strukturmekaniska modellen.
Fyra olika verktyg har utvecklats för att beräkna utmattning med både amplitud och frekvens metoder, där amplitud verktygen även kan estimera extremlaster. Utmattningsverktygen har jämförts med varan- dra på ett enkelt exempel vilket har visat att den estimerade ackumulerade utmattningsskadan kan reduceras genom att använda flera variabler. En tillämpning av verktygen har gjorts på ett kritiskt del- system av CorPower Oceans vågkraftverk. I denna tillämpning så har ett antal kritiska punkter för extrem och utmattningsbelastning identifieras i en hållfasthetsanalys. En ny design har föreslagits baserat på resultaten från hållfasthetsanalysen. Genom att öka antalet variabler i hållfasthetsanalysen och använda verktygen framtagna i detta examensarbete kan utmattningsuppskattningarna på vågkraftverket förbät- tras markant. Vilket av de fyra utmattningsverktygen som bör användas beror på omständigheterna av varje fall.
1 Background 1
1.1 Wave energy . . . 1
1.2 System overview . . . 2
1.3 CorPower Oceans design process . . . 3
1.4 Purpose & objectives . . . 4
1.5 Disposition of report . . . 4
2 Theory 6 2.1 Definitions and background theory . . . 6
2.1.1 Coordinate system . . . 6
2.1.2 Froude scaling laws. . . 6
2.1.3 Irregular waves . . . 7
2.1.4 Design load cases . . . 8
2.1.5 Reduction factors. . . 9
2.2 Extreme loading . . . 10
2.2.1 Ultimate tensile strength . . . 11
2.2.2 Deflection & buckling . . . 11
2.3 Fatigue . . . 12
2.3.1 Fatigue and how to estimate it . . . 12
2.3.2 Calculating fatigue . . . 13
2.3.3 Von Misses or Principal stress. . . 17
3 CPO models 19 3.1 Hydrodynamic model. . . 19
3.1.1 Hydrodynamic forces. . . 20
3.1.2 Machinery forces . . . 21
3.1.3 Validation & output . . . 21
3.2 Solid mechanical model . . . 22
3.2.1 FEA in general and SolidWorks. . . 22
3.2.2 Fatigue in SolidWorks . . . 23
3.2.3 Validation & accuracy . . . 23
4 Fatigue methods & tools 25 4.0.1 Wave scatter redistribution . . . 26
4.1 Single variable - Amplitude method. . . 27
4.1.1 Sensitivity analysis of input variables. . . 28
4.2 Multiple variable - Amplitude method . . . 30
4.2.1 RP direct . . . 30
4.2.2 RP two step. . . 32
4.3 Spectral . . . 33
4.3.1 Sensitivity analysis spectral methods . . . 33
4.4 Simple test . . . 35
4.4.1 Results simple test . . . 35
4.5 Discussion - Fatigue methods & tools. . . 36
5.2 Load analysis . . . 39
5.3 Strength analysis . . . 41
5.4 Redesign. . . 42
5.5 Discussion application . . . 44
6 Conclusion 46 6.1 Fulfilment of purpose and objectives . . . 47
6.2 Future work . . . 47
Appendices 51 A Hydrodynamic model. . . 52
A.1 Hydrodynamic forces. . . 52
A.2 Machinery forces . . . 54
B Rainflow Projection . . . 56
B.1 RP direct method . . . 56
B.2 RP two step method . . . 57
B.3 Position factor in RP two Step . . . 61
C Frequency methods . . . 63
C.1 Power spectral density . . . 63
C.2 Stress transfer function . . . 64
C.3 Damping . . . 64
C.4 Damage calculations . . . 65
C.5 Multiple loads . . . 67
D Amplitude methods - detailed methodology . . . 68
D.1 Input/output for tools . . . 68
D.2 Data inspection and correction . . . 68
D.3 Rainflow filter & Rainflow Cycles . . . 69
D.4 Rainflow Matrix . . . 70
D.5 Merging RM from different sea states . . . 71
D.6 Regeneration of RM . . . 73
D.7 Control . . . 75
D.8 Random selection of RFC - Multiple variables . . . 75
D.9 Results from FEA studies - Multiple variables. . . 76
D.10 Finding extreme nodes - Multiple variables . . . 76
E FEA verifications . . . 77
E.1 Convergence study . . . 77
E.2 Displacement check. . . 78
ALS = Accidental limiting states CPO = CorPower Ocean
DLC = Design load case DOF = Degrees of freedom FEA = Finite element analysis FFT = Fast Fourier transformation FLS = Fatigue limiting states
JONSWAP = Joint north sea wave project LC = Linear combination
LCD = Lowest common denominator PSD = Power spectral density PTO = Power take of device RFC = Rainflow Cycle RM = Rainflow Matrix RP = Rainflow Projection
SLS = Serviceability limiting states ULS = Ultimate limiting states WED = Wave energy device WEC = Wave energy converter
α = Material property, based on starting point of SN curve.
β = Material property, based on slope of SN curve.
γ = Safety factor
γU LS= Safety factor against ultimate limiting states γF LS= Safety factor against fatigue limiting states δ = Deflection
ζ = Modal damping
η = Position of piston rod in application θ = Wave direction
λ = Wave length ρ = Density of fluid
ρf = Density of full scale model ρm= Density of scale model σ = General sign for stress
σ1, σ2, σ3= First principle stresses
σcorr = Strength including correction factors.
σvM = von Mises stress
σtensile = Ultimate limiting tensile strength of material σyielding = Yielding strength of material
τ = Shear stress
Ω = Scaling factor between scale and full size model A = Wave amplitude
C = Load stress constant CD = Buoy drag coefficient D = Accumulated fatigue damage
Dp = Accumulated pseudo fatigue damage E = Mean energy flux in waves¯
E = Young’s modulus F = General symbol for force g = Gravitational constant
g(s) = Rayleigh probability density function of the stress amplitude Hσ = Stress transfer function
Hs= Wave height h = Water depth
I = Area moment of inertia M = General symbol for moment mb = Mass of the buoy
mP T O = Mass of the PTO system m∞= Infinite frequency mass matrix
NT = Total number of load cycles in design life Tp = Wave period
S = Amplitude of stress cycle Sη = Power spectral density Sσ = Stress energy spectrum K = Stiffness matrix
Kf = Reduction factor
Kcorr = Reduction factor due to corrosion in material Kweld= Reduction factor due to welded geometries k = Wave number
Vg = Wave group velocity u = Displacement vector x = Position vector of WED
˙
x = Velocity vector of WED
¨
x = Acceleration vector of WED
Background
1.1 Wave energy
With a steadily growing energy demand and an increasing concern over different climate issues, consum- ing clean energy has become increasingly important. Where the sector of renewable energy has grown rapidly during the last decades. Large amount of research have been put into solar cells increasing their efficiency. Wind turbines have been able to reduce their cost of producing energy and prototypes of offshore floating wind turbines are being tested which opens up for large possibilities of harvesting energy at deeper waters [1],[2]. Other renewable energy concepts such as tidal turbines and wave energy devices are also being developed to use the large potential of ocean energy, [3].
It is the sun that creates the wind and the wind that creates the waves. In the classical theory of entropy this would mean that the sun has higher entropy than wind, which has higher entropy than the waves.
This could perhaps explain the complexity of taking energy from the different sources. Solar energy has been used for millenniums by the plants on the earth. Windmills and wind turbines have been used for centuries in the modern world. While taking out the energy from ocean waves has until today not been made economically feasible. The idea of entropy can of course not explain all of this but it is an interesting aspect to keep in mind when developing wave energy. Harvesting the energy in waves has a large potential. Waves are relatively stable, contains a large amount of energy and are often located far away from land. The mean energy flux in a wave can be calculated by,
E =¯ 1
2ρgA2Vg.
Where ρ is the density, g the gravitational constant, A the wave amplitude, and Vg is the wave group velocity [4]. Having a 1 km2of deep water ocean with A=3m, Vg=3.5 m/s the waves on this area contains
∼11 MWh. This is approximately what 161 Swedish households uses daily,[5]. With a total world ocean area on approximately 361.900.000 km2it is clear that the potential for wave energy is large, [6].
The machine catching energy of waves is often called a wave energy device (WED) or wave energy con- verter (WEC). There are and have been many prototypes of different devices. These prototypes can broadly be sorted into three types as displayed in Figure(1.1.1), 1) overtopping devices, 2) oscillating water column, 3) point absorbers,[7]. This thesis have been done in collaboration with CorPower Ocean (CPO) where the WED they are designing is a mechanical point absorber.
Figure 1.1.1: Wave energy devices can be broadly sorted into three types, overtopping devices, oscillating water columns and point absorber. Picture from [7].
1.2 System overview
The WED developed by CPO is a complex system containing several different modules. Some of the main modules of the WED have been illustrated in Figure(1.2.1). The WED is a deep water buoy oscillating at the surface, where this wave induced motion is transformed to electricity. The centre of the WED consist of the PTO system which takes the translational motion from the buoy to rotational motion for the generators and the wave springs that make the buoy oscillate out of phase with the waves. The buoy and floating systems are connected through the piston rod to the mooring line which in return are connected to the anchor keeping the buoy in place during operation.
WEC unit
Mooring line
Mooring
PTO system Wave springs
Irregular waves
Piston rod
Figure 1.2.1: System overview of the wave energy device developed by CorPower Ocean.
1.3 CorPower Oceans design process
The WED developed by CPO has not yet been produced in full scale, but a 1:12 and 1:16 scale models have been created and tested. At the moment a 1:2 scale model is being designed. In this design phase a large amount of work have been done to create a hydrodynamic simulation model. Where this model describes the behaviour and forces on the system in irregular waves. The hydrodynamical model has been verified by experimental tank testing on the 1:12 and 1:16 models. The hydro mechanical model is used as basis for the design, it gives motions and forces as input for the solid mechanical strength analysis of the system. The design process at CPO can be seen as an iterative process, where results from the hydrodynamic model will give input for the solid mechanical model. Where the result from the solid mechanical model might led to a new iteration of the design. This design process have been illustrated in Figure(1.3.1). Where depending on the evaluation/redesign a new design can either lead to a large iteration where a new hydrodynamic simulation needs to be done or to a small iteration using the solid mechanic models. The large iteration is done if the redesign have an effect that might change the behaviour and forces on the WED in the hydrodynamic model.
Figure 1.3.1: A schematic picture of the design process used at CPO.
During its life time the WED is going to be exposed to a number of different load cases. Where some of these will lead to extreme loads on the WED, and others will lead to fatigue loads. The system therefore needs to be designed to withstand and survive both extreme and fatigue loads. As of today CPO does not have an efficient process of extracting extreme loads and does not have a specific method of analysing fatigue loads obtained from the hydrodynamic model.
Today the fatigue of the WED is calculated by taking the worst load case scenario and applying this load with a constant amplitude for 106times (infinite lifetime). This is a conservative way of dimensioning for fatigue where the average load most likely is far from the maximum load. Designing the structure like this probably means that the structure is over dimensioned, meaning extra weight and cost. A more accurate way of calculating the fatigue damage of the WED would be to take into account the irregularities from the waves and by so also more irregularities in the loads on the WED.
For the WED developed by CPO it is important to have a low inertia for the system to be able to effi- ciently extract energy from the waves. Reducing or limiting weight of the structure is thus an important factor when designing the WED. Reducing weight could also lead to a more cost efficient system, with less material, easier handling etc.. Something that is crucial when trying to prove that the concept of wave energy works and can become cost efficient.
1.4 Purpose & objectives
The Purpose of this thesis have been to develop a tool that takes the varying forces from irregular waves that the WED might encounter and creates extreme and fatigue loads from these that can be used in the strength analysis. This in practice have meant using the large amount of data produced by the CPO hydrodynamic model developed in Matlab/Simulink, processing it and export extreme and fatigue loads that can be used in the solid mechanical model SolidWorks-Simulation. As described by Figure(1.4.1), this is a tool between the hydrodynamic and solid mechanical model. A tool to simplify and improve the current design process.
Figure 1.4.1: The purpose of this thesis is to create a tool that takes the data from the hydrodynamic model processes and creates extreme and fatigue loads that are usable in the solid mechanical model.
To reach the purpose of the thesis a number of objectives have been set that are listed below.
1. Create a tool for a general load case that can be used with small modifications for other Matlab users. The tool should take the data from the hydrodynamic model, reduce it and format extreme and fatigue loads in the way that it can be used for strength analysis in SolidWorks. (Chapter 4) 2. Develop and use fatigue tools based on different methods for a simplified example. Compare and
evaluate the methods against each other and current methods. (Chapter 4)
3. Do a load and strength analysis on some subsystem of the WED with results from the developed tool. Compare this results with current methods used by CPO. Based on the outcome of the analysis do a redesign of the part to improve the system. (Chapter 5)
1.5 Disposition of report
This report consist of seven separate parts Background, Theory, CPO models, Fatigue methods & tools, Application , Conclusion and Appendices. The theory chapter first handles the background theory that have been used throughout the thesis, but also important theory for extreme loading and fatigue. Some of the more detailed theory of fatigue for multiple variables and in the frequency domain is described in the Appendices(B)-(C). In the CPO models some details about the models and system used by CPO is described, this is since the models have had a large influence on how the tools in the thesis have been formed. The fatigue methods & tools chapter presents and explains the methodology of four different fatigue tools that have been created for estimating fatigue loads, 1) Single variable method 2) RP direct method 3) RP two step method 4) Spectral method. The four different fatigue tools have been evaluated on a simple geometry with some loads from the hydrodynamic model developed by CPO. With this simple example the fatigue methods have been compared to each other.
The Application chapter have been aimed to do a case study on a subsystem of the WED developed by CPO. Hot spots caused by fatigue and extreme loads have been determined and a redesign of the subsystem is suggested and analysed. A conclusion chapter have been created to show the main con- clusions of the thesis, but also how the purposes have been fullfilled and suggestion for future work. In the Appendices some more details about the hydrodynamic model developed by CPO is given, more described theory for fatigue using Rainflow Projection and spectral methods, methodology for each step in the amplitude methods and finally an appendix describing how the FEA verifications have been done.
Theory
2.1 Definitions and background theory
This section goes through some definitions and general naval architecture theory such as scaling laws and wave scatters that have been central through out the thesis. It also goes through more background theory specific for wave energy and how design load cases can be defined for this. It also presents theory about reduction factors that are used in strength analysis for both extreme loads and fatigue.
2.1.1 Coordinate system
The coordinate system used to throughout the project contains three translational degrees of freedom (Heave, Sway and Surge) and three rotational (Roll, Pitch, and Yaw) as described in Figure(2.1.1).
Figure 2.1.1: The coordinate system used for the WED. Picture from [8].
The WED developed by CPO is axially symmetric on the outside meaning that the the two rotational motions roll and pitch and the two translational motions surge and sway would be the same. This is true for the outer hydro mechanical loads, however for the inner loads the WED is not axi-symmetric.
Meaning that some simplifications have to be done when calculating the structural mechanical loads in the unsymmetrical inner part.
2.1.2 Froude scaling laws
The data from the hydrodynamic model is for the full scale (1:1) WED, and the structure analysed is
1:2 model. This scaling can be done by using the Froude scaling laws. These laws allows to transform different physical parameters depending on the difference in scale of the models (Ω). Scaling values for some parameters have been listed below in Table(2.1.1),[9]. Where ρρf
m is the density quota between the full (ρf) and scale model (ρm), set to 1.026 for the two models developed by CPO.
Table 2.1.1: Froude scaling table used for scaling properties between full scale and half scale model, [9].
Physical Parameter Unit Multiplication factor
Length [m] Ω
Structural mass [Kg] Ω3·ρρf
m
Force [N] Ω3·ρρf
m
Moment [Nm] Ω4·ρρf
m
Acceleration [m/s2] x¨F = ¨xM
Time [s] √
Ω
Pressure N/m2 ΩρρF
M
2.1.3 Irregular waves
The waves at sea and that will also act on the WED are irregular and are more or less stochastic as they are in nature. As can be seen in Figure(2.1.2) an irregular sea can be described as a superposition of many different regular waves.
Figure 2.1.2: A irregular sea can be seen as composition of many regular waves. Picture from [10].
The irregular seas can be described by different wave spectra based on two statistical parameters of the sea state, wave height(Hs) and wave period (Tp). The wave spectrum used in this analysis is derived from the the Joint North Sea Wave Obesrvation Project (JONSWAP). With this wave spectrum a random sea can be built up with the two parameters Hsand Tp. The JONSWAP is a extension of the Bretschnider spectrum, what spectrum to use depends on the location of the WED, [10].
As described by the JONSWAP spectrum the state of the sea can be described by two parameters Hs and Tp. The occurrence of certain combination of Hs and Tp can be defined by a statistic wave scatter, as can be seen in Figure(2.1.3). Where each box represents a separate sea state. The figure does not include all relevant parameters such as wave direction (θ) that might also be important when looking on wave loads. Wave scatters are usually site specific, where a wave scatter in the pacific ocean is different from one in the Baltic ocean. The wave scatter in Figure(2.1.3) is for the sight Yue outside the coast of France.
Wave scatter, YUE
4 6 8 10 12 14 16
Wave period [s]
1
2
3
4
5
6
7
Wave height [m]
10 20 30 40 50 60
occurence
Figure 2.1.3: A statistic wave scatter from two years measurement, where the irregular ocean waves are defined by two parameters significant wave height(Hs) and wave period (Tp). Measured data from Yue, outside the coast of France.
2.1.4 Design load cases
The WED will be exposed to several different design load cases (DLC), where each load case can be defined by two main factors, 1) Operational mode of the WED, including factors such as the power pro- duction, faults and emergency modes. 2) Environmental mode including factors such as wave height and ice in the water. Essentially combining these two factors one can obtain almost an infinite amount of load cases. Where analysing all these fast becomes unpractical if not impossible. To limit the number of load cases and make this practically possible regulatory bodies suggest a number of operational modes for the DLC that needs to be calculated. Where these load cases typically handles cases as, power production, start up, emergency shut down, survival, damage stability, transportation etc., [11]. The environmental factors would typically be determined by variables such as wave height(Hs), wave period (Tp) and the direction of the waves(θ).
Guidelines for offshore designs [12][13], typically divide the loads into ultimate limit states (ULS) which can be seen as states limited by extreme loads, loads that might break the WED after one impact. Fatigue limit states (FLS) which are the many loads occurring during a longer time leading to fatigue damage.
Accidental limit states (ALS) and serviceability limit states (SLS). Only ULS and FLS have been treated in this project. In this thesis the extreme loads have been based on simulations from a 50 year’s storm, but also the loads that might occur during the more common operational modes. The fatigue loads have been based on simulation data from 25-108 sea states including normal, detuned and survival modes for a fully functional WED as operational modes.
The operational profile of a WED, depending on wave height and period could typically look like in Fig- ure(2.1.4). Where three operational modes are displayed in colours, all modes assuming that the WED will be fully functional.
Figure 2.1.4: Typical operational profile for a wave energy device, where the mode of operation can be divided into three modes; normal operation, detuned and survival depending on wave period and height.
When choosing what DLC to analyse judgement needs to be done of what modes that are critical for the part being analysed and what load data that is available. For estimating ULS in different components the worst sea state in case of loads on the WED would typically be used as a determinative load. For fatigue and FLS this would probably not be determined by one sea state but a combination of different operational modes and sea states, since all of theses would contribute to the accumulated fatigue damage.
2.1.5 Reduction factors
Depending on certain properties and field of application of the part there might be need to add a re- duction factor (Kf) in the strength analysis. These correction factors could be applied in both the ULS and FLS cases, where the allowed stress then would be reduced. These reduction factors can typically be added for weaker parts such as welds, different geometries, or environment of operation where a corrosive environment would reduce the strength of the material gradually. The correction factors are often based on semi-empirical equations or experience and are defined in different rules and regulations by third party classification agencies such as ABS, [14]. For fatigue these reduction factors can sometimes lead to new SN-curves that are reduced compared to the original one. These reduced SN curves can be found for specific materials in classification rules, [14].
For welding the correction factors typically depend on a number of different factors, such as the thickness of plates, direction of load contra direction of weld etc.. Based on these measurements a reduction factor can typically be estimated. A weld that is perpendicular to the force could typically be a problem while a weld that is parallel to the loading is usually not, this is because the parallel weld will not be critical in the way that it does not need to carry any load.
(a) Weld parallel to load. (b) Weld perpendicular to load.
Figure 2.1.5: Depending on weld and load direction correction factors might be added in the strength analysis.
Figure from [14].
If there are several reduction factors on a geometry a total reduction factor can be calculated by multi- plying the different correction factors with each other,
Kf = Kweld· . . . · Ki. (2.1.1)
Where this correction factor can be used when analysing extreme and fatigue loads on a structure. For the extreme loads the allowed strength can be directly reduced by this factor. In the case of the fatigue the SN curve can be shifted downwards with this factor leading to a new SN-curve.
2.2 Extreme loading
The extreme loads are the highest load that occur during the whole life time of the WED. Depending on what force that is considered the extreme loads can occur in different sea states, it might be during a 50 year storm but it might also be during a normal operational sea state. The goal of this WED is to withstand a 50 year storm and the loads that might occur on it during this storm, but also the extreme loads that might occur in other sea states. A 50 year storm implies the worst weather that the WED will encounter during a 50 operational period, at the specific location. These states that are limited by one set of extreme loads also refereed to as ultimate limiting states.
The extreme load on a component is determined by taking the maximum load that occurs on the com- ponent during its life time, including 50 years storms and different operational conditions. When having multiple loads, the maximum of each separate load occurring on the component during its lifetime are determined, applying these different maximum load together then becomes the extreme loads. For multi- ple loads applying all the maximum loads at the same time could be considered a conservative estimation since it might very well be that theses loads do not happen at the same time and therefore applying theme all on the same time would lead to higher stress than would happen in reality.
When designing against extreme loads there are usually a few different criteria that needs to be fulfilled in the design. Where typical criteria are maximum tensile (σallowed) and shear (τallowed) stress, maximum allowed displacement of the component (δallowed), safety against buckling etc. These criteria change de- pending on material and component that is analysed. Whatever the criteria are they need to be fulfilled, where often only a few of the many criteria are dimensioning. Since wave energy is a relatively new area complete standards are not jet fully developed, typical guidelines can be found in texts like Guidelines on design and operation of wave energy converters by DNV GL, [12]. Where in these document many of the guidelines are based on experience from floating offshore wind turbines and other offshore structures where there are more experience.
2.2.1 Ultimate tensile strength
It is important that the WED is designed for holding against extreme loads, where the structure would be subjected to a one time maximum load Fm,max. Where this load or combination of loads (m) would lead to a maximum stress in the material σmax. This σmax is in most cases not to exceed the yielding stress limit of the material (σyiels), where the material starts to deform permanently. This permanent deformation might cause severe problems or failure of the whole WED if at a critical area. Yielding is a phenomena that can be seen in materials such as metals or plastics, however it does not occur in many composite materials because of the brittle properties of glass and carbon fibre, [15]. If the stress level in the component exceeds the ultimate tensile strength σtensile this would lead to fracture of the material leading to definitive failure of the WED. The stress can be in two main directions, compres- sion and tension. The strength in these two directions can vary a lot depending on the material, where non-isotropic materials like fibre reinforced composites or concrete have different properties in compres- sion and tension. Depending on the role of the structural element σallowedcould be based σyieldor σtensile. Depending on the geometry of the structure subjected to load and the properties of the material there are other stress components than only the normal stresses that will determine if the material will break or not. These are the shear forces (τ ). To be able to evaluate the different stress components and compare them to the material σtensile a equivalent resulting stress needs to be defined. One approximation that will be used in this thesis is the von Mises stress (σvM). Which is often used as an approximation when calculating stresses in structures, [16]. It is also recommended by classification societies such as DNV GL in [13]. It is calculated by,
σvM = 1
√2
p(σ1− σ2)2+ (σ2− σ3)2+ (σ3− σ1)2. (2.2.1)
Where σ1, σ2, σ3 are the main principle stresses that can be solved by a eigenvalue problem containing the different stress components of σi and τij [16].
2.2.2 Deflection & buckling
It is not certain that the strength will be the limiting criteria for the structure, where other criteria such as the deflection or stiffness of the structure might be limiting as well. The deflection criteria is important because a too large deflection might cause collision or interference with other components in the structure. Where a collision with other parts might cause a destructible failure of the WED and therefore should be avoided. The deflection of a component can usually be defined as non dimensional value δ = def lection
Lengthof structuralelement.
Another phenomena that can occur on the WED due to extreme loads is buckling. This is usually a case for long slender beams or thin plates exposed to compression forces as can be seen in Figure(2.2.1).
Long slender beams and thin plates, could typically occur when trying to create a lightweight structure using plates and stiffeners, which also exists on the WED structure. There are several different kinds of buckling, such as Euler buckling, torsional buckling, local buckling or combined buckling, [17]. If buckling will occur or not mainly depends on the geometry and Young’s modulus (E) of the material and of course the load on the structure.
Figure 2.2.1: A thin plate subjected to compression force(Nx) along the edges, a typical example of where buckling could occur. Picture from [17].
2.3 Fatigue
This section first describes the concept of fatigue and approaches of how to calculate it, then it describes a bit more in detail the SN-approach of calculating fatigue that have been used in this thesis. The section also goes through how to calculate the fatigue first in the time domain using one variable and then mul- tiple variables. Also how to calculate fatigue in the frequency domain is explained. Where more theory about Rainflow Projection and calculating fatigue in the frequency domain can be found in Appendix(B) and (C).
2.3.1 Fatigue and how to estimate it
The most probable cause of fracture in a construction is in many applications especially metal not the once in a lifetime extreme load but the many time occurring lower load causing the phenomena called fatigue. Fatigue can thus be defined as a failure that occurs due to repeated loads, where non of the loads gives higher stresses than the ultimate stress for the material. The process of fatigue can be said to contain of three stages [18],
1. Crack initiation, a crack is created in the material.
2. Crack propagation, the already existing crack already continuous to grow from the stress cycles it is subjected to.
3. Final fracture, the crack has grown so big that there will be a final fracture in the component.
When estimating the fatigue life or damage of a component there are three main approaches to calculate the fatigue. Stress life (SN), Strain life (EN) and Linear Elastic Fracture Mechanics (LEFM), [18]. These approaches are shortly described below. The one that have been used in this thesis is the SN approach.
1 SN approach. This approach is based on how many stress cycles that the material can withstand before failure. Where the fatigue properties of the material, usually are obtained from experimental testing and varies from material to material. The material data are usually presented in so called SN-curves (also called Whöler curves) describing how many cycles the material can withstand be- fore fracture. An example of a SN-curve can be seen in Figure(2.3.1). The SN method is the most frequently used method for high cycle fatigue, [18].
2 EN approach. This approach calculates the plastic stresses and it possible to use for both crack prediction and total life estimations. It is a common method for low cycle fatigue. This method is not as common as the SN method and thus there are far less material information in this method, [18].
3 LEFM approach. This approach assumes that there already is a crack in the material and then tries to calculate the growth rate of the crack. Crack growth is estimated by the stress intensity at the tip of the crack. This is a fracture mechanics approach, it is often used in applications where cracks already exist and needs to be monitored and the growth rate can be estimated,[18].
Fatigue can be defined into high and low cycle fatigue, where low cycle is less than 103 cycles and high cycle is more than 103 cycles. Having a average wave period on 5s the number of wave encounters on a year would be ∼6.3 millions and on 20 years ∼126 millions. If every wave encounter would lead to a stress cycle this would lead to that the dimensioning would be against high cycle fatigue. However if only a few of this wave encounters would lead to significantly high stress in the material this would lead to a low cycle fatigue.
Figure 2.3.1: Typical SN curve of a material, with number of cycles before failure at the horizontal axis at each load level on the vertical axis. Figure from [18].
Using the SN approach for calculating the fatigue damage the important information is the stress cycles in the material, more exact the peaks and valleys of the stress cycles. It is the amplitude and frequency of these cycles that will determine if there will be fatigue or not. Having a load signal from a simulation or experiment this usually contains a lot of data with thousands of values from a long measuring period and high sampling frequency. Where the important data for fatigue are the one determining stress cycles, the peaks and valleys. Thus reducing the data to only contain the information of peaks and valleys of the stress are an important part of calculating the fatigue. Where by having a data set only containing the information about of peaks and valleys in the stress fatigue can be calculated relatively easy.
2.3.2 Calculating fatigue
In this thesis two main ways of calculating fatigue have been studied, amplitude and spectral methods.
Amplitude method can be used for single variables using Rainflow Cycles (RFC) to calculate fatigue, they can also be further developed for multiple variables using Rainflow Proejection (RP) methods. The spectral methods calculates the fatigue in the frequency domain by using a so called Power Spectral Density (PSD) of the load signal. The spectral method can be used for single but also multiple variables if the correlation between the loads are known. The main specifics of spectral and amplitude methods for single and multiple variables have been summarized in Table(2.3.1).
Table 2.3.1: Summary of different methods of calculating fatigue.
Method Single variable Multiple variable
Amplitude This method is based on reducing the load series to only contain the peaks and valleys. This information is stored in so called Rainflow Cycles. These are then combined with the SN-curve and Palmgren Miner’s rule to calculate ac- cumulated fatigue damage.
For multiple variables Rainflow Projec- tion can be used to find the stress peaks based on several loads. Rainflow Pro- jection is based on the same methods used in the single variable but has a special methodology when it comes to finding the Rainflow Cycles.
Spectral Fatigue in the frequency domain can be done by calculating a PSD of the load and a stress transfer function for the analysed part. A method often used for offshore applications exposed for stochastic loading or for structures that are in or close to resonance with the loads.
Creating several PSD does not lead to any more complications than creating one. Using multiple variables the cor- relation between loads needs to be de- termined.
The fundamental principles of the amplitude (single and multiple variables) and the spectral theory are briefly explained in the proceeding sections. More detailed theory about Rainflow Projection and using the frequency domain for fatigue calculations can be found in Appendix(B) and (C).
Single variable fatigue
The theory described in this section for calculating the single variable fatigue is based on theory from, Guide to load analysis for durability in vehicle engineering by P. Johannesson, [19]. As mentioned earlier the important information when calculating fatigue is the peaks and valleys in the stress. However the data from the hydrodynamic model does not provide information about the stress σ(t) but only the load F (t). With a single load and assuming a linear elastic material σ(t) can be assumed to be proportional to F (t) with the constant C. This is mathematically described by,
F (t) · C = σ(t). (2.3.1)
Where the peak in F (t) appear at the same time as the peak in σ(t), thus F (t) can be used to find the corresponding peaks and valleys in σ(t). Since only the peaks in the stress signals are important the turn- ing points of a load signal can be filtered out. Where these turning points then can be used to calculate the RFC of the load signal. These RFC contains all the information important for fatigue calculations and can be used as they are or be gathered in a Rainflow Matrice (RM). An example of a force signal F (t) and the points used for creating the RFC can be seen in Figure(2.3.2a). Having the RFC these can be discretized into different bins and stored in a RM. Where two points creating a RFC turns into one point in the RM, with the valley load on the horrisontal axis and the peak load at the vertical axis. A example of the RM can be seen in Figure(2.3.2b). In the RM the RFC looses its sequential order and its re- lation to time. The methodology for finding turning points, RFC and RMs can be found in Appendix(D).
time [s]
820 840 860 880 900 920 940 960 980
Force [N]
×105
-4 -3 -2 -1 0 1 2 3 4 5
Filtered load signal and removed points
Filtered F RFC points
(a) RFC turnig points (red) from a load signal. (b) Peaks and valleys of RFC, sorted as a RM.
Figure 2.3.2: By calculating turning points and RFC, they can be discretized and gathered in a RM only containing the information important for fatigue.
With the RFC or a RM containing the load cycles this can be used to calculate the accumulated dam- age of a structure. Calculating the accumulated fatigue damage consists of three theories, 1) The RFC and their stress amplitude, 2) SN-curve estimated by the basquins equation, 3) Palmgren Miner’s rule to finally calculate the accumulated damage. A way of estimating the SN curve is the basquins equation,[19],
N = αS−β. (2.3.2)
Where α ans β are material properties based on the SN-curve, S is the amplitude of the stress (valley to peak in the RFC) while N is the number of cycles. One way to calculate the accumulated damage is then to use the Palmgren-Miner rule which adds up the damage of all the RFC,
D = 1 α
X
i
Siβ. (2.3.3)
For some applications for example when the material data or the constant C is not available it can be practical to calculate pseudo damage from a stress series. This pseudo damage can be calculated with,
Dp=X
i
Siβ. (2.3.4)
Where this also can be useful for comparing different load series where the force amplitude between a peak and valley can be used instead of the stress amplitude. But it is important to point out that this does not calculate accumulated damage only pseudo damage.
Multiple variable fatigue
For the single variable load where there is only one load acting on the structure it is relatively easy to pick out the extreme loads and RFC since they are only depending on that one input signal as was described in the previous section. In reality however there are often several components that acts on a structure making the load case far more complicated. The extreme load can be determined by the worst combination of the two or more loads that will occur on the structure on the same time. For fatigue with a theory that is based on RFC it becomes more complicated. For in phase loads the stress peaks could probably be found relatively easy since they probably occur when both load peaks. But when looking on two or more out of phase loads it is not as easy, the peak stress could occur with any one of the loads
peaks, or maybe somewhere in between the peaks of the several load signals. For example having two out of phase forces F1(t) and F2(t) each creating a contribution to the total stress σ(t) with the constant C1, C2 this could be formulated as,
F1(t) · C1+ F2(t) · C2= σ(t). (2.3.5)
Where in this case a peak in F1(t) does not ensure a peak in σ(t) because σ(t) is also influenced by F2(t) and C2. The peaks and valleys of σ(t) caused by F1(t) and F2(t) is therefore unknown if the constants C1 and C2 is unknown. An example of two out of phase loads can be seen in Figure(2.3.3). Where the load combination leading to a stress peak is unknown.
5 6 7 8 9 10 11 12
Time [s]
-1.5 -1 -0.5 0 0.5 1 1.5
Force [N]
Two out of phase loads
F1
F2
Stress peak?
Stress peak?
Stress peak?
Figure 2.3.3: For two out of phase forces the stress peaks important for fatigue are not as easily found as for the single force case.
Having an arbitrary structure and two irregular out of phase loads the problem of finding the peaks and valleys of the stress becomes even more problematic. A theory to cope with multiple input loads is called Rainflow Projection (RP). RP is a method based on finding the peaks and valleys in σ(t) based on the different loads(m) and combinations of Cm, Fm(t) [19]. The RP methods builds on the theory of RFC and RM. This thesis have led to two different sorts of RP methods. 1) RP direct where the load is taken out directly from the load signals based on different linear combinations (LC) of the loads. 2) RP two step, where the constants Cmare calculated as an extra step in a FEA software and then used to find the peaks in σ(t). Having the RFC of the stress, accumulated damage can be calculated in the same way as in the single variable method with the Basquins approximation and the Palmgren Miner’s rule. Theory behind the two RP methods developed in this thesis can be found in Appendix(C).
Fatigue in the frequency domain
An other approach for calculating fatigue is the spectral method, which in some literature is also refereed to as the frequency based method. In this method the accumulated fatigue damage is calculate in the frequency domain instead of the time domain as is done in the amplitude methods. This method was first developed in the 1980’s for simplifying the calculations on oil rigs. It is a frequently used method for system exposed to random loads such as in the offshore industry, [20]. It is also recommended for analysis where the loading frequencies is close to the systems eigenfrequencies, meaning that the reso- nant fatigue is of importance, [21]. This method calculates the fatigue using a power spectral density (PSD) of the load signal and the stress transfer function of the analysed part. The reasons for using the frequency domain for calculating fatigue are mainly two points 1) The load on the structure might be random in nature and then the best approach for calculating the loads is a statistical method which this method is by using the PSD. 2) Calculating the fatigue with the frequency domain might reduce the calculation time compared to a long time history of data that is used when in amplitude methods [20].
Using the frequency methods there are a few underlying assumptions [22], some of which are listed below.
• Load analysis and associated structural analysis is assumed to be linear.
• The loading vector is randomly distributed according to the PSD.
• The load sequence is time independent.
For calculating fatigue in the frequency domain there are as mentioned above two things that are initially needed, 1) PSD of the load signal (Sη), 2) the stress transfer function (Hσ(w)). Where these two can be put together to create the stress energy spectrum (Sσ) as described by equation(2.3.6),[14]. This stress transfer function is used when calculating the number of stress cycles at the different amplitudes, and finally for calculating the fatigue in different structures. The accumulated fatigue damage is calculated by using the SN-curve of the material and adding the separate load cycles.
Sσ(ω) = |Hσ(ω)|2Sη(ω) (2.3.6)
Using the Sσ this can be combined with a number of statistical parameters based on the PSD of the load signal and Sσ to calculate the accumulative damage of the WED. This method of calculating fatigue is described by ABS in Fatigue assessment of offshore structures, [14]. Where the accumulative damage (D) finally can be calculated with,
D = NT α
Z ∞ 0
sβg(s)ds. (2.3.7)
Where α, β are material properties, NT total number of cycles in the design life, s is the specific value of the stress range and g(s) is the long term probability density function of the stress distribution. These pa- rameters and the method behind the expression in equation(2.3.7) have been described in Appendix(C.4).
Using the frequency domain for calculating the fatigue it is important selecting the right damping of the structure for having a accurate result, something that is also more thoroughly explained in Appendix(C.3).
Knowing the correlation between several loads the fatigue can also be calculated for multiple loads in the frequency domain, where more information about this can be found in Appendix(C.5).
2.3.3 Von Misses or Principal stress
When calculating fatigue there are a few different stresses that can be used, two common ones are von Mises stress and Principal stresses. If the weakest link is a crack or weld with known geometry this crack would probably be most vulnerable for stress in one direction. Where the principal stress then would be a good option to use. If the geometry of the weakest link is unknown typically a pore or
something in the material, the most sensitive stress direction is not known. Not knowing the sensitive direction this would imply that the principle stress is not as suitable since this calculates the stress in a sensitive direction. In that case the signed von Mises might be a better option for calculating the stress.
CPO models
This thesis have been done in collaboration with CPO, and thus the tool developed have been matched to the hydrodynamic and solid mechanic model used by CPO. These two models have had large influence in the way that the tools have been built. To show how these two models have put out some constraints for the tools these two models are described in this chapter. However even if the tools, code and methods developed in this thesis are made to fit the models used by CPO they can most likely be used directly or with small adjustments for other cases and models as well.
3.1 Hydrodynamic model
As the hydrodynamic model is the basis for the rest of the system the fundamentals of the model are described in this section. To first understand the fundamentals of the hydrodynamics, the wave loads of floating offshore structures should be explained. Wave loads as illustrated in Figure(3.1.1), can be seen as two parts first excitation forces and then the restoring loads from the added mass, restoring forces, put these together and you will get the motions and behaviour of the body.
Figure 3.1.1: Wave loads can be seen as two parts, the exicitation forces from waves and internal systems and the responses of the body with properties like added mass, hydrostatic forces. Picture from [23]
The hydrodynamic model at CPO is developed in Matlab/Simulink. The model created by CPO can analyse the forces and motions of the system in both regular and irregular waves, the irregular waves are defined by two parameters, significant wave height Hsand peak period Tp of the waves. These two parameters can then be combined by the JONSWAP (Joint north sea wave project) wave spectrum to create the wave spectrum of the waves. The hydrodynamic model has been validated by comparing the results from simulations with results from experimental tank testing of scale models.
The Simulink model developed by CPO is formulated assuming linear wave theory. Linear wave theory assumes as non linear wave theory ideal fluid (no viscosity) and that there is no surface tension at the free surface. The linear theory also neglects the higher order terms of the boundary conditions allowing for a
simpler wave equation and practical relationships, as e.g. the dispersion relation (k = ω2/gtan(kh)), that for deep water waves becomes k = ω2/g. The linearisation is usually a good approximation for waves where the wave height(H) is much lower than the wave length(λ), H/λ << 1. Linear wave theory allows for summing up the different waves by superposition,[4]. The linearised theory also allows for analysing the properties of the buoy in frequency domain, as is done in WAMIT [24].
The current Simulink model is restricted to two degrees of freedom (DOF), surge and heave. These two motions are coupled to each other, meaning that the motions in heave direction does have an affect on motion in the surge direction. However this coupling effect is usually much smaller than the direct effect.
The model is based on the Newtons second law formulated as,
(I(mb+ mP T O) + m∞)¨xbuoy = FP T O(x, ˙x) + Fhydro(x, ˙x) (3.1.1)
Where I is the Identity matrix ¨x is the acceleration vector, mbis the mass of the buoy, mP T Ois the mass of the PTO system and m∞ is the infinity frequency added mass matrix. Fhydro is the hydrodynamical force vector coming from waves, currents and other body fluid interactions. FP T O is the machinery force vector coming from the power take off (PTO) system. These equations can when solved give the total motion of the WED as described by the most right part of Figure(3.1.1). The forces and what they represent are presented in the proceeding sections. A more thorough description of how the loads are calculated can be found in Appendix(A), however a complete derivation of the loads is not given in this thesis because it considered out of scope.
3.1.1 Hydrodynamic forces
The Simulink model developed by CPO uses WAMIT, a wave interaction analysis program for a some of their calculation. WAMIT uses linear and second-order potential theory for calculating forces on floating or submerged bodies in waves. It uses the so called panel method to solve for the velocity potential of the fluid and by so also the pressure distribution on the bodies, [24]. WAMIT is run as a preprocessor to the Simulink model to get the hydrodynamic coefficients of the model in advance. The results are then combined with particular environmental conditions and a model of the PTO/mooring system to simulate the complete behaviour of the WED. The hydrodynamical or outer forces (Fhydro) on the buoy can be divided into four different parts,
Fhydro= Fexe(x, ˙x) + FRad( ˙x) + Fhyst(x) + FDrag( ˙x). (3.1.2)
Where Fexe are the excitation forces from the wave, Frad are the radiation forces from the buoy, Fhyst
are the hydrostatic forces and Fdrag are the drag forces on the buoy. Fradcan physically be explained as the force that are generated on the buoy when it is moving up and down creating a wave. To calculate this WAMIT calculates a number of radiation state vectors for the geometry that then can be used in the CPO Simulink model to calculate the radiation force in an efficient way. Fexe are the forces on the buoy coming from the wave and is also calculated with the help of WAMIT data and depends on the angular frequency of the incoming waves.
Fhyst is calculated by multiplying the submerged volume and multiplying it with the water density and gravity. Fdrag is calculated by estimating the drag coefficient CD of the body in different directions and then multiplying it with the projected area and the velocity of body or surrounding fluid. A more detailed description of how the hydrodynamical forces are calculated in the Simulink model can be found in Modeling, Simulation and Dynamic control of a wave energy converter, by Maria Bånkestad [25]. It follows the classical hydrodynamical theory that also can be found in books like Marine Hydrodynamics of J.N. Newman,[4].
3.1.2 Machinery forces
Except forces from waves and hydrodynamics of the buoy there are also several forces coming from machinery of the WED, also referd to as the PTO system. It can be seen as the part that takes out the energy from the waves and turns it into the electrical energy. The PTO force (FP T O) is divided in to the following components,
FP T O= Ftrans( ˙x) + Fgas,pretension(x, ˙x) + Ff riction( ˙x) + Fgas,wavespring(x). (3.1.3)
Where Ftrans are the forces going of to the transmission system turned into rotational motion and then energy. Ff riction is the friction force, Fgas,pretension is the pretension force. Fgas,wavespring is the force from the wave spring. By having a active control system the Ftrans, Fgas,wavespring, Fgaspretension can be adjusted during the simulation, in the same way as it would be in the real world, to get the buoy to behave as wanted. The forces and where they are acting on the WED can be seen in Figure(3.1.2). All machinery force are represented in 1 DOF.
Buoy hull
Wave spring
Gear box and generator
Rack
Pretension spring and Piston rod
F_PTO F_trans + F_fric
F_gas,wavespring
F_gas,pretension
Figure 3.1.2: Description of WED and the machinery forces that are implemented in the hydrodynamic model.
3.1.3 Validation & output
A mathematical model does rarely show exactly what happens and the Simulink model developed by CPO is no exception from this. The question is rather how well it does represent the motions and forces that will act on the WED. The radiation damping forces, drag forces and other crucial parts of the model have been validated, with experimental results from two different laboratory testing 1:30 and 1:16 scale
models, [26]. However a certain percentage error marginal have not been determined on the hydrody- namic model compared to the reel model. The models for machinery forces have not yet been validated.
A test rig have been built and is running where experiments have been made on the machinery forces.
However so far the results have not yet been analysed for validation of the model.
The output of the hydrodynamic model are for example the forces described in the previous sections, positions, velocities and accelerations of different parts of the buoy, energy output etc.. These are saved as time series of 1800 seconds with a iteration time step on 0.001s, leading to 1 800 000 points or values for each variable and sea state. A time step in the simulation model on 0.001s is relatively small which leads to large result series as output from the model. Enlarging this time step could reduce the data amount significantly. However this simulation time step have been determined by CPO to capture certain effects in the hydrodynamic model that might be lost if the simulation time step was larger. One data file is produced for every sea state sea state simulated. The output data is based on the full scale model of the WED, and thus needs to be scaled using Froude scaling laws when estimating the forces and accelerations on the 1:2 scale model. All forces are not given from the hydrodynamic model, therefore some of the loads needed might need to be derived by approximations of other values such as weight and accelerations.
3.2 Solid mechanical model
The second model is the solid mechanical model that calculates the stresses and fatigue damage of the structure and PTO system. These stresses have been calculated by so called Finite Element Analysis (FEA) where the geometric and material properties are combined to calculate the stress and fatigue damage of the components. The program used in this thesis is SolidWorks but any other FEA program such as Ansys, SolidEdge or Abaqus could probably been used.
One of the large advantages with working in this kind of programs are that the interface is user friendly where the stress and accumulated fatigue damage of the different parts can be seen directly in the graph- ical interface of the program. Also applying forces and boundary conditions to the model can be done easily, and the results can be seen in a easy way. However SolidWorks have some limitations and to do some of the more developed fatigue studies such as RP two step as have been developed in this thesis, the data from the SolidWorks FEA analysis needs to be exported and used in Matlab where larger freedom of use is possible compared to in SolidWorks.
3.2.1 FEA in general and SolidWorks
FEA analyis devides an object into many small elements connected by nodes, where the stress and deformation is calculated for every separate node. The general FEA method is based on the virtual work method (strong form), which can then be translated in to a physical interpretation in the integral form (weak form). By doing the Galerkin approximation the matrix formulation used in FEA can be formulated,[27],[28]. Using the matrix form the translational and rotational displacement of a structure (u) can be described as the force and moments applied to the structure (F) and the stiffness matrix of the structure (K), as described in equation(3.2.1),[28],
F1
...
Mn
=
k11 . . . k1n ... . .. ...
knn . . . knn
u1
...
θn
. (3.2.1)
Which can also be written in matrix notation as,
F = Ku. (3.2.2)
