Non-linear Ultimate Limit State Analysis of a Wind Turbine Blade

Contents

Figures

Tables

Notation

1 Introduction

2 State-of-the-Art
2.1 Standards and Guidelines in Wind Energy
2.2 Design Philosophy
2.3 Design Process
2.3.1 Process Description
2.3.2 Partial Safety Factors due to GL .
2.3.3 Analysis due to GL
2.4 Analysis Methods due to Eurocode

3 Theoretical Background
3.1 Rotor Blade Loads
3.2 Composite Laminates
3.2.1 Macromechanics
3.2.2 Classical Laminate Theory
3.2.3 Unsymmetrically Cross-Ply Laminated Plates
3.3 Numerical Analysis
3.4 Non-linear Effects
3.5 Analytical Analysis

4 Methods
4.1 Finite Element Model Setup
4.1.1 Software Tools
4.1.2 Geometry
4.1.3 Materials and Layup
4.1.4 Mesh
4.1.5 Load Application and Constraints
4.1.6 Solver Setup
4.1.7 Output Request
4.1.8 Groups
4.1.9 Model Singularity
4.2 Numerical Computations
4.2.1 Global Blade Deflection
4.2.2 Strain Distribution
4.2.3 Local Displacement of the Cap .
4.2.4 Buckling Resistance of the Cap .
4.2.5 Strength Analysis
4.2.5.1 Characteristic Resistance
4.2.5.2 Design Resistance
4.2.6 Stability Analysis
4.2.6.1 Characteristic Resistance
4.2.6.2 Design Resistance
4.2.7 Stimulation of Imperfections
4.2.7.1 Global Scaling
4.2.7.2 Local Modification
4.3 Analytical Computations
4.3.1 Global Deflection of the Blade
4.3.2 Strain Distribution
4.3.3 Local Displacement of the Cap
4.3.4 Buckling Resistance of the Cap

5 Results
5.1 Plausibility of the FEA Results
5.2 Strength Analysis
5.2.1 Characteristic Resistance
5.2.2 Design Resistance
5.3 Stability Analysis
5.3.1 Characteristic Resistance
5.3.2 Design Resistance
5.4 Stimulation of Imperfections

6 Discussion
6.1 Methods
6.1.1 Finite Element Model Setup
6.1.2 Numerical Computations
6.1.3 Analytical Computations
6.2 Results
6.2.1 Plausibility Check
6.2.2 Non-linear Effects
6.2.3 Load Carrying Capacity
6.2.4 Verification of Analysis
6.2.5 Stimulation of Imperfections

7 Conclusions

Bibliography

Appendices
A SSP 34 m Blade
B Simplified GL Approach

Figures

1 Terms of wind turbine

2 Blade coordinate systems

3 Model coordinate systems

1.1 Similar structures bent due to wind

2.1 Frequency density distribution of load and resistance functions
2.2 Design process for wind turbine rotor blades

3.1 Wind turbine blade loads and its sources
3.2 Loads on an airfoil
3.3 Unidirectionally reinforced lamina
3.4 Geometry of an N-layered laminate
3.5 Rectangular plate under longitudinal load
3.6 Contrast between symmetric, antisymmetric and unsymmetric cross-ply lam- inates
3.7 Simply supported plate condition
3.8 Rectangular plate under transversal load
3.9 Non-linear material behavior of composites
3.10 Geometrically non-linear bifurcation buckling
3.11 Geometrically non-linear limit point buckling
3.12 Geometrically non-linear contact of cross sectional shells
3.13 Nonlinear boundary conditions
3.14 The rotor blade as discretized beam under its load distribution and an in- finitesimal piece of continuous beam
3.15 Hollow shell structured beam element under bending load

4.1 Numbering of the cross sectional faces and the key points
4.2 Blade geometry, constraints and concentrated forces
4.3 20-node hexahedron continuum composite brick element
4.4 Linear interpolation element
4.5 Key-Point-Path-Group of the model seen from the tip
4.6 Side view of a draped element
4.7 Exclusion of draped elements
4.8 Stress peaks caused by geometrical uncertainties in FE model
4.9 Derivation of SRF and MoS for strength verification
4.10 Derivation of SRF and MoS for stability verification
4.11 Application of the imperfection
4.12 Application of Strain-Reversal Method
4.13 Transv. and long. stresses of critical element
4.14 Loading over local displacement of critical element
4.15 Distances and locations for the determination of the bending stiffness
4.16 Computation scheme of local cap displacement
4.17 Suction side cap and the plate model of the cap
4.18 Computation scheme for the buckling resistance of the cap

5.1 Global Deflection of the Blade
5.2 Longitudinal strain distribution on the suction side cap
5.3 Local displacement of the suction side cap
5.4 Buckling resistance of the suction side cap
5.5 Characteristic strength resistance
5.6 Critical locations at 1.63 · Ld
5.7 Design strength resistance
5.8 Characteristic buckling resistance
5.9 Buckling mode shapes
5.10 Design buckling resistance (variant A and B)
5.11 Design buckling resistance (bifurcation analysis)
5.12 Strength resistance due to failure index
5.13 Buckling resistance due to SRM
5.14 Characteristic strength resistance due to local modification of imperfections
5.15 Dimple outwards buckling mechanism at 110% design load
5.16 Buckling mechanism of the outward dimple and the critical location

B.1 GL verification with the application of the safety factor on the external load.

Tables

2.1 Partial safety factors for the loads according to IEC
2.2 Safety factors for strength analysis according to GL
2.3 Safety factors for stability analysis according to GL
2.4 Safety factors for test load according to GL
2.5 Analysis Methods according to Eurocode

4.1 Numerical computation process

5.1 Verification of the characteristic strength
5.2 Verification of the design strength
5.3 Verification of characteristic buckling resistance due to bifurcation analysis. .
5.4 Verification of design buckling resistance due to variant A and B
5.5 Verification of design buckling resistance due to bifurcation analysis
5.6 Global scaling of mode shape imperfections
5.7 Local imperfection modification

6.1 Verifications due to GL guideline

Notation

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Abbreviations

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Terms and Definitions

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Fig. 1: Terms of the rotor blade (a), the turbine (b) and the the blade cross section (c).

Coordinate Systems

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Fig. 2: Blade coordinate systems.

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Tab. 3: Load case definitions.

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Fig. 3: Model coordinate systems.

1 Introduction

Imagine there is a flexible cornstalk in the corn field bending under the influence of the wind as shown in Fig. 1.1a. When you cut the stalk and look inside, you come across a cylindrical thin-walled, hollow structure. Evolution decided the stalk to become as it is, because in this way it could withstand the wind, and thus large displacements. When you compare the stalk with a blade of a wind turbine as shown in Fig. 1.1b they have some structural similarities: Wind turbine blades are slender and have to be light and thus flexible, thin-walled, and hollow. Both structures underly large displacements caused by wind forces, where the cornstalk and the blade bend due to drag and lift force respectively. The structural difference to the stalk is, that the shape is not cylindrical, but rather a rectangular hollow beam encased with an airfoil shape. A wind turbine blade structure is slightly weaker than the cylindrical shell, but still relatively strong[28]. The metaphor shows, that engineers have reached a state of optimization in the compromise between load carrying and weight that was already created in nature. But this work reveals the design problems that engineers encounter dealing with thin-walled structures.

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Fig. 1.1: Similar structures bent due to wind.

To describe the behavior structures under bending with computational methods, we have two opportunities: (1) linear analysis and (2) non-linear analysis. A linear analysis underlies the simplification that loading conditions in a blade are deformation independent. Imagine a wind gust loads the blade from a wind speed of 1 km/h to 100 km/h. To obtain the displacement of the structure at 100 km/h, the displacement of the structure at 1 km/h is simply multiplied by a factor of 100. That is, the deformation history between both considered conditions plays no role. Whereas for a non-linear analysis, the deformation history plays a major role. The shape or the stiffness of the blade at 100 km/h wind speed may be different to 1 km/h wind speed. In other words: In a non-linear analysis the deformation history as function of load is split into small load steps between both wind speeds. If the non-linearity is caused by the change of structural shape, the behavior is defined as geometric non-linearity. In structural engineering, there is a rule of thumb that suggests the requirement of a non-linear analysis if the displacements are larger than 1/20th of its largest dimension. This rule of thumb is tested for wind turbine blades in this work with both described analysis methods and leads to the first formulation of a question:

1. Is there a need of a geometrically non-linear analysis method in wind turbine blade design?

To get a building permit for the wind turbine, a certificate for its design and thus for the blade is required. The certificate is delivered, when the design requirements due to standards and guidelines used in wind energy are fulfilled. One such guideline is published by Germanischer Lloyd (GL)[13], that contains as lowest requirement a linear analysis for the verification of the ultimate limit states: strength and stability. A certification via geometrically non-linear approach is also possible, but not mandatory. Both ways will be investigated in this work, that leads to a second question that will be answered:

2. Does the linear GL approach reveal more conservative results than the geometrically non-linear GL approach?

Thin-walled structures as introduced above are highly sensitive to geometric imperfec- tions[32]. Since real blades that depart the manufacturer’s mold are never perfect, imper- fections have to be considered during analysis. Small dimples on the surface or variations in wall thickness may occur due to manufacturing uncertainties. Another standard used for the structural design of similar shaped structures like a stalk mentioned above, namely cylindrical shells made from steel, is Eurocode[6]. This standard is based on empiric investigations and years of experience with steel structures. Comparing the requirements for the application of imperfections in both, GL guideline and Eurocode, the second con- tains more detailed descriptions. Thus, a third question to be answered in this work can be stated:

3. How accurate is the GL approach to apply an imperfection compared to the approach in Eurocode?

Non-linear analysis are usually conducted as numerical finite element analysis. Compu- tational costs for linear analysis are relatively low, whereas costs for non-linear analysis are depending on the degree of detail and the size of the model. For a long time, this was an exclusion criterion for engineers to conduct non-linear analysis. But today, computational costs decreased and the computational power had risen. Computer clusters are available and affordable for everyone. Why should we not use the resources that are well devel- oped to benefit from greater understanding of design problems via non-linear finite element analysis?

In this work, a geometrically non-linear structural response of a 34 m wind turbine blade under flap-wise loading is compared with a linear analysis. Non-linear effects revealed from a numerical finite element model are compared with an analytical plate model of the cap under compression load. The plate model is subjected to a transversal load due to the non- linear Brazier effect[35], and to in-plane loading to reveal the local cap displacement and the buckling resistance respectively. Moreover, focus is on the verification of characteristic and design resistance in the ultimate strength and stability limit state. For the design resistance, I use an analysis due to the lowest GL requirements and compare the results with a non-linear approach proposed by GL that required the application of an imperfection to the model. As well I apply imperfections due to Eurocode that seem more realistic compared with GL’s approach.

The structure of this work is built up as follows. The second chapter state-of-the-art introduces the philosophy and terms of limit state design used in standards and gives an overview about essential analysis methods needed to understand the conducted verifications in this work. Further, GL’s material safety factors and the description of linear and non- linear analysis are cited from GL guideline. Chapter 3 gives the theoretical background, that is needed to understand the theory in this work. In chapter 4 the methods, that I used for numerical and analytical computations, are described. Further the finite element setup is presented. That is, the way how I applied the material safety factors for different analysis are explained, as well as the derivation of the analytical cap model. In chapter 5, the results of the analytical in comparison with numerical computations and two full-scale tests are presented, as well as the results of the verifications due to GL and Eurocode. Further, the influence of the imperfections is described. In chapter 6, the methods and the results are discussed and future investigations are proposed. Finally, the main conclusions are presented in chapter 7.

Numerous simulations reveal the requirement of non-linear analysis methods in design of wind turbine blades. A linear analysis due to the lowest GL requirements yields less conservative results than a non-linear approach proposed by GL. That leads to the fact, that the investigated blade designed after the lowest requirements would pass the certi- fication, whereas the same blade design verified with the non-linear approach would not. The non-linear structural response is significantly dependent on the scaling of an applied imperfection.

2 State-of-the-Art

This chapter describes the state-of-the-art of standards and guidelines regarding rotor blade design and certification. The design philosophy, the standards are based on, is introduced. Further, the focus is on the design process: The required analyses and the safety factors for analyses due to GL are introduced. In the last section a scheme is shown how to categorize different analysis methods due to Eurocode.

2.1 Standards and Guidelines in Wind Energy

The Technical Committee 88 (TC88) of the International Electrotechnical Commission (IEC) has published a compendium of wind energy specific standards, that relate to different sectors in wind energy. Besides IEC standards, certification bodies such as Det Norske Veritas (DNV)[9]and Germanischer Lloyd (GL)[13], and Danish Energy Agency (DEA)[8], publish their own guidelines that give details for the structural design of wind turbines and wind turbine blades. Further, there is a joint effort between DNV and Risø DTU (since 2012 DTU Wind Energy) that publishes another guideline called Guidelines for Design of Wind Turbines[11].

The fundamental elements of these guidelines - that must be followed to attain certifica- tion - are based on the IEC standards, but they describe the design and analysis procedures for the components of a wind turbine in more detail[12]. A certification scheme is defined as a consistent system of a guideline together with its references to standards. The man- ufacturer decides, which scheme is taken for a wind turbine, because these may differ in safety factors. However, the manufacturer has to select one scheme to follow and is not al- lowed to merge them.[12]Also a scheme in different nations has to follow different national requirements.

In general a certificate indicates that a product is in conformity with defined requirements. A certificate for a wind turbine is required for several reasons, e.g. to get a building permit, to be approved for credit, and also for the insurance of the wind turbine. The certification bodies are third parties separate from the turbine manufacturer and the purchaser.

In this work, the scheme of the GL Guideline for the Certification of Wind Turbines[13]will be followed for verification, because the considered blade (SSP 34 m) has been certified according to that scheme.[14]

2.2 Design Philosophy

In structural engineering there are two design philosophies existing: (1) ’Permissible or Al- lowable Stress Design’ (PSD/ASD) and (2) ’Limit State Design’ (LSD) also called ’Load and Resistance Factor Design’ (LRFD). Where PSD is based on a simple safety factor between load and resistance, LSD is based on probabilistic calculation methods. The structural reliability plays a central role and is calculated for several limit states of the structure.[24]

The IEC 61400 standards and guidelines are based on LSD such as described in the Eurocode or the ISO 2394[17](General Principles on Reliability for Structures). The scheme of Eurocode or ISO 2394 can not always be directly applied on the GL Guidelines [13].

For each component of the wind turbine the ultimate, fatigue¹, and serviceability limit state must be analyzed. Therefore, the internal loads, such as bending moments and shear forces, acting at one component are considered. There are partial safety factors defined on the load side and on the material side that account for uncertainties. Finally, the uncertainties in the analysis methods are taken into account.

The partial safety factor for the internal loads γF accounts for possible unfavorable deviations of the load and uncertainties in the loading model.[16]Multiplying γF with the characteristic internal load value from calculations Fk yields the design value Fd:

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The partial safety factor for materials γm accounts for unfavorable deviations of the strength, possible inaccurate assessment of the resistance of sections or load-carrying ca- pacity of parts of the structure, uncertainties in: (1) the geometrical parameters, (2) in the relation between the material properties in the structure and those measured by tests on control specimens, and (3) in conversion factors.[16]Dividing characteristic material properties such as material strength or moduli by γM yields the design property fd:

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A partial safety factor for the consequence of failure γn that depends on the component class is defined. This factor distinguishes between different component classes from 1 to 3. A class 1 component is used for ’fail-safe’ structural components whose failure does not result in a failure of a major part of the wind turbine, for example replaceable bearings. The class 2 represents ’non fail-safe’ structural components - like rotor blades - whose failure may lead to the failure of a major part. Class 3 components are defined as ’non fail-safe’ mechanical components that link directly to actors of the safety system. For verification the internal load and resistance are compared in the same quantity, such as force, strain, or stress. For this, to achieve comparison internal load and resistance functions are defined. Verification is successful when the internal load function S yields a lower or equal value than the resistance function R:

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where the index d stands for the design point.

A principle diagram that illustrates the correspondence between the above derived internal load and resistance functions is shown in Fig. 2.1. Since internal loads as well as resistance are subjected to statistical uncertainty, the frequency density distributions fS and fR of load and resistance are shown respectively. The mean values are indicated by the index m. Considering the 95% quantile of the load function yields the characteristic load Sk, and the 5% quantile of the resistance yields the characteristic resistance Rk. When the safety factors are applied as described above, we obtain the design load Sd and the design resistance Rd. The design goal is to meet both at one point (Sd = Rd). However, if there is a deviation between both we have a safety reserve.

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Fig. 2.1: Frequency density distribution of load and resistance functions.

For a given design two indicators for ’load carrying capacity’ can be determined: (1) safety reserve factor and (2) margin of safety. The safety reserve factor is defined as ratio between design resistance and design load:

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When we consider the external load history of the external load L, the external design load is defined as [illustration not visible in this excerpt] and the external failing load, where the structure actually fails is considered as [illustration not visible in this excerpt]. In other words: The external failing load is reached, when the internal load F reaches the design resistance [illustration not visible in this excerpt] or [illustration not visible in this excerpt]. Hence the margin of safety is defined as:

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2.3 Design Process

2.3.1 Process Description

A principle overview of the rotor blade design process is given in Fig. 2.2. In this thesis the process steps in bold boxes are treated. An aeorelastic simulation reveals the loads that are considered for the different analyses in different limit states. The full-scale blade test is performed as static test and as fatigue blade test. A verification of the analysis in the sense of Eq. 2.3 is conducted.

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Fig. 2.2: Design process for wind turbine rotor blades.

2.3.2 Partial Safety Factors due to GL

In accordance to the GL scheme the partial safety factors for strength and stability for fiber reinforced plastics (FRP) are considered. The safety factors for the loads are shown in Tab. 2.1. They depend on the design situation and on the design load case (DLC) that is a combination of different impacts.

Tab. 2.1: Partial safety factors for the loads according to IEC[16].

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The material safety factors are defined in general as:

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According to GL Guideline the indices for strength analysis are x = a and x = c for stability analysis. The partial material safety factors [illustration not visible in this excerpt] are shown in Tab. 2.2. Further, it is given what uncertainty they take into account. The factors are to be applied on the characteristic material strength.

Tab. 2.2: Safety factors for strength analysis according to GL[13].

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Example: A material safety factor for the strength of a post-cured laminate yields with Eq. 2.6:

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The partial safety factors for stability analysis and the considered aspect are shown in Tab. 2.3. They are to be applied on the mean values of the material stiffness. The verification of the stability resistance can be provided in form of strain or stress analysis.

Tab. 2.3: Safety factors for stability analysis according to GL[13].

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Since design codes for the design of rotor blades are available, the partial safety factor for the consequence of failure for a component class 2 - such as a wind turbine blade - is defined according to IEC[16]as:

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The external certification load for the ultimate test is obtained by:

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where [illustration not visible in this excerpt] and [illustration not visible in this excerpt] can be taken from Tab. 2.4. The values for [illustration not visible in this excerpt] between 20° C and −30° C may be interpolated linearly.

Tab. 2.4: Safety factors for test load according to GL[13].

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2.3.3 Analysis due to GL

This section describes the analysis requirements in accordance with GL guideline.

The ultimate strength and stability analyses can be conducted as a linear static analysis (LA) and linear bifurcation analysis (LBA) respectively. As the failure index the Tsai-Wu failure criterion can be used together with the maximum stress criterion.[36]The stability analysis can be conducted in terms of a geometrically non-linear analysis with imperfections (GNIA). Therefore, the following recommendations have to be considered:

- The first critical linear buckling mode has to be applied globally to the perfect model as predeformation.
- The maximum buckle height of the modes’ critical buckle is scaled by a dimple tolerance factor U0 = 1/400 of its largest horizontal dimension. A smaller predeformation is permitted, if the height is verified.
- This predeformed model has to undergo a fibre failure analysis.
- A material safety factor [illustration not visible in this excerpt] has to be applied on the material strength.
- The partial material safety factor for stability [illustration not visible in this excerpt] may be applied either on the load or on the material stiffness.
- Additionally a linear bifurcation analysis (LBA) is required to verify that the first bifurcation load is larger than the characteristic load.

2.4 Analysis Methods due to Eurocode

Since the GL guideline is based on Eurocode series - a collection of codes for design of steel structures - excerpts are presented here. A General overview of analysis methods according to[6]is given in Tab. 2.5. With each row the calculation cost rises because of the increasing degree of detail. The methods this thesis focuses on are described in more detail below. Moreover, the way Eurocode 3 proposes to apply imperfections is shown.

Tab. 2.5: Analysis Methods according to Eurocode[6].

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Linear Analysis (LA) The structural behavior is described on the basis of linear bending and stretching with small displacement applied to the perfect geometry of the shell mid surface. Its linearity follows from the assumptions of linear-elastic material law and small linear displacements. The theory of small displacements implies that the geometry under load is identical to the undeformed structure.[29]

Linear Bifurcation Analysis (LBA) LBA corresponds to the solution of an eigenvalue problem for a thin-walled shell structure on the basis of linear bending and stretching with small displacement applied to the perfect geometry of the shell mid surface. Results are eigenmodes and eigenvalues. This analysis is used to calculate the linear elastic buckling resistance.[29]

Geometrically Non-linear Analysis (GNA) Herein the structural behavior is described as in LA. But, instead of the assumption of small linear deflections, a non-linear theory of large displacements is used. Thereby, changes in geometry - induced by external loads - are embraced. Characteristic is the use of GNA buckling analysis, i.e. the determination of the bifurcation load, along the non-linear load-displacement path (GNA+LBA) or in other words at each load level increment of the load history.[29]If pressure loads are dominant in certain parts of the shell, GNA provides the critical buckling load of the perfect structure.[6]

Geometrically Non-linear Analysis with Imperfections (GNIA) This type of analysis is equal to a GNA. But two different kinds of imperfections are added: (1) Geometrical imperfections such as shape and thickness uncertainties, load and support eccentricity.

(2) Structural imperfections, such as milling, forming, straighting and welding internal stresses and material irregularities.[28]As in GNA, a geometrically non-linear analysis with imperfections combined with a linear bifurcation analysis (GNIA+LBA) is also possible.

MNA, GMNA and GMNIA Tab. 2.5 shows further forms of analysis. These are combinations of the explained analysis from above with material non-linearity.[29]These are not considered in this work.

Imperfections Imperfections can be applied as global predeformation affine to the first critical eigenmode due to a LBA. Imperfections may be excluded when they are unrealistic due to manufacturing. As well, a simple dimple can be applied as imperfection. The amplitude of the dimple [Abbildung in dieser Leseprobe nicht enthalten] can be derived in two ways:

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where [Abbildung in dieser Leseprobe nicht enthalten] is the relevant length of the dimple, [Abbildung in dieser Leseprobe nicht enthalten] is the dimple tolerance factor, [Abbildung in dieser Leseprobe nicht enthalten] is factor to reach an appropriate tolerance level, and t is the thickness of the shell.

[...]

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¹ Fatigue is sometimes treated as separate type of limit state.[11]