Searching for an improvement in fatigue assessment to increase lifetime expectancy - Het zoeken naar verbeteringen in de vermoeiing berekening om een langer levensduur te berekenen

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Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Maritime and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

This report consists of 99 pages and 5 appendices. It may only be reproduced literally and as a whole. For commercial purposes only with written authorization of Delft University of Technology. Requests for consult are only taken into consideration under the condition that the applicant denies all legal rights on liabilities concerning

Specialization: Transport Engineering and Logistics

Report number: 2014.TEL.7911

Title:

Searching for an improvement in

fatigue assessment to increase

lifetime expectancy

Author:

Richard Karssen

Title (in Dutch) Het zoeken naar verbeteringen in de vermoeiing berekening om een langer levensduur te berekenen

Assignment: Master thesis Confidential: yes

Initiator (university): Prof. dr. ir. G. Lodewijks Initiator (company): ir. M. de Jongh

Supervisor: ir. W. van den Bos

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Delft University of Technology

FACULTY OF MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

Student: R. Karssen Assignment type: Master thesis

Supervisor (TUD): Ir. W. van den Bos Creditpoints (EC): 35 Supervisor (VIRO) Ir. M. de Jongh Specialization: TEL

Report number: 2014.TEL.7911 Confidential: yes

Subject:

Searching for an improvement in fatigue assessment to increase

lifetime expectancy

VIRO is a Dutch engineering company focusing on the mechanical analyses of steel structures, for instance in offshore- and amusement ride engineering. Since most steel structures are welded and endure long lifetimes, fatigue has become an important design parameter. The current calculated fatigue life expectancy according the European standard; Eurocode 3 is considered to be short. Therefore, the aim of this research is to improve the fatigue assessment and exclude any conservative assumptions.

The intention is to increase life expectancy by improving the assessment, without changing current design of structures. This research is divided in three parts. The first part is based on stress criteria; where the goal is to decrease the stress (range) used for analyses. The second part is fatigue evaluation method based, where the goal is to find higher allowable stresses using different fatigue evaluation methods. In the third part, four design codes are considered where different approaches are compared to find methods that increase lifetime expectancy. Together, improvements from these three parts will result in a longer life expectancy, as is shown in a test case in the final chapter.

The report should comply with the guidelines of the section. Details can be found on the website.

The professor,

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Master Thesis Richard Karssen

Preface

This thesis will represent my graduation research in the field of fatigue engineering, fulfilled at VIRO B.V. Schiedam. It is the result of a life-long enthusiasm for amusement rides, combined with knowledge of mechanical engineering acquired at Technical University Delft and VIRO.

Fatigue is a general phenomenon in engineering and finds a great field of application in the amusement ride industry. Amusement rides endure high loads and long lifetimes, and require similar specialization as cranes and other lifting equipment which is a specialty at the Transportation Engineering department at the Technical University Delft. For the knowledge I have gained here, I would like to thank Prof. dr. ir. G. Lodewijks and ir. W. van den Bos from the department of Transportation Engineering for their guidance and supervision.

Furthermore, I would like to thank the team of engineers at VIRO for their time spend teaching me during this graduation. Especially ir. M. de Jongh, who made this graduation research possible, and for being the person for my daily supervision during my graduation at VIRO.

Richard Karssen, December 2014

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Searching for an improvement in fatigue assessment to increase lifetime expectancy

Summary

Fatigue is a serious design parameter in general engineering. It arises in many materials and applications, yet the combination with welding’s has become a critical point of interest. For this, many researches have been performed which has led to a good understanding of fatigue and resulted in the formation of methods and codes. The current method has resulted in short fatigue life expectancies and the wish of VIRO is to investigate if this lifetime can be increased by a better understanding of the fatigue assessment. Therefore, this research was initiated. To aid in finding any improvement, this research is divided into three parts: stress criteria, evaluation methods and fatigue codes.

Research

The first part focusses on the stress criteria that is used for the fatigue assessment. Multiaxial stress criteria such as the Von Mises equivalent stress and the (maximum) principal stress are often used to find the stress in complex geometries. Questions as which of these criteria should be used and how to account for a stress range, arise and answers are desirable.

Secondly, an introduction into three popular fatigue evaluation methods is presented. These evaluation methods are mainly used to calculate the fatigue life of welded structures. The methods under scope are; the Nominal Stress Method, the Hot Spot Stress Method and the Effective Notch Stress Method. Each method requires more effort and the idea is that each method results into a longer lifetime as effect of increased accuracy. Yet proof of this idea is desired which will be given, using experimental data from the test case in the last chapter.

The third part of this research focusses on four design codes that are relevant for the fatigue assessment of amusement rides. The documents under scope are: 1) Eurocode 3 for fatigue, 2) the IIW guidelines for fatigue, 3) NEN-EN 13001 standard for lifting equipment and 4) the DNV guidelines for fatigue. The documents share similar approaches for fatigue, for instance the introduced fatigue evaluation methods. Yet small differences exist which are investigated, for example the values of different welding categories and the calculation of S-N curves which is code dependent. The aim is to find which code (guideline or standard) gives the longest lifetimes.

Applying improvements

The three investigated parts are examined in the last chapter of this thesis, where a test case is performed, implementing the improvements on an amusement ride construction using finite element analyses:

Firstly, the principal stress criterion is to be used for the fatigue assessment which gives the correct stress, in contrast with the equivalent (Von Mises) stress where the latter lead to unreliable results. In

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addition, the calculation of the correct stress range between two load cases is given, which rules out a conservative approach when the principal stress directions are not in-line.

Secondly, the Nominal Stress Method and the Hot Spot Stress Method are used solely. The Effective Notch Stress Method should be a promising method, yet not all codes recognize this method. Secondly, it requires increased engineering skills to create a highly detailed FEM model, increasing calculation effort for large structural models. Therefore only the Nominal Stress Method and Hot Spot Stress Method are used for the experimental test case. The fact is that the Hot Spot Stress Method is a more accurate method compared with the Nominal Stress Method, and should therefore result into longer lifetimes. Yet it appears from experimental data that this is not always the case. When assuming that both methods use the same stress input which results from FEM (structural stress), the FAT-values for an improved welding category according to the Nominal Stress Method easily reaches values higher than those for the Hot Spot Stress Method. Nominal Stress Methods FAT-values easily reach FAT 125 where those of the Hot Spot Stress mainly reach FAT 100. The input stress for the two evaluation methods differ significantly in theory; in practice the structural stress is found using FEM, instead the nominal stress which is normally found using analytical (hand) calculations. An advantage of the Hot Spot Stress Method is the possibility of stress extrapolation. This is a feature that decreases the found stress with an average of 6% (according the test case) which has beneficial effect on the lifetime expectancy.

Thirdly, the differences between the four codes are found to be significant. Though it was assumed that Eurocode 3 was the most conservative code; from the test case it appeared that this was not the case. Basically, the lifetime expectancy given by this code is in certain cases larger than calculated using others codes. The reason of this remarkable conclusion depends on a combination of aspects, including; FAT-values, evaluation methods, calculation of the S-N curve and the allowable Palmgren-Miner summation. Where other codes fail at one of these aspects, Eurocode 3 does not.

Conclusion

The conclusion that follows from the test case is that it is possible to increase the lifetime expectancy of the amusement ride. Unfortunately, the increase is not large enough to meet the demanded lifetime that is required for certification. The reason is that there are flaws in the mechanical design of the (welded) structure which remain critical, even when the welding in question is improved with post-weld treatment. The increase in lifetime expectancy from the test case for Eurocode 3 and IIW is 44% and 46% which is significant, yet the default fatigue life is 8,5 and 4 years respectively. The estimated lifetime needed for certification is 21 years, therefore is 44% and 46% increase not sufficient.

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Samenvatting

Vermoeiing is een belangrijk uitgangspunt voor vele constructies. Het treedt op in vele materialen en de combinatie met gelaste constructies is zeer kritisch. Om deze reden is er veel onderzoek gedaan naar vermoeiing, wat heeft geleidt tot eenduidige methodes en standaarden. De huidige methode heeft als resultaat dat de levensverwachting te laag is en de wens vanuit VIRO is een onderzoek naar een verbeterde vermoeiingsberekening. Hieruit is dit onderzoek voort gekomen, welke is onderverdeeld in drie onderdelen: Stress criteria, evaluatie methodes en vermoeiing codes.

Onderzoek

Het eerste gedeelte richt zich op de stress criteria die geldt voor de vermoeiingsberekening. Aangezien multi-axiale spanning criteria zoals de Von Mises vergelijk spanning en de (maximale) hoofdspanning vaak gebruikt worden, is er een doorslag nodig in het te gebruiken type spanning. Welke van deze de criteria correct is, en hoe omgegaan moet worden met een spanningsbereik (range), zijn vragen die beantwoord moeten worden.

Het tweede gedeelte introduceert drie, voor de vermoeiingsberekening vaak gebruikte evaluatie methodes. Deze evaluatie methodes worden in vele codes en standaarden veel gebruikt wanneer gelaste constructies doorgerekend moeten worden. Het gaat hier om de Nominaal Spanning Methode, de Hot Spot Spanning Methode en de Effectieve Kerf Spanning Methode. Deze methodes verschillen in nauwkeurigheid en het idee is dat iedere methode een langere levensduur oplevert, maar hiervoor is bewijs nodig dat in het laatste hoofdstuk gegeven wordt.

Het derde gedeelte van dit onderzoek richt zich op vier codes die relevant zijn voor de vermoeiingsberekening van attracties. Deze zijn: 1) de Eurocode 3 voor vermoeiing, 2) de IIW richtlijnen voor vermoeiing, 3) NEN-EN 13001 standaard voor hijswerktuigen en 4) de DNV richtlijnen voor vermoeiing. Elk document levert een vergelijkbare aanpak voor vermoeiing (zoals de geïntroduceerde evaluatie methodes) maar bevatten ook kleine verschillen, zoals verschillende waarden voor las categorieën en de berekening van de S-N lijnen. Het doel is om te onderzoeken of één van deze codes een langere levensduur dan andere codes bepaald.

Toepassen van verbeteringen

Samen worden de drie onderdelen gebruikt in een testcase in het laatste hoofdstuk van dit onderzoek. Hier wordt met de hulp van de eindige elementen methode de verbeteringen uitgevoerd op een attractie.

Ten eerste, uit literatuur en data blijkt dat de hoofspanningen gebruikt horen te worden voor de vermoeiingsberekening. Dit in tegenstelling tot de vergelijkspanning, welke een onbetrouwbaar uitkomst kan geven. Daarnaast wordt er een correcte berekening van de spanningsbereik gegeven

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welke er voor zorgt dat er geen conservatief bereik wordt meegenomen indien de hoofdrichtingen niet in elkaars verlengde liggen.

Ten tweede, alleen de Nominaal Spanning Methode en de Hot Spot Spanning Methode zijn gebruikt voor dit onderzoek. De Effectieve Kerf Spanning Methode zou een veelbelovende methode zijn maar niet alle codes erkennen deze methode en deze vraagt om een uitgebreid FEM model wat leidt tot veel rekentijd voor grotere constructies. Het idee van de overige twee methodes is dat de Hot Spot Spanning Methode nauwkeuriger is en daardoor een langere levensduur zou kunnen berekenen. Maar uit de test blijkt dat dit niet altijd het geval is. Er van uitgaande dat wanneer beide methodes dezelfde ingaande spanning gebruiken (wat theoretisch niet zo is), kan de FAT-waarde voor sommige lassen hoger uitkomen bij de Nominaal Spanning Methode dan bij de Hot Spot Spanning Methode. Hier is dan direct een nadeel te zien van de Nominaal Spanning Methode. De spanning die deze methode gebruikt, dient eigenlijk de hand berekenende nominaal spanning te zijn, maar in de praktijk is het de structurele spanning uit het eindige elementen model dat gebruikt wordt. Duidelijk verschil is dus niet aan te tonen. Ten slotte, een groot voordeel van de Hot Spot Spanning Methode is dat deze gebruik kan maken van extrapolatie van de structurele spanning. Hierdoor kan de piek spanning buiten de berekening gehouden worden en wordt in dit onderzoek gemiddeld 6% lagere spanningen gevonden.

Ten derde, de gevonden, onderlinge verschillen tussen de codes blijken significant. Hoewel oorspronkelijk er van uit is gegaan dat de Eurocode 3 de kortste levensduur berekend, blijkt uit de test dat deze juist een lange levensduur geeft, vaak langer dan andere codes. De oorzaken hier achter zijn terug te vinden in de las categorieën van de evaluatie methodes, de berekening van de S-N lijnen en de Palmgren-Miner summatie. Alle codes presteren slecht op een enkel punt en verlagen daardoor de levensduur verwachting, behalve voor Eurocode 3.

Conclusie

De conclusie is dat het inderdaad mogelijk is om de levensverwachting te vergroten voor vermoeiing, getest op de attractie uit de test case. Maar deze vergroting is helaas niet in staat een dusdanig levensverwachting te behalen die nodig is voor certificatie. De reden is dat er door slecht ontwerp, locaties in de constructie zitten waardoor hoge spanningen ontstaan, ook al neemt men de mogelijkheid van lasverbetering mee. De verlenging van levensduur door lasverbetering levert voor Eurocode 3 en de IIW een verlenging van respectievelijk 44% en 46% op. Dit verschil is significant, maar doordat de beginwaarde van de levensverwachting voor iedere code slechts 8,5 en 4 jaar is, levert dit een kleine vooruitgang op. De benodigde levensverwachting voor een certificatie is 21 jaar en de verbetering van 44% en 46% is niet voldoende.

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Nomenclature

Symbol Description Unit

∆𝜎

𝑟𝑎𝑛𝑔𝑒 Stress range measured [MPa]

∆𝜎

_𝐹𝐴𝑇 Stress range at N=2·106 _[MPa]

𝑛

_𝑖 Occurring life N for stress range 1 [N]

𝑛

̅

_𝑖 Allowable life N for stress range 1 [N]

m Slope coefficient -

𝜏

Shear [MPa]

𝜎

Stress [MPa] F force [Newton] A Area [m/s²]

[𝜎]

Stress tensor -

𝜎

𝑣𝑀 Von Mises Equivalent stress [MPa]

𝜎

𝑚𝑎𝑥𝑝𝑟𝑖𝑛 (Maximum) Principal stress [MPa]

𝜎

𝑝1 Eigen value [MPa]

𝑛

𝑝1

⃑⃑⃑⃑⃑⃑

Eigen vector -

𝛼

Angle of principal direction -

λ Biaxiality ratio -

∆𝜎

Stress range [MPa]

𝜎

⊥ Stress perpendicular [MPa]

𝜎

⫽ Stress parallel [MPa]

𝜏

_⫽ Shear parallel [MPa]

a Acceleration [m/s²]

ω Rotational velocity [rad/s]

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List of abbreviations

Abbreviation Composition

S

Stress

N

Life

D

Damage parameter

FAT-value

Fatigue value at 2·10

6

_N

FAT-class

Welding category

FEM

Finite Element Method

S-N curve

Stress-Life curve

FW

Fillet weld

PP

Partial penetration weld

FP

Full penetration weld

NSM

Nominal Stress Method

HSSM

Hot Spot Stress Method

ENSM

Effective Notch Stress Method

EC3

Eurocode 3

IIW

International Institute for Welding

DNV

Det Norske Veritas

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1. Introduction

1.1 Introduction to this research

This thesis focusses on the fatigue phenomenon which is a serious design parameter in transportation engineering. Mechanical constructions that witness many large stress fluctuations such as bridges, excavators and cranes suffer from fatigue and failure must be prevented. Research into this phenomenon has led to a better understanding and (inter) national codes have been published that should guarantee a correct fatigue assessment. Yet there is such a large amount of codes and fatigue assessment methods that the question rises which of these methods and codes give a reliable fatigue life expectation for a specific field of technology.

VIRO is a Dutch engineering company that witnesses fatigue assessment in the design and analyses of mechanical constructions. A major role is found in offshore engineering constructions though amusement rides are frequently assessed constructions as well. This variety of projects has resulted that VIRO became familiar with many international codes that focus on offshore engineering. The fatigue assessment of amusement rides is prescribed to follow Eurocode (specifically NEN-EN1993-1-9). The methods used for fatigue assessment in this standard are comparable on the findings of the International Institute of Welding (IIW) [1]. IIW has published its own guidelines for the assessment for fatigue as well, and it can be said that this latter is significantly more enhanced than Eurocode 3. Its approach for a fatigue assessment consists of many (extra) methods and the collection of methods differs per code as well.

Amusement ride projects of VIRO require a fatigue assessment for already-built and operating rides. This assessment is necessary for certification according newer editions of standards. It frequently appears that fatigue is a significant factor in the product lifetime expectancy calculations. The reason is that with the current fatigue assessment, short lifetime are calculated and the idea is that the current method is simply conservative. This is a problem for VIRO since it cannot deliver the lifetime expectancy clients demand for their constructions.

1.2 Goal of this research

The fatigue assessment plays a great role in the engineering process because of the combination of large stresses and a high number of stress cycles. It is the wish of VIRO to have a better understanding in the fatigue assessment since this frequently leads to a decrease in lifetime expectancy. Often, the fatigue life expectancy according to calculations based on Eurocode 3 is not sufficient and the operational lifetime of a construction is reduced significantly. VIRO wants to know the origin of this low outcome and would like to see what the effect is when other methods in combination with other codes are used. The goal is to find methods that increase lifetime expectancy based on the assessment, such

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that no alterations in the already-built constructions are necessary and still a longer lifetime is to be expected.

1.3 Structure of this research

The described goal is approached by dividing this research into three parts.

The first part focusses on the type of stress (range) which is accountable for the fatigue assessment. Since amusement rides hardly follow rectangular shapes, finding the correct stress direction with FEM software becomes complex. Nominal component stresses in x- and y-direction are not so easily found and the stress cycles are not always ‘in the same direction’. Therefore, a multiaxial stress state is assumed, leading into errors while determining correct stress range. Certain stress criteria are considered and applicability will be tested. The goal is to find methods that can be used in a multiaxial stress state that delivers a correct stress range that can be used in fatigue assessment. With a correct approach for finding the correct stress range, no conservative assumptions have to be made.

The second part focusses on the fatigue evaluations that are frequently used in codes. Three evaluation methods are considered; the Nominal Stress Method, the Structural Stress Method and the Effective Notch Stress Method. These methods are primarily based on welded structures, since fatigue damage is often found to originate in welds. The assumption is that one evaluation method is more accurate than the other, and that the more accurate method calculates a longer lifetime. The goal is to find if this is true and basically: which evaluation method gives the longest lifetime expectancy.

The third part focusses on four fatigue assessment codes. The Eurocode for steel, the IIW recommendations for fatigue, crane standard NEN-EN 13001 and the recommended practice for fatigue by DNV. The assumption is that the Eurocode for steel has a conservative approach and the effect of this will be compared with other codes. Subjects are; the fatigue evaluation methods; post weld improvement methods and fatigue life calculations. The goal is to find methods in the fatigue assessment that increase life expectancy by using arguments that are found by comparisons.

The decision to divide this research into three parts is because this gives a simple yet understanding idea of the synergy between these parts. Interesting questions are based on two individual parts overlapping which creates typical issues between the two. For example; ‘which stress is used according which code’ is an issue of ‘stress criteria’ combined with the chapter on standards and guidelines. Figure 1.1 shows the dividing in a graphical manner. A combination of these three parts are used to create an improvement in the fatigue assessment for VIRO. The idea is that the stress criteria part will result into a decreased stress range, the best evaluation method is used and the highest values for resistance from a certain code are used. An exercise will be performed on an amusement ride in the final chapter. A

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Figure 1.1 The three parts of this research, including their main topics

1.4 Thesis boundaries

Because fatigue is a broad subject, certain boundaries must be set up to clearly distinct which parameters are considered. First of all, this thesis focusses on steel constructions such that aluminum, composites and high strength steel (yield ≥ 355 [MPa]) are not considered. In offshore and onshore engineering holds that the usage of high strength steel in not unique. Yet for fatigue assessment according Eurocode 3 holds that the steel type used should comply at least with NEN-EN 1993-1 [2].

Secondly, the determination of load cases in this thesis are predetermined using the Rain-flow method [3]. The stress spectrum ( based on actual measured data) is not considered for fatigue assessment. Even though the spectrum has great effect on fatigue assessment and would improve the calculation if this is predicted more accurately, it is not considered for this thesis. The reason is that there is no available data and the load history at cycles from N=107_{is never 100% accurate. According to} literature, 80% of fatigue induced failures occurred due to under-estimating fatigue load [4] and therefore a conservative stress spectrum is assumed.

Thirdly, the variety of safety factors and reliability is not considered in this report. This means that the stress values in the test case do not include safety factors. The reason for this is that these factors will disturb the abstract differences between codes since each code assumes other values.

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2. Introduction to fatigue

The basics of fatigue will be rehearsed in this chapter. Secondly, the effect of welding and fatigue is introduced. Thirdly, the application of using FEM software for fatigue assessment is introduced.

2.1 The fatigue phenomenon

Dynamically loaded materials do not have infinite life and will show cracks after being stressed a large number of times. This process is called fatigue, where microscopically small cracks are formed by numerous repetitions of stress cycles. These cracks will grow with each stress cycle until critical lengths are reached. Eventually, a structure will fail because the stiffness of the structure is completely lost due to that crack. The danger of fatigue is that fatigue cracks are not always clearly visible and failure can act suddenly, resulting in the collapse of an entire construction.

The danger of fatigue

Failure modes such as yielding and buckling are visible by the deformation of the material. Fatigue is less visible since it can occur inside materials (and welding’s) and the stress needed for cracks to grow does not has to be large: nominal stresses which are 50% lower than yield stress can be sufficient to create fatigue fracture. Yield stress is for many designs the limiting factor and not the ‘fatigue stress’. Therefore is the fatigue phenomenon often responsible for failure. Research by the UK Health and Safety Executive showed that 25% of all structural damages on offshore constructions on the North Sea from 1974 to 1992 is caused by fatigue [5]. Therefore it is important to find this ‘fatigue stress’.

Early research

In the 19th_{century German railway engineer August Wöhler noticed that steel suffers from aging and} investigated the number of stress repetitions needed before failure occurred. His findings where plotted and the graphs (originally called Wöhler curves) are known nowadays as the S-N curves (S = stress, N = cycles, see Figure 2.1) and are well-used in fatigue engineering. These curves show the relation between the allowable number of stress cycles for a certain stress range and can include properties such as stress concentrations, production process, type of material and welding configurations. Predefined welded constructions such as T-joints are assigned to certain S-N curves such that finding the nominal stress is sufficient to calculate lifetime expectancy.

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Figure 2.1 An example of S-N curve of UNS G41300 Steel [6]

Fatigue is nowadays a well-understood subject. Countless fatigue tests of all types of components have been performed and researched. However, a major problem in the outcome of test data is the statistical behavior of fatigue, since the number of repetitions are in the order of 106_{to 10}7_{. Deviations of 10%} easily mean one million stress cycles less (or more) before failure.

̅̅̅ = ∆𝜎

₁ _𝐹𝐴𝑇𝑚

∙ 2 × 10

6 _(2.1)

𝑛

₁

̅̅̅ = (

∆𝜎𝐹𝐴𝑇 ∆𝜎𝑟𝑎𝑛𝑔𝑒

)

𝑚

∙ 2 × 10

6 _(2.2)

The subscript ‘m’ is the value of the slope of the S-N curve and often 3 or 5. This number is also called the Basquin coefficient, named after the discoverer of the linear Log/Log relation of these curves [8]. Other calculation methods, for example using modification factors [6] are available yet most standards and guideline use this calculation principle, called the ‘Nominal Stress Method’.

A frequently used method for the calculation of variable stress is the Palmgren-Miner rule. This is a linear damage rule for bearings which was firstly proposed by Palmgren in 1924 and further developed for fatigue by Miner in 1945. The main principle of this rule is given in Figure 2.2 showing two load cases with certain stress ranges;

∆𝜎

1and

∆𝜎

2 and certain number of stress cycles,

𝑛

1and

𝑛

2. A S-N

curve with FAT-value of 100 is given, assuming that the stressed component is classified as FAT-class 100 [8], [9]. Using equation 2.1 and 2.2, a maximum number of cycles,

𝑛

̅

𝑖 and

𝑛

̅̅̅

2, is found for the two

occurring stress ranges.

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Figure 2.2 shows two stress ranges and their number of occurring cycles 𝑛𝑖 plotted on the horizontal axis, showing the allowable number of cycles

𝑛

̅

𝑖 as well. The fraction of ‘occurring’ divided by

‘allowable’, 𝑛1

𝑛1

̅̅̅̅, for each stress range is calculated and summed up according to the Palmgren-Miner

Damage summation [8]:

𝐷

_𝑀

= ∑

𝑛1 𝑛1 ̅̅̅̅

=

𝑛1 𝑛1 ̅̅̅̅

+

𝑛2 𝑛2 ̅̅̅̅

≤ 1

(2.3)

The sum is supposed to be equal or lower than unity (=1) and deviating values can be given by standards and guidelines. Some opinions state that this rule is inaccurate and both longer and shorter lifetimes are found using the Palmgren-Miner summation during tests [8]. Research is performed trying to find more reliable or accurate methods that also accounts for instance mean stress effects and cycling hardening. This phenomenon increases fatigue life by overloading material in front of a crack with compressive stress which is the result of a sudden large stress cycle [8].

2.3 Fatigue in welded constructions

Fatigue in welding’s are covered in chapter four yet a short introduction is desirable. Though fatigue cracking also appears in non-welded materials, it frequently occurs at welding’s. A combination of stress concentrations, weld irregularities, residual mean stress and poor manufacturing contribute greatly to the growth of fatigue cracks. In section 4.1 is explained how these effect the weldments. Also, three well-known fatigue evaluation methods are treated. These are applicable for bolted or non-welded material as well, though the major role in these methods is dedicated to weldments.

Cracks at welding’s originate mainly in two distinct locations; the weld toe and the weld root. The weld toe is the most common ‘hot spot’, induced by a combination of stress concentrations and residual tensile stress. Cracks starting at the root of a weld occur less frequent but are hazardous since they are invisible for the naked eye and only visible when the crack has grown through the weld into the surface.

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Three main weld types are distinguished, having different effect on fatigue resistances;

 Filled weld (FW)

 Partial penetrating weld (PP)

 Full penetrating weld (FP)

Since fillet welds and partial penetration welds do not penetrate completely, there is always a location (the root) where there is a lack of fusion. Fillet welds are simple and hardly ever need preparation of the parent material, where partial and full penetrating welds need preparation such as beveling. For fatigue resistance holds that full penetrating welding’s have the preference, since weld toe cracking can easily treated (see section 5.2) which is not the case for weld-root failure. According to Nieme et al (2006), the variation in geometry of the weld toe is the main reason for scatter in fatigue test results [11]. Post-weld treatment can exclude the low reliability and therefore longer lifetimes can be expected and calculated. Benchmarking these welds and their improvements can be found in section 5.3 as well.

2.4 Using FEM in the fatigue assessment

A great tool in general mechanical engineering is found in the finite element method (FEM). A finite element analysis (FEA) helps the engineer by solving constructions which are time consuming when solved using analytical methods (hand calculations). Many commercial software packages such as ANSYS, FEMAP and NX are available.

The principle of FEM is that a structure is simulated using a finite number of elements assorted in a mesh. Considering the analysis, these elements can be one dimensional (line element, axles, beams, shafts), two dimensional (plates, shells) and three dimensional (solids). For fatigue assessment all three type of elements can be used. Even 1D elements can be used because if the moment of inertia and cross section are known, the membrane and bending stresses can be found, resulting in nominal stress. The fatigue assessment based on nominal stress (section 4.2) suffices using this membrane and

bending stresses, with the condition that load direction is mainly parallel or perpendicular and details of the weld configuration is found in a weld atlas (FAT-class or S-N curves).

In Figure 2.4 is a T-joint shown, made from 2D and 3D elements. When local stress instead of nominal stress is to be measured, for instance at the weld seam between the two plates in perpendicular direction, the 1D model does not satisfies anymore. The Hot Spot Stress Method and Effective Notch Stress Method (both treated in section 4.2) can be used for measuring the stress in all three directions. Calculating the stress perpendicular to the weld toe via these two methods is considered to be a more accurate calculation process and the idea is that the fatigue life found is larger than found using the Nominal Stress Method.

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Figure 2.4 Welding of two plates, modeled as 2D elements (left) and 3D solids (right) [12]

A difficulty can lie in the situation when the mesh does not coincide with the coordinate system of the weld. At oblique, cylindrical or circular constructions, the coordination of the mesh does not always coincides with that of the weld or the global coordinate system. Therefore two options are possible.

The first option is to adapt the mesh such that the coordination of the mesh coincides with a coordinate system, yet manually meshing can be a time consuming operation. Another option is possible, by changing the coordinate system of the mesh by introducing a native (or local) coordinate system. In this case is the Cartesian coordinate system of an element is replaced with a predefined coordinate system. This allows to locally apply a coordinate system that coincides with the direction of a load (or stress) or any preferred direction (the seam of a weld). Depending on the model this can still be time consuming yet automated functions such as the Weld Finder function by SDC Verifier [13] for FEMAP allows to create a local coordinate system that is transverse to a weld. Creating a Material coordinate system in NX allows to (manually) created local coordinate systems which has similar effect.

The second option is to accept that the component stresses (x, y and z or; parallel, perpendicular and transverse) do not coincide with a coordinate system and to use a multiaxial stress criterion that combines two or three component stresses plus shear stress. FEM software is able to give results in component stress directions (x, y and z) yet multiaxial criteria is included as well. Two well-known multiaxial criteria are Von Mises’ equivalent stress and Rankine’s principal stress, treated in chapter three.

Creating a good mesh is a specialty which not only applies for fatigue engineering but for engineering in general. The experience of an engineer using the finite element methods has direct effect on the quality of the analysis. Good workmanship in FEM is an expertise: without reliable FEM-results the fatigue assessment is useless.

This tensor can be simplified if an element is considered thin, as in a plate or plain. In this case, the tensor will be simplified to the following two dimensional version:

[𝜎] = |

𝜎

_𝑦𝑦

|

(3.3)

This stress tensor is a convenient method to represent the stress and shear for a given element. In the case of simple configurations, the stresses are found via an analytical calculation. When using FEM, the stress tensor is retrievable as well and can be altered from the global Cartesian coordinate system to any coordinate system.

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𝜎

0

𝑥𝑥

0 0

0 0 0

|

(3.4)

In linear elastic engineering this stress is limited by the yield limit (|𝜎𝑥𝑥| ≤ 𝜎𝑦𝑖𝑒𝑙𝑑). This yield limit is assumed to act on a plane in one direction; for 1D bodies this is the stress acting over the cross section, see Figure 3.3A. A question arises when a second (component) stress is introduced:

[𝜎] = |

𝜎

0

𝑥𝑥

𝜎

0

_𝑦𝑦

0

0 |

(3.5)

This is technically a biaxial stress situation where uniaxial approaches are not sufficient anymore. To state the stress in a biaxial element (Figure 3.3B) one parameter is not sufficient. When a third stress is introduced (Figure 3.3C), a multiaxial stress state is created. For this situation holds that the stress cannot be given by one value, since each direction has its own stress value.

Figure 3.3 A) Uniaxial stress state. B) Biaxial stress state. C) Multiaxial stress state

Two methods that can handle 1D, 2D and 3D stress states are now introduced: the equivalent stress method and the principal stress method. These two are called multiaxial stress criteria since these methods tend to give the state of stress in one parameter.

3.2 Equivalent stress

The stress in practical mechanical engineering requires a multiaxial approach for many reasons. Firstly; an element is hardly stressed in one direction only; a second stress perpendicular to the main stress is often present due to the Poisson’s ratio. Secondly, shear stress also arises in elements which can be the result of shear loading. Thirdly, component stresses are not easily acquired when the geometry of an element does not follows the global coordinate system. See the next figure for example.

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Figure 3.4 Elements in line having a:

A) Cartesian coordinate system B) Cylindrical coordinate system C) Random coordinate system

For 2D planes or 3D solids holds that the magnitude of the stress and its direction are not always straightforward. If the load direction is not in-line with the component direction, it is difficult to find it. This demands a multiaxial stress parameter which always gives the highest occurring stress, despite the direction of the stress and element. Two well-known equivalent stress criteria are known; the Tresca criterion and the Von Mises criterion.

3.2.1 Tresca criterion

The Tresca criterion is used to state the value of shear stress in an element induced by a biaxial or multiaxial stress state. The stress between two planes that resists sliding is given in Figure 3.5. Brittle materials that have little resistance to shear often fail by shear which is witnessed by a failure plane of 45° degrees. Further introduction into the Tresca criterion can be found in elemental engineering handbooks as Shigley’s [6].

Figure 3.5 Tresca shear stress acting between two planes

The equation for the Tresca stress is given as following:

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3.2.2 Von Mises criterion

The following explanation is based on the article of Jong and Springer [15]. The Von Mises criterion is a frequently used stress criterion to state the magnitude of multiaxial stresses acting on a element by a single parameter. Accordingly, an element having component stresses below the yield stress could have an equivalent stress larger than yield stress. It is suggested by Richard von Mises that a material will yield when the equivalent stress reaches yield stress as well. Therefore, the Von Mises stress criterion is a popular multiaxial stress criterion for static engineering. The equation for an element in 3D Cartesian coordinate system is given by:

𝜎

𝑣𝑀

=

1 √2

=

√2 ∙ (𝜎

_𝑥𝑥2

₎

√2

=

√2

∙ √𝜎

𝑥𝑥 2

_{= 𝜎}

𝑥𝑥 (3.8) The Von Mises criterion is based on a distortion-energy theory that predicts yielding when the distortion strain energy per unit volume reaches the same distortion energy per unit volume needed for that element to yield in a uniaxial tensile test. The Tresca criterion focusses on the Maximum Shear Stress theory, which implies that the maximum yield stress by shear is half of the normal yield stress. The Von Mises criterion is therefore 15% more allowable than the Tresca criteria which often is shown in Figure 3.6 using the failure envelope often given in textbooks [6] [15].

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3.3 Principal stress

Principal stress is a frequently used stress criterion to state the stress in uniaxial, biaxial and multiaxial elements. It has a magnitude and a certain direction, in which shear stress is included. The origin of principal stress has a mathematical explanation, the principal stresses

𝜎

𝑝 are found to be the Eigen

values of the Cauchy stress tensor [

𝜎

]. Each Eigen value

𝜎

_𝑝 has an Eigen vector

𝑛

⃑⃑⃑⃑⃑⃑

_𝑝1 which states the direction of that principal stress. solving the tensor using matrix operations will give the values of the principal stresses and their directions. This will be demonstrated with an example case.

3.3.1 Calculating the principal stress using matrix operations

. The principal stress direction is given by the Eigen

vector

𝑛

⃑⃑⃑⃑⃑⃑

_𝑝1:

Figure 3.7 Example of an element loaded with stress only and no shear acting

Then the Cauchy tensor for matrix operations will be:

([𝜎] − [𝐼]𝜎

𝑝

) = |

𝜎

𝑥𝑥

− 𝜎

𝑝

𝜏

𝑥𝑦

= 0

𝜏

𝑥𝑧

= 0

𝜏

𝑦𝑥

= 0

𝜎

𝑦𝑦

− 𝜎

𝑝

𝜏

𝑦𝑧

= 0

𝜏

𝑧𝑥

= 0

𝜏

𝑧𝑦

= 0

𝜎

𝑧𝑧

− 𝜎

𝑝

| = |

10 − 𝜎

_𝑝

0

0 −5 − 𝜎

𝑝

0

0 5 − 𝜎

_𝑝

1

0 }.

The vectors describe the x-direction, z-direction and the y-direction respectively.

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Solving a 3x3 matrix means that the part under scope is 3D as in a solid. This does not applies for two dimensional elements such as plates, which require a 2x2 matrix. The math necessary for solving a 2x2 reduces matrix significantly, becoming a second order differential equation using the ABC formula [6]:

𝜎

_{𝑝𝑚𝑎𝑥}

=

𝜎𝑥𝑥+ 𝜎𝑦𝑦 2

+ √(

𝜎𝑥𝑥− 𝜎𝑦𝑦 2

)

2

+ 𝜏

_𝑥𝑦2 _(3.10)

𝜎

𝑝𝑚𝑖𝑛

=

𝜎𝑥𝑥+ 𝜎₂ 𝑦𝑦

_𝑝2 only in 3D elements) is the stress that remains. The maximum principle stress is the most significant for fatigue since this is the most tensile stress, therefore when the term ‘principal stress’ is used without a prefix, it automatically assumes the maximum principal stress.

3.3.2 Mohr’s circle

A method to graphically represent the principal stresses is the well-known circle of Mohr. Using this circle, one can find the directions and magnitudes of the principal stresses. This holds for 2D and for 3D elements, yet this research will only focus on 2D elements.

The circle is named after German civil engineer Otto Mohr who developed the graphical technique in 1882. It provides a clear picture of an otherwise complicated analyses. The circle in Figure 3.8 represents an element having the following stress tensor:

[𝜎] = |

𝜎

_𝜏

𝑥𝑥

𝜏

𝑥𝑦

𝑦𝑥

𝜎

𝑦𝑦

| = |10 3

3

2 |

(3.12)

The component stress direction (or global coordinate system) is given by the blue line. The principal direction is given by the horizontal line and the shear stress is plotted at the vertical axis. A convention is followed for shear stress since shear cannot technically be negative yet the direction can differ for two situations. Shear stress that tends to rotate the element clockwise is plotted above the

𝜎-

axis and shear stress tending to rotate the element counter-clockwise is plotted below the

𝜎-

axis. Further information for the drawing of Mohr’s circle is given in literature [6].

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Figure 3.8 Example of the circle of Mohr

Solving the Eigen values using matrix operation or differential equations 3.10 and 3.11 result in Eigen values 11 and 1, leading to a maximum principal stress of 11 [MPa] and a minimum principal stress of 1 [MPa]. The direction of these principal stresses can be found by plotting the Eigen vectors or by the following analytical formula, where

𝛼

is the angle between the first component direction (in this case

𝜎

_𝑥𝑥

)

and the principal direction:

tan 2𝛼 =

𝜏𝑥𝑦

𝜎𝑥𝑥− 𝜎𝑦𝑦 (3.13

)

It is this direction that makes the principal stress criterion suitable for the fatigue assessment since in this direction lies the highest stress of an element. Yet the question arises what the difference is compared with the Von Mises equivalent stress since this also gives ‘the highest stress’ of an element. This is treated in the next section.

3.4 Difference between principal stress and Von Mises criteria

The two multiaxial stress criteria are commonly used in fatigue engineering without a clear designation about when to apply the one or the other. Design codes for fatigue suggest using principal stress for welds (Eurocode 3 [16], IIW [9], DNV [17]) for certain evaluation methods (chapter four), though in literature the Von Mises equivalent stress is used for fatigue analyses [18]. Arguments for choosing which criterion to use can also depend on material type. The Von Mises criterion is often referred to ductile materials which are characterized by ductile failure such as yielding. In contrary, the principal stress is referred to brittle materials, that show fracture as its failure mode. In the case of welded connections (chapter four), no distinction based on material can be made. Steel is ductile, thus Von Mises should be appropriate, though weldments show brittle failure such that principal stress should hold. Another argument is based on the fact that Von Mises is scalar, which has no direction and thus no accurate assumption can be made.

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To continue the debate between these two criteria, certain cases will be tested. Both criteria are calculated for the so-called ‘biaxiality ratio’ which states the ratio between two component stresses:

Biaxiality ratio

λ =

𝜎𝑦𝑦

𝜎𝑥𝑥 (3.14)

The two criteria will be tested for their outcome for a certain input. Shear is considered zero in the first comparison such that the component stresses are equal to the principal stresses. The principal stress and Von Mises stress are calculated with stress component

𝜎

𝑥𝑥=100 [MPa] and

𝜎

𝑦𝑦 varying from 100

[MPa] to -100 [MPa], resulting into a biaxiality ratio of = −1 ≤ λ ≤ 1. The values for the multiaxial criteria for varying biaxiality are calculated, both analytically and numerically. Equations 3.7 and 3.10 are used for the analytical calculation and the numerical calculation is solved with finite element software FEMAP. The analytical results are given in Figure 3.9 and the numerical results can be found in the appendices.

Figure 3.9 shows that there is a significant difference between the principal stress and Von Mises stress depending biaxiality. The maximum principal stress for all biaxiality ratio’s remains 100 [MPa] since no shear is added and only the minimum principal stress varies. When

𝜎

𝑦𝑦=0, the uniaxial situation is

found and both principal and Von Mises give the same output. When

𝜎

𝑥𝑥

= 𝜎

𝑦𝑦

,

both criteria give the

same output. The interresting area lies for biaxiality values between 0 < 𝜆 < 1 and 𝜆 > 0. For situation where 𝜆 = 0,5 the measured Von Mises stress is 13,4% lower than the maximum principal stress. When 𝜆 becomes negative, the oposite is measureable: the Von Mises stress exceeds the principal stress. For 𝜆 = −1 the Von Mises stress is 73.2% higher than the principal stress. It can be concluded that, when the biaxiality is known, an increase or decrease in output is possible only by choosing a stress criterion. Therefore the choice of using which criteria is used, influences the outcome of stress significantly!

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Figure 3.9 Criteria compared for analytical results, without shear

Figure 3.10 Criteria compared for analytical results, including shear

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

140

160

180 -1

-0,8

-0,6

-0,4

-0,2

0 0,2

0,4

0,6

0,8

1 Str

ess

Biaxiality

Stress criteria versus biaxiality

Von Mises

Max Principal

Min Principal

-100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180 -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1

Str

ess

Biaxiality

Stress criteria versus biaxiality, with shear

Von Mises + τ

MaxPrin + τ

MinPrin + τ

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Figure 3.10 shows the results when 𝜏𝑥𝑦= 20 [MPa] shear is added to both criteria. The result leads to similar graphs, only a jump in stress is visible. Similar to the first comparison, it appears that when the biaxiality is 𝜆 < 0, the principal stress gives a lower output than the Von Mises stress. New is that when 𝜆 > 0, the principal stress gives a constantly higher output than Von Mises. In Figure 3.11 is the situation with and without shear plotted. It is visible that adding shears stress rises the value of the stress criteria, and the order of the curves is not altered.

Figure 3.11 Criteria compared with and without shear

Apparently, when biaxiality is known, it appears that by choosing a certain stress criteria, the results will be higher or lower. Fatigue life expectancy can be increased just by choosing a preferable stress criterion if the biaxiality is known. In practice, there is mainly one major stress and a smaller minor stress in another direction, leading to biaxiality ratio’s between λ= -0,5 and λ= 0,5. Biaxiality ratios near λ= 1 are, for instance, occurring in pressure vessels where using Von Mises will always result in lower values.

The origin of the difference between the two criteria is created by the difference in stress on the two planes. This is the Tresca stress

𝜎

𝑇𝑟𝑒𝑠𝑐𝑎

= 𝜎

𝑥𝑥

− 𝜎

𝑦𝑦, which is incorporated partially by the Von Mises

equation. The principal stress also acknowledges the Tresca stress, though not in the direction of the principal stress but on a plane of 45 degrees. This is found in the circle of Mohr, on an angle of 90° from the principal axis. Thus, technically the Tresca stress is not taken into account by the maximum principal stress and therefore the principal stress gives a lower value than the Von Mises stress when the biaxiality in an element is λ < 0.

0 20 40 60 80 100 120 140 160 180 -1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1

Str

ess

Biaxiality

Stress criteria versus biaxiality, with and without shear

Von Mises

Max Principal

Max Principal + τ

Von Mises + τ

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3.5 Multiaxial stress range

In the previous sections it was made clear how stresses are calculated in a multiaxial stress state. For fatigue assessment, also the stress range is important; the difference in stress between two load cases. The calculation of a stress range in a multiaxial stress state is demonstrated in the following section with the goal of finding an efficient method to find the correct stress range.

3.5.1 Determine uniaxial stress range

A stress range arises from two load cases, resulting in an upper limit and lower limit. This is clarified in the next figure where one load case is tensile (blue) and a second is compressive (orange). Both act on an element that shows a crack like defect.

Figure 3.12 In literature and codes, uniaxial stress ranges are assumed

The stress range for a certain component direction (xx or yy) is determined by:

𝜎

_𝑀𝑎𝑥

− 𝜎

_𝑀𝑖𝑛

= ∆𝜎

_{𝑅𝑎𝑛𝑔𝑒} (3.15) And for shear stress range:

𝜏

𝑀𝑎𝑥

− 𝜏

𝑀𝑖𝑛

= ∆𝜏

𝑅𝑎𝑛𝑔𝑒 (3.16)

Determining the stress range according component stress (1D) is the most straight forward method. This component stress range is simple, easy and gives accurate values for analyses if only one component is acting. The functionality is lost when the direction of the stress does not coincides for the two load cases, or; - when the coordinate system does not allow for the component stress to retrieve, or; - when multiple component stresses act simultaneously, in other words: the multiaxial stress states make it difficult to determine a stress range.

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3.5.2 Determine multiaxial stress range

When a combination of two component stresses (or shear) arise in an element, it becomes multiaxial loaded. The stresses combined will behave differently, resulting in a multiaxial stress situation. This is shown for example in the following figure, where a 2D element is loaded with a tensile load in one load case (blue) and loaded with compression in a second load case (orange), yet the loads are not in-line but under an angle. If a crack (or welding) exists, it faces a multiaxial stress range. The question rises how to define a stress range in a situations when both load cases are multiaxial.

Figure 3.13 Two load cases with individual directions acting on an 2D element

Von Mises equivalent stress range:

When the Von Mises equivalent stress is calculated for both load cases, subtraction will give a stress range. First, the equivalent stress for the 2D stress situation is calculated based on equation 3.7:

𝜎

𝑣𝑀

=

1 √2

∙ √(𝜎

𝑥𝑥

−𝜎

𝑦𝑦

)

2

_{+ 𝜎}

𝑦𝑦2

+ 𝜎

𝑥𝑥2

+ 6𝜏

𝑥𝑦2

= √𝜎

𝑥𝑥2

+ 𝜎

𝑦𝑦2

− (𝜎

𝑥𝑥

∙ 𝜎

𝑦𝑦

) + 3𝜏

𝑥𝑦2

And the stress range:

σ_vM1− σ_vM2= ∆σvM (3.17)

Example:

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For load case 1

[

𝜎

_𝜏

𝑥𝑥1

𝜏

𝑦𝑥1

𝑥𝑦1

𝜎

𝑦𝑦1

] = [50 15

(3.19)

When the above calculations are used, a possible error in this process is evident. Calculating stress ranges based on the maximum principal stress only can lead to a fault in the determination of the correct stress range. For example, for a completely reversed load case, the principal stress directions look unchanged. However, the type of principal stress is changed. Where the maximum principal stress always gives the ‘most tensile’ principal stress, in the compressive load case, the minimum principal stress is the prominent principal stress. The stress range is to be determined not only based on the largest values, it should be in the same direction as much as possible. To show the fault in this process, an example is given. The load cases from Figure 3.14 are used again. The presence of shear lets the principal directions shift from the component direction to a new direction. The stress tensor for each load case is given:

Searching for an improvement in fatigue assessment to increase lifetime expectancy - Het zoeken naar verbeteringen in de vermoeiing berekening om een langer levensduur te berekenen

Specialization: Transport Engineering and Logistics

Report number: 2014.TEL.7911

Title:

Searching for an improvement in

fatigue assessment to increase

lifetime expectancy

Author:

Richard Karssen

Searching for an improvement in fatigue assessment to increase

lifetime expectancy

Preface

Summary

Samenvatting

Nomenclature

∆𝜎

∆𝜎

𝑛

𝑛

̅

𝜏

𝜎

[𝜎]

𝜎

𝜎

𝜎

𝑛

⃑⃑⃑⃑⃑⃑

𝛼

∆𝜎

𝜎

𝜎

𝜏

List of abbreviations

Abbreviation Composition

S

Stress

N

Life

D

Damage parameter

FAT-value

Fatigue value at 2·10

N

FAT-class

Welding category

FEM

Finite Element Method

S-N curve

Stress-Life curve

FW

Fillet weld

PP

Partial penetration weld

FP

Full penetration weld

NSM

Nominal Stress Method

HSSM

Hot Spot Stress Method

ENSM

Effective Notch Stress Method

EC3

Eurocode 3

IIW

International Institute for Welding

DNV

Det Norske Veritas

Contents

1. Introduction

1.1 Introduction to this research

1.2 Goal of this research

1.3 Structure of this research

1.4 Thesis boundaries

2. Introduction to fatigue

2.1 The fatigue phenomenon

2.2 Fatigue life calculation

∆𝜎

∆𝜎

∆𝜎

_N

_𝜎