Analysis of Fatigue Life in Two Weld Class Systems.pdf

7/22/2019 Analysis of Fatigue Life in Two Weld Class Systems.pdf

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Analysis of Fatigue Life

in Two Weld Class Systems

Master Thesis in Solid Mechanics

Niklas Karlsson, Per-Henrik Lenander

LITH-IKP-EX--05/2302--SE

November 2005


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Master Thesis in Solid Mechanics

Niklas Karlsson, Per-Henrik Lenander

Analysis of Fatigue Life in Two Weld Class Systems

LITH-IKP-EX--05/2302--SE

Department of Mechanical Engineering

Linkping University

SE-581 83 Linkping, Sweden

Printed in Sweden by UniTryck, Linkping, 2005


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Abstract

In current welding standards, there is a lack of connection between acceptance limits andfatigue life. In an ideal standard there should be a clear and consistent connection, assuringthat a certain welding class always implies a certain fatigue life of the welded joint. VolvoConstruction Equipment is currently involved in reworking the company welding standardSTD5605,51, aiming at introducing such a fatigue connection in the standard.

The objective of this thesis work is to provide the basic data for reworking the standard, i.e. tocalculate fatigue lives for the defect types in the current welding standard. To extend thestudy, the corresponding ISO standard ISO5817 is studied as well.

For the fatigue life calculations, FEM (finite element method) and LEFM (linear elasticfracture mechanics) are used. A few other methods are briefly described and quantitativelycompared.

The results show a very scattered acceptance limit dependence for the fatigue lives in thedifferent defect cases. This implies that the acceptance limits in most cases need to be revised.Furthermore, some cases should be removed from the standard and some cases from the ISO

standard could be included in the Volvo standard.


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Preface

This thesis has been performed as the final assignment for the examination as Master ofScience at Linkping University. The thesis work was initiated by and carried out at Volvo

Articulated Haulersin Bras, Sweden, between June and November of 2005.

The intention is to provide a foundation for further work on reviewing the companys weldingstandard. Finite element modelling and calculation of the fatigue lives of all applicable defecttypes included in the Volvo standard, as well as in the corresponding ISO standard, has been

performed.

Applied theories and methods are thoroughly described in theory chapters, but the reader isassumed to have basic knowledge in solid mechanics.

We would like to thank the following people at Volvo Articulated Haulersin Bras, who haveall contributed to our thesis work: Our supervisor M.Sc. Bertil Jonsson, for invaluablesupport; weld auditor Stefan Stlberg, for hands-on experience on the shop-floor; Qualityengineer Stig Malmqvist, for sharing his expertise on welding standards; everybody at

Helfordonsgruppen, for a memorable time.

We would also like to thank our examiner Prof. Tore Dahlberg at the Division of SolidMechanics, Department of Mechanical Engineering, Linkping University, for reading andcommenting the entire thesis, and our opponents Mr Mats Andersson and Mr MattiasDanielsson for their valuable opinions.

Bras in November 2005

Niklas Karlsson

Per-Henrik Lenander


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Notations

The following constants, functions and variables are used in this thesis. The constants,functions and variables not listed here are explained in the text. Throughout the thesisvariables are written in italicand constants in plain text.

Variables, functions and symbols

A Acceptance limit (in STD5605,51)a Crack lengthC Material parameterC FAT-value in the (Swedish BSK)c Crack length

rkf Stress range (in the Swedish BSK)

h Acceptance limit (in ISO5817)

IK Stress intensity factor in mode I

IcK Fracture toughness in mode IeffK Effective stress intensity factor

tK Stress concentration factor

IK Stress intensity range in mode I

thK Threshold valuem SlopeN Number of cyclesn Material parameter

tn Number of cycles (in the Swedish BSK)

t Sheet thickness

R RadiusW Sheet widthz Utilization factor (in the Swedish BSK) Angle Angle

Q Probability factor (in IIW)

t Thickness factor Poissons ratio Stress

nom Nominal stress

geo Geometrical stress

hs Hot spot stress

notch Notch stress

UTS Ultimate strength

FL Fatigue limit

Y Yield limit Stress range

Shear stress


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Contents

Abstract..................................................................................................................................... vPreface .....................................................................................................................................viiNotations .................................................................................................................................. ixContents.................................................................................................................................... xi1 Introduction......................................................................................................................1

1.1 The company..................................................................................................................11.2 Background .................................................................................................................... 11.3 Objectives....................................................................................................................... 11.4 Procedure........................................................................................................................1

1.4.1 Restrictions............................................................................................................21.5 Outlines of the report...................................................................................................... 2

2 Fatigue life calculation..................................................................................................... 52.1 Basic theory on LEFM................................................................................................... 5

2.1.1 Loading of cracks.................................................................................................. 5

2.1.2 Stress intensity factor ............................................................................................ 52.1.3 Paris law ..............................................................................................................62.1.4 Requirements......................................................................................................... 72.1.5 Plane stress versus plane strain ............................................................................ 7

2.2 Other methods ................................................................................................................72.2.1 Type of fracture..................................................................................................... 92.2.2 S-N curves .............................................................................................................92.2.3 The Swedish standard.......................................................................................... 102.2.4 The nominal stress method.................................................................................. 112.2.5 Example of the nominal stress method................................................................ 122.2.6 Comparison with the Swedish standard.............................................................. 132.2.7 Problems with interpreting the standard ............................................................13

2.2.8 The Hot spot method ........................................................................................... 152.2.9 Example of the Hot spot method ......................................................................... 162.2.10 The effective notch method..................................................................................182.2.11 Example of the notch method ..............................................................................182.2.12 Linear Elastic Fracture Mechanics..................................................................... 20

2.3 Comparison of methods ............................................................................................... 202.3.1 Advantages with LEFM....................................................................................... 212.3.2 Disadvantages with LEFM.................................................................................. 21

3 Procedure and methods ................................................................................................. 233.1 Modelling ..................................................................................................................... 23

3.1.1 Basic modelling ................................................................................................... 233.1.2 Boundary conditions ...........................................................................................23

3.1.3 Influence of boundary conditions ........................................................................ 253.2 Meshing........................................................................................................................ 26

3.2.1 Achieving a meshable model...............................................................................263.2.2 Elements ..............................................................................................................273.2.3 The crack tip ........................................................................................................273.2.4 The box ................................................................................................................293.2.5 The centre lines ................................................................................................... 303.2.6 Remaining parts of the model..............................................................................313.2.7 Meshing problems ............................................................................................... 31


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3.2.8 Convergence........................................................................................................323.2.9 Macros................................................................................................................. 33

3.3 Postprocessing.............................................................................................................. 343.3.1 How stress intensity factors are calculated in ANSYS........................................ 343.3.2 Numeric integration ............................................................................................ 36

3.4 Verification of integration method...............................................................................38

3.4.1 Approximation with upper and lower summations ............................................. 403.4.2 Spline values integrated in Excel ........................................................................ 41

3.5 Analytical case for an internal crack ............................................................................ 423.5.1 Crack growth rate ............................................................................................... 433.5.2 Algorithm............................................................................................................. 443.5.3 Discussion about integration limits..................................................................... 443.5.4 Comparison with AFGROW................................................................................ 46

4 Weld auditing.................................................................................................................. 474.1 Overall on weld auditing.............................................................................................. 474.2 Review of previous weld audits ...................................................................................474.3 Weld audit on a rear frame........................................................................................... 49

4.3.1 Profile projector measuring of silicone impressions .......................................... 49

4.3.2 Comments ............................................................................................................505 Compilation of STD5605,51 .......................................................................................... 51

5.1 About the standard ....................................................................................................... 515.2 Modelling ..................................................................................................................... 525.3 Results .......................................................................................................................... 52

6 Compilation of ISO5817 .............................................................................................. 1456.1 About the standard ..................................................................................................... 1456.2 Modelling ...................................................................................................................1456.3 Results ........................................................................................................................ 145

7 Study of sheet thickness dependence.......................................................................... 2217.1 Introduction ................................................................................................................2217.2 Studied geometry - STD5605,51 Case 20.................................................................. 221

7.2.1 Requirements..................................................................................................... 2227.3 Modelling ...................................................................................................................223

7.3.1 Dimensions ........................................................................................................ 2247.4 Influence of sheet length ............................................................................................2247.5 Results ........................................................................................................................ 2267.6 Comments on the results ............................................................................................ 230

7.6.1 The thickness effect............................................................................................ 2317.6.2 Effect of absolute acceptance limits.................................................................. 2317.6.3 Sheet length ....................................................................................................... 2317.6.4 The bending case............................................................................................... 231

7.7 Conclusions ................................................................................................................ 232

7.8 Theory and calculations on the thickness effect.........................................................2328 Conclusions and discussion .........................................................................................2378.1 Conclusions ................................................................................................................ 2378.2 General discussion......................................................................................................2378.3 Proposals for revising the standards...........................................................................238

8.3.1 Primary proposal for guidlines......................................................................... 2398.3.2 Secondary proposal for guidelines.................................................................... 2398.3.3 Other proposals for STD5605,51 ...................................................................... 2398.3.4 Other proposals for ISO5817............................................................................239


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8.4 Discussion for STD5605,51.......................................................................................2398.4.1 Cases which could be added or removed from STD5605,51 ............................2408.4.2 Recommendations for achieving proposals for STD5605,51............................ 241

8.5 Discussion for ISO5817 ............................................................................................. 2428.5.1 Cases which could be added or removed from ISO5817 .................................. 2428.5.2 Recommendations for achieving proposals for ISO5817..................................243

8.6 Recommendations for further studies ........................................................................ 244References ............................................................................................................................. 247Appendix A ........................................................................................................................... 249Appendix B diagrams for STD5605,51............................................................................ 251Appendix C diagrams for ISO5817 .................................................................................271


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1 Introduction 1

1 Introduction

1.1 The company

The thesis work was carried out at Volvo Articulated Haulers in Bras, 30 kilometresnortheast of Vxj in southern Sweden. It has recently been merged with Volvo Wheel

Loaders into Volvo HLBL (Hauler Loader Business Line), which is a part of VolvoConstruction Equipment.

In the Bras factory articulated haulers are designed and manufactured. An articulated hauleris most commonly used to transport gravel and rocks in rough terrain during road and

building construction, but also other applications are available. Volvo CE markets articulatedhaulers in a payload range of 25 to 40 tons. In Bras the 650 employees currently produceover 2 000 machines yearly. There is also a factory situated in Pederneiras, Brazil. Volvo CEhas approximately 40% of the world market for articulated haulers.

1.2 Background

The background of this thesis is the ongoing efforts to review the Volvo company weldingstandard, STD5605,511[1]. An important objective of this is to achieve a clear and consistentconnection between the acceptance limits in the standard and the fatigue lives of the weldedstructures.

The welding standard contains descriptions of a number of possible weld defects, with limitsfor the accepted dimensions of the defects for each welding class. In the Volvo standard, fourwelding classes, A to D, can be used for assigning suitable requirements when designing aweld joint. They also contain additional designations, for example U for fatigue loaded welds.

Reworking the welding standard requires a large amount of quantitative data on currentfatigue lives to be calculated. This thesis was initiated to produce all this data as well asadditional knowledge on fatigue life issues for weld joints.

1.3 Objectives

The main objective of the thesis work is to provide life calculations for all interesting defecttypes, in all welding classes, both for the Volvo standard STD5605,51 and for thecorresponding ISO standard ISO5817 [2]. The ISO standard is studied in order to takeadvantage of possibly useful features. The fatigue life data is presented in tables anddiagrams, and also thoroughly commented in the text.

As an effect of the extensive work on modelling and calculating all cases, a lot of knowledgeof various factors affecting the fatigue life has been achieved and will be presented in thereport. The influence of sheet thickness and bending loads will be separately investigated.

It is the authors objective to provide general recommendations for how the standards couldbe revised; primarily for the Volvo standard, but since ISO5817 has also been thoroughly

reviewed, it will be discussed as well.1.4 Procedure

A specified set of interesting defect cases in the two standards are being investigated. Foreach case the geometry is modelled, stress intensity factors are obtained, and finally thefatigue life is calculated.

1Previously STD5605,51 was called 5.501E. It is now also known as STD181-0001. Throughout this work theVolvo CE weld standard will be referred to as STD5605,51


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2 Analysis of Fatigue Life in Two Weld Class Systems

For a normal type of defect (a few exceptions exist), the weld joint geometry with a defectof the current type and an initial crack is modelled in the FE programme ANSYS. A macro isthen used to let a crack propagate in certain steps into the model. For every crack length theFE problem is solved and the stress intensity factors are calculated. The results are used todetermine the stress intensity factor as function of crack length for the particular case. Thenthis function is used when calculating the fatigue life by use of fracture mechanics

(integration of Paris law). The life is assumed to be finished when the crack reaches half thesheet thickness.

All steps of the procedure are described thoroughly in the following chapters.

In a few cases, especially for some internal defects, special methods for estimating the crackpropagation have to be applied. This is further discussed in theory chapters and casedescriptions.

1.4.1 Restrictions

It should be emphasised that all results in this report are theoretical, based on the exactconditions that are given in the description of each case. In reality, for example, geometriesare never perfect, material quality varies and loading conditions can be very complex.

All geometries have been modelled in 2-D. Thus, the depth direction is not considered,except in the analytical case used for internal cracks.

An important restriction is that an initial crack length of 0.1 mm is assumed. This means thatno life during the crack initiation phase is taken into account.

All calculations are performed on 10 mm thick sheets. Applying the results to otherthicknesses can not be done without consideration. This is investigated and discussed inChapter 7.

Occasionally it is commented in the report that some issue is not fully investigated(normally due to lack of time or due to the problem lying outside the scope of this work). Inthese cases conclusions are based on reasonable assumptions, given in the text.

1.5 Outlines of the report

In Chapter 2, basic theory on methods for fatigue life calculation is presented. The focus lieson LEFM (linear elastic fracture mechanics), which is used in this thesis. This is followed bya brief description of a few other methods for fatigue calculations and a comparison of all themethods for two particular cases.

The procedure of work used in the thesis is thoroughly described in Chapter 3; modelling, FEanalysis and integration of Paris law. An analytical solution method used for internal cracksis also described.

An introduction to weld auditing is given in Chapter 4.

Chapters 5 and 6 contain the results, descriptions and comments for all cases that have beencalculated, for Volvo STD5605,51 and for ISO5817 respectively. For each of the standards ashort introduction is followed by a compilation of the results. Detailed information on eachdefect type is then presented in a commentary page followed by one data sheet per weldingclass for the defect.

Chapter 7 describes a special study of the sheet thickness effect on the fatigue life,investigating the applicability of the results on different sheet thicknesses.


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1 Introduction 3

Conclusions and discussion are presented in Chapter 8. The authors give theirrecommendations for how the standards could be reworked to obtain a better connection

between welding class and fatigue life. A few other important observations and conclusionsare also provided, as well as suggestions for further research in the area.

Appendix A contains the MATLAB program for calculating stress intensity factors for theanalytical case with an elliptical inner crack.

In Appendices B and C, tables and diagrams of results for STD5605,51 and ISO5817 can befound respectively.


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2 Fatigue life calculation 5

2 Fatigue life calculation

This chapter contains theory on LEFM as well as other methods for predicting the fatigue life

of a structure subjected to cyclic loading. There is also a comparison between the four

methods brought up in this chapter.

2.1 Basic theory on LEFMIn order to calculate the number of cycles to fracture,Linear Elastic Fracture Mechanics(LEFM) is used. When using LEFM one assumes that the stresses at the crack tip tend toinfinity. The theory in this chapter is taken from the referenceFailure, Fracture, Fatigue An

Introduction [3].

2.1.1 Loading of cracks

A crack can be loaded in three different ways, see Figure 2.1.

Figure 2.1. Three different modes a crack can be loaded in.

In Mode I the crack is opened. Mode I is the most dangerous way to load the crack becausethis loading case generates the greatest stress intensity of all loading cases.

In Mode II and III the crack is sheared in two different planes.

2.1.2 Stress intensity factor

Thestress intensity factoris defined as

afK = nomI

where 0 is the nominal stress, athe crack length, andfis a function of geometry and

loading. Numerous analytical cases have been derived giving thef-function. Some of themcan be found in, for example,Formelsamling i hllfasthetslra, reference [4].

The range of the stress intensity factor can be calculated as

minImaxII KKK =

Mode I Mode II Mode III


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where

maxII KK = if 0minI K

minImaxII KKK = if 0minI >K

0I =K if 0maxI K

The stress intensity range KIis used inParis lawwhen calculating the number of cycles tofailure, see Equation (2.1).

Stress intensity factors in the same mode can be added by superposition, i.e.,

...CIBI

AI

totalI +++= KKKK

If a crack is loaded in several modes at the same time an effective stress intensity factor, effK ,

can be calculated as

2III

2II

2I

2eff

1

4KKKK

+++=

where

43 = for plane strain

+

=1

3 for plane stress

and isPoissons ratio.

There are also other suggestions for calculating the effective stress intensity factor.

2.1.3 Paris law

When plotting the logarithm of the crack propagation rate versus the logarithm of the range ofthe stress intensity factor, the following graph is achieved (Figure 2.2).

Figure 2.2. The crack propagation rate versus the stress intensity range.

dNdlog a

0 Ilog K thlog K IclogK


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For thI KK


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linear elastic fracture mechanics (LEFM). Theory on all methods except LEFM are takenfrom the reference Svetsutvrdering med FEM[5]

All methods in this chapter use data from the International Institute of Welding (IIW) [6] forcalculating the life of the weld. Data from IIW are based on real welded structures which have

been exposed to cyclic loading. Values from these tests have been plotted in an S-N-diagramand a FAT value (FATigue) of the stress for sustaining 2 million cycles has been calculated.This FAT value gives a failure probability of 2.3% for the weld.

Two examples are used when describing the methods. Both are taken from IIWsFatiguedesigns for welded joints and components[6]. The first example is a transverse butt weld,Case 213 (see Figure 2.3) and the second example is a cruciform joint, Case 413 (see Figure2.4).

Figure 2.3. Case 213, transverse butt weld.

Figure 2.4. Case 413, cruciform joint.

Both cases are modelled with a transition radiusR= 1 mm. This is the smallest radius whichcan be expected for a normal weld without any subsequent machining [5].

A comparison with the Swedish standard is performed.

t R

w

crack

t 0.15t

t

h

R

crack

w


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2.2.1 Type of fracture

In Case 413, with the cruciform joint, a crack could propagate both from the weld toe andfrom the root. All calculation will be made on cracks propagating from the toe.

A toe crack is advantageous because it can more easily be detected with the eye and steps torepair the weld can be taken before fracture.

To detect a root crack some kind of instrument, for example ultrasound, must be used. This isa much more time consuming process and a toe crack is therefore favourable.

2.2.2 S-N curves

In an S-N curve, also known as a Whler curve, the logarithm of the stress is plotted versusthe logarithm of the number of cycles (see Figure 2.5).

Figure 2.5. S-N curve.

August Whler found that the stress amplitude described the fatigue life better than themaximum stress. The stress amplitude is defined as

2minmax

a

=

The stress range is defined as

minmax =

For welds it is a more appropriate approach to use the stress range for fatigue analysis.

For stresses above the ultimate strength, UTS , the component fractures immediately while for

stresses below the fatigue limit, FL , the component has a theoretical infinite life at constantamplitude loading.

UTSlog

FLlog

log

Nlog 0

slope mConstant amplitudefatigue limit

slope for varying amplitude


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is used instead of a for welds. The reason for this is that residual stresses close to the

yield limit Y can be expected in both tension and compression when the metal cools in theweld. This means that if, for example, compressive stresses are applied to a weld where theresidual stress is near the yield limit in tension, the weld will endure a positive stress rangeeven though weld is loaded in compression. The root for fillet welds often have negativeresidual stresses while toe cracks have positive residual stresses, but this may vary.

A conservative approach when calculating the fatigue life of welds, is thus to use the stressrange .

When the residual stresses are known, correction factors for compensating the worst casescenario can be found in, for example, [6]. The correction factors are functions of the stressratio, which is defined as

max

min

=R

These correction factors are greater than one and are multiplied with the FAT-value in orderto receive a longer fatigue life. This may be used for the ground material and to some extent

on simple welds. The correction factors are not used for complex structures.To calculate the number of cycles to failure, the straight line with slope mcan be used. Theequation

CmN logloglog +=

gives the life

NC

CN m

m =

=

The IIW uses aFAT-value of the stress which has been selected so that a certain componentshould sustain 2 million cycles before failure. This gives the following equation for

calculating the number of cycles to failure for a certain range of the stress:

=

=

=NFAT

NC

FATCmm

m

m

66

102102

6102

=m

FATN (2.2)

Equation (2.2) makes it possible to use the tested cases in the IIWs handbook [6] to calculatethe life of a component.

2.2.3 The Swedish standard

The Swedish BSK [7], [8] (translated: Regulations for Steel Constructions) standard uses thesame methods as the IIW standard. The following equation can be found in [7]

6

3

rk

t

3/1

t

6

rk 102102

=

=

f

Cn

nCf (2.3)

When Equation (2.3) is compared to (2.2) one can see that it is the same equation, only thenotations differ.Nis the same as tn , rkf equals and Ccorresponds to theFAT-value.


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Fatigue lives for a number of cases, as for the IIW standard, have been tested. The majordifference is that the Swedish standard also deals with three different welding classes from Ato C, where class A gives the longest life. However the descriptions of the classes are vague,without any actual acceptance limits. This makes it difficult to define which class thecomponent belongs to. Therefore the predicted number of cycles may differ a lot, dependingon which class is chosen.

The BSK also offers the opportunity to predict the fatigue life for varying stress amplitudewith a special equation which takes the new slope of the curve into consideration, see Figure2.5.

The BSK can only be used to evaluate the fatigue life for nominal stresses, i.e. it does notsupport the hot spot and effective notch methods.

2.2.4 The nominal stress method

In the nominal stress method the nominal stress, nom , is used to calculate the life of the

welded structure. This is a simple method which usually can be carried out with onlyhandbook results if the geometry is simple. For more complex structures the nominal stress isoften hard to find in an FE model.

To be able to use the nominal stress method the current structure must be similar to one of thestructures available for the method. Misalignments and defects must lie within the weldclasses.

The nominal stress is defined as the global stress, for example the stress applied far awayfrom the weld in Case 213 and 413, and is mostly perpendicular to the weld. If the structure isnot loaded solely in tension, for example when there is a bending moment present, thenominal stress can be extrapolated to the weld toe, see Figure 2.6.

Figure 2.6. The nominal stress, nom , extrapolated from the true stress at the surface.

0 x

nom


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For evaluation of fatigue life with nominal stress a large number of cases with associatedFAT-values are available from the IIW, [6].

2.2.5 Example of the nominal stress method

The following first principal stress curve is achieved for Case 213 and 413 (see Figure 2.7 and

2.8). Both cases have a transition radius of 1 mm.

0 2 4 6 8 10 120

20

40

60

80

100

120

140

160

180

200

220

x [mm]

1stprincipalstress[MPa]

Nominal method for Case 213

Start of transistion radius

End of transist ion radius

nom

Figure 2.7. Nominal method for Case 213 in IIW.

0 2 4 6 8 10 12 140

50

100

150

200

250

300

x [mm]

1stprincipalstress[MPa]

Nominal method for Case 413

Start of transist ion radius

End of transistion radiusnom

Figure 2.8. Nominal method Case 413 in IIW.

Since there is no bending moment here, the nominal stress is equal to the applied stress, i.e.104 MPa for Case 213 and 82 MPa for Case 413. TheFAT-value is 80 MPa for Case 213respectively 63 MPa for Case 413 but both these values have been multiplied with a factor of


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1.3 ( Q ) to achieve 50% probability of failure instead of 2.3% which is the case for theFAT-

value.

The life for the welded structure can be calculated with the modified Equation (2.2) as

6

3

nom

Q

nom 102

=

FATN (2.4)

which gives a life of 6102 cycles for both Case 213 and 413.

2.2.6 Comparison with the Swedish standard

If the Swedish BSK [7], [8] standard is used with welding class B on similar cases the fatiguelife (Equation 2.3 with probability factor as in Equation 2.4) becomes 61040.1 for Case 213(Case 12 WB in [7]) and 61040.1 for Case 413 (Case 30 WB in [7]) i.e., the fatigue life isabout 30% lower than for the IIW. The BSK standard is in general more conservative thanIIW.

An interesting notation can be made when predicting the life for Case 213. Two cases are

available in the BSK; Case 12, which is used here, deals with rewelded root, while Case 11does not. The differences between the welds are might seem small for an untrained eye and ifCase 11 is used, with the same welding class, the fatigue life becomes 61085.2 cyclesinstead of the 61040.1 cycles it was for Case 12, i.e., more than twice the life. This is quiteremarkable and describes the difficulties with predicting fatigue life.

The achieved results illustrate the problems with predicting the life of welded structures. Eventhough almost the same fatigue data have been used, interpretation of the data leads todifferent results due to different approaches.

2.2.7 Problems with interpreting the standard

There is no straightforward description in the Swedish BSK standard of how the standardshould be interpreted. This could cause some problems. As an example the fatigue life forCase 413 will be calculated. The easy way is to use the nominal stress at the edges. Then enter

82rk=f MPa ( ) in Equation 2.3. This gives for the life61040.1 for Case 30 WB [7], as

mentioned above. Another way to calculate the stress range is described below. This methodof solving fatigue problems was developed atBombardierin Kalmar, Sweden, by Per-OlofDanielsson [9] and Anders Lindstrm. The first step is to make a free-body diagram of thecruciform joint in Case 413, see Figure 2.9.


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Figure 2.9. Free-body diagram of the cruciform joint in Case 413.Note that the possible misalignment has been disregarded.

The following stresses can be derived

MPa585

10

22

82

22:

:

nom =

==

=

a

t

When the stresses, ( rd ) and ( rd ) are known, these can be used to calculate theutilization factor for multiaxial stress state, which can be calculated with the followingequation

1.1

2

allowed,

2

allowed,

+

=

z

56allowed, = MPa ( rdf ) for Case 30 WB [8] and allowed,allowed, 6.0 = ( rvdf ). Both

allowed, and allowed, are multiplied with the factor 3.1Q = for 50% probability of

failure, which gives

18.2566.03.1

58

563.1

5822

+

=z

The corresponding uniaxial stress can now be calculated as

111561.118.21.1rk == Czf MPa

This stress is considerably higher than the nominal stress, 82nom = MPa. Now the fatigue lifecan be calculated as

66

3

6

3

rk

t 1025.0102111

56102

=

=

f

Cn cycles

t

nom nom

a


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which is considerably lower than the 61040.1 cycles achieved by use of nominal stress. Thisis a very conservative way to calculate the life but enlighten the problems which arises whenthere are almost no instructions given to the user of the standard.

If the probability factor, Q , is inserted when the number of cycles are calculated instead of

in the utilization factor,z, as described below

01.2566.0

58

56

5822

+

=z

102561.1

01.2

1.1rk == C

zf

66

3

6

3

rk

Q

t 1072.0102102

563.1102

=

=

f

Cn

the fatigue life becomes almost three times as high. Again this confirms the difficulties withpredicting the life.

2.2.8 The Hot spot method

The Hot spot method was originally designed to be used within the offshore industry usingmeasured strains. This made the method applicable in situations were no stresses had beencalculated. Later, FE-analysis have been used to calculate the stresses near the weld and

predict the life of a given structure.

This method can be used where the local geometry disturbs the nominal stress or where noIIW case describes the particular structure. In practice, only one S-N diagram is needed formost welds regardless of the defects in the geometry.

Disadvantages with the hot spot method are that only toe cracks can be evaluated. The stressneeds to be almost perpendicular to the weld, the density of the mesh must be fine close to the

weld toe, and all geometrical defects near the toe must be modelled. Another disadvantage isthat the method is not accurate for thick plates.

The Hot spot is the toe of the weld. A Hot spot stress or a geometrical stress is extrapolatedfrom points on the 1stprincipal stress curve close to the weld (see Figure 2.10).


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Figure 2.10. The Hot spot stress, hs , linear extrapolation.

Different equations are used to extrapolate the stress. The following equation describes linearextrapolation.

1.0t0.4tnom 67.067.1 = (2.5)

where 0.4t is the stress at 0.4 times the thickness from the toe, etc. The first extrapolation

point, 0.4t , has been chosen since the stress here is not affected by the weld toe geometry.

2.2.9 Example of the Hot spot method

The same example as for the nominal stress gives the following graphs for Case 213 and 413(see Figure 2.11 and 2.12).

0

x

hs

Extrapolation points

0.4t

1.0t


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0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160

180

200

220

x [mm]

1stprincipalstress

[MPa]

Hot spot method for Case 213


End of transistion radius

hs


Figure 2.11. The hot spot method for Case 213.

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

300

x [mm]

1s

tprincipalstress[MPa]

Hot spot method for Case 413

Start of t ransistion radius


hs


Figure 2.12. The hot spot method for Case 413.

According to [6] the FAT value for flat butt welds is the same as used for the nominal stressmethod i.e., 80 MPa, while it should be 100 MPa for fillet welds with crack at toe ground.

With these FAT values the equation

6

3

hs

Q

hs 102

=

FATN

gives a life of 61090.1 cycles for Case 213 and 61016.9 for Case 413. The reason why thefatigue life for Case 413 is remarkably long is that the FAT value is relatively large comparedto the extrapolated hot spot stress.


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In both examples the 1stprincipal stress has decreased to a value close to the nominal stress atthe distance 0.4tfrom the toe.

2.2.10 The effective notch method

In the effective notch method the notch is replaced by a radius of 1 mm more than the realcase. For a welded condition all notches are modelled with a radius of 1 mm (see Figure 2.13)

implying that the real radius is zero.

Figure 2.13. All notches modelled by a radius.

The FE-model or a handbook with the radii gives the stress concentration at the notch and thestress is then used to calculate the life in the same way for the nominal stress method as forthe hot spot method.

The effective notch method is advantageous if root cracks are to be evaluated or if differentgeometries are to be compared.Disadvantages are that the method has not been verified for thicknesses less than 5 mm andthe stress must be perpendicular to the weld. Since the stress must be perpendicular to theweld, the 1stprincipal stress is commonly used in order to simulate the worst case scenario.

2.2.11 Example of the notch method

The maximum stress at the notch in both Case 213 and 413 is given in Figures 2.14 and 2.15respectively.

radii


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0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

140

160

180

200

220

x [mm]

1stprincipalstress[

MPa]

Notch method for Case 213



notch

Figure 2.14. Effective notch stress, notch , for Case 213.

0 1 2 3 4 5 60

50

100

150

200

250

300

x [mm]

1stprin

cipalstress[MPa]

Notch method for Case 413

Start of transist ion radius


notch

Figure 2.15. Effective notch stress, notch , for Case 413.

Note that only the toe notch is considered in Case 413.For the effective notch method only oneFAT-value is available in [3] for modelled radii of 1mm. This singleFAT-value, 225 MPa, does not treat possible misalignment.

The same equation as in previous examples

6

3

notch

Qnotch 102

=

FATN


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gives the life 61045.5 cycles for Case 213 and 61097.1 cycles for Case 413.

2.2.12 Linear Elastic Fracture Mechanics

Theory about LEFM was given in chapter 2.1 (Basic Theory). How LEFM is applied for Case213 and 413 is given in chapter 3 (Procedure and methods).

2.3 Comparison of methodsAll four methods described above are compared qualitatively in Figure 2.16.

Figure 2.16. The diagram describes a qualitative comparison of accuracy and modelcomplexity for the four methods.

Figure 2.16 is taken fromModelling and Fatigue Life Assessment of Complex Fabricated

Structuresby Marquis and Samuelsson [10] and describes the accuracy in calculated lifecompared to the complexity of the model for the four above described methods. Example ofcomplexity could be whether or not it is a simple 2D model of a fillet weld or an advanced 3Dmodel of the rear frame in an articulated hauler. It can be seen in Figure 2.16 that LEFM is avery accurate method which also requires a lot of work while the other, simpler methods, areranked depending on accuracy as the notch, hot spot and nominal stress method. The nominalstress method is the least accurate. The notch method gives relatively good result at a lowerworking effort. The nominal stress method can not always be used at very complex structuressince it requires non disturbed nominal stresses [5].

Results for all four methods are shown in Table 2.1 and 2.2 below.

Table 2.1. Results for Case 213.

Effective FractureNominal Hot spot

Notch Mechanics

Life (cycles): 2 106

1.90 106

5.45 106

1.48 106

% of 2 106cycles: 100% 95% 273% 74%

Accuracy

Complexity0

LEFM

Notch method

Hot spot method

Nominal stress method

Working effort


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Table 2.2. Results for Case 413.

Effective FractureNominal Hot spot

Notch Mechanics

Life (cycles): 2 106

9.16 106

1.97 106

0.90 106

% of 2 106cycles: 100% 458% 99% 45%

As can be seen in the tables, the hot spot method have a life close to the expected 2 millioncycles for Case 213 while it divert extremely much for Case 413. This result must bequestioned. The reason why the life it is extremely long is that the prescribed FAT value forfillet welds is very high compared to the extrapolated hot spot stress.

The effective notch method seems to agree for Case 413 while it diverts a lot for Case 213.This is because the stress concentration is quite small compared to the FAT value, whichresults in a longer life. Also this result must of course be questioned.

Fracture mechanics gives a shorter life than all the other methods. When using fracturemechanics and Paris law the life has been integrated from an initial crack length of 0.1 mm.

Linear elastic fracture mechanics (LEFM) is not applicable on small cracks, which is aproblem since most defects have values below 0.1 mm. To be able to use fracture mechanicsone has to assume that there is a known crack which is not too small.

If one does not want to assume that there is a crack from the beginning other methods must beused during the initial stage of the formation of the crack, before LEFM can be used.However in reality there are always flaws and pores which serves as initial cracks.

2.3.1 Advantages with LEFM

Linear elastic fracture mechanics (LEFM) will be used to evaluate the life of the weldedstructures in the Volvo CE standard STD5605,51 and in the ISO 5817 standard. The reasonwhy LEFM is chosen instead of the other methods is that the conditions change from the

initial stage as the crack grows down into the material, and LEFM offers a chance to catchthese changes. All the other methods look only upon the initial state. This may result in a toosimplified picture of reality because all cases in the welding classes have to be compared withonly two cases from the IIW.

For example, if the stress concentration is the same for an undercut (see Chapter 5,STD5605,51 Case 6) as for a penetration bead (see Chapter 5, STD5605,51 Case 12) and theeffective notch method is used, both cases will get the same life. This, however, may give atotally wrong picture of the reality since the crack in the case with the undercut may grow 3mm while the crack in the penetration bead can grow 5 mm before reaching half the thicknessof the material. The extra 2 mm which the crack in the penetration bead can grow results in alonger life. However, this will not be seen in the results if, for example, the effective notchmethod is used.

2.3.2 Disadvantages with LEFM

When LEFM is used several different FE-simulations must be performed in order to haveenough points to describe the curve to be integrated in Paris law. When using any of the threeother methods to calculate the life, only one FE-simulation has to be performed. Thus, LEFMneeds more working effort.

Another problem which arises is to mesh the area around the crack. Relatively small elementsmust be used to get a good result. Small elements results in many elements, which further


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results in larger models and longer simulation times. Another problem is the transition regionbetween relatively small element near the crack tip and larger elements far away, which oftencontains badly shaped elements that are not desirable. Badly shaped elements are rectangularelements which have a large quotient between the long and short side or contain large blunt orsmall sharp angles.

One further disadvantage with LEFM is that a curve of the achieved results from the FE-simulation must be approximated and later integrated numerically. Errors can occur in boththese steps.

Though there are some disadvantages with LEFM, especially the many extra simulations andthe more time-consuming steps, the advantages of more geometry dependent and accurateresults outweigh the disadvantages.


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3 Procedure and methods 23

3 Procedure and methods

This chapter contains description about the overall analytical and FE modelling, which gives

the stress intensity factors. There is also information about how the stress intensity factors

with corresponding crack length are integrated numerically in order to determine the fatigue

life.

3.1 Modelling

In this chapter questions about the modelling of the crack and the rest of the geometry will bediscussed.

All cases in the Volvo CE standard STD5605,51 and the ISO5817 are modelled in theFE-program (Finite Element) ANSYS [11], [12], except a handful of cases which are solvedanalytically.

3.1.1 Basic modelling

In ANSYS, like in most FE-programs, the geometry is built by use of keypoints, which areplaced in space. Lines are drawn between the keypoints and the lines are used to create areas.

Areas are the highest order in 2D simulations. If 3D modelling is performed, the areas areused to build up volumes.

To model the crack a small cut in the geometry is performed. This means that there is arelatively small distance between the points at the opening of the crack. In all models (in thisstudy) the growth of the crack is perpendicular to the horizontal sheet. Crack growth issimulated by moving the crack tip keypoint into the material.

3.1.2 Boundary conditions

Both for transverse butt welds and for cruciform joints three types of boundary conditions aretested. One way is to apply stress, , in both ends see Figure 3.1. To prevent the model from

performing rigid body motions one point in the middle of the model is locked in both degrees

of freedom. To prevent the model from rotation, one more point is locked in thex-direction.

Figure 3.1. Natural boundary conditions for a butt weld.

Another set of boundary conditions was tested, for the left side and the right side respectively.One end of the model was supported in thex-direction and stress was applied at the other end.To prevent the model from rigid body motion, one node on the side where the model issupported in thex-direction is also locked in they-direction (see Figure 3.2 and 3.3).

y

0

0 0


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Figure 3.2. Left side of the welded sheet is supported.

Figure 3.3. Right side of the weld is supported.

Results for all three types of boundary conditions are given in Table 3.1.

Table 3.1. Results for different boundary conditions.

K_eff [MPa(mm)]

Crack length [mm]: Natural: Left side locked: Right side locked:

0,0000 0,00 0,00 0,000,0025 19,92 19,82 19,92

0,0050 28,13 27,87 28,13

0,0100 39,50 39,48 39,50

0,0200 54,83 54,63 54,83

0,0400 74,59 74,55 74,59

0,0800 98,67 98,72 98,67

0,1600 125,77 126,32 125,77

0,3200 158,98 158,77 158,98

0,6400 203,56 204,93 203,56

1,2800 279,48 280,22 279,48

2,5600 456,16 456,51 456,16

5,1200 1182,89 1183,06 1182,89

Max difference: - 0,9% 0,0%

x

y

0

0

x

y

0

0


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As can be seen in Table 3.1, the types of boundary conditions do not influence the results verymuch. The maximum difference compared to the boundary conditions in Figure 3.1 is around0.9%. Henceforth boundary conditions given in Figure 3.1 will be used.

Reaction forces at the supports have been controlled for all types of boundary conditions. Forthe boundary conditions selected for further use, the reaction forces were negligible both inx-and they-direction. For the other types of boundary conditions, the total reaction force wasthe same as the applied force in thex-direction, while it was negligible in they-direction asexpected.

3.1.3 Influence of boundary conditions

The geometry is modelled with a 200 mm long horizontal sheet. Boundary conditions areapplied at the ends of this sheet. To study the effects of the boundary conditions and toinvestigate the influences of bending at the crack, the model has been prolonged 100 mm atone end at the time, see Figure 3.4 and 3.5. The results are presented in Table 3.2 below.

Figure 3.4. Left end prolonged.

Figure 3.5. Right end prolonged.

0

0


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Table 3.2. Results when one end of the sheet is prolonged.

K_eff [MPa(mm)]

Crack length [mm]: Original: Elongated left: Elongated right:

0,0000 0,00 0,00 0,000,0025 19,92 19,82 19,92

0,0050 28,13 27,87 28,13

0,0100 39,50 39,48 39,50

0,0200 54,83 54,63 54,83

0,0400 74,59 74,55 74,59

0,0800 98,67 98,72 98,67

0,1600 125,77 126,32 125,77

0,3200 158,98 158,77 158,97

0,6400 203,56 204,93 203,55

1,2800 279,48 280,22 279,48

2,5600 456,16 456,51 456,16

5,1200 1182,89 1183,06 1182,89

Max difference: - 0,9% 0,0%

As can be seen in Table 3.2 the boundary conditions in the original model have no decisiveeffect on the result and can be used further on without complications. The maximumdifference according to boundary conditions in Figure 3.1 is around 0.9%.

3.2 Meshing

Meshing of the model turned out be one of the largest challenges in this work. How themeshing problems were solved is described in this chapter.

3.2.1 Achieving a meshable model

A model with a crack can be difficult to mesh because the elements must be relatively smallnear the crack tip. In order to get around that problem the total area of the sheet is divided intosmaller areas.

Since the elements around the crack tip must be relatively small and it is extremely timeconsuming to use these small elements on the whole model small-size elements are usedaround the crack and larger elements are used far away from the crack. Problems may occurin the transition area between smaller and larger elements. Badly shaped elements i.e.,

rectangular elements that have a large ratio between the long and the short side or elementswith large blunt or small sharp angles, are often generated.

Dividing a complex area into more simple areas makes the model easier to mesh. It alsomakes it possible to choose a different element size on a specified area. This method is usedto create a fine mesh around the crack tip. This is done by creating a box around the crack tip,see Figure 3.6., containing small elements.


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Figure 3.6. Four areas create a box around the crack tip.

There is also an area division from the crack tip vertically through the material, see Figure3.6. This facilitates the meshing.

3.2.2 Elements

The elements used when meshing the models are plane stress eight node serendipity elementsand six node triangular elements. Each node has two degrees of freedom; one in thex- andone in they-direction.

3.2.3 The crack tip

When meshing the model, relatively small elements are preferred close to the crack tip. Pieelements are created around the tip (see Figure 3.7 ).

Figure 3.7. Pie elements around the crack tip.

crack opening

crack tip

1

2 4

3 5

crack tip


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To obtain stresses going to infinity at the crack tip, the side nodes on the triangular pieelements are moved to the quarter point of the element towards the crack tip.

Figure 3.8 shows an actual ANSYS plot of the crack tip elements.

Figure 3.8. Screenshot from ANSYS of the crack tip.

The first two rows of elements around the crack tip have the length 3105.0 mm.

3105.0 mm

crack tip


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3.2.4 The box

The box around the crack consists of four areas. The element length is 3102 mm in the box.An ANSYS plot over the box can be seen below (Figure 3.9).

Figure 3.9. The mesh in the box around the crack tip. Screenshot from ANSYS.

The mesh in the box is of high quality with no badly shaped elements.

crack tip


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3.2.5 The centre lines

The centre line and the crack surfaces are divided into two lines outside the box. The part ofthe line nearest to the box receives a gradually decreasing node density while the remaining

part of the lines get an element side length of 0.1 mm (see Figure 3.10).

Figure 3.10. The centre line below the box.

The same situation is applied to the two crack surfaces above the box.

One could say that the size and number of elements near the crack tip is overkill but for manycases the crack is set to move from 0.1 mm down to 5 mm. In order to build a general macrowhich can take care of the entire crack growth, this solution was regarded as a straightforwardmethod.

graduallyincreasingelement size

0.1 mm


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3.2.6 Remaining parts of the model

The remaining lines of the model get an element size of 1 mm. This is shown in Figure 3.11.

Figure 3.11. The global model is meshed with larger elements.

3.2.7 Meshing problems

Problems with badly shaped elements, i.e. rectangular elements with a large ratio between thelong and the short side or elements with large blunt and small sharp angles between the sides,appears in the transition area between the box and the global mesh. Smaller element size andgradually larger elements on the centre lines is one way of solving the problem. Another wayis to create more areas around the crack and gradually increase the element size further awayfrom the crack in these areas. This is however time consuming to carry out.

The chosen solution, as described above, is a relative fine mesh on the centre lines whichmoves the problem to the mesh generator. On some geometries so called guiding lines aredrawn in the immediate surroundings to the crack. This helps the mesh generator to create

good elements.Although large efforts have been made to control the mesh, sometimes some bad elementsappear around the box. One example of this is shown in Figure 3.12, where the light greyelements are badly shaped.

crack tip


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Figure 3.12. The light grey elements are badly shaped.

When using a smaller element side length on the global areas, 0.2 mm instead of 1 mm, thebadly shaped elements disappear. The calculation time however is about 10 to 20 times longerand the calculated stress intensity factors improve less than 1.5%. Since more than 100geometries are to be calculated the coarser mesh is used to save time.

3.2.8 Convergence

In order to guarantee convergence, the two prior test examples - the transverse butt weld andthe cruciform joint (Case 213 and 413) - are calculated with the above mentioned mesh and amesh that has half the element size on all elements except the crack elements.

Results in Table 3.3, for Case 213 and Table 3.4, show that the solution has converged. As

mentioned before, lack of time is the main factor for choosing the coarser mesh, because overone hundred geometries are to be calculated. The calculation time increase 10 to 20 timeswith the finer mesh, and that justifies the decision to use the coarser mesh.


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Table 3.3. Convergence results for transverse butt weld, Case 213.

K_eff [MPa(mm)]

Crack length [mm]: Coarse mesh: Finer mesh: Deviation:

0,0000 0,00 0,00 0,00%

0,0025 19,92 19,77 0,74%

0,0050 28,13 27,86 0,97%

0,0100 39,50 39,02 1,22%

0,0200 54,83 54,17 1,23%

0,0400 74,59 73,99 0,81%

0,0800 98,67 98,43 0,25%

0,1600 125,77 126,75 0,77%

0,3200 158,98 159,85 0,55%

0,6400 203,56 204,32 0,37%

1,2800 279,48 279,68 0,07%

2,5600 456,16 456,26 0,02%

5,1200 1182,89 1183,02 0,01%

Table 3.4. Convergence results for cruciform joint, Case 413.

K_eff [MPa(mm)]

Crack length [mm]: Coarse mesh: Finer mesh: Deviation:

0,0000 0,00 0,00 0,00%

0,0025 26,32 26,56 0,90%

0,0050 37,18 37,43 0,69%

0,0100 52,11 52,48 0,71%

0,0200 72,44 72,94 0,69%

0,0400 98,99 99,73 0,75%

0,0800 131,58 132,55 0,74%

0,1600 167,51 168,73 0,73%

0,3200 203,79 205,04 0,61%

0,6400 243,29 245,09 0,74%

1,2800 303,09 304,72 0,54%

2,5600 431,45 433,52 0,48%

5,1200 947,84 949,44 0,17%

3.2.9 Macros

After the geometry has been modelled, all of the remaining work - crack growth, areasubdivision, meshing, solving and calculation of stress intensity factors - is the same for allmodels. Therefore generic macros are developed for ANSYS.


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When the crack has propagated a small distance into the material, a generic macro looks thesame for any geometry. The differences between different geometries occur in the beginningof the crack growth, when geometrical differences like undercuts, transition radii, sharptransitions and flat sheets affect the area subdivision. Therefore all the macros with the abovementioned geometries look basically the same and differ only for the initial state of the crack.

The advantage of the macros are that they need only three keypoints as input and thenautomatically move the crack tip into the geometry, divide it into areas and put the correctmesh size on all lines. After solution, the stress intensity factors are calculated and written to afile containing the results for all the steps during the crack growth. The macros also save themeshed models for each crack depth, making it possible to review the mesh afterwards.

3.3 Postprocessing

This chapter deals with how the ANSYS calculates the stress intensity factors and how theresults from the ANSYS are treated in order to calculate the fatigue life.

3.3.1 How stress intensity factors are calculated in ANSYS

There are various ways to calculate the effective stress intensity factor for a crack, for

example the J-integral, energy release rate, nodal displacement near the crack, etc.. The latteris used to calculate the stress intensity factors in ANSYS.

Paris and Sih [13], derived the displacement near a crack for each mode. The equations belowdescribe the total displacements and have been compiled in theANSYS, Inc Theory Reference[14].

( ) ( ) ( )

( ) ( ) ( )

( )

+

=

+

+

+

=

+

+

+

=

3):(3.12sin2G

2

2):(3.12

3cos

2cos32

24G2

3sin

2sin12

24G

1):(3.12

3sin

2sin32

24G2

3cos

2cos12

24G

III

III

III

r

rK

w

rrKrK

v

rrKrK

u

where u , v , w are displacements and r, are the coordinates in the two localcoordinate system given in Figure 3.13 below.

Figure 3.13. Local coordinate systems at crack front. Picture taken fromANSYS, Inc TheoryReference[14].


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Further IK , IIK and IIIK are the stress intensity factors for the three different modes, and

+

=

stressplanefor

1

3

caseicaxisymmetranorstrainplanefor43

Gis the shear modulus, i.e.

( )+=

12

EG

whereEis Youngs modulus and is Poissons ratio. ( )r are terms of order ror higher.

The stress intensity factors are interesting when 0r and = 180, therefore the givenangle is inserted into equations (3.1:1-3) and the higher order terms are neglected. It gives

( )

( )

=

+

=

+

=

=

+

=

+

=

3):(3.222

2):(3.21

2

1):(3.21

2

2G

2

122G

122G

III

I

II

III

I

II

r

wGK

r

vGK

r

uGK

rKw

rKv

rKu

In all equations (3.2:1-3) the following limit values must be evaluated

rr

0

lim

where is either u , v or w . This is done by using the 5 nodes in the path in ANSYS,

mentioned above, see Figure 3.7 and 3.14.

Figure 3.14. (a) displays the crack tip for a half-crack model, while (b) displays the crack tipfor a full crack model, which is used in this study. Picture taken from [14].

The following curve is fitted for v .


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( ) ( ) ( )

rr

rvrrrv BABA +=

+= (3.3)

The displacement of the crack tip node is said to be zero, while the known displacement inpoints K and J are used to determine the constants A and B. The limit value becomes

( )

( ) ABAlimlim 00 =+=

rr

rv

rr (3.4)

The same thing is performed for mode II and III, which gives corresponding constants.Combining Equation (3.4) and Equation (3.2) finally gives the stress intensity factors below.

=

+=

+=

1):(3.52A2

1):(3.51

A2

1):(3.51

A2

IIIIII

IIII

II

GK

GK

GK

3.3.2 Numeric integration

As described above, the macros create a result file with the effective stress intensity factor.The results are transferred to the calculation program MATLAB [15]. In MATLAB theeffective stress intensity factor is plotted as a function of the crack length and a curve isapproximated with cubic splines [16].

The advantage of using splines is that the slope of the curve is the same for the incoming andoutgoing line of each point. This gives a smoother curve.

Efforts were made to fit a curve with different degrees of polynomial functions. Degree 3gave the best curve fit but the problem was that the curve did not go through the origin. Thisis very important when calculating the life of a structure (i.e. when performing the numericintegration) since a great deal of the life refers to small crack lengths. Therefore the curvefitted to the points needs to be more exact for small crack lengths. It is not so important whenthe crack has grown over, say, approximately 2 mm for a 10 mm sheet, since the crack

propagates rapidly at this stage and not many percent of the life is left. This is why splines arechosen to approximate the curve; they give a better curve at small crack lengths.

One other effort that was made to better describe the curve at small crack lengths was to use ageometrical series for the crack length when calculating the stress intensity factor. This meansthat many stress intensity factors were calculated for small crack lengths and only a few forlonger crack lengths. The series used was

1002

1

=

m

a [mm], 10,...,3,2,1,0,1=m

i.e., 12 stress intensity factors were calculated for each model. Ten integration points wereused below 1 mm and only 3 points above 1 mm. The extra point comes from the stressintensity factor being zero when the crack length is zero. Figure 3.15 displays the stressintensity factor as a function of crack length for Case 213.


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0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

a [mm]

K

eff

[MPa(m)]

Keff(a)

Figure 3.15. The stress intensity factor as a function of crack length for Case 213.

The lines in Figure 3.15 at 0.1 mm and 5 mm mark the integration limits. Numericalintegration in MATLAB with Simpson quadrature [16] for this case gives a life of 61048.1 cycles. The dots in Figure 3.15 mark the crack lengths where the stress intensity factor has

been calculated. The splines are based on these points.

To rule out the possibility of errors in life due to too few points for longer cracks, a test withone value for each half mm was carried out. This means that 9 integration points instead of 3are used (see Figure 3.16). A comparison of the curves and lives shows that differences in lifeare negligible.


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0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

a [mm]

Keff

[MPa(m)]

Keff(a)

Figure 3.16. The difference between many (squares) and fewer (dots) integration points forlonger cracks.

The dotted curve and the boxes in Figure 3.16 display the previous results, which werepresented in Figure 3.15. The solid curve and the squares in Figure 3.16 display what happensif more integration points are used for longer crack lengths. Between 1 and 5 mm in cracklengths, the small difference between the two curves gives a difference in life of 0.2% only.

3.4 Verification of integration method

The life is calculated using Paris law. This is done numerically by use of the function( )aKeff .

In order to verify the use of splines to approximate the ( )aKeff -curve and the numericalintegration technique, a few different approximations are made in Excel [17]. All data is takenfrom Case 213 and can be found in Table 3.5 below.


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Table 3.5. Data from Case 213, ANSYS results

a [mm]: Keff[MPa(m)]:

0.0000 0.00

0.0025 0.63

0.0050 0.89

0.0100 1.25

0.0200 1.73

0.0400 2.36

0.0800 3.12

0.1600 3.98

0.3200 5.03

0.6400 6.44

1.2800 8.84

2.5600 14.43

5.1200 37.41

The values in Table 3.5 are plotted together with the approximated splines in Figure 3.17.

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

a [mm]

K

eff

[MPa(m)]

Keff(a)

Integration points

Curve approximation with splines

Curve with straight lines

Figure 3.17. Plotted data from Case 213.

As can be seen in Figure 3.17, the difference between the spline approximated curve and thestraight line curve becomes greater for larger crack depths.


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No integration point is available for the crack length 0.1 mm, which is used as start defectwhen integrating with Paris law. However there is a point at a= 0.08 mm which will be usedin the verification of the calculated life.

MATLAB gives the life 61060.1 cycles when Simpson quadrature [16] is used to integratethe life over the interval 12.508.0 a mm.

3.4.1 Approximation with upper and lower summationsParis law gives

( )a

KCN

a

a

n

N

d1

dc

i I0

=

The curve ( )( )( )naKC

afI

1

= is to be integrated numerically in the interval of

12.508.0 a mm. Since there is not the same distance between any of the integrationpoints, the integration is done with both over and under sums. The fatigue life is thenapproximated with the average of the over and under sums. In Figure 3.18 ( )af is described.

Figure 3.18. Upper and lower summations.

The upper sum becomes

( ) ( )ii

n

in

i

aaKC

N

= += 1

1 I

u

1

and for the lower sum

a

upper sumlower sum

( )af

0

( )af based on spline values

( )af with straight lines betweenthe integration points

ia 1i+a 1ia


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( ) ( )ii

n

in

i

aaKC

N

= += +

11 1I

l

1

An approximation of the fatigue life is

2

lu NNN +

=

This is the same as a Riemann integral. If this is done for integration points 12.508.0 a mm, the fatigue life becomes 61078.1 cycles compared to 61060.1 cycles for the splineapproximation. It should be mentioned that this method overestimates the fatigue life sincethe spline based curve always will have less or equal values than the average line. However itshows that the value calculated with splines in MATLAB is approximately good.

3.4.2 Spline values integrated in Excel

A spline curve for ( )aK is created in MATLAB. The distance is 0.001 mm in crack lengthbetween each point which builds the curve, so from 0 mm to 5.12 mm it contains more than5 000 integration points. The crack length vector and the corresponding stress intensity factor

vector is transferred to Excel and integrated as a Riemann integral, according to

( ) 21 11

1 I

+

=

= ii

n

in

i

aa

KCN

Description of the Riemann integral can be found in Figure 3.19.

Figure 3.19. Approximation with Riemann sums.

( )af

a 0

( )af based on splines

ia 1i+a 1-ia


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The approximation with Riemann sums gives 61060.1 cycles i.e., the same as with theSimpson quadrature in MATLAB, which means that the integration method is verified.

3.5 Analytical case for an internal crack

A few cases in the Volvo STD5605,51 standard handle internal defect. These cases are;internal crack (Case 25), lack of fusion (Case 26) and lack of penetration (Case 27). When the

defect is set to have a maximum length in the depth direction, the analytical case for anelliptical buried flaw in a flat plate from [18] can be used. Descriptions of the case can befound in Figure 3.20 and 3.21.

Figure 3.20. An internal elliptical crack under tension.

Figure 3.21. Denotations for the crack.

This case, taken from [10], is programmed in Matlab [7]. The Matlab program can be foundin the Appendix A. The stress intensity factor is calculated as

Q

aFK

= mI

whereFcan be calculated as

w

4

4

2

21

22fgf

t

aM

t

aMMF

+

+=

i.e.,Fis a function of the angle , which means that IK can be calculated at an arbitraryposition along the crack tip. In these cases IK will be calculated for the angle 0 ( IcK ) and

90 ( IaK ), i.e. where the two extreme values of the stress concentration factor tK can be

expected. Further

2a

2c

t

2a

2c

2W

t


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=

t

a

W

cf

2

2secw

5.12

11.0

05.0

+

=

c

aM

5.14

23.0

29.0

+

=

c

aM

+

= cos41

26.2

1

4

c

a

t

a

t

a

g

The following conditions apply depending on in which direction, aor c, the ellipse is larger.For 1ca :

65.1

464.11

+=c

aQ

4/1

22

2

sincos

+

=c

af

11=M

For 1>ca :

65.1

464.11

+=a

cQ

4/1

22

2

cossin

+

=a

cf

a

cM =1

3.5.1 Crack growth rate

The crack growth is not the same in the a-direction as in the c-direction. To calculate the nextcrack length the following equation is used

+=

+=

+

+

Md

d

Md

d

i1i

i1i

N

ccc

N

aaa

(3.6)

where M is a dimensionless constant. Paris law was given in Chapter 2:


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( )nKN

aIaC

d

d= (2.1)

Paris law (2.1) can be inserted into (3.6:1) and the corresponding for (3.6:2), which gives

( )

( )

+=

+=

+

+

MC

MC

Ici1i

Iai1i

n

n

Kcc

Kaa (3.7)

The next crack lengths which the program uses to calculate the stress intensity factors in thea- and c-directions, depend on the stress intensity factor in the given direction. The constant

Mdetermines the resolution i.e., how many integration points that will be used. The greaterMis, the fewer integration points.

The stress intensity factors and corresponding crack lengths are later integrated with Parislaw, in the same way as described in Chapter 3.3.2, in order to achieve the fatigue life.

3.5.2 Algorithm

The Matlab program is based on the following algorithm:1. Start lengths of the crack in the a- and the c-direction ( ia and ic ) are given from the

acceptance limits in the weld classes.2. ia and ic are used to calculate ( )iIaK and ( )iIcK .3. The new crack lengths, 1i+a and 1i+c , are calculated based on ( )iIaK and ( )iIcK 4. The interruption criterion is checked. In this case the crack is set to grow up to a givenlength in the a-direction. If the criterion is fulfilled, all necessary stress intensity factors have

been calculated and the fatigue life can be integrated, if not, the program starts over from 2.

The interruption criterion is based on the length in the a-direction. The crack grows faster inthis direction, and in all other cases in the standard the fatigue life has been integrated fromthe initial crack length until the crack length is half of the sheet thickness. Therefore thecalculations are interrupted when the crack length ais half the sheet thickness.

3.5.3 Discussion about integration limits

It can be discussed whether the crack should be integrated to half of the sheet thickness in thea-direction. The sheet often has a larger depth in the c-direction, implying that lots of thefatigue life is left after the crack has reached the surface in the a-direction. This can be seen ifthe crack length is plotted versus the number of cycles. For cracks with the same depth as thesheet, the curve tends rapidly towards infinity when half of the sheet is left in the a-direction,implying that the rest of the life is negligible (see Figure 3.22).


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0 2 4 6 8

x 105

0

2

4

6

8

10

N [-]

2acrack

[mm

]

acrack

(N)

Figure 3.22. Crack length a(N) for an infinitely deep crack. STD5605 Case 27 D.

For an internal elliptical crack the same curve goes more slowly towards infinity at the samecrack length, basically because the crack can grow a very long distance in the c-direction (seeFigure 3.23).

0 2 4 6 8 10

x 10

5

0

2

4

6

8

10

N [-]

2acrack

[mm]

acrack

(N)

Figure 3.23. Crack length a(N) for an internal elliptical crack. STD5605 Case 27 C.

However, in order to be consistent compared to other cases, the fatigue life is integrated tohalf of the thickness also for this case.


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3.5.4 Comparison with AFGROW

AFGROW [19] is a freeware program used by the United States Department of Defence andthe aircraft industry to validate and predict the fatigue life of new and old structures. The

program is developed within the United States Air Force.

Many elementary cases are available in the program and one of its strengths is that the fatigue

life for a large number of load spectra can be calculated. There are also five different materialmodels to choose between, with lots of material parameters to use.

The elliptical internal crack is defined in AFGROW and is here used together with the Walkerequation, to confirm the values from the MATLAB program.

Comparison with AFGROW for a sheet with thickness t= 10 mm and width W= 1000 mmgives the same result for a crack which grows from a= 1 mm and c= 5 mm to a= 4.6 mm.The fatigue life becomes 1.4 million cycles. The curves from AFGROW and MATLAB have

been plotted in Figure 3.24.

Documents

Analysis of Fatigue Life in Two Weld Class Systems.pdf