Fatigue Life Analysis of Weld Ends. Comparison between finite

Fatigue Life Analysis of Weld Ends Comparison between finite element analysis and testing

Marcus Ramström

Mechanical Engineering / Solid Mechanics

Division of Solid Mechanics

Master thesis

Department of Management and Engineering

LIU-IEI-TEK-A--15/02300—SE

Fatigue Life Analysis of Weld Ends Comparison between finite element analysis and testing

Marcus Ramström

Supervisor: Maria Nygren, Toyota Material Handling, Mjölby

Richard Fyhr, Toyota Material Handling, Mjölby

Daniel Leidermark, Linköping University, Linköping

Examiner: Bo Torstenfelt, Linköping University, Linköping

Student reviewer: Hugo Inglehammar

LIU-IEI-TEK-A--15/02300—SE

Linköping University Electronic Press

M. Ramström ǀ Fatigue life analysis with respect to weld end modelling

I

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© Marcus Ramström, 2015



I


II

ABSTRACT

This master thesis investigates the fatigue life of weld ends. Fatigue life analyses have been an important

subject in many industries but the research for weld ends is limited, and often the weld ends are the most

critical parts of the weld.

The main purpose of this thesis is to find a calculation method for prediction of fatigue life of weld ends.

Today’s computational methods for fatigue life calculations are based on continuous welds. The methods

assume that the weld does not have any start or end. Therefore, focus is on how to model the weld ends

accurate in a finite element (FE) analysis, since it affects the prediction of the fatigue life. The Effective

notch approach is the calculation method at focus for this research in order to develop a new value of the

fatigue class (FAT) adjusted for weld ends. The Effective notch method is only applicable for a

continuous weld. Therefore, a comparison between the continuous weld and discontinuous weld end is

made.

The project is divided into several parts, the main parts are; fatigue testing between a continuous and

discontinuous weld end, validation against strain gauge measurements, calibration of the FE-models,

comparison between the computational methods and develop a new FAT-value for weld end based on the

Effective notch approach.

The tests result indicates higher fatigue strength for the discontinuous weld end compared with the

continuous weld. The fatigue testing is compared with FE-analysis, with focus on the Effective notch

approach. The Effective notch method with standard FAT225 MPa, estimates the fatigue life close to the

discontinuous test results. The adjusted FAT-value to match the discontinuous weld end is in the interval

194-269 MPa, dependent on actual test results or International Institute of Welding (IIW) standard norm.

The conclusion is that it is possible to simulate the discontinuous weld end with a continuous FE-model.

The continuous model can be evaluated with the developed FAT-value for the Effective notch approach.

The FE-model should include the notch radius and the standards IIW recommend.

III


IV

PREFACE AND ACKNOWLEDGEMENT

The work presented in this master thesis has been carried out at Toyota Material Handling, Mjölby and at

the Division of Solid Mechanics, Linköping University, during the spring semester 2015. The project has

been performed at the CAE-group. This thesis is a research study of fatigue life analysis of weld ends. By

this thesis, I will conclude my Master of Science in Mechanical Engineering at Linköping University,

Sweden.

First, I would like to thank my supervisors, for all their support and guidance during the project.

Maria Nygren Specialist, Finite element analysis and fatigue analysis.

Toyota Material Handling

Richard Fyhr Manager CAE-group

Toyota Material Handling

Daniel Leidermark Associate, Division of Solid Mechanics.

Linköping University. Linköping Institute of Technology.

I would also like to thank the CEA-group and the coworkers I meet at Toyota Material Handling for their

support and interesting discussions and my opponent, Hugo Ingelhammar, for rewarding opinions and

discussions.

Mjölby 2015-06-04

Marcus Ramström

V

M. Ramström ǀ Chapter 1 - Introduction

VI

NOMENCLATURE

𝐸 = 𝑌𝑜𝑢𝑛𝑔′𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 [𝐺𝑃𝑎] 𝜈 = 𝑃𝑜𝑠𝑠𝑖𝑜𝑛′𝑠 𝑟𝑎𝑡𝑖𝑜 [−] 𝜌 = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 [𝑘𝑔/𝑚3] 𝑅𝑝0.2 = 0.2 % 𝑜𝑓𝑓𝑠𝑒𝑡 𝑦𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ [𝑀𝑃𝑎]

𝑅𝑚 = 𝑈𝑙𝑡𝑖𝑚𝑎𝑡𝑒 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ [𝑀𝑃𝑎] 𝜎𝑟 = 𝐶𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑛𝑔𝑒 [𝑀𝑃𝑎] ∆𝜎𝑟𝑑 = 𝐷𝑒𝑠𝑖𝑔𝑛 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑛𝑔𝑒 [𝑀𝑃𝑎] 𝜎𝑛𝑜𝑚 = 𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 [𝑀𝑃𝑎] 𝜎ℎ𝑠 = 𝐻𝑜𝑡 𝑠𝑝𝑜𝑡 𝑠𝑡𝑟𝑒𝑠𝑠 [𝑀𝑃𝑎] ∆𝜎𝑚𝑎𝑥 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑛𝑔𝑒 [𝑀𝑃𝑎] 𝜎𝑚𝑎𝑥 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑡𝑟𝑒𝑠𝑠 [𝑀𝑃𝐴] 𝜎𝑚𝑖𝑛 = 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑠𝑡𝑟𝑒𝑠𝑠 [𝑀𝑃𝐴] 𝐹𝐴𝑇 = 𝐹𝑎𝑡𝑖𝑔𝑢𝑒 𝑐𝑙𝑎𝑠𝑠 [𝑀𝑃𝑎] 𝑅 = 𝑆𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑡𝑖𝑜

𝑎 = 𝑇ℎ𝑟𝑜𝑎𝑡 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 [𝑚] 𝑁 = 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑓𝑎𝑡𝑖𝑔𝑢𝑒 𝑙𝑖𝑓𝑒 [𝑐𝑦𝑐𝑙𝑒𝑠] 𝑁𝑡 = 𝐷𝑒𝑠𝑖𝑔𝑛 𝑙𝑖𝑓𝑒, 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑦𝑐𝑙𝑒𝑠 [𝑐𝑦𝑐𝑙𝑒𝑠] 𝑛𝑡 = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑦𝑐𝑙𝑒𝑠 [𝑐𝑦𝑐𝑙𝑒𝑠] 𝑛𝑖 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑦𝑐𝑙𝑒𝑠 𝑎𝑡 𝑙𝑜𝑎𝑑 𝑙𝑒𝑣𝑒𝑙 𝑖 [𝑐𝑦𝑐𝑙𝑒𝑠] 𝜑𝑡 = 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟

𝜑𝑚 = 𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑓𝑎𝑐𝑡𝑜𝑟

𝜑𝑒 = 𝑆𝑡𝑟𝑒𝑠𝑠 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟

𝜑𝑄 = 𝑅𝑖𝑠𝑘 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

𝛾𝑚 = 𝐹𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑛𝑠𝑒𝑞𝑢𝑛𝑐𝑒 𝑓𝑎𝑐𝑡𝑜𝑟

𝑠𝑚 = 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟

∆𝜎𝑖 = 𝑆𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑛𝑔𝑒 𝑎𝑡 𝑙𝑜𝑎𝑑 𝑙𝑒𝑣𝑒𝑙 𝑖 [𝑀𝑃𝑎] ∆𝜎𝑟𝑒𝑓 = 𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑡𝑟𝑒𝑠𝑠 𝑟𝑎𝑛𝑔𝑒 [𝑀𝑃𝑎]

𝑘𝑚 = 𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑣𝑒 𝑓𝑎𝑐𝑡𝑜𝑟

𝑚 = 𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡, 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑆/𝑁 𝑐𝑢𝑟𝑣𝑒

𝑓 = 𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑢𝑠𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟

∆𝐿 = 𝐷𝑒𝑙𝑡𝑎 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑙𝑒𝑛𝑔𝑡ℎ 𝑐ℎ𝑎𝑛𝑔𝑒 𝑓𝑟𝑜𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑙𝑒𝑛𝑔𝑡ℎ. [𝑚] 𝐿 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ [𝑚] 𝛼 = 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 [1/°C]

∆𝑇 = 𝐷𝑒𝑙𝑡𝑎 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑓𝑟𝑜𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 [°𝐶] 𝐷 = 𝑃𝑎𝑟𝑖𝑠′𝑙𝑎𝑤 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑔 = 𝑃𝑎𝑟𝑖𝑠′𝑙𝑎𝑤 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

ABBREVIATIONS

𝐶𝐴𝐸 = 𝐶𝑜𝑚𝑝𝑢𝑡𝑒𝑟 𝑎𝑖𝑑𝑒𝑑 𝑒𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝐹𝐸 = 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡 𝐹𝐸𝑀 = 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡 𝑀𝑒𝑡ℎ𝑜𝑑

𝐼𝐼𝑊 = 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐼𝑛𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑜𝑓 𝑊𝑒𝑙𝑑𝑖𝑛𝑔

𝑑. 𝑜. 𝑓 = 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚

𝑠𝑡𝑑𝑣. = 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛


VII

Table of Contents

CHAPTER 1 ................................................................................................................................................ 1

INTRODUCTION ....................................................................................................................................... 1

1.1 Background ................................................................................................................................... 1

1.2 Toyota Material Handling ............................................................................................................. 2

1.3 Problem description ....................................................................................................................... 3

1.4 Purpose .......................................................................................................................................... 4

1.5 Goal ............................................................................................................................................... 4

1.6 Question at issue ............................................................................................................................ 4

1.7 Delimitations ................................................................................................................................. 5

1.8 Method........................................................................................................................................... 7

1.8.1 Literature study ...................................................................................................................... 7

1.8.2 Finite element analysis .......................................................................................................... 7

1.8.3 Fatigue testing ....................................................................................................................... 7

1.9 Finite element software ................................................................................................................. 8

1.10 Other considerations ...................................................................................................................... 8

CHAPTER 2 ................................................................................................................................................ 9

THEORY ...................................................................................................................................................... 9

2.1 Introduction of fatigue ................................................................................................................... 9

2.1.1 Fatigue class ........................................................................................................................ 10

2.1.2 S/N Curve ............................................................................................................................ 10

2.2 Welds ........................................................................................................................................... 13

2.2.1 Welding classes ................................................................................................................... 14

2.3 How the material is affected by welding ..................................................................................... 14

2.3.1 Residual stresses .................................................................................................................. 14

2.3.2 Deformation due to weld shrinkage..................................................................................... 15

2.3.3 Crack initiation .................................................................................................................... 15

2.4 Welds affect in structures ............................................................................................................ 16

2.5 Welds effect on fatigue life ......................................................................................................... 16

2.6 Fatigue design of welded joints and components ........................................................................ 17

2.6.1 Nominal stress method ........................................................................................................ 18

2.6.2 Geometrical stress / Hot-spot method ................................................................................. 20

2.6.3 Effective notch method ........................................................................................................ 21

2.6.4 Generally for the methods ................................................................................................... 23

2.6.5 Fracture mechanics .............................................................................................................. 24

2.6.6 Evaluation of test data ......................................................................................................... 26


VIII

CHAPTER 3 .............................................................................................................................................. 27

FATIGUE TESTING ................................................................................................................................ 27

3.1 Fatigue testing ............................................................................................................................. 27

3.1.1 Weld properties ................................................................................................................... 27

3.1.2 Measuring before testing ..................................................................................................... 29

3.1.3 Test setup ............................................................................................................................. 30

3.2 Strain gauge test .......................................................................................................................... 30

3.2.1 Strain gauge test results ....................................................................................................... 32

Test program............................................................................................................................................ 35

3.3 Results from testing ..................................................................................................................... 35

CHAPTER 4 .............................................................................................................................................. 39

FINITE ELEMENT ANALYSIS ............................................................................................................. 39

4.1 FE-model ..................................................................................................................................... 39

4.2 Model Description ....................................................................................................................... 39

4.2.1 Material ............................................................................................................................... 40

4.2.2 Analysis setup ...................................................................................................................... 40

4.3 Mesh convergence study ............................................................................................................. 40

4.3.1 Elements .............................................................................................................................. 40

4.3.2 Convergence study .............................................................................................................. 40

4.4 Boundary conditions and Load cases study ................................................................................. 43

4.4.1 Comparison study for nonlinear geometric ......................................................................... 45

4.5 Validation of FE-model ............................................................................................................... 47

4.5.1 Boundary condition, Load case and load level .................................................................... 47

4.5.2 Simulation of the pre-stress ................................................................................................. 47

4.6 Numerical analysis ...................................................................................................................... 49

4.6.1 Effective notch model.......................................................................................................... 49

4.6.2 Hot-spot model .................................................................................................................... 51

4.6.3 Nominal stress method ........................................................................................................ 53

CHAPTER 5 .............................................................................................................................................. 54

RESULTS ................................................................................................................................................... 54

5.1 Test results ................................................................................................................................... 54

5.2 Comparison of the computational methods ................................................................................. 55

5.3 Comparison study between FE-analysis and tests ....................................................................... 56

5.4 Developed FAT-value ................................................................................................................. 57

5.4.1 Discontinuous weld end ...................................................................................................... 58

5.4.2 Continuous weld end ........................................................................................................... 59


IX

CHAPTER 6 .............................................................................................................................................. 61

CONCLUSIONS ........................................................................................................................................ 61

6.1 Tests ............................................................................................................................................ 61

6.2 Comparison of computational methods and tests ........................................................................ 61

6.3 Comparison of the Effective notch method and tests .................................................................. 61

6.4 Developed new FAT-value.......................................................................................................... 62

CHAPTER 7 .............................................................................................................................................. 63

DISCUSSION............................................................................................................................................. 63

CHAPTER 8 .............................................................................................................................................. 65

FUTURE WORK ...................................................................................................................................... 65

BIBLIOGRAPHY ..................................................................................................................................... 67

APPENDIX ................................................................................................................................................ 69

APPENDIX A ......................................................................................................................................... 69

Drawings for the specimens ................................................................................................................ 69

APPENDIX B.......................................................................................................................................... 74

Strain Gauges placement ..................................................................................................................... 74

APPENDIX C.......................................................................................................................................... 76

Nominal stress method ........................................................................................................................ 76


X

List of Figures Figure 1. a. Continuous T-joint fillet weld. b. Zoom in on continuous fillet weld. c. Discontinuous T-joint

fillet weld. d. Zoom in on discontinuous fillet weld. ..................................................................................... 5

Figure 2. Specimens dimension. a) Discontinuous weld end. b) Continuous weld end. ............................... 6

Figure 3. Flowchart that describes the projects procedure. ........................................................................... 7

Figure 4. Abaqus work procedure. ................................................................................................................ 8

Figure 5. S/N curve with key positions. [13] ............................................................................................... 11

Figure 6. Weld zones. [15] .......................................................................................................................... 13

Figure 7. To the left is a real fillet weld profile, [7] to the right is a simplified fillet weld profile, with

marked throat thickness, referred as a, and 45 degree angles. ..................................................................... 14

Figure 8. Fatigue fracture modes in a fillet weld. [7] .................................................................................. 16

Figure 9. Comparison between the different methods used for fatigue life design. .................................... 17

Figure 10. To the left, points for linear extrapolation and to the right, points for quadratic extrapolation. 21

Figure 11. Fictitious notch radii. ................................................................................................................. 22

Figure 12. FE-model with notch radius. ...................................................................................................... 22

Figure 13. Illustrates the crack propagation rate as a function of stress intensity range. [12] ..................... 25

Figure 14. Robot welding. To the left a discontinuous string. .................................................................... 27

Figure 15. Weld drawing for the remade continuous weld end. .................................................................. 28

Figure 16. End sides for the continuous weld design. To the left side A, and to the right side B. .............. 28

Figure 17. Continuous weld, to the left the first string, to the right the second string. ............................... 28

Figure 18. End sides for the discontinuous weld end design. ...................................................................... 29

Figure 19. Specimens during measurements and markup. .......................................................................... 29

Figure 20. The heat affects. The above is the discontinuous end, and the below is the continuous end. .... 29

Figure 21. Tensile test machine. .................................................................................................................. 30

Figure 22. Red lines indicate strain gauge placements. Left, continuous weld end. Right, discontinuous

weld end. ..................................................................................................................................................... 31

Figure 23. Strain gauge placements for the discontinuous weld end. ......................................................... 32

Figure 24. Strain gauge placements for the continuous weld end. To the left side A and to the right side B.

..................................................................................................................................................................... 32

Figure 25. The recorded pre-stresses for the discontinuous weld end. Four gauges in the figure; OS_AH_3,

OS_AV_3, OS_BH_3 and OS_BV_3. ........................................................................................................ 34

Figure 26. The start of the load level. Four gauges in the figure; OS_AH_3, OS_AV_3, OS_BH_3 and

OS_BV_3. ................................................................................................................................................... 34

Figure 27. Overview for the discontinuous weld end, all four load levels. Four gauges in the figure;

OS_AH_3, OS_AV_3, OS_BH_3 and OS_BV_3. ..................................................................................... 35

Figure 28. Discontinuous weld end. ............................................................................................................ 36

Figure 29. Continuous weld end. ................................................................................................................. 36

Figure 30. Example of cracks. To the left a continuous, and to the right a discontinuous. ......................... 37

Figure 31. a. T-joint fillet weld with continuous weld end b. T-joint fillet weld with discontinuous weld

end. .............................................................................................................................................................. 39

Figure 32. The path starts in the weld toe and continuous 14 mm perpendicular against the weld. ........... 40

Figure 33. Illustrates Mesh 1, Mesh2, Mesh 3 and Mesh 4. The grey area indicates stresses above 225

MPa. ............................................................................................................................................................ 41

Figure 34. Result for the different mesh qualities. ...................................................................................... 42

Figure 35. a. B.C. and load case 1. b. B.C. and load case 2. c. B.C. and load case 3. d. B.C. and load case

4. e. B.C. and load case 5. ........................................................................................................................... 44

Figure 36. Results for B.C. and load case study. (F = 120 kN) ................................................................... 45

Figure 37. Path for Nlgeom study. (F= 240kN, R=0.1, mesh size = 0.5mm) ............................................. 46

Figure 38. Comparison study between Nlgeom and strain gauge measurements. ...................................... 46

Figure 39. Base plate with curvature. .......................................................................................................... 47


XI

Figure 40. Boundary condition for simulating mounting in the test machine. ............................................ 48

Figure 41. Pre-stress due to mounting in the test machine. ......................................................................... 48

Figure 42. Deformation scale factor is set to 25. ......................................................................................... 48

Figure 43. Sub-model dimensions and notch refinement. Notch radius = 1 mm, (toes and root), Green

radius = 2 mm. ............................................................................................................................................. 50

Figure 44. Sub-model dimensions. .............................................................................................................. 50

Figure 45. Sub-model mesh, element size 0.25 mm in the notches. ........................................................... 50

Figure 46. Effective notch result. Grey area indicates stress above 225 MPa. (F=120, Mesh size=0.25

mm) ............................................................................................................................................................. 51

Figure 47. Mesh Hot-spot model. ................................................................................................................ 52

Figure 48. Hot-spot results. Result points are marked with red dots, distance 4, 9, 10 and 14 mm from the

weld toe. ...................................................................................................................................................... 52

Figure 49. Comparison between the continuous and discontinuous test results. ......................................... 54

Figure 50. Comparison between the computational methods for the continuous weld end. ....................... 55

Figure 51. Comparison between the Effective notch approach and the tests. ............................................. 56

Figure 52. Developed S/N curve discontinuous, characteristic curve with slope 3 and stdv. 0.17. ............ 58

Figure 53. Developed S/N curve discontinuous, characteristic curve with slope 3 and stdv. 0.034. .......... 58

Figure 54. Developed S/N curve discontinuous, characteristic curve with slope 3.13 and stdv. 0.034. ..... 59

Figure 55. Developed S/N curve continuous, characteristic curve with slope 3 and stdv. 0.17. ................. 59

Figure 56. Developed S/N curve continuous, characteristic curve with slope 3 and stdv. 0.052. ............... 60

Figure 57. Developed S/N curve continuous, characteristic curve with slope 3.42 and stdv. 0.052. .......... 60

List of Tables Table 1. Risk of failure coefficient with slope, m = 3. (Konstruktionshandbok, 2014, [9]) – Table 9-6. ... 12

Table 2. Recommended values of n according to SSAB's research. (SSAB, 2011) – Table 5.9. .............. 19

Table 3. Stress variation factor. (SSAB, 2011) – Eq. 5.12 -5.14. ............................................................... 19

Table 4. IIW's proposal of values for the partial coefficient in different situations. [3] ............................. 19

Table 5. k-value dependent of number of test results. [3] ........................................................................... 26

Table 6. Stresses from the strain gauge tests. .............................................................................................. 33

Table 7. Test program. ................................................................................................................................ 35

Table 8. Discontinuous test data. ................................................................................................................. 36

Table 9. Continuous test data. ..................................................................................................................... 36

Table 10. Test results. .................................................................................................................................. 37

Table 11. Material properties. ..................................................................................................................... 40

Table 12. Mesh levels for the mesh study. .................................................................................................. 41

Table 13. Load levels. (B.C. and load case 3) ............................................................................................. 47

Table 14. Effective notch results. ................................................................................................................ 51

Table 15. Hot-spot results. Linear and Quadratic. ....................................................................................... 52

Table 16. Nominal stress method results, Continuous weld end. ................................................................ 53

Table 17. Nominal stress results, Discontinuous weld end. ........................................................................ 53

Table 18. Test data. ..................................................................................................................................... 55

Table 19. FAT-value. .................................................................................................................................. 55

Table 20. Cycles to failure. Effective notch calculations and test data. ...................................................... 56

Table 21. Data for new developed FAT-value, discontinuous. ................................................................... 58

Table 22. Data for new developed FAT-value, continuous. ........................................................................ 59


1

CHAPTER 1

INTRODUCTION

This chapter is an introduction to the project, describing the background, problem, purpose and method.

1.1 Background

Most failures in load carrying structures designed with welds are caused by fatigue failure. This is due to

the strength against cyclic loading, particularly around the welds, which is significantly lower than the

static strength. This ratio is difficult to consider but important when designing welded structures.

Fatigue is a process where the material gradually breaks down by cyclic loading and eventually fails. The

process starts with crack initiation and further crack propagation in areas with high local stress. Fatigue

cracks can be difficult to detect, because it is a local phenomenon characterized by small plastic

deformations. [1]

A welded design’s fatigue life is affected by varying factors whose effects are difficult to predict. Fatigue

strength is therefore regarded as an empirical science and knowledge in the subject has been built through

extensive testing. The factors with the greatest impact on the fatigue strength are; number of cycles,

distribution and intensity of loads and the degree of notch action. Other factors are the steels material’s

static strength, surface quality, weld defects, mean stress, residual stresses, plate thickness, temperature

and corrosion. [1] Fatigue failure in welded designs is usually initiated in the transition zone between the

base material and the weld, but also in the weld root, weld end and by defects in the weld. [2]

For fatigue design of steel structures there are some developed standards. These are primarily for

structures, such as steel building constructions, cranes, ships and pressure vessels. The standards are

developed by the International Institute of Welding, IIW and American Welding Society, AWS. These

standards can only be used to a certain extent, and can for instance not be used to calculate fatigue life of

weld ends. Many industries lack standards for fatigue design of welded structures. [1] This is a major

research area which needs to be investigated more deeply, especially regarding weld ends.

It has been shown in research that a weld with a strict perpendicular end, a so called discontinuous end,

provide approximately equivalent fatigue life as a continuous end. This similarity is investigated deeper in

this thesis through comparison studies between finite element (FE) analysis and tests. [3]


2

1.2 Toyota Material Handling

Toyota Material Handling Europe (TMHE) is the European part of the world wide concern Toyota

Material Handling Group (TMHG). TMHG is today the world leader in material handling. [4]

BT Products was founded in 1946 by Ivan Lundqvist in Stockholm, with the main focus to simplify the

movement of the raw material at workplaces. The workplaces in focus were then new production of

houses, apartments and production facilities. The company’s name was originally Bygg- och

Transportekonomi, BT, (Construction and Transport economics). In the beginning the company started to

import forklifts from United States (Clark counterbalanced forklifts) for the Swedish construction

industry. The market needed these new products and the production started. The first BT hand pallet

forklift left the factory in 1948. The market was in need of this new product and the company continued to

grow. In 1949 BT Products created the BT-pallet, together with Statens Järnvägar, SJ, due to the big

amount of material moved by railway. The purpose of the cooperation was to construct a design that could

carry as much material as possible and fit in the freight trains. Then the same pallet standard begun to be

used all over Europe and the BT-pallet were renamed to Euro-pallet. [4] In 1952 BT Products moved their

organization to Mjölby, Sweden. Larger facilities were needed and good connection to the railway was

important. New products were developed to satisfy customer’s demands. This good development of the

company opened up for new opportunities for an increasing market. BT Products were then established all

over Europe, but it was still difficult to reach the market outside Europe, especially in United States. [4] A

major step in the development was in 1978 when BT Products introduced the first driverless forklift. [4]

BT Products bought up Raymond in the United States and BT thereby expanded its market.

In 2000 BT Products became a part of the Toyota family after a two years long cooperation with Toyota.

BT Products have since then belonged to the Toyota Industry Corporation (TICO). Today TICO has four

primary areas; Automotive, Material handling, Electronics and Textile machines. The material handling

organization is mainly composed of the Toyota Material Handling Group (TMHG), where Toyota

Material Handling Europe represents TMHG in Europe.

BT Products also bought up CESAB and their main suppliers LTE in Italy and further expanded its

market.

TMHG has development and research in Japan, Sweden, USA and Italy. TMHE has three production

centers in Europe, placed in Sweden, France and Italy. The main office for TMHE is in Mjölby, Sweden.

Toyota Material Handling Europe is the new name of the organization in Mjölby, but the name BT

Products lives on, largely because of the reputation BT Products has built up. [5]


3

1.3 Problem description

Welded joints are commonly used in BT Product's forklifts. The welded joints are designed against both

fatigue failure and static failure. BT Product works with accurate fatigue analysis including IIW standards,

FE-analysis, load analysis and fatigue testing.

There are mainly four methodologies for calculating the fatigue life; Nominal stress method, Geometrical

method also called Hot-spot method, Effective notch method and Fracture mechanics. The Nominal

method and the Hot-spot method are used for simple weld geometry and are only applicable for estimating

the fatigue life of the weld toe. The Effective notch method can manage more complex geometries and

evaluates both the weld toe and root. The Effective notch method is classified as a more accurate method

then the Nominal method and Hot-spot method. The problem with the Effective notch method is that

singular points should be avoided in the FE-model, singularities are avoided by a fictitious radius in the

toe and root. However, there is no method for avoiding singularities in the weld ends. [6] The fracture

mechanics approach is often the most accurate method but it requires significant effort and knowledge,

such as crack length, weld defects and material properties [7]. The weld can be modelled in different ways

in FE-analysis, for example with different detail accuracy such as triangle or convex/concave weld profile,

with/without notch radius and rounded end. The most critical areas in the FE-model are the weld’s start

and end, due to the singularities. The four methods handle this problem differently, but no method is

suitable for weld ends. The Nominal stress method only considers the nominal stress and has different

factors depending on the case. The Hot-spot method considers the strain on two or three points in front of

the weld toe, usually calculated through a FE-analysis.The Effective notch method considers the stress in

the transition zone between the weld and the base material. The transition is rounded with a fictitious

radius recommended by International Institute of Welding, IIW. The Effective notch and the Hot-spot

require that the max principle stress should be perpendicular to the weld toe, but it gives valid results for

max principle stress direction oriented ± 60°, illustrated in Figure 1. Since, this project is a research for

weld ends with the stress direction 90° to the weld toe the original method cannot be used.

Figure 1. The stress direction oriented ± 60° perpendicular to the weld toe.

Today at BT Products the weld is modelled with an isosceles triangle profile with a strict perpendicular

end, so called discontinuous end. No modifications are made at the start or end of the weld. The Effective

notch method with fictitious radius is mainly used at BT Products. They have developed a method to

evaluate the result, where the crucial maximum principle stress is taken in the notch radius at the third

corner node from the welds start or end, notice the node distance is fixed. The problem is that it does not

always provide an accurate stress level, depending on weld geometry, load direction etc. Therefore, the

aim of this project is to develop a modeling technique and evaluation method for fatigue life prediction of

weld ends.


4

1.4 Purpose The main purpose of this thesis is to develop a method for prediction of fatigue life of weld ends. There

are accurate methods and recommendations from IIW and AWS, for how to compute fatigue life of welds,

but for weld ends these methods are not applicable [1].

1.5 Goal The project’s objective is to develop a modeling technique and evaluation method for fatigue life

prediction of weld ends. The focus is on how to model the weld ends correct in a FE-analysis, since the

modeling affects the prediction of the fatigue life, and to find the right FAT-value for the weld ends, FAT-

value is described in chapter 2.2.1 Fatigue class,. The aim is to guarantee that the calculation method used

to estimate fatigue life match the welds produced in BT Product’s production. This enables more accurate

fatigue life calculations at the dimensioning level in the design process. Overall, this leads to better

product quality, lower development cost, lower claim cost, shorter development time and increased safety

in design.

1.6 Question at issue

Q1 - How exact are the methods estimating the fatigue life of the weld ends?

Q2 - What affect the differences of the results from the methods?

Q3 - Is there a similarity between discontinuous and continuous weld ends regarding fatigue

strength?

Q4 - Can the similarity be used to calculate the fatigue life of weld ends?

Q5 - Does the FAT-value need to be adjusted for dimensioning weld ends?


5

1.7 Limitations

All the specimens investigated in this project have the same welded geometry, T-joint fillet weld, shown

in Figure 2. The difference is two types of weld design; discontinuous and continuous weld ends. The first

welding case is a continuous weld end, shown in Figure 2 a. and Figure 2 b. The second case is a

discontinuous weld end, shown in Figure 2 c. and Figure 2 d. That is the original weld design used in BT

Products’.

Load case is limited to alternating tensile load in the base plate’s horizontal direction, z-direction in Figure

2 a. and c. The tests are limited to one load case at four different load levels.

a. b.

c.d.

Figure 2. a. Continuous T-joint fillet weld. b. Zoom in on continuous fillet weld. c. Discontinuous T-joint fillet weld.

d. Zoom in on discontinuous fillet weld.


6

Geometry The dimensions of the specimen are shown in Figure 3. The fillet weld is made as two types, continuous

respectively discontinuous weld ends. The welds throat thickness is 5 mm and welding class C.

Figure 3. Specimens dimension. a) Discontinuous weld end. b) Continuous weld end.

a.

b.


7

Assumptions - The material properties are assumed isotropic and linear elastic in the FE-analysis. Thus, no

plasticity is considered. This is also how the evaluation methods are designed. Since it is the stress

range that gives fatigue life and not the stress level.

- The weld and the base material have matching material properties in the FE-analysis. The applied

material properties are for the flat iron steel used in the specimens. Material data is taken from BT

Product’s material data base.

- The FE-models mesh convergence is not investigated. The element size is according to IIW

recommendations for each method.

- No properties from the welding operation are considered in the FE-models.

- The residual stresses and the pre-stresses due to the welding process and deformed test specimens

are not considered, since they have very little influence at fatigue testing. The pre-stresses can be

seen in Figure 27.

1.8 Method This project includes three main parts; literature study, FE-analysis and fatigue testing. The thesis

procedure is described in the flowchart shown in Figure 4.

1.8.1 Literature study The project starts with a pre-study of literature, articles and further research in the subject. It is performed

to see where the development is today and how the problem could be tackled. It will also provide a

theoretical knowledge and understanding of the subject. The theories used for validation of the FE-model

are described in chapter 2.6 Fatigue design of welded joints and components.

1.8.2 Finite element analysis The simulations are performed with a FE-program. Different weld designs are analyzed with varying load

levels. The FE-models are analyzed with both ideal geometrical conditions and with misalignments. The

FE results are used in the different evaluation methods to calculate fatigue life of the weld. The focus has

been on the Effective notch method, but the Nominal stress method and Hot-spot method are also evaluated

to receive a comparison. The simulation results are then compared with the test results plotted in an S/N

curve.

1.8.3 Fatigue testing The testing is performed by clamping the welded specimens in a fatigue test machine. The specimens are

loaded with a repeating tensile load until fracture. The machine is set up with a maximum force and a fix

stress ratio. The specimens are tested at four different load levels and several specimens are tested at each

load level. The test results are used for a comparison study with the FE-analysis plotted in an S/N curve. A

Literature

study

FE-

analysis

Fatigue

testing

Calibration

/

Validation

Evaluation

Figure 4. Flowchart that describes the projects procedure.


8

specimen for each weld design, continuous and discontinuous weld end, sets up with strain gauges. This

test results are used to validate the FE-model.

1.9 Finite element software In general, the finite element simulation procedure can be divided in three main stages; a pre-processor

where the model’s geometry and setup is defined, a solver where the calculations are performed and a

post-processor where the results can be visualized and studied.

The software used in this project is Abaqus CAE version 6.14-1. Abaqus CAE procedure includes these

three stages, as illustrated in Figure 5. First geometry is created in, or imported to, the pre-processor. In

this stage different conditions are assigned to the model, for example; initial conditions, material

properties, boundary conditions, mesh, etc.

After the pre-processor is done, the job is submitted for analysis and an input file is created. The input file

is named; jobname.inp. In the input file all data are defined in option blocks, option blocks include two

main types of data; Model data and History data. Model data describes specific aspects of the problem

definition, such as initial conditions, element and boundary conditions, etc. History data describes the

option blocks; analysis procedure, loading, output request. Abaqus solver reads the input file and performs

the analysis. The solver sends back an Abaqus Output Database-file, jobname.odb. The output can then be

presented in Abaqus post-processor, named Abaqus Viewer. In the Abaqus Viewer the OBD-file can be

processed with different plug-ins. Furthermore, results can also be directly extracted using Python scripts

from the output-file.

1.10 Other considerations No ethical or gender specific issues are raised or discussed in the project. Nor is it directly related to

questions concerning the environment or sustainable development.

Pre-processor

Abaqus/CAE-file (jobname.cae)

Solver

(Abaqus)

Post-processor

(Abaqus Viewer or Abaqus CAE)

INP-file (jobname.inp)

ODB-file (jobname.odb)

Figure 5. Abaqus work procedure.

M. Ramström ǀ Chapter 2 - Theory

9

CHAPTER 2

THEORY

This chapter describes the basic theory of fatigue, welds and calculation methods.

2.1 Introduction of fatigue Fatigue is a localized process when cracks initiate and grow to failure under cyclic and fluctuating loads.

Fatigue begins at stress concentrations. In general, a design is exposed for fatigue if the number of load

cycles is above 1000 repetitions. [8]

A design can be exposed to several million cycles before failure eventually occurs. Fatigue of this kind is

a significant problem in areas around rotating machines such as propellers, and other areas exposed to

constant vibration. Fatigue is also a problem in structures exposed to dynamic and fluctuating loads over a

long period, which is often the case for components in construction machines such as forklifts. [9] 70 %

of all machine failure is due to fatigue failure [10].

High cycle fatigue includes analyzes of infinite and finite fatigue life, typically with number of load cycles

over 100 000. The loading only gives elastic responses. Low cycle fatigue includes analyzes for only finite

failure, typically with number of load cycles lower than 100 000. Cyclic stresses are close to or at the

materials yield limit, thus significant degree of global and local plasticity. [11]

The fatigue process contains three main phases: crack initiation, crack propagation and residue fracture.

Crack initiation and further crack propagation begins when the initiated crack is subjected to dynamic

loadings. The first phase is crack initiation, where microscopic cracks occur in the material’s microscopic

structure. Cracks can be initiated from loads, damages and/or machining etc. High cycle fatigue has two

main initiation causes; stress concentrations and pile-up of dislocations, both are connected to the

metallurgical microstructure. Stress concentrations are caused by inclusions, initial cracks, pores, grains

and corrosion pits etc. This leads to increased stress level locally. The mechanism pile-up of dislocations

is because of persistent slip bands that become the nucleus of cracks. This results in a local decrease in

fatigue strength. The dominating initiation cause depend on the purity of the material, loading, surface

roughness and surface defects/scratches etc. [12]

When the crack is initiated the fatigue process will continue into the next phase, crack propagation. The

three most important factors for crack propagation are: stress intensity range, stress intensity ratio and

stress history. Fracture mechanics analysis can be performed to predict the crack propagation, for

example with Paris’ law. With Paris’ law number of cycles before failure, maximum crack length for the

corresponding load or the specific threshold stress intensity can be calculated. More information about

crack propagation and calculation methods can be found in the chapter 0


10

Fracture mechanics. [12]

Last phase is residue fracture, which is a type of ultimate failure/fracture. Fracture occurs when a crack

propagates through the material or the weld. This will result in one or several break points.

Fatigue life is affected of environmental conditions, primarily by corrosion, known as corrosion fatigue,

and temperature. Corrosion fatigue will initiate cracks due to corrosion pits because of stress

concentrations. Corrosion fatigue tests indicate a significant reduction of fatigue strength. High service

temperature also reduces the fatigue strength. [8]

Fatigue cracks are sensitive and occur mostly during tensile stress or alternating stresses, but fatigue

cracks have occurred during compressive loads due to positive residual stresses [12]. The fatigue process

is cumulative, the material do not recover when rested. [9]

2.1.1 Fatigue class Fatigue class (FAT) also known as property classes, is defined as the characteristic fatigue strength at

constant stress range at 2∙106 load cycles. The characteristic fatigue strength has the unit N/mm

2. The

fatigue class value can be referred to as FAT or C, dependent on literature. Property classes have been

developed through fatigue testing of various welded joints. Test results have been complied over time,

which has formed the property classes. Property classes are mainly used for fatigue life calculations with

the Nominal stress method. [1]

2.1.2 S/N Curve The S/N curve, also called Wöhler curve, is a diagram that illustrates fatigue strength, see Figure 6. The

S/N curve shows the number of load cycles to failure as a function of certain stress range. The stress range

is more appropriate to use for welded joints than the stress amplitude, and therefore, stress range is used

hereafter. The S/N curve is created based on test results plotted in a logarithmic diagram. In order to

obtain reliable values, at least five identical specimens should be tested at each stress range. [9]

The assumption of using stress range instead of stress amplitude for welds is because of the residual

stresses in the weld. Residual stresses can occur in both tension and compression, which leads to difficult

calculations of how the loading and the residual stress counteract or contribute to each other. Calculations

made with the stress range give a more conservative fatigue life result. [13]


11

Figure 6. S/N curve with key positions. [13]

The quantities in Figure 6 are, the number of load cycles to failure, 𝑁𝑓, the fatigue limit, 𝜎𝐹𝐿 , the ultimate

strength, 𝜎𝑈𝑇𝑆 , the stress range, ∆𝜎 , and the curve slope, m.

The S/N curve is used to design the fatigue life of a certain component, either dependent on dimensioning

for a certain fatigue life, e.g. infinite life, or for a given stress range. [1]

The fatigue limit is a stress limit where the detail has a theoretical infinite life at constant stress range, thus

no fatigue failure occurs. When the fatigue limit is reached, this varies between different standards, but is

usually around 107 cycles for welded joints and 10

8 cycles for base material not affected by welding.

These values are correct according to IIW recommendations. [9]

Characteristic fatigue strength is used to take into account some scatter in the test samples. It is often used

when dimensioning with FAT-values. If a large number of tests are carried out, the characteristic fatigue

strength can be determined by the risk of failure coefficient, 𝜑𝑄 , see Table 1.

The risk of failure coefficient, 𝜑𝑄 , in Table 1 is only valid in the area where the S/N curve has the gradient

-1/3, m=3. See values for gradient -1/5, m=5 in the book; Konstruktionshandbok för svetsade produkter i

stål, 2014, page 291. [9]

For example, the stress range for the mean fatigue strength minus two standard deviations, stdv.,

calculated in log (N) will correspond to a probability of failure of 2.3%, see Table 1.


12

Table 1. Risk of failure coefficient with slope, m = 3. (Konstruktionshandbok, 2014, [9]) – Table 9-6.

Approximated risk of failure 50 % 15.9 % 2 % 1 % 1.35 ‰ 10⁻³

Number standard deviations 0 1 2 2.33 3 3.1

Risk of failure coefficient 1.32-1.40 1.15-1.19 1 0.95 0.84-0.87 0.83-0.86

A correlation of the characteristic curves gradient has been found, both for welds and base material. This

correlation is a simplification of the curve to a straight line with the gradient -1/m in a logarithmic

diagram. For welds loaded in normal stress the characteristic curve has often a gradient of -1/3, m=3, but it

may vary for different cases. For welds affected in shear stress and base material not affected of welding

the gradient is usually -1/5, m=5. This value is used in most standards and applies to constant stress range.

There are different perceptions of the fatigue limit and some argue that there is no absolute fatigue limit,

and consider instead a change in the assumed fatigue limit point to a new gradient of -1/22, m=22. This

slope is shown in Figure 6 marked as Slope for varying amplitude. [9]

The S/N curves slope can be described by Equation (1).

∆𝜎𝑚 𝑁 = 𝐶 (1)

Where 𝑁 is number of cycles, ∆𝜎 is the stress range, 𝑚 and 𝐶 are material parameters.

For calculating number of load cycles to failure Equation (1) is rewritten by Equation (2).

𝑁 =

𝐶

∆𝜎𝑚

(2)

If the stress range versus number of load cycles to failure are plotted with logarithmic scale a linear

relationship is received. The S/N curves linear relationship is described in Equation (3).

log 𝑁 = −𝑚 log ∆𝜎 + log 𝐶 (3)

The stress amplitude is defined in Equation (4).

𝜎𝑎 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛

2 (4)

The stress range is defined in Equation (5).

∆𝜎 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛 (5)

The mean stress is defined in Equation (6).

𝜎𝑚 =

𝜎𝑚𝑎𝑥 + 𝜎𝑚𝑖𝑛

2 (6)

The stress ratio defined in Equation (7).

𝑅 = 𝜎𝑚𝑖𝑛

𝜎𝑚𝑎𝑥 (7)


13

The parameter with the highest impact on the fatigue life calculations is the stress amplitude and stress

range. The load variation can be pulsating-, alternating- or fully reversed loading. Pulsating loading occur

if the stress ratio is positive. Alternating loading leads to a negative stress ratio corresponding to both

tensile and compressive stresses. Fully reversed loading includes both tension and compression, and it has

a mean stress of zero. The loading variation does not affect the number of cycles to failure significantly

and it is often assumed to be a sinusoidal loading. [14]

There are some general effects regarding an S/N curve for a welded joint, these are stated below:

- The sharper notch, the steeper is the slope of the curve.

- The sharper notch, lead to less scatter in the curve, less scatter of the test results.

- The fewer cycles to failure, lead to less scatter of the test results.

The S/N curve is limited and mainly valid for uniaxial loads, primary normal and shear stress. The curve

is also only valid for loads with a certain R-ratio. [9]

There is usually a spread of test results in the S/N diagram, due to the welding process which cause

different residual stresses, initial cracks, discontinuities etc. When testing, the failure probability is 50 %.

Therefore, a lower quartile is normally selected when dimensioning for the characteristic fatigue strength,

often a quartile corresponding to 2.3% failure probability. The quartiles are calculated by the standard

deviation. [1]

2.2 Welds Welding is a process for bonding parts together. This is performed by melting the base material and

adding a filler material to form a pool of molten material. When it cools down a strong bond between the

materials is formed. Welding can be performed by robot or manually.

Welds consists of three zones, illustrated in Figure 7. The three zones are; weld metal, heat affected zone

and base material. Each zone has different metallurgical structure and the heat affected zone can be

subdivided into several zones with different metallurgical structures. There are essential load transferring

and non-load-transferring welds. [15]

Figure 7. Weld zones. [15]


14

The throat thickness, referred as a, is a measure of the welds thickness. The dimensions of a weld are

usually defined with the throat thickness and the length of the weld. The throat thickness depends also on

the penetration into the base material. The nominal throat thickness is the largest triangle that can be

formed inside the weld’s profile.

In reality welds have a complex geometry, illustrated in Figure 8. Both the profile and the start and end

have complex shapes. In the FE-model the fillet weld is assumed to be an isosceles triangle.

2.2.1 Welding classes General welding classes available are B, C, and D. The weld class indicates how much defects the weld

contains and also misalignments of the base material components, both internal and external. Welding

Class B is the class with minimum amount of misalignments and requires, in principle, post-treatment.

Class B is used for high fatigue stresses, high risk zones and risk of brittle fracture. Class C does not

require post-treatment after the welding. It is less time consuming than class B. Welding class C is

standard at BT Products. Class D is used for welds not subjected to high loads, for example, mounts for

cables and hoses. The cost and time for processing a weld increase with quality.

2.3 How the material is affected by welding

2.3.1 Residual stresses Residual stresses in welds, also called initial-, rest- or internal stresses, always occurs after the welding

process. The reason for the residual stresses is that of the melted material wants to shrink during cooling,

while the adjacent materials prevent shrinking. When producing a welded joint high temperature is

applied, in general so high that the adjacent materials begin to melt. When a metal is heated it will expand

due to its metallurgy. The expansion is dependent of the materials’ thermal expansion coefficient and

temperature difference. Heat expansion is defined in Equation (8). [9]

(8)

Figure 8. To the left is a real fillet weld profile, [7] to the right is a simplified fillet weld profile, with marked throat

thickness, referred as a, and 45 degree angles.


15

∆𝐿 = 𝐿 ∗ 𝛼 ∗ ∆𝑇

Where ∆𝐿 is the length change from initial reference length, 𝐿 is the initial length, 𝛼 is the thermal

expansion coefficient and ∆𝑇 is the temperature change. For steel the thermal expansion coefficient is

about 13x10-6

/ (°C) at 20 degrees. [16]

Since the heating is so high that plasticity occurs, tensile stresses will remain in the material. The result

leads to longitudinal tensile stresses in and around the weld. Equilibrium in the section must be fulfilled

and compression residual stresses arise some distance from the weld. The magnitude of the residual

stresses also depends on the material’s mechanical properties. A high strength material receives higher

residual stresses then a low strength material. This relationship is not linear. For a low strength material

with low yield strength the residual stresses usually exceed the yield stress, but not for a high strength

material. The area affected with residual stresses is dependent of the applied heat expressed in [MJ/m]

when welding. The applied energy depends on the welding method. The residual stresses, especially the

longitudinal tensile stress in and around the welded joints have a negative effect on the fatigue resistance.

It cause fatigue cracks for both tensile- and compression stress. Usually fatigue failure only occurs under

tensile stresses. Residual stresses increase the possibility for crack initiation and propagation both in and

near the weld. It is assumed in many standards that the longitudinal tensile stress in and around all welds

is equal to the material’s yield stress. If it is not assumed to be equal then a correction factor is used.

2.3.2 Deformation due to weld shrinkage During the welding process deformation occurs due to the cooling process. Generally, the interaction is;

smaller deformation leads to higher stresses and vice versa. Different deformations that may occur are;

cross shrinkage, deformation angle, length contraction, bending and buckling, depending on the base

material’s geometry, weld design and weld sequence.

An increase in throat thickness or in joint volume results in increased deformations, due to increased

amount of heat. The size of the deformation is important when tight tolerances are needed. Effective

welding techniques such as symmetrical welding, welding sequence of components and placement near

the center of gravity can reduce deformations.

2.3.3 Crack initiation After welding, the crack initiation phase is already fully or partially passed due to the residual stresses.

When calculating the weld’s fatigue life the crack initiation phase is usually ignored. This is a

conservative assumption for the fatigue life of the weld toe. An initial crack will occur between the

bonded plates. The cracks length and placement are dependent on the welded design etc. Fatigue fracture

possibilities are shown in Figure 9. Statistics indicate that the fractures occur more often in the weld toe

than in the root. [7]


16

Figure 9. Fatigue fracture modes in a fillet weld. [7]

2.4 Welds affect in structures Welded joints strength at static load is generally regarded and assumed as the base materials’ properties.

One important exception is if the weld is exposed to an instability phenomenon, e.g. buckling. The

problem is the residual stresses in the welded area. It causes decreased stress limits for an instability

phenomenon. Dynamic loads have a significant effect on the welds fatigue life and it needs to be

considered while dimensioning. [1]

After the welding process the base material’s properties have been changed and cracks have been initiated

in the welded area. Different types of defects and discontinuities are also formed in and around the weld.

Welds have often naturally sharp notch radii, which cause high notch stresses in the transition zone

between the weld and the base materials. Therefore, a welded design will never achieve the same fatigue

life as the unprocessed base material. [1]

2.5 Welds effect on fatigue life It is in reality hard and almost impossible to precisely determine the fatigue life of a welded joint. The

main reason for this is that there are never two welds that are equal. The welded joints never obtain

exactly the same geometry and properties. The weld receives custom characteristics due to various factors

that affect each other in a complex way and affecting the fatigue life of the weld.

The five factors with the greatest impact on the fatigue life are:

- Stress range

- Mean stress and residual stresses

- Material properties (Crack propagation/Crack initiation)

- Geometric stress concentrations

- Size and location of weld discontinuities

The factors effect depending on welded geometry, weld dimensions, the residual stresses and

discontinuities size and type. [15]

The weld has geometric details that cause stress concentrations, especially at the weld ends, toes, root and

internal defects. These stress concentrations leads to crack propagation and eventually fracture [1]. Tensile

residual stresses can occur even in compressive loads due to micro cracking. Therefore, crack propagation

and fracture also occur in compression due to positive residual stresses [12]. The crack propagation rate is

basically independent of the material’s strength. Thus, the weld’s fatigue life is almost independent of the

material’s strength [1].


17

2.6 Fatigue design of welded joints and components It is complicated to calculate the fatigue life of a weld. Fatigue life is affected by a number of factors,

where many of them are difficult to derive physically. [1]

There are mainly four methods for calculating the fatigue life; Nominal stress method, Geometrical

method also called Hot-spot method, Effective notch method and Fracture mechanics. Fracture mechanics

also includes different evaluation methods. [6]

The methods are appropriate for different types of problem, depending on the methods limitations and

assumptions. The methods have different pros and cons, and the results of the methods do not always

correspond to each other.

The Nominal stress method is used for basic geometry, where the nominal stress is defined. The Hot-spot

method can be used for basic and a bit more advanced geometries, but it loses accuracy for more complex

geometries. The method evaluates results from tests or field measurements or from a FE-analysis. The

Effective notch method and Fracture mechanics are appropriate for more complex problems and are

primary used with FE-analysis. The methods require more effort and give a result with higher accuracy.

Generally, the methods requiring less effort are applicable for basic geometries and vice versa, illustrated

in Figure 6. Today Fracture mechanics and Nominal stress method are often based on FE-analysis,

although the methods originally were based on analytical calculations. [1]

Based on a research project investigating the different evaluation methods made by Volvo VCE, 2006, the

conclusion was that the Effective notch method should be used for the weld toe analysis and Fracture

mechanics for the root analysis. [7]

Figure 10. Comparison between the different methods used for fatigue life design.

Calculations for welds are normally made with the assumption of elastic conditions, and that the throat

thickness includes the weld’s penetration into the base material.

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

Acc

ura

cy

Problem complexity

Calculation methods for fatige life of welds

Fracture mechanics

Effective notch

Hot spot

Nominal stressEffort


18

2.6.1 Nominal stress method The first developed method for fatigue life calculations of welds was the Nominal stress method, the

method was developed through testing of different weld designs. The method is based on and interacts

with fatigue classes. More about fatigue classes can be found in chapter 2.1.1 Fatigue class. The Nominal

stress method only considers the nominal stress in the calculation. Due to this, the method is suitable for

calculations without FE-analysis. The notch stresses and local stress concentrations are considered in the

FAT-value. The FAT-value depends mainly on the welded geometry, weld’s design and the welding class,

where a higher FAT-value indicates higher fatigue strength. Tables with FAT-values can be found in

literature about the field, i.e. Konstruktionshandbok för svetsade produkter i stål, Claes Olsson [9]. The

method is limited to only evaluate the weld toe, so it has no capability of evaluating the weld root.

Calculation of the fatigue life is defined by Equation (9) . (SSAB, 2011) – Eq. 5.22.

𝑁 = 𝑁𝑡 (

∆𝜎𝑟𝑑

∆𝜎)

𝑚

(9)

Where 𝑁 is calculated fatigue life, 𝑁𝑡 is the designed life, ∆𝜎𝑟𝑑 is the allowed stress range, ∆𝜎 is stress

range, 𝑚 is the gradient of the S/N curve.

The maximal stress range is calculated with Equation (10). Note that if the minimum stress is negative, the

negative sign is taken into account. (SSAB, 2011) – Eq. 5.19.

∆𝜎𝑚𝑎𝑥 = 𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛 (10)

Equation (9) can be rewritten if the number of cycles is given and the stress range is sought.

Then, the allowed stress range can be determined by Equation (11). (SSAB, 2011) – Eq. 5.15.

∆𝜎𝑟𝑑 =

𝐹𝐴𝑇 𝜑𝑡 𝜑𝑚 𝜑𝑒

𝛾𝑚 √𝑆𝑚𝑚

(11)

Where 𝜑𝑡 is the thickness factor, 𝜑𝑚 is the material factor, 𝜑𝑒 is the stress variation factor (residual

stresses), 𝛾𝑚 is a failure impact factor and 𝑆𝑚 is the cumulative stress parameter.

Equation (11) contains of partial factors and the FAT-value. Additionally, other correction factors can be

used, for example for elevated temperature or for corrosion in air or seawater. See IIW recommendations

for more information.

The base material’s thickness has an impact on the fatigue strength. Reduced plate thickness increases

fatigue strength (for the same stress level over the cross-section). The reason for this is that a smaller

volume will be exposed for higher stresses, and the probability of failure then decrease which results in a

higher fatigue strength. The thickness factor is applied when welds are loaded perpendicular to the weld

direction, and when a fracture is assumed to occur in the weld toe and not in the weld root. The thickness

factor is defined in Equation (12). (SSAB, 2011) – Eq. 5.11.

𝜑𝑡 = (

𝑡0

𝑡)

𝑓

(12)

Where 𝑡 is the material thickness, 𝑡0 is the reference thickness; 15 mm. The exponent 𝑓 is dependent on

the joint type and varies between 0 and 0.15. The 𝑛-value is selected according to Table 2. [1]


19

Table 2. Recommended values of n according to SSAB's research. (SSAB, 2011) – Table 5.9.

Joint type Class f

Fillet weld, transverse T-weld, sheets with transverse junction, longitudinal

stiffeners

Untreated

weld

0.15

Fillet weld, transverse T-weld, sheets with transverse junction, longitudinal

stiffeners

Treated weld 0.1

Transverse butt weld Untreated

weld

0.1

Treated butt, rye, transverse welds or weld junctions All 0

Non welded material All 0

The material factor, 𝜑𝑚, is dependent on the surface roughness and the yield stress. For welds the fatigue

strength is almost independent of the static yield stress. Hence the material factor is assumed to equal to

unity.

The stress variation factor, 𝜑𝑒, is dependent of the stress variation and the stress ratio. The stress variation

factor is used because of the residual stresses occurring in and around the weld. The factor is defined in

Table 3. Table 3. Stress variation factor. (SSAB, 2011) – Eq. 5.12 -5.14.

Stress variation factor Stress ratio Max stress

Weld 𝜑𝑒 = 1 0 ≤ R ≤ 0.5 𝜎𝑚𝑎𝑥 > 0

Base material 𝜑𝑒 = 1 − 0.3𝑅 0 ≤ R ≤ 0.5 𝜎𝑚𝑎𝑥 > 0

Weld 𝜑𝑒 = 1 − 0.2𝑅 -1 ≤ R ≤ 0

Base material 𝜑𝑒 = 1 − 0.25𝑅 -1 ≤ R ≤ 0

𝜑𝑒 = 1.3 𝜎𝑚𝑎𝑥 < 0

When the applied stress variation is unknown the factor is set to unity, which gives a conservative result.

A safety factor, 𝛾𝑚, is included in Equation (11). The safety factor is often connected to fracture

consequences. Safety factors are either the partial coefficient, 𝛾𝑚 , or the risk of failure coefficient, 𝜑𝑄.

IIW (24) gives the following recommendations of the factor in Table 4. [9]

Table 4. IIW's proposal of values for the partial coefficient in different situations. [3]

Consequence Fail safe, damage resistance design Safe life, infinite life

Failure in secondary design details 1.00 1.15

Loss of vital structures 1.15 1.30

Loss of human life 1.30 1.45

In certain cases the mean fatigue strength is more interesting than the characteristic fatigue strength. For

example, if the calculations should be compared with testing or when the purpose is to design with a

specific risk of failure. Then it is useful to take advantage of the risk of failure coefficient rather than

partial coefficient. [9] More information about risk of failure coefficient is in chapter 2.1.2 S/N Curve. The

Cumulative stress parameter, 𝑆𝑚 , defines in Equation (13). (SSAB, 2011) – Eq. 5.10.


20

𝑆𝑚 =

𝑁𝑡 𝑘𝑚

2 ∗ 106 (13)

Where 𝑘𝑚is the collective factor, defined in Equation (14). (SSAB, 2011) – Eq. 5.8.

𝑘𝑚 = ∑ (

∆𝜎𝑖

∆𝜎𝑟𝑒𝑓)

𝑚

∗ 𝑛𝑖

𝑛𝑡𝑖

(14)

Where ∆𝜎𝑖 is the stress range on level i, 𝑛𝑖 is number of cycles at level i, 𝑛𝑡 is total number of cycles, and

∆𝜎𝑟𝑒𝑓 is the reference value for the stress range. With constant stress range, 𝑘𝑚 = 1 and with varying

stress range, 𝑘𝑚 < 1.

2.6.2 Geometrical stress / Hot-spot method The Geometrical stress also known as Hot-spot approach was developed for estimate fatigue life through

tests and field measurements. The method calculates a geometric stress, also known as Hot-spot stress, hs,

by extrapolating strain/stress measurements at the weld toe. The measurements are usually performed with

strain gauges and are picked in specific reference points depending the method; linear extrapolation or

quadratic extrapolation. With the development of computers and availability of FE-software, FE-analysis

became more frequently used for Hot-spot calculations. The method has high accuracy for simple

geometries, but for complex geometries the accuracy decreases.

The method is only suitable for evaluation of fracture at the weld toe for fillet and butt welds. Therefore,

one limitation is that there is no capability of evaluating the root. Ideally, the max principle stress should

be perpendicular to the weld, but it gives valid results for max principle stress direction oriented ± 60°

perpendicular to the weld toe.

The Hot-spot method requires fewer S/N curves than the nominal stress method. Since the Hot-spot stress

itself takes account of the welded geometry. For example, the same S/N curve can be used for all fillet

welds without post-treatment or large misalignments, regardless the welds geometry. The method has no

simple way to consider geometrical misalignments and eccentricities in the weld design. One way is to

model the misalignments and eccentricities in the FE-model. [6]

The fatigue life is calculated with the Hot-spot stress together with a FAT-value suitable for the Hot-spot

approach, see Equation (15). For example, recommends IIW FAT-values in the report; “Fatigue design of

welded joints and components”. [3]

𝑁 = 𝑁𝑡 (

𝐹𝐴𝑇 𝜑𝑄

𝜎ℎ𝑠)

𝑚

(15)

Where 𝑁 is calculated fatigue life, 𝑁𝑡 is the required life, FAT is the FAT-value, 𝜑𝑄 is the risk of failure

coefficient, 𝜎ℎ𝑠 is the Hot-spot stress and 𝑚 is the gradient of the S/N curve.

The Hot-spot stress is calculated through Hooke’s law, since the stress is approximated to a linear elastic

uniaxial stress state. The uniaxial orientation is perpendicular to the weld. Equation (16) is Hooke’s law

with the Hot-spot stress and strain.

𝜎ℎ𝑠 = 𝐸 𝜀ℎ𝑠 (16)


21

The stress/strain is calculated in specific points dependent on linear or quadratic extrapolation. The

specific points are recommendations from IIW. The linear extrapolation is calculated through two points,

at distance 0.4t and 1.0t in front of the weld toe, see Equation (17) and Figure 11. The factor t is the

thickness of the base material. The quadratic extrapolation is calculated through three points, at distance

0.4t, 0.9t and 1.4t, see Equation (18) and Figure 11.

𝜎ℎ𝑠 = 1.67 𝜎0.4𝑡 − 0.67 𝜎1.0𝑡 (17)

𝜎ℎ𝑠 = 2.52 𝜎0.4𝑡 − 2.24 𝜎0.9𝑡 + 0.72 𝜎1.4𝑡 (18)

Linear extrapolation is suitable in cases where there is no significant nonlinear behavior (e.g. low stress

gradients) or the mesh is coarse (i.e. element size > 0.2t), otherwise quadratic extrapolation is

recommended. [6]

2.6.3 Effective notch method Calculations with the Effective notch method are based on the stresses in the local weld radius, referred as

notch stresses. Notch stress is considered the max principle stress in a certain notch, under the assumption

of linear elastic material. This notch stress has been used successfully in fatigue calculations of non-

welded structures. [6]

The notch stress is obtained by FE-analysis. In the FE-model the entire design's form is considered, with

transition radii, fillets etc. A fictitious radius is used at the weld toes and/or root, see Figure 12. In more

comprehensive FE-models sub models are effective to use to get a refined element size. The element size

should be analyzed with a mesh convergence study or follow standards for the method. This results in

accurate solutions.

When the notch stress is determined a dense mesh is required, because the fictitious notch radius is small

and the stress gradient is usually very large near the notch. An example of fictitious notch radii is

illustrated in Figure 12. For base material thickness greater or equal to 5 mm, IIW [4] recommend using a

notch radius of 1 mm and a FAT-value of 225 MPa. The previous recommendations are connected with an

element size. The recommended element size around the notch is; notch radius divided by four. [1] For

example, notch radius 1 mm should have element size 0.25 mm.

Figure 11. To the left, points for linear extrapolation and to the right, points for quadratic extrapolation.


22

Figure 12. Fictitious notch radii.

The effective notch approach is an effective method when a number of weld geometries or penetration

depth should be compared with each other. Besides evaluating the weld toes, the weld root can also be

evaluated, this makes it possible to dimension the weld for both toe and root fractures.

The method has the ability to evaluate complex designs where the nominal stress is not possible to use.

The method can be applied to major misalignments, such as angular errors and eccentricity. These errors

can be modelled in the current FE-model or applied with a scale factor. Figure 13 illustrates a FE-model

with notch radius 1 mm in toe and root.

Figure 13. FE-model with notch radius. Disadvantages and limitations

- For valid results the principle stress direction needs to be oriented ± 60° to the weld toe and root.

- The method may require an advanced FE-model with a large number of elements and it may need

sub models. This lead to increased work effort and calculation time.

- The problem with Effective notch method is that singular points should be avoided in the FE-

model.


23

Calculation system

The fatigue life is calculated by Equation (19).

𝑁 = 𝑁𝑡 (

∆𝜎𝑟𝑑

𝜎𝑟)

𝑚

(19)

Where ∆𝜎𝑟𝑑 is the design normal stress range, also called allowed stress range, 𝑁𝑡 is the required life, ∆𝜎

is the stress range and 𝑚 is the gradient of the S/N curve.

The stress range is calculated with a FE-model. It is the largest principle stress value during the load cycle.

∆𝜎𝑟𝑑 =

𝜑𝑄 𝐹𝐴𝑇

√𝑁𝑡

2 ∗ 106

𝑚

(20)

Equation (19) combined with Equation (20) can be rewritten to Equation (21).

𝑁 = 𝑁𝑡 (


𝜎𝑟)

𝑚

(21)

Risk of failure is 2.3% and the design life is 2 million cycles correspond the safety factor, 𝜑𝑄 = 1.

𝑁 = 2 ∗ 106 (

𝐹𝐴𝑇

𝜎𝑟)

3

(22)

Median fatigue life: FAT-value = 225 MPa, 𝜑𝑄 = 1.3, 𝑁𝑡 = 2*106 (IIW). Use max principal stress for 𝜎𝑟.

𝑁 = 2 ∗ 106 (

225 ∗ 1.3

𝜎𝑟)

3

(23)

2.6.4 Generally for the methods The design is approved if Equation (24) and Equation (25) are satisfied.

𝑁 ≥ 𝑁𝑡 (24)

Where 𝑁 is the fatigue life and 𝑁𝑡 is the designed fatigue life.

∆𝜎𝑟 ∗ 𝛾𝑓 ≤ ∆𝜎𝑟𝑑 (25)

Where 𝛾𝑓 is a safety factor, 𝛾𝑓 = 1

𝜑𝑄 ,


24

2.6.5 Fracture mechanics Fracture mechanical is mainly used in areas where the crack propagation dominates, for example fatigue

calculations of welds.

2.6.5.1 Stress intensity factor One way to calculate the crack propagation and failure state is with the use of the stress intensity factor.

Equation (26) defines the stress intensity factor for a Mode I condition. 𝐾𝐼 is the predominant failure

mode.

𝐾𝐼 = 𝜎𝑛𝑜𝑚 𝑓 √𝜋𝑎 (26)

Where 𝑎 is the crack length, 𝑓 depends of geometry and 𝜎𝑛𝑜𝑚 is the nominal stress.

The 𝑓-function is computed through derivation of numerous analytical cases or numerically through FE-

analyses. Derived cases can be found in literature regarding fatigue and fracture mechanics, e.g. in the

book: Failure, Fracture, Fatigue by Tore Dahlberg [17].

Equation (27) defines the range of the stress intensity factor and Equation (28) defines the stress intensity

ratio. The stress intensity range can be used in Paris’ law when estimating the number of cycles to failure.

∆𝐾𝐼 = 𝐾𝐼 𝑚𝑎𝑥 − 𝐾𝐼 𝑚𝑖𝑛 (27)

𝑅 =

𝐾𝐼 𝑚𝑖𝑛

𝐾𝐼 𝑚𝑎𝑥 (28)

Where 𝐾𝐼 𝑚𝑎𝑥 is the maximum intensity factor and 𝐾𝐼 𝑚𝑖𝑛 is the minimum intensity factor.

There are also other methods to calculate the stress intensity factor.

Fracture toughness is a mechanical property that describes the material’s point of fracture and is denoted

as KIC.

2.6.5.2 Paris’ law, also known as Paris-Erdogan law Paris’ law is a method developed for calculations of crack propagation. It is the most commonly used

method. Tests have shown how crack propagations depend on the stress intensity factor [12]. Paris’ law is

defined in Equation (29).

𝑑𝑎

𝑑𝑁= 𝐷(∆𝐾𝐼)𝑔 (29)

Where 𝑁 is the number of load cycles, 𝑎 is the crack length, ∆𝐾𝐼 is the stress intensity range and 𝐷 and 𝑔

are material parameters describing the crack propagation behavior. The left hand side in Equation (29) is

known as the crack propagation rate.


25

Figure 14. Illustrates the crack propagation rate as a function of stress intensity range. [12]

Paris’ law describes the crack propagation rate as a function of the stress intensity factor. With logarithmic

scale it is a linear relationship between the variables. The linear part is illustrated in zone II in Figure 14.

There is a threshold value, ∆Kth, located as zone I in Figure 14. If the stress intensity range is below the

threshold value the initial crack will not start to propagate. For example, if the crack is short, see Equation

(26).

Fracture toughness, KIC, is set as an above limit in Paris’ laws diagram, located as zone III in Figure 14. Fracture criteria at plane strain condition is KIC, fracture occur when KI = KIC. [13]


26

2.6.6 Evaluation of test data For evaluating the fatigue tests with the FE-analysis, the test results characteristic fatigue strength is

calculated. The aim is to compare the tests with the FE-analysis in an S/N curve. The results validate the

use of the methods for weld ends. The fallowing calculation method for determined the S/N curve is taken

from IIW recommendations [3].

From the fatigue tests, the number of load cycles to fracture is received for a specific stress range. With

logarithmic scale there is a linear relationship defined in Equation (30). This equation forms the S/N

curve.

log 𝑁 = log 𝐶 − 𝑚 log ∆𝜎 (30)

Where 𝑁 is the number of cycles, 𝐶 is the constant, 𝑚 is the slope of the S/N curve and ∆𝜎 is the stress

range.

The slope, m, and the constant C can be determined from Equation (30). The characteristic value is

defined in Equation (31).

𝐶 = 𝑥𝑚 − 𝑘 ∗ 𝑠𝑡𝑑𝑣 (31)

Where 𝐶 is the characteristic fatigue strength, 𝑥𝑚 is the mean stress value, 𝑘 is dependent of specimens

and taken from Table 5 and 𝑠𝑡𝑑𝑣 is the standard deviation.

To investigate the characteristic fatigue strength and the scatter in the test results, the standard deviation is

used. The standard deviation is defined in Equation (32).

𝑠𝑡𝑑𝑣 = √∑(𝑥𝑚 − 𝑥𝑖)2

𝑛 − 1 (32)

Where 𝑥𝑚 is the mean value, 𝑥𝑖 is the test samples and 𝑛 is the number of test results.

The mean value is defined in Equation (33).

𝑥𝑚 =

∑ 𝑥𝑖

𝑛 (33)

Table 5. k-value dependent of number of test results. [3]

Number of test results, 𝑛 10 15 20 25 30 40 50 100

𝑘 2.7 2.4 2.3 2.2 2.15 2.05 2.0 1.9

To produce the characteristic curve in this project, two standard deviations have been applied (i.e. 2.3 %

failure probability).

M. Ramström ǀ Chapter 3 – Fatigue testing

27

CHAPTER 3 FATIGUE TESTING

This chapter describes all facts about the testing including the strain gauge measurements.

3.1 Fatigue testing The specimens have plate thicknesses 10 mm and throat thickness 5 mm. The choice of plate and throat

thickness is based on common material that is used in the forklifts at BT Products. Dimensions are

illustrated in APPENDIX A - Drawings for the specimens.

3.1.1 Weld properties The specimens were robot welded, with welding class C and no post-treatment. The weld power was set to

295 ampere and the feeding was constant. Robotic welding was chosen since it is the most common

method used in the production at BT Products. Robot welding gives less scatter and an increased

credibility to the results compared with manual welding. The welding process is shown in Figure 15.

Figure 15. Robot welding. To the left a discontinuous string.

The continuous weld design where remade after the machine operators suggestions. The problem in the

first design was the demand of no stop and start around the corners. According to the operator it could not

be welded continuously around the corner, because of the burn and melting in the base material. The

suggested design was two continuous strings, the first string contained the long side and both the short

sides, and the second string were the other long side, see drawing in Figure 16. This influences the

continuous welds short ends, a visual difference is noted and shown in Figure 17. It also affects the

penetration at the weld end. In the weld start, the penetration is less than in the weld ending. The end sides

are called side A and side B. Side A is the start side. The discontinuous weld design is made with

continuous strings along each side of the vertical plate. The strings ends are similar with one start and stop

at each end side, see Figure 19.


28

Figure 16. Weld drawing for the remade continuous weld end.

Figure 17. End sides for the continuous weld design. To the left side A, and to the right side B.

Figure 18. Continuous weld, to the left the first string, to the right the second string.


29

Figure 19. End sides for the discontinuous weld end design.

3.1.2 Measuring before testing Markup and measuring of the specimen’s misalignments and deformation caused by the welding was

performed before testing, see Figure 16. The welding process showed a significant effect on the base plate,

especially for the specimens with continuous weld. For the continuous weld, the outer plate edge’s was

raised up about 1.2-1.5 mm on each side, and for the discontinuous welded specimens, the outer edge

raised about 0.15-0.30 mm on each side. In Figure 21 the heat effect on the underside is shown, where the

continuous welding shows a larger heat affected area due to longer welding strings. The measuring data

was registered in a document to interpret the test results.

Figure 21. The heat affects. The above is the discontinuous end, and the below is the continuous end.

Figure 20. Specimens during measurements and markup.


30

3.1.3 Test setup The testing was performed in a tensile test machine, shown in Figure 22. The specimens were mounted in

the machine’s fixture with hydraulic grips, the grip fixture dimensions were width 50 mm and depth 65

mm. First, one side was mounted in the hydraulic grip, and then the other side was steered into the other

grip. During mounting the specimen were straightened out from the curvature produced by the welding

process. The criterion for when the tests are finished is when the deformation increased 5 % with respect

to the undeformed state, according to BT products test standard.

Figure 22. Fatigue test machine.

3.2 Strain gauge test In order to validate the FE-model strain gauges where placed on one specimen of each weld end design,

19 gauges on the discontinuous respectively 18 on the continuous. The strain gauge placements depend on

the welds geometry. Gauges are placed around the area of interest, which is around the weld toes and the

center of the plate, see Figure 23. Figure 23 shows only half of the specimens. Gauges are placed

symmetrical on both sides. Also three gauges are placed on the underside of the base plate. Figure 24

show the gauge placement for the discontinuous weld end. For the continuous end, 5 respectively 6 were

mounted on each side around the weld. The difference in gauges are caused by the weld ends local

geometry, therefore, one extra gauge were mounted on one side, see Figure 25. The used strain gauges

model is Quarter Bridge, which evaluate a single axis in tension or compression. The gauge measures the

changes in resistance, which unbalances the bridge and produces an output voltage. More information can

be found in APPENDIX B - Strain Gauges placement.


31

Each specimen were loaded at the four load levels, see Table 7, for a short period. The period was long

enough to stabilize the cycles, approximately 150-200 cycles, see Figure 27. The strain gauges were set to

zero before the specimen was mounted in the machine. During the mounting, the specimen was

straightened out which cause pre-stresses. The pre-stresses were measured, but not part of the fatigue test.

Figure 23. Red lines indicate strain gauge placements. Left, continuous weld end. Right, discontinuous weld end.


32

Figure 24. Strain gauge placements for the discontinuous weld end.

3.2.1 Strain gauge test results The results from the strain gauge measurements contained pre-stresses and fatigue stresses for the

different load levels. The results for the pre-stresses and the load level, 200 kN, are summarized in Table

6. The occurred pre-stresses where different for the weld designs, the continuous weld end received higher

stresses mainly affected by the significant curvature and misalignments in the baseplate. The pre-stresses

indicate both compressive and tension stresses, which leads to a stress ratio below zero at some locations.

The software used for the test data is FAMOS. The strain measurements were transformed to stresses

through Hooke’s law in one dimension.

From the test results, the magnitude of the stresses for load case, 200 kN, was about 330-338 MPa for the

continuous end, and about 266-314 MPa for the discontinuous end, measured in z-direction in the center

gauges 2.5 mm from the weld toe. The pre-stress was 58 and 15 MPa for the continuous end, and 15 and

-18 MPa for the discontinuous end, measured on the upper surface of the baseplate in z-direction in the

center gauges 100 mm away from the weld toe. The stress ranges for load case 200 kN and the pre-

stresses are shown in Table 6.

Figure 25. Strain gauge placements for the continuous weld end. To the left side A and to the right side B.


33

Table 6. Stresses from the strain gauge tests.

Continuous weld end Discontinuous weld end

Strain

gauge

Pre-stress

[MPa]

Stress range

[MPa]

200 kN

Strain

gauge

Pre-stress

[MPa]

Stress range

[MPa]

200 kN

OS_A_1 -6 53 OS_AC_1 15 200

OS_A_2 50 140 OS_AH_1 4 49

OS_A_3 112 330 OS_AH_2 13 96

OS_A_4 38 141 OS_AH_3 6 266

OS_A_5 -8 51 OS_AH_4 3 220

OS_AC_1 54 230 OS_AV_1 -6 48

OS_AC_2 58 200 OS_AV_2 -15 101

OS_B_1 1 50 OS_AV_3 -2 308

OS_B_2 42 203 OS_AV_4 3 217

OS_B_3 55 338 OS_BC_1 -18 223

OS_B_4 18 138 OS_BH_1 -3 49

OS_B_5 6 42 OS_BH_2 -22 139

OS_B_6 -5 59 OS_BH_3 -20 314

OS_BC_1 28 237 OS_BH_4 -12 230

OS_BC_2 15 226 OS_BV_1 6 56

US_AC_1 -48 222 OS_BV_2 8 120

US_BC_1 -5 197 OS_BV_3 -9 282

US_CC_1 2 163 OS_BV_4 -11 220

US_AC_1 -2 207

US_BC_1 30 190

US_CC_1 4 163

The measured pre-stresses for some gauges are shown in Figure 26. The start of the first load level,

120 kN, including the pre-stresses and the stabilization of the gauges are shown in Figure 27. The four

load levels and the total measurements for the discontinuous weld end are shown in Figure 28.


34

Figure 26. The recorded pre-stresses for the discontinuous weld end. Four gauges in the figure; OS_AH_3, OS_AV_3,

OS_BH_3 and OS_BV_3.

Figure 27. The start of the load level. Four gauges in the figure; OS_AH_3, OS_AV_3, OS_BH_3 and OS_BV_3.


35

Figure 28. Overview for the discontinuous weld end, all four load levels. Four gauges in the figure; OS_AH_3, OS_AV_3,

OS_BH_3 and OS_BV_3.

Test program The testing was performed externally at Volvo trucks in Göteborg. The external testing included 16

specimens, eight with continuous and eight with discontinuous weld end, shown in Table 7. The frequency

doesn’t affect the fatigue result but reduce testing time.

Table 7. Test program.

Specimen

Continuous

Specimen

Discontinuous

Force max

[kN]

Force min

[kN] R

Stress range

[MPa]

Frequency

[Hz]

2 2 240 24 0.1 216 8

2 2 200 20 0.1 180 20

2 2 160 16 0.1 144 20

2 2 120 12 0.1 108 20

The stress ratio, R, is set to 0.1, since the test object should never be exposed to compression and be more

stabilized during the test.

3.3 Results from testing The testing includes the two different weld ends. Each weld end design has been tested on four different

load levels. Every load level is tested for two specimens. The test results are marked with crosses in

Figure 29 for discontinuous and Figure 30 for continuous. The three lines in the figure below represent,

mean curve, results mean curve and the characteristic curve. The results mean curve is determined with

the least square method and receives a specific slope shown in Table 8 and Table 9 and 50 % risk of

failure. The mean curve and the characteristic curve have slope 3 and 50 % risk of failure respectively

2.3 % risk of failure. The test results are summarized in Table 10. The results indicate higher fatigue

strength for the discontinuous weld end.


36

Figure 29. Discontinuous weld end.

Table 8. Discontinuous test data.

Slope Standard deviation

Slope mean curve and characteristic curve 3.00 0.170

Slope results mean curve 3.13 0.034

Figure 30. Continuous weld end.

Table 9. Continuous test data.


Slope mean curve and characteristic curve 3.00 0.170

Slope results mean curve 3.42 0.052

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles

S/N curve - Discontinuous weld end

Mean curve

Characteristiccurve

Results meancurve

Results

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles

S/N curve - Continuous weld end

Mean curve

Characteristiccurve

Results meancurve

Results


37

Table 10. Test results.

F amp. [kN] Fmax [kN] Fmin [kN] Mean cycles

Continuous

Mean cycles

Discontinuous

Load case 1 54 12 120 776,000 1,452,000

Load case 2 72 16 160 289,000 598,000

Load case 3 90 20 200 133,000 326,000

Load case 4 108 24 240 73,000 159,000

The occurred crack after the fatigue test for the different weld designs are shown in Figure 31. The results

for the continuous weld end occurs always at the same side, side A. This is an interesting notification

which is assumed to be caused by the less penetration compared with the other side. For the discontinuous

weld no similar pattern can be seen, since the sides are similar. Unfortunately no note where made of

which side of the specimens where pressed in the test machine which may have had an increased effect of

the pre-stresses.

Figure 31. Example of cracks. To the left a continuous, and to the right a discontinuous.

38

M. Ramström ǀ Chapter 4 - Finite Element Analysis

39

a. b.

CHAPTER 4 FINITE ELEMENT ANALYSIS

This chapter includes a mesh convergence study, a boundary condition study, a nonlinear geometric

study, a Pre-stress study and FE-models for the different methods. The results from the strain gauge test

are used for calibration and validation.

4.1 FE-model The aim with the FE-analysis is to simulate reality to provide accurate estimations of the failure

probability. FEM calculate a numerical solution to the problem defined in the pre-processor and are

mainly dependent on geometry, boundary conditions, mesh, etc. The more accurate the boundary

conditions are, the more accurate and comparable the FE-analysis will be.

4.2 Model Description The model investigated is a T-joint fillet weld with two different weld ends, continuous and discontinuous,

illustrated in Figure 32. The dimensions are equal in the specimens and can be seen in APPENDIX A -

Drawings for the specimens.

Figure 32. a. T-joint fillet weld with continuous weld end b. T-joint fillet weld with discontinuous weld end.


40

4.2.1 Material The material used for the test objects are SS2134 with properties according to Table 11 from BT Product’s

material library. The material in the FE-simulation is assumed to have isotropic and linear elastic

properties.

Table 11. Material properties.

General structural steel, SS2134

Young’s modulus, E 210 GPa

Poissions’s ratio, ʋ 0.30

Density, ρ 7850 kg/m3

Yield strength, Rp0.2 355 MPa

Ultimate tensile strength, Rm 490 MPa

4.2.2 Analysis setup All simulations are made in Abaqus 6.14-1. The FE-analysis is a general static analysis. The solution

technique is Full Newton analysis.

4.3 Mesh convergence study A mesh convergence study is performed to make sure that the solution is independent of the mesh

resolution.

4.3.1 Elements The element type used is ten-node quadratic tetrahedral elements, known as C3D10. Each face of the

element is defined by six nodes and this element has curved faces. The elements have 4 integration points.

This quadratic element type will represent the displacement and boundary conditions better than linear

elements, with the same amount of elements. [18]

4.3.2 Convergence study The models main aim is to produce accurate stresses at the region around the weld. The elements at larger

distance from the region of interest is only representing geometry and transmitting load. Therefore, the

convergence study is made through a local mesh refinement, performed through successive level of local

mesh refinement. The study is made for the continuous weld end model. The convergence is investigated

for a path started from the weld toe and along 14 mm perpendicular to the weld. The investigated output is

the maximum principal stress, and the paths node values include intersections from the connected nodes,

see Figure 33. Four different mesh levels are compared and described in Table 12 and illustrated in Figure

34. The convergence study is made for Load level 1, 120 kN, and B.C and load case 1, simple support

condition. B.C. and load case 1 are described in chapter 4.4 Boundary conditions and Load cases study.

Figure 33. The path starts in the weld toe and continuous 14 mm perpendicular to the weld.


41

Table 12. Mesh levels for the mesh study.

Mesh1 Mesh2 Mesh3 Mesh4

Number of nodes: 122,000 290,000 567,000 798,000

Number of elements: 81,000 201,000 400,000 564,000

Element types: C3D10 C3D10 C3D10 C3D10

Element size ca: 2 mm 1-2 mm 0.5-1 mm 0.5 mm

Figure 34. Illustrates Mesh 1, Mesh2, Mesh 3 and Mesh 4. The grey area indicates stresses above 225 MPa.


42

Figure 35. Result for the different mesh qualities.

The results from the mesh convergence study are shown in Figure 35. It shows that convergence occurs at

about 400,000 elements with element size of about 1 mm close to the weld. The difference is about 1.5 %

between mesh 3 and mesh 4, which indicates that convergence is reached and the solution is independent

of further refinements. This element size, mesh 3, will be used in verification of the B.C and calibration

analysis for the specimens and strain gauge measurements.

Generally, every problem needs a convergence study to ensure that the solution is independent of the

mesh. Already made mesh studies can also be used for similar problems. There already exists mesh

standards for the Effective notch and the Hot-spot method. Therefore, no mesh studies are performed for

these models and the mesh size are taken from IIW [3], when these methods are used.

100

150

200

250

300

350

400

0 1 2 3 4 5

Stre

ss, M

ax. P

rin

cip

el [

MP

a]

Distance along path [mm]

Mesh convergence study

Mesh 1

Mesh2

Mesh 3

Mesh4


43

4.4 Boundary conditions and Load cases study A comparison study of the models boundary conditions and the applied loads are performed due to

investigation of the effect on the solution and understanding how to simulate reality in a reliable way. The

solutions are compared with the strain gauge measurements. The investigation includes five different

variants of B.C. and load cases, named case 1-5, where case 1-4 includes a symmetry-plane condition and

case 5 includes no symmetric condition. The load magnitude used in the study is 120 kN.

B.C. and Load case 1 – Simple support condition at the outer surface and z-symmetry condition,

see Figure 36 a. The load is applied as a negative pressure on the surface. The pressure is evenly

distributed over the surface nodes and oriented in z-direction in Figure 36 a.

B.C. and Load case 2 – Clamped condition at the outer surface and z-symmetry condition, see

Figure 36 b. The clamped condition is applied on a reference point connected through a rigid body

with tie constraint to the outer surface and locked in all directions and rotations except the load

direction, z-direction. The load is applied as a negative pressure on the outer surface. The

pressure is even distributed over the surface nodes and oriented in z-direction in Figure 36 b.

B.C. and Load case 3 – Clamped condition at the upper and under surfaces with similar size as the

grip fixture in the test machine and z-symmetry condition, see Figure 36 c. The clamped condition

is applied on a reference point connected through a rigid body with tie constrain to the grip fixture

surface and locked in all directions and rotations without the load direction, z-direction. The load

is applied as a concentrated force in a reference point connected to the upper and under surfaces

with similar size as the grip fixture with a rigid body tie constraint. The force are evenly

distributed over the surface nodes and orientated in z-direction in Figure 36 c.

B.C. and Load case 4 – Simple support condition at the centrum edges and z-symmetry condition,

see Figure 36 d. The load is applied as a concentrated force in a reference point connected to the

upper and under surfaces with similar size as the grip fixture with a rigid body tie constraint. The

force are even distributed over the surface nodes and orientated in z-direction in Figure 36 d.

B.C. and Load case 5 – No symmetry plane, se Figure 36 e. Clamped condition at upper and

under surface with similar size as grip fixture at one side where the surfaces are locked in all

directions and rotations. At the other side is a concentrated force in a reference point connected to

the upper and under surfaces with similar size as the grip fixture with a rigid body tie constraint.

The force are even distributed over the surface nodes and orientated in z-direction in Figure 36 e.

These different boundary conditions restrain the model to avoid rigid body motions.


44

a. b.

c. d.

e.

Figure 36. a. B.C. and load case 1. b. B.C. and load case 2. c. B.C. and load case 3. d. B.C. and load case 4. e. B.C.

and load case 5.


45

Figure 37. Results for B.C. and load case study. (F = 120 kN) The result shows different maximum principle stress around the weld toe. The stress varies in the range of

420 MPa to 470 MPa.

B.C and load case 3 and 5 simulate reality and the strain gauge measurements best. Therefore, they are

chosen to be used in further fatigue life analysis. The maximum principal stress between them differs

about 1.3 %. For the Effective notch and the Hot-spot models, B.C. and load case 3 is used, due to the

reduced number of elements.

4.4.1 Comparison study for nonlinear geometric analysis Nonlinear geometric, called Nlgeom in Abaqus. Nlgeom is a setup alternative that considers nonlinear

effects for large deformation. A comparison study has been performed to consider this effect in the results.

It has been done for the continuous fillet weld along the path perpendicular to the weld toe, and the

measured points are located at 2.5 mm and 14 mm from the weld, illustrated in Figure 38.

The applied boundary condition and load case is B.C. and load case 3 from chapter 4.4 Boundary

conditions and Load cases study. The nonlinearity affects the result more with larger load level. The used

mesh size, mesh 3, is similar to the mesh size used in the mesh convergence study.


46

Figure 38. Path for Nlgeom study. (F= 240kN, R=0.1, mesh size = 0.5mm) The Figure 39 shows that analysis with Nlgeom is more close to the strain gauges measurements.

Therefore, FE-analyses compared with the strain gauges measurements and test results are done with

Nlgeom.

Figure 39. Comparison study between Nlgeom and strain gauge measurements.

200

250

300

350

400

450

500

550

600

650

0 2 4 6 8 10 12 14

Stre

ss r

ange

, Max

Pri

nci

pe

l [M

Pa]

Z-distance along path [mm]

Nlgeom study

Nlgeom on

Nlgeom off

Strain gauge 2,5 side A

Strain gauge 2,5 side b

Strain gauge 14 side A

Strain gauge 14 side B


47

4.5 Validation of FE-model

4.5.1 Boundary condition, Load case and load level B.C. and load case 3 is applied on the two different weld ends. The load levels match the testing and load

data can be found in Table 13.

Table 13. Load levels. (B.C. and load case 3)

Load level Magnitude and Direction, max [kN] Magnitude and Direction, min [kN]

Load level 4 [0,0,240] [0,0,24]

Load level 3 [0,0,200] [0,0,20]

Load level 2 [0,0,160] [0,0,16]

Load level 1 [0,0,120] [0,0,12]

4.5.2 Simulation of the pre-stress This FE-model simulates the pre-stress caused by the mounting in the test machine. The pre-stress

analysis is performed for the continuous weld end, where the base plates curvature were significant larger

than with the discontinuous weld end.

The geometry of the model includes the baseplate’s deformation caused by the welding. The deformation

is measured and modelled in twelve points around the baseplate, see Figure 40. All the points around the

base plates outer edges are bent up, the vertical distance varies between 0.3-1.2 mm.

The boundary conditions should simulate the tests mounting process in the test machine. One side is

clamped and the other side is exposed for a vertical displacement, seen in Figure 41. The clamped

condition is applied on upper and under surface with the same area as the grip fixture. The displacement

condition has the same surface size as the grip fixture and is applied on the upper surface. The applied

vertical displacement is the specific pre measured distance so the specimen is straightened out.

Figure 40. Base plate with curvature.


48

Figure 41. Boundary condition for simulating mounting in the test machine.

The results from the pre-stress simulation are shown in Figure 42 and Figure 43. The FE-model and the

strain gauge measurements show similar stress field for the pre-stress. The clamped side’s upper surface is

exposed for a tensile stress, and the side with applied displacement is exposed for a compressive stress.

The base plate’s underside is exposed for compressive stress at the clamped side and tension stress at the

side with applied displacement. The result from this test corresponds to the results from the pre-stress in

the strain gauge measurements.

Figure 42. Pre-stress due to mounting in the test machine.

Figure 43. Deformation scale factor is set to 25.

Ca: - 60 MPa

Ca: 60 MPa


49

4.6 Numerical analysis The numerical analysis is performed on the test objects geometry and compared with the test results. The

numerical analysis is made for the three commonly used calculation methods for fatigue life estimations.

The three methods are Effective notch, Hot-spot and Nominal stress.

Boundary condition and load case 3 is used and load levels are similar with tests. The FE-analysis is

performed with the min and max force in Table 13, the min forces are calculated with the stress ratio

factor 0.1. Then the stress range is calculated by the difference of the max and min results.

4.6.1 Effective notch model The fatigue life estimated with the Effective notch approach is calculated from the FE-analysis. The

Effective notch approach evaluates the notch stress, and includes both the toe and the root. The largest

notch stress is used for calculating the fatigue life of the weld. The method has a standard FAT-value 225

MPa. [3]

The estimated number of cycles is calculated with equation (34). The number of cycles is calculated with

50 % risk of failure due to later comparison with the tests.

𝑁 = 𝑁𝑡 (𝐹𝐴𝑇 𝜑𝑄

𝜎𝑟)

𝑚

(34)

Where 𝑁𝑡= 2,000,000, 𝜑𝑄 = 1.3, 𝑚 = 3.

The Effective notch model may need a global model and a local sub-model, where the global model has a

coarse mesh, due to computer limitations. The Effective notch approach can only be applied on the

continuous FE-model.

4.6.1.1 Sub-model For the Effective notch method a sub-model is used to refine the mesh around the weld’s notches. With

notch radius 1 mm, the model needs an element size about 0.25 mm in the notch. [3] Therefore, the mesh

around the notches are refined in a radius of two 2 mm, see Figure 44, and with a smooth growth rate.

Dimensions of the sub-model are shown in Figure 44 and Figure 45.


50

Figure 44. Sub-model dimensions and notch refinement. Notch radius = 1 mm, (toes and root), Green radius = 2 mm.

Figure 45. Sub-model dimensions.

Figure 46. Sub-model mesh, element size 0.25 mm in the notches. The results from the FE-analysis are summarized in Table 14. Figure 47 shows the results for one specific

load level.


51

Table 14. Effective notch results.

Force [kN] Stress range, σtoe [MPa] Cycles

12-120 313 1,630,000

16-160 424 656,000

20-200 534 329,000

24-240 644 187,000

Figure 47. Effective notch result. Grey area indicates stress above 225 MPa. (F=120, Mesh size=0.25 mm)

4.6.2 Hot-spot model The fatigue life estimated with the Hot-spot approach is calculated from the FE-analysis. FAT-value 100

is recommended from IIW for a non-load transferring fillet weld. [3] Mesh size is from the same

recommendations, and the mesh size are 1 mm in a radius 14 mm around the weld, see Figure 48. The Hot

spot approach can only be applied on the continuous FE-model.


50 % risk of failure.

𝑁 = 𝑁𝑡 (


𝜎ℎ𝑠)

𝑚

(35)

Where 𝑁𝑡= 2,000,000, 𝜑𝑄 = 1.3, 𝑚 = 3.


52

Figure 48. Mesh Hot-spot model.

The results from the FE-analysis are summarized in Table 15. The result from a specific load level is

shown in Figure 49. The Hot-spot measuring points are marked with red dots at distance 4, 9, 10 and 14

mm perpendicular to the weld toe.

Table 15. Hot-spot results. Linear and Quadratic.

Continuous weld end, FAT100

Stress range Linear Quadratic

Force

[kN]

σ0,4t

[MPa]

σ0,9t

[MPa]

σ1,0t

[MPa]

σ1,4t

[MPa] 𝜎ℎ𝑠

[MPa]

Cycles 𝜎ℎ𝑠 [MPa]

Cycles

12-120 149 135 133 128 159 1,092,000 164 997,000

16-160 200 181 178 172 214 449,000 221 410,000

20-200 251 227 223 215 269 226,000 277 206,000

24-240 302 274 270 259 324 130,000 335 117,000

Figure 49. Hot-spot result. Result points are marked with red dots, distance 4, 9, 10 and 14 mm from the weld toe.


53

4.6.3 Nominal stress method The fatigue life estimated with the nominal stress approach is calculated with the analytical stress range.

The FAT-value 71 is used for a continuous fillet weld end and the FAT-value 80 is used for a

discontinuous fillet weld end. Both the values are from the book; Konstruktionshandbok, Claes Olsson,

Welded joint numbers: 57, weld class: Fillet weld, non-load transfer connections. [9]


50 % risk of failure.

𝑁 = 𝑁𝑡 (

𝐹𝐴𝑇 𝜑𝑄 𝜑𝑡

∆𝜎)

𝑚

(36)

Where 𝑁𝑡= 2,000,000, 𝜑𝑄 = 1.3, 𝜑𝑡 = 1.0627, 𝑚 = 3.

Table 16. Nominal stress method results, Continuous weld end.

Continuous weld end, FAT71

Force [kN] Stress range, ∆σ [MPa] Cycles

120 112 1,343,000

160 144 632,000

200 180 323,000

240 216 187,000

Table 17. Nominal stress results, Discontinuous weld end.

Discontinuous weld end, FAT80

Force [kN] Stress range, ∆σ [MPa] Cycles

120 112 1,922,000

160 144 904,000

200 180 463,000

240 216 268,000

M. Ramström ǀ Chapter 5 - Results

54

CHAPTER 5

RESULTS

This chapter presents the testing and the FE-analysis results and comparison studies.

5.1 Test results The results for the comparison between continuous and discontinuous weld end from the tests are

presented below. The S/N curves based on the test results are shown in Figure 50. Each curve in the S/N

diagram is based on eight specimens. The testing include four load levels in the interval 120-240 kN and

for each load level two specimens have been tested. The curves equation is a linear correlation between

the test data according to the least square method. According to IIW, [3], since the number of test results

are less than 10 test objects, 0.17 standard deviation has been used instead of the actual 0.052 (Continuous

weld end) respectively 0.034 (Discontinuous weld end). To produce the characteristic curve, two standard

deviations have been applied (i.e. 2.3 % failure probability). The actual and characteristic slopes are

presented in Table 18. The results indicate that the discontinuous end obtains significant higher fatigue

strength. The both slopes are slightly higher than three, but the discontinuous end receives a slope closer

to three. The FAT-value determined from the tests are FAT66 MPa for discontinuous and FAT51 MPa for

the continuous weld end, see Table 19.

Figure 50. Comparison between the continuous and discontinuous test results.

10

100

1000

10000 100000 1000000 10000000

Stre

ss r

ange

σr [

MP

a]

Cycles

S/N-Curve

Continous - Characteristic curve - 2,3% Continuous - Test results- 50%DisContinous - Characteristic curve - 2,3% DisContinous - Test results - 50%


55

Table 18. Test data.


Continuous test results 3.42 0.052

Continuous characteristic 3.00 0.170

Discontinuous test results 3.13 0.034

Discontinuous characteristic 3.00 0.170

Table 19. FAT-value.

Discontinuous Continuous

FAT-value [MPa] 66 51

Stdv. 0.17 0.17

Slope, m 3.00 3.00

Cycles [N] 2x106 2x10

6

5.2 Comparison of the computational methods The results for the comparison between the computational methods are presented in Figure 51. The

comparison is between the Effective notch method, the Hot-spot method and the Nominal stress method

for the continuous weld end. The curves are characteristic curves (i.e. 2.3 % failure probability) and have

slope -1/3 in the log-log diagram, m=3.

Figure 51. Comparison between the computational methods for the continuous weld end.

10

100

1000

10000 100000 1000000 10000000

Forc

e [

kN]

Cycles

S/N curve - Computational method

Effective notch

Hot Spot

Nominal stressContinuous (FAT71)


56

5.3 Comparison study between FE-analysis and tests The aim of the project is to find a method to calculate the fatigue life of the discontinuous weld end with

the Effective notch method. The FE-model used for the Effective notch approach is the continuous weld

end, due to the methods limitations and avoid singularities. The results for the comparison between the

computational Effective notch method and the tests are presented in Figure 52. The comparison includes

the tests for both the continuous and discontinuous weld end. The curves are characteristic curves (i.e. 2.3

% failure probability). To produce the tests characteristic curves, two standard deviations of the actual test

results have been applied. The actual standard deviations are 0.052 for continuous and 0.034 for

discontinuous weld end. The slopes are also the actual slope determined by the test results, 3.13 for

continuous and 3.42 for discontinuous weld end. Slope 3 and 2.3 % failure probability is used for the

Effective notch curve. The data for the S/N curve are summarized in Table 20.

Figure 52. Comparison between the Effective notch approach and the tests.

Table 20. Cycles to failure. Effective notch calculations and test data.

Stress range [MPa] Cycles,

Effective notch

Cycles,

Discontinuous weld end

Cycles,

Continuous weld end

108 1,630,000 1,452,000 776,000

144 656,000 598,000 289,000

180 329,000 326,000 133,000

216 187,000 159,000 73,000

10

100

1000

10000 100000 1000000 10000000

Forc

e [

kN]

Cycles

S/N curve - Effective notch compared with physical tests

Effetive notch

Continuous testdata

Discontinuoustest data


57

5.4 Developed FAT-value The results of the tests and the Effective Notch approach are analyzed further, in order to investigate and

develop a new FAT-value for weld ends which are simulated continuous. The develop FAT-value are

made as an S/N-curve determined by Equation (37). The S/N curves vertical axis is the notch stress from

the FE-model and the horizontal axis is the tests result, the method is illustrated in Figure 53. This forms

an Effective notch curve suited for discontinuous ends simulated with a continuous FE-model. The factors

that affect the new FAT-value are standard deviation and slope, shown in Table 21. The FAT-value is

calculated for 2,000,000 cycles. The constant C is determined by Equation (38) and to produce the

characteristic curve, where two standard deviations have been applied (i.e. 2.3 % failure probability). The

constant, m, is determined by the least square method.

The new FAT-value calculations are made for both the discontinuous and continuous test results.

log 𝑁 = log 𝐶 − 𝑚 log ∆𝜎 (37)

𝐶 = 𝑥𝑚 − 2 ∗ 𝑠𝑡𝑑𝑣 (38)

Figure 53. Method to develop FAT-value. The circular dot is a mark for the calculated FAT-value.


58

5.4.1 Discontinuous weld end For the discontinuous test results the new FAT-value is in the interval of 194-269 MPa, see Table 21. The

actual value from the test results indicates a FAT-value about 269 MPa. According to the norm that IIW

recommend the FAT-value 194 MPa. Figure 54, Figure 55 and Figure 56 show the develop S/N curves

with the new FAT-value at 2,000,000 cycles, marked with a dot.

Table 21. Data for new developed FAT-value, discontinuous.

FAT-value [MPa] 194 263 269

Stdv. 0.17 0.034 0.034

Slope, m 3 3 3.13

Cycles [N] 2x106

2x106 2x10

6

Figure 54. Developed S/N curve discontinuous, characteristic curve with slope 3 and stdv. 0.17.

Figure 55. Developed S/N curve discontinuous, characteristic curve with slope 3 and stdv. 0.034.

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles

S/N curve - Discontinuous compared with Effective notch method

Results curve

Characteristiccurve

Results

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles

S/N curve. Discontinuous compared with Effective notch method

Results curve

Characteristiccurve

Results


59

Figure 56. Developed S/N curve discontinuous, characteristic curve with slope 3.13 and stdv. 0.034.

5.4.2 Continuous weld end For the continuous test results the FAT-value is in the interval of 151-218 MPa, see Table 22. The actual

value from the test results indicates a FAT-value about 218 MPa. According to the norm that IIW

recommend the FAT-value is 151 MPa. Figure 57, Figure 58 and Figure 59 show the determined S/N

curves for continuous weld with the FAT-value at 2,000,000 cycles, marked with a dot.

Table 22. Data for new developed FAT-value, continuous.

FAT-value [MPa] 151 196 218

Stdv. 0.17 0.052 0.052

Slope, m 3 3 3.42

Cycles [N] 2x106

2x106 2x10

6

Figure 57. Developed S/N curve continuous, characteristic curve with slope 3 and stdv. 0.17.

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles

S/N curve. Discontinuous compared with Effective notch method

Results curve

Characteristiccurve

Results

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr [

MP

a]

Cycles

S/N curve - Continuous compared with Effective notch method

Results curve

Characteristiccurve

Results


60

Figure 58. Developed S/N curve continuous, characteristic curve with slope 3 and stdv. 0.052.

Figure 59. Developed S/N curve continuous, characteristic curve with slope 3.42 and stdv. 0.052.

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles


Results curve

Characteristiccurve

Results

10

100

1 000

10 000 100 000 1 000 000

Stre

ss r

ange

σr

[MP

a]

Cycles


Results curve

Characteristiccurve

Results

M. Ramström ǀ Chapter 6 - Conclusions

61

CHAPTER 6

CONCLUSIONS

In this chapter conclusions of the results are summarized. The conclusion involves the test results and the

different methods with focus on the Effective notch approach.

6.1 Tests The testing is an important part of the project to carry out a comparison with the calculation methods.

Number of specimens was eight for each weld end design, distributed in four load levels. This caused

problems to statistically ensure the results. Recommended are 5-7 specimens on each load level, therefore,

more specimens are needed to ensure the results [9]. The calculated standard deviation of the test results is

small compared with IIW recommendations. The slopes is slightly over three for both designs, three are

standard for welded joints subjected to normal stress [3].

The results from the tests indicate significant higher fatigue strength for the discontinuous weld end. The

Nominal stress method, which can be used for both the continuous and discontinuous weld end, estimates

the same outcome, but with different results in fatigue life. The Effective notch and the Hot-spot method

cannot be applied on the discontinuous weld end.

6.2 Comparison of computational methods and tests The three computational methods investigated are compared for the continuous weld end. The Effective

notch indicates highest fatigue strength, the nominal stress method indicates the second highest fatigue

strength. The Hot-spot approach differs most and indicates lower fatigue strength.

No method shows conservative results compared with the tests for the continuous weld end. The Hot-spot

approach estimates the continuous weld best and is the most accurate method for the continuous case.

However, the Nominal stress and the Hot-spot approach cannot be applied in complex load cases and for

that reason are the Effective notch method was used in further investigations.

The Effective notch and the Hot-spot approach cannot evaluate the discontinuous FE-model in a reliable

way. They are only validated for max principle stress oriented ± 60° perpendicular to the weld toe.

The nominal stress method can only give a coarse estimation of the weld fatigue life (if the nominal stress

is known). It can be a useful method for a coarse estimation between different welded joints.

6.3 Comparison of the Effective notch method and tests The Effective notch method can only be applied on the continuous FE-model, the model overestimate the

fatigue life of the continuous and match relative well with the test results of the discontinuous weld end.

The results indicate that the method needs some adjustment to match the discontinuous weld end, and

don’t match the continuous weld end.

M. Ramström ǀ Chapter 6 - Conclusions

62

6.4 Developed new FAT-value The developed FAT-value for the discontinuous weld end is in the interval 194-269 MPa dependent on

actual results or IIW standard norm. The interval is dependent of the standard deviation and the S/N

curves slope. These values can be determined from test results or taken from IIW recommendations. In

this case, the obtained interval is mainly dependent on the standard deviation, 0.034 or 0.17. FAT269 is

the value that matches BT Products production, but for a more reliable result are more specimens needed.

The overall conclusion is that it is possible to simulate the discontinuous weld end with a continuous FE-

model. The continuous model can be evaluated with the developed FAT-value for the Effective notch

approach. The FE-model should include the notch radius and the standards IIW recommend. Adjustments

for the new FAT-value need further investigations and cases to ensure a more general and reliable value.

The standard value, FAT225, are in the interval determined in this project, and the value for weld ends

simulated with a continuous weld is assumed to be close to this value.

M. Ramström ǀ Chapter 7 - Discussion

63

CHAPTER 7

DISCUSSION

In this chapter the test results and the different methods including the developed FAT-value are discussed.

The aim of this thesis was to find a calculation method to estimate fatigue life of weld ends. Since, the

Effective notch approach is the most common used and appropriate method, the focus was on developing

a new FAT-value for weld ends. The Effective notch approach is only applicable to the continuous FE-

model and the weld ends of interest are the discontinuous ends. Therefore, a comparison study between

the continuous FE-model and the test results for the continuous and discontinuous is made.

The test results indicate higher fatigue strength for the discontinuous than the continuous weld end. The

Nominal stress method which can be applied in both cases, estimates the same outcome, but with higher

fatigue life and different interrelationship. This is due to the coarse accuracy of the method and need of

scale factors, such as misalignment factor etc., see chapter 2.6.1 Nominal stress method. Also the two

cases used don’t match perfect with the tested specimens design and the cases contain parameters with

large intervals, see cases in APPENDIX C - Nominal stress method. In the calculations for the nominal

stresses was only the first term in equation (39) considered, and the effect of the moment was neglected.

This can be a reason for the overestimating of the calculations compared with the tests. In reality, there

was a moment, caused by the plate curvature, which increased the nominal stress. The pre-stresses from

the mounting in the test machine could also be an affecting factor. However, this method is useless if the

nominal stress is not defined and if the weld don’t match the cases available.

𝜎𝑛𝑜𝑚 =

𝐹

𝐴+

𝑀

𝐼 𝑧

(39)

The fatigue test results indicates that the continuous specimens where affected by factor that decrease

fatigue life, such as mean stress and residual stresses, geometrical stress concentrations and

discontinuities. Other affecting factors such as stress range and the material properties are the same.

The strain gauge measurements indicate higher stresses 2.5 mm from the weld toe for the continuous weld

compared with the discontinuous. This higher stress concentration is partially due to the pre-stresses and

may also depend on geometrical stress concentrations.

The Effective notch method with standard FAT225 MPa, overestimates the fatigue life for the continuous

and the discontinuous weld ends but the calculated result is close to the test result with the discontinuous

weld. The adjusted FAT-value to match the discontinuous weld end is in the interval 194-269 MPa

dependent on actual results or IIW standard norm. The interval is dependent of the standard deviation and

the slope. The actual result, determined from the test results indicates FAT269 MPa, but since only two

specimens were used at each load level, there are uncertainties about the actual standard deviation and

slope. If the number of specimens is less than ten the standard deviation should be estimated to 0.17 for

M. Ramström ǀ Chapter 7 - Discussion

64

geometrically simple structures according to IIW [3]. The difference between standard deviation 0.034

and 0.17 gives the large interval.

For the calculated result of the continuous weld compared with the test result of the continuous weld, FAT

225 should match. The results however indicate a FAT-value in the interval 151-218 MPa.

There is no mention of if the standard deviation is for robot or manually welded joints, which affect the

weld quality and standard deviation. More tests are required to ensure the actual scatter in BT Products

production. The actual standard deviation should be used in further adjustments for the FAT-value for

weld ends.

The slope of the characteristic curve is -1/m in a log-log diagram, were m is 3 for a welded joint affected

by normal tension according to IIW [3]. The two weld end designs received both an m-value above 3 and

the calculated FAT-values increase with a higher m-value. The m-value is also dependent on the number

of tests, and more tests will therefore ensure a correct m-value.

The occurred pre-stresses from the mounting in the test machine can be a factor that decreases the fatigue

life. The pre-stresses for the continuous where significant high compared with the discontinuous weld end.

The difference is caused from the curvature and misalignments in the baseplate, where the continuous was

affected considerably more. The stresses perpendicular to the continuous weld toe was 112 MPa at one

side and 55 MPa at the other side. The pre-stresses increase the mean stress, which affects the fatigue life.

A few of the strain gauges showed compressive strains (stresses). This leads to a stress ratio with changing

signs, R<0, in that locations. Fatigue and also welded joints fatigue, are more sensitive to tension than

compression stresses, but fatigue cracks can occur during compressive stresses due to positive residual

stresses. The occurred tensile stresses increase the risk of reaching the yield limit, due to the increased

mean stress. At that yield limit the material will plastically deform and the methods do not consider this.

The test results indicate interesting similarities and dissimilarities. The crack propagation for the

continuous weld end where at the same end side, side A. Side A where the start side for the weld string.

This similarity may be caused by the less penetration into the base material compared with the other side.

For the discontinuous weld end no similarities can be seen, since to the sides are similar. Unfortunately no

note where made of which side of the specimens where pressed in the test machine and this may have had

an increased effect of the pre-stresses. This could be an interesting parameter in further studies and testing.

The testing shows nothing about when, where and how the crack initiates and propagates to fracture, than

a crack initiation analysis is required. Another important factor is the local geometry that affects the stress

concentrations near the weld ends. For example, a larger radius and a smoother transition increase the

fatigue strength. The main affecting local geometry parameters are radius, undercut and lack of fusion,

illustrated in Figure 60. The effect of these parameters needs further investigations with microscopic

studies of the weld ends.

Figure 60. Weld profile, local geometry parameters.

M. Ramström ǀ Chapter 8 - Future work

65

CHAPTER 8

FUTURE WORK

In this chapter some suggestions of future work are presented. This thesis should be regarded as a

continuous study of weld ends calculations and further investigations are recommended.

This thesis contains the theory of fatigue life calculations of welds and an introduction to weld end

calculations. The most common method today, Effective notch, should be developed and adjusted to weld

ends. The conclusion is to simulate the discontinuous weld end with a continuous weld end with a

modified FAT-value. The aim is to develop and ensure a general FAT-value for weld ends.

The adjusted FAT-value is determined through tests compared with FE-analysis. Therefore, the number of

specimens is an important factor to statistically ensure the test results. To ensure the standard deviation

and the S/N curves slope, data from at least 5-7 specimens at each load level is required. [9] Therefore,

there is a need for more specimens.

Since the method should be adjusted for weld ends simulated with a continuous weld more cases is

interesting. In this thesis only one load case is investigated. The case is a non-load-transferring fillet weld.

More and varied load cases are also an interesting factor. For example, load-transmitting welds and

bending stresses are two interesting cases. Other parameters are throat thickness, plate thickness and other

types of welds. This first case showed results useful in further investigations.

It would also be interesting to investigate where and how the crack propagates through a visual crack

initiation investigation.

In the future, it is proposed that BT Products creates an internal database for fatigue testing and

determines their own standard deviation that matches their welded joints.

66

M. Ramström ǀ Bibliography

67

BIBLIOGRAPHY

[1] Svenskt Stål AB, SSAB, Plåthandbok, att konstruera och tillverka i höghållfast stål, 2 ed., Nyköping:

Österbergs, Sörmlandstryck AB, 2011.

[2] Boverkets handbok om stålkonstruktioner, BSK 99, Stålkonstruktioner, Karlskrona: Boverket,

Byggavdelningen, 2004.

[3] International institute of welding, IIW, "IIW-1823-07, Recommendations for fatigue design of

welded joints and components," IIW, 2008.

[4] Toyota Material Handling, "http://www.toyota-forklifts.eu," Toyota forklifts, [Online]. Available:

http://www.toyota-forklifts.eu/en/company/tmhephilosophy/Pages/History.aspx. [Accessed 11 02

2015].

[5] Toyota Material Handling, "http://www.toyota-forklifts.eu," Toyota forklift, [Online]. Available:

http://www.toyota-forklifts.eu/en/company/Pages/About-TICO.aspx. [Accessed 11 02 2015].

[6] Åsa Eriksson, Anna-Maria Lingnell, Claes Olsson, Hans Spennare, Svetsutvärdering med FEM, third

ed., Malmö: Liber AB, 2002.

[7] Bertil Jonsson, "Assessment of fatigue life," Volvo VCE division HL, Göteborg, 2006.

[8] F. V. Lawrence, "Mechanisms of Fatigue Crack Initiation and Growth," University of Illinois,

DEpartment of Mechanical Science and Engineering, 2014-07-02.

[9] C. Olsson, Konstruktionshandbok, för svetsade produkter i stål, 5 ed., Onsala: TechStrat Publishing,

2014.

[10] LIU, Ru Lin Peng, "http://www.iei.liu.se/kmt/education/undergraduatecourses-tmmi18?l=sv,"

[Online]. Available: http://www.iei.liu.se/kmt/education/tmmi18-intranet/coursemap-

tmmi18/forelasningar/1.612690/TMMI18_FL1_2015.pdf. [Accessed 17 02 2015].

[11] LIU, Johan Moverare, "Deformation and Fracture TMKM90. Chapter 9: Cyclic Stress and Strain

Fatigue," 2013. [Online]. Available: http://www.iei.liu.se/kmt/education/tmkm90-

intranet/coursemap-tmkm90/slides/1.512500/SlidesChapter9.pdf.

[12] A. Ekberg, "http://www.am.chalmers.se," [Online]. Available:

http://www.am.chalmers.se/~anek/teaching/fatfract/98-6,7,8,9,10,11.pdf. [Accessed 18 02 2015].

[13] Niklas Karlsson, Per-Henrik Lenander, "Analysis of Fatigue Life, LITH-IKP-EX--05/2302--SE,"

Division of Solid Mechanics, Linköping, 2005.

[14] A. Göransson, "Fatique analysis of weld ends - Comparison between testing and FEM-calculations,"

BT Products, Mjölby, 2014.

[15] S. Bergdahl, "An Examination of Welded Joints Regarding Weld Geometry, Weld Discontinuities

and the Effects of Shot Peening," LITH-IKP-EX--05/2315--SE, Linköping, 2004.

[16] Hyperphysics, [Online]. Available: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thexp.html.

[Accessed 16 03 2015].

[17] Tore Dahlberg, Anders Ekberg, Failure, Fracture, Fatigue: An introduction, Lund: Studentlitteratur,

2002.

[18] Robert D. Cook, David S. Malkus, Michael E. Plesha, Robert J. Witt, Concept and Applications of

Finite Element Analysis, fourth ed., John Wiley Sons, 2002.

68

M. Ramström ǀ Appendix

69

APPENDIX

The appendix includes the documentation considered not relevant to present in the main report, but still

fills an important purpose.

APPENDIX A

Drawings for the specimens


70


71


72


73


74

OS-A-1:OS-A-5

OS-CA-1

US-CA/B/C-1

OS-CA-2

2.5 mm

APPENDIX B

Strain Gauges placement

Strain gauges placement for continuous weld end


75

2,5 mm

OS-AV-1:OS-AV-3 and OS-AH-1:OS-AH-3

OS-AV-4 and OS-AH-4

US-CA/B/C-1

OS-CA-1

Strain gauges placement for discontinuous weld end


76

APPENDIX C

Nominal stress method

Discontinuous weld end.

FAT80 MPa is used.

Continuous weld end.

FAT71 MPa is used. (l=100 mm)

The Figures and the nominal stress data are from Internal Institute of welding, IIW. [3]

Documents

Fatigue Life Analysis of Weld Ends. Comparison between finite