Master Thesis
Analysis on Steel Composition and its
E↵ect on Weld Quality
A Case-Study Done on EB Welded Diaphragms at Siemens SIT
Author:
Idunn Arnard´ottir
Supervisor: Lorenzo Daghini
A thesis submitted in fulfilment of the requirements for the degree of Master of Science
in the
Department of Production Engineering
I, Idunn Arnard´ottir, declare that this thesis titled, ’Analysis on Steel Composition and its E↵ect on Weld Quality’ and the work presented in it are my own. I confirm that:
⌅ This work was done wholly or mainly while in candidature for a research degree
at this University.
⌅ Where any part of this thesis has previously been submitted for a degree or any
other qualification at this University or any other institution, this has been clearly stated.
⌅ Where I have consulted the published work of others, this is always clearly
at-tributed.
⌅ Where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this thesis is entirely my own work.
⌅ I have acknowledged all main sources of help.
⌅ Where the thesis is based on work done by myself jointly with others, I have made
clear exactly what was done by others and what I have contributed myself.
Signed: Date:
Abstract
School of Industrial Engineering and Management Department of Production Engineering
Master of Science
Analysis on Steel Composition and its E↵ect on Weld Quality by Idunn Arnard´ottir
The repair rate on electron beam welded diaphragm hasn’t been at the desired level at Siemens SIT for several years. An improvement program the past five years has reduce the repair rate from 60% to 12-15% but the goal of 5% repair rate hasn’t been met. Collection of diaphragm weld and repair data started in the fall 2011 and in this thesis the material composition of the materials used for the production will be analysed from a statistical perspective. The thesis includes a comprehensive research of the non parametric statistical methods suitable for non normally distributed, highly kurtotic and skewed data. Unfortunately a lot of statistical tests loose their power to correctly reject a false hypothesis with this kind of data. All of the elements in the material composition and the mechanical properties were analysed individually. In some of the cases it was possible to use statistical methods but in other it was not possible to conclude anything with statistics. Every case of outliers was evaluated individually.
The main conclusions are that in all of the four materials there are some elements and mechanical properties outside of the material specifications. A number of cases also had outliers inside of the material specification and in most cases those were causing the variability in the data and had higher repair rates than the overall repair rate. Some trends were found, for example the weld quality was better for lower yield strength in all materials and higher chromium content in material A produced better quality. The first steps to improvement for Siemens are to find out why materials outside of the material specifications are getting all the way to the production without anyone noticing. A simple material process control chart could visually notify if a material is outside of the specification limits or even just outside of the usual. Knowing exactly how the material is before starting the production will give time and space for preventive measures if they are necessary and could improve lead times and decrease costs.
Abstract
School of Industrial Engineering and Management Department of Production Engineering
Master of Science
Analysis on Steel Composition and its E↵ect on Weld Quality by Idunn Arnard´ottir
F¨or n˚agra ˚ar har Siemens hatt kvalitetsproblem med Electron Beam svetsing av mel-lanv¨aggar. F¨orb¨attringsprojekt har reducerat reparationer intensivt men f¨ortfarande finns det problemer med svetsningen. Inga mellanv¨aggar l¨amnar produktionen utan att ha bra kvalitet s˚a att f¨orb¨attringarna ska fokusera p˚a att reducera produktion-skostnader och ledtider. H¨osten 2011 b¨orjade Siemens att samla reparation data f¨or mellanv¨aggarna och i det examens arbetet blir materialsamans¨attningen analyserad med statistiska metoder. Arbetet inneh¨aller ¨overgripande analysis p˚a non-parametriska statistiska metoder och hur dom funkerar p˚a skevat data med h¨og kurtosis. Tyv¨arr finns det inte m˚anga metoder som har h¨og chans att f¨orkasta falska hypoteser n¨ar man jobbar med s˚ant data. Alla elementer och mekaniska egenskaper av materialet blev analyserade, men tyv¨arr inte alltid med statistiska metoder. Alla extremv¨arden blev analyserade in-dividuellt. I alla fyra materialer finns det elementer och mekaniska egenskaper utan f¨or materialets specifikationer. I vissa fall finns det outliers inom materialets specifikationer men som har h¨ogre reparation procent ¨an det vanliga och orsakar variationen i datat. Vissa tendenser kunne hittas, till exempel att det g˚ar b¨attre att svetsa materialer med l˚agre str¨ackgr¨ans och h¨ogre kromhalt g˚ar b¨attre f¨or material A. Simpelt processstyrning diagram skulle kunna visa om material ¨ar utanf¨or material specifikationen men ocks˚a om materialet ¨ar utanf¨or det ”vanliga”. Att veta exact hur materialet ¨ar innan produktion kan sjunka ledtider och produktionskostnader.
I would like to thank my supervisors, Lorenzo Daghini at KTH and Ronny Larsson at Siemens for all their help and advices throughout the whole process of this thesis. I would also like to thank Lars Mattsson, Ronny Norsberg, and Peter L. Ekberg for their advice on the data analysis aspect of the thesis.
My friends and family get a special thank you for keeping me sane in moments of despair.
Declaration of Authorship i
Abstract ii
Samanfattning iii
Acknowledgements iv
Contents v
List of Figures viii
List of Tables ix 1 Introduction 1 1.1 Structure . . . 1 1.2 Research Question . . . 2 1.3 Background . . . 2 1.3.1 Siemens . . . 2
1.3.2 Siemens Industrial Turbomachinery . . . 3
1.3.3 Diaphragms . . . 3
1.3.4 The Process of Producing Diaphragms . . . 4
1.3.5 Electron Beam Welding . . . 4
1.3.6 Materials . . . 6 1.3.7 Welding Alternatives. . . 8 2 Literature Review 10 2.1 Material . . . 10 2.1.1 Elements . . . 10 2.1.2 Mechanical Properties . . . 12 2.1.3 Weld Composition . . . 12 2.2 Statistical Methods . . . 13 2.2.1 Descriptives . . . 13 2.2.2 Analysis of Variance . . . 15 2.2.3 Normality . . . 16 2.2.4 Homogeneity of Variances . . . 19 v
2.2.5 Non-Parametric Comparison Between k Groups. . . 21
2.2.6 Transformations . . . 23
3 Methods 25 3.1 Database . . . 25
3.1.1 The Quality Factor . . . 26
3.1.2 Valitity & Reliability. . . 27
3.2 Analysis . . . 27
3.2.1 Assumptions . . . 27
3.2.2 Step-By-Step Description of the Analysis . . . 28
4 Results 30 4.1 Material A . . . 30 4.1.1 Material Composition . . . 30 4.1.2 Mechanical Properties . . . 30 4.1.3 Schae✏er Diagram . . . 31 4.1.4 Hypothesis Testing . . . 32 4.2 Material B. . . 39 4.2.1 Material Composition . . . 39 4.2.2 Mechanical Properties . . . 39 4.2.3 Hypothesis Testing . . . 40 4.3 Material C. . . 48 4.3.1 Material Composition . . . 48 4.3.2 Mechanical Properties . . . 49 4.3.3 Hypothesis Testing . . . 49 4.4 Material D . . . 57 4.4.1 Material Composition . . . 57 4.4.2 Mechanical Properties . . . 57 4.4.3 Hypothesis Testing . . . 58 4.4.4 Schae✏er Diagram . . . 58 5 Conclusions 65 5.1 Material A . . . 65 5.2 Material B. . . 66 5.3 Material C. . . 68 5.4 Material D . . . 69
6 Discussions & Future Steps 71
Bibliography 73
A Process Flow - BAPS Diaphragms 76
B Process Flow - Band Diaphragms 79
D Histogram for Silicon Content in Material A 83
E Histogram for Phosphorus Content in Material A 84
F Histogram for Yield Strength in Material A 85
G Histogram for Tensile Strength in Material A 86
H Histogram for Molybdenum Content in Material B 87
I Histogram for Nickel Content in Material C 88
J Histogram for Chromium Content in Material C 89
K Histogram for Molybdenum Content in Material C 90
L Histogram for Silicon Content in Material D 91
M Histogram for Yield Strength in Material D 92
1.1 Example of a Band Diaphragm . . . 4
1.2 Beam Deflect . . . 7
2.1 Normal distribution . . . 17
2.2 Skewed distribution . . . 17
2.3 Normal Probability Plot . . . 18
3.1 Defect Zone - Diaphragm . . . 26
4.1 Position of Material A . . . 32
D.1 Silicon Content . . . 83
E.1 Phosphorus content - Material A . . . 84
F.1 Yield Strength - Material A . . . 85
G.1 Tensile Strength - Material A . . . 86
H.1 Molybdenum Content - Material B . . . 87
I.1 Nickel Content - Material C . . . 88
J.1 Chromium Content - Material C . . . 89
K.1 Molybdenum Content - Material C . . . 90
L.1 Silicon Content - Material D. . . 91
M.1 Yield Strength- Material D . . . 92
N.1 Tensile Strength - Material D . . . 93
1.1 Chemical composition requirements for Material A . . . 6
1.2 Chemical composition requirements for Material B . . . 6
1.3 Chemical composition requirements for Material C . . . 7
1.4 Chemical composition requirements for Material D . . . 8
4.1 Statistical Descriptives for Material A . . . 30
4.2 Mechanical Properties of Material a . . . 31
4.3 Nickel Equivalent - Material A . . . 31
4.4 Chromium Equivalent - Material A . . . 31
4.5 Reheat Cracking Factor - Material A . . . 33
4.6 Carbon Content - Material A . . . 33
4.7 Manganese Content - Material A . . . 34
4.8 Silicon Content - Material A. . . 34
4.9 Silicon Content - Material A- Without Outliers . . . 35
4.10 Phosphorus content - Material A . . . 35
4.11 Phosphorus content - Material A - Without Outliers . . . 35
4.12 Sulphur content - Material A . . . 36
4.13 Sulphur content - Material A- Without Outliers. . . 36
4.14 Chromium Content - Material A . . . 37
4.15 Molybdenum Content - Material A . . . 37
4.16 Yield Strength - Material A . . . 38
4.17 Tensile Strength - Material A . . . 38
4.18 Statistical Descriptives for Material B-1 . . . 39
4.19 Statistical Descriptives for Material B-2 . . . 39
4.20 Mechanical Properties of Material B . . . 40
4.21 Carbon Content - Material A . . . 40
4.22 Manganese Content - Material B . . . 41
4.23 Silicon Content - Material B. . . 41
4.24 Phosphorus Content - Material B . . . 42
4.25 Sulphur Content - Material B . . . 42
4.26 Nickel Content - Material B . . . 43
4.27 Chromium Content - Material B . . . 43
4.28 Molybdenum Content - Material B . . . 44
4.29 Molybdenum Content - Material B - Without Outliers . . . 44
4.30 Vanadium - Material B. . . 45
4.31 Vanadium - Material B - Without Outliers. . . 45
4.32 Yield Strength - Material B . . . 46
4.33 Yield Strength - Material B - Without - Outliers . . . 46 ix
4.34 Tensile Strength - Material B . . . 46
4.35 Tensile Strength - Material B- Without Outliers . . . 47
4.36 Hardness - Material B . . . 47
4.37 Statistical Descriptives for Material C . . . 48
4.38 Statistical Descriptives for Material C . . . 48
4.39 Mechanical Properties of Material C . . . 49
4.40 Carbon Content - Material C . . . 50
4.41 Manganese Content - Material C . . . 50
4.42 Silicon Content- Material C . . . 51
4.43 Silicon Content- Material C - Without Outliers . . . 51
4.44 Phosphorus Content - Material C . . . 51
4.45 Sulphur Content - Material C . . . 52
4.46 Nickel Content - Material C . . . 52
4.47 Chromium Content - Material C . . . 53
4.48 Chromium Content - Material C - Without Outliers . . . 53
4.49 Molybdenum Content - Material C . . . 53
4.50 Molybdenum Content - Material C- Without Outliers . . . 54
4.51 Vanadium Content - Material C. . . 54
4.52 Yield Strength [MPa] - Material C . . . 55
4.53 Tensile Strength - Material C . . . 55
4.54 Tensile Strength - Material C- Without Outliers . . . 55
4.55 Hardness - Material C . . . 56
4.56 Statistical Descriptives for Material D . . . 57
4.57 Mechanical Properties of Material D . . . 58
4.58 Nickel Equivalent - Material D . . . 59
4.59 Chromium Equivalent - Material D . . . 59
4.60 Carbon Content - Material D . . . 59
4.61 Manganese Content - Material D . . . 60
4.62 Silicon Content - Material D. . . 60
4.63 Phosphorus Content - Material D . . . 61
4.64 Sulphur - Material D . . . 61
4.65 NIckel Content - Material D . . . 61
4.66 Chromium Content - Material D . . . 62
4.67 Yield Strength - Material D . . . 62
4.68 Yield Strength - Material D . . . 63
4.69 Tensile Strength - Material D . . . 63
4.70 Tensile Strength - Material D . . . 63
Introduction
This thesis project was initiated by the department of Production Development at Siemens SIT. For some years, the quality of the electron beam welded diaphragms hasn’t been at a desired level. In 2008 repair rate was approximately 60% and that same year an improvement project was initiated to reduce the repair rate down to 5 %. That goal has still not been met, and the repair rate remains around 12-15 %. The diaphragms are never released unless the desired quality is met, so this improvement project is mainly about reducing production lead times and cost. The improvement project has corrected around twenty di↵erent measures to improve the quality, but the remaining 12-15 % seem to happen randomly. Some measures that have already been taken are:
• Making standards on how to approve new suppliers. • Optimize the deepness of the weld.
• Implement attenuation measurements before the assembly of the diaphragm. • Standardize cleaning instructions.
• Changing of detergent.
In 2012 Siemens started to collect repair data about the diaphragms and now it is time to put this data to use.
1.1
Structure
First in section 1.3 the background of the product, the diaphragm, is explained. The steels used for the production, the process flow and the most common welding alterna-tives used are introduced. In chapter2the e↵ect of the di↵erent elements in the steels is
explained briefly as well as di↵erent factors a↵ecting the weld composition and quality. Furthermore the statistical methods used in the analysis are explained. In chapter3the collection of data and the database used in the analysis are explained. As well there is a step-by-step explanation of how the analysis was performed. In chapter 4 the raw data and results from the statistical analysis are presented, they are then analysed and discussed in chapter 5. Future research and recommended next steps for Siemens are presented in the chapter6.
1.2
Research Question
The initial goal of the improvement project that was started in 2008 was to lower the repair rate down to 5%. Now, about six years later the repair rate remains around 12 15%. The aim of this thesis is to see if there are any factors influencing the weld quality, from a statistical perspective. The focus will be on the chemical composition on the steels used in the production as well as their mechanical properties. This is because the repair rates seems to follow a pattern. The diaphragms are produced for one turbine order (can be more than one turbine in an order) at a time and all of the materials are bought for each order. The repair rate also seem to follow that steam turbine order pattern where some orders have high repair rate and others have zero repair rate. The main hypothesis and its alternative hypothesis are therefore
H0: The steel composition and mechanical properties are the same no matter the quality
of the weld.
H1: The steel composition and mechanical properties di↵er between the di↵erent stages
of quality.
The di↵erent stages of quality will be explained in details in section3.1.1.
1.3
Background
1.3.1 Siemens
Telegraphen Bauanstalt von Siemens & Halske was founded in 1847 by the thirty-one-year old Wernar von Siemens and the mechanical engineer Johann Georg Halske. For the first decades the majority of sales were electrical telegraphs. In 1848 Siemens & Halske won the contract to build a 500 km telegraph line from Berlin to Frankfurt. That line was the first long-distance route in Europe. In continuation of this, Siemens
& Halske began expanding the Russian telegraph network and in 1855 they opened their first foreign subsidiary in St. Petersburg. A Siemens Alternator supplied the power to the world’s first street lightning in the United Kingdom in 1881, and in the 1920s and 1930s Siemens started to manufacture radios, television set and electron microscopes. It was also in the 1920s that Siemens constructed a Hydro Power station in the Irish Free State, a world first for its design. After a rapid development and globalization Siemens is now present in almost every country in the world with around 362,000 employees. The core values, responsible, excellent and innovative are still, more than 160 years later, the basis of Siemens’ success. Siemens focus business areas are Energy, Healthcare, Industry, Infrastructure & Cities with a broad product group for example Energy, Financial Solutions, Mobility and Automation. [1]
1.3.2 Siemens Industrial Turbomachinery
Siemens Industrial Turbomachinery (Siemens SIT) is located in Finsp˚ang, Sweden. Like the name suggest, the production at Finsp˚ang is first class Steam- and Gas- Turbines. Though, Finsp˚ang only produces diaphragms for steam turbines that are then assembled with other turbine components in Germany. [2]
Swedish turbomachinery goes back to the year 1893 when Gustaf De Laval started his business in Stockholm. His company merged with Svenska Turbinfabriks Aktiebolaget Ljungstr¨om - STAL in the 1950s, and the production was moved to the industry area in Finsp˚ang, where STAL had been located since 1911. In Finsp˚ang there has been production of steam turbines for ships, nuclear power plants and later for solar power plants. Along with the steam turbines, development of gas turbines started in 1944. STAL-LAVAL changed names and ownership several times before the company was bought by Siemens in 2003. [2]
Now, Siemens produces gas turbines with a capacity from 5 MW to 375 MW and steam turbines with power output range of 45kW to 1,900 MW. [1]
1.3.3 Diaphragms
Diaphragm is a vital part of every steam turbine, as they guide the inflow of the stream as well as transform the enthalpy of the stream into kinetic energy [3]. At Siemens SIT, there are two di↵erent kinds of diaphragms, Band diaphragms and Blade Advanced Pro-cess System (BAPS) diaphragms. The Band diaphragm consists of an inner ring, inner band, guide vanes, outer band and an outer ring. An example of the Band diaphragm can be seen in figure 1.1. The BAPS diaphragm consists of an inner ring, BAPS guide vanes and an outer ring. The inner and outer rings can be from two di↵erent materials,
Material A and Material B. The guide vanes are from Material C and the bands are from Material D. The di↵erent materials, their chemical composition and mechanical properties will be discussed in chapter1.3.6. The joints are welded together with elec-tron beam welding, using one weld and a support edge or using two welds from opposite sites. There are a several di↵erent welding alternatives used at Siemens SIT, depending on what kind of a diaphragm is being welded, its size and what material the diaphragm consists of. The two most common welding alternatives are discussed in chapter 1.3.7. Every diaphragm is made upon customer order and is therefore not standardized. Both the sizes and materials vary according to the customers steam data.
Figure 1.1: Example of a Band Diaphragm.
1.3.4 The Process of Producing Diaphragms
The process of producing the Band and BAPS diaphragms is similar, but detailed de-scription on the process flow for each type can be seen in AppendicesA &B.
1.3.5 Electron Beam Welding
Electron beam (EB) welding is used for creating reliable, narrow and parallel weld seams. EB welding can be applied to very fine films as well as joining work pieces of more than 200mm section thickness in one operation [4]. Electron beam welding has a wide range of application and is used in many industrial areas, including automotive industries, nuclear power industries, power plants and tool construction [5]. It is also suitable for
many di↵erent materials including almost all steels, aluminium and its alloys, titanium and gold [5]. Generally, the weld beam is held stationary while the workpiece is moved [5].
Advantages of Electron Beam Welding [4][5]
• Extremely high power density at the focus of the beam (107W ⇤ cm2).
• Energy transfer occurs within the workpiece itself. • No filler metal is needed.
• No edge preparation is needed.
• Narrow welds and narrow heat a↵ected zones (HAZ) due to high welding speeds. • The inertia free oscillation makes it possible to join material that are otherwise
not considered suitable for welding.
• The welding distance can be varied which allows a variety of shapes to be welded. • EB welding is performed in vacuum to exclude oxygen from contact with the
cathode and to provide optimal conditions for the electric gun.
• Electrical and mechanical parameters are highly reproducible and consistent. Limitations of Electron Beam Welding [4][5]
• High cooling rates can lead to hardening that can lead to cracks. • The beam can be deflected by magnetism.
• Size of the workpiece is limited by the size of the vacuum chamber. • High investment.
• High precision of seam preparation is required.
1.3.6 Materials
Material A - Inner and Outer Diaphragm Rings
Material A is a ferrous alloy special steel with specific temperature properties for pres-sure equipment. The material requirements can be seen in table 1.1
Table 1.1: Chemical composition requirements for Material A
% wt C Si Mn P S Cr Mo Min 0.08 - 0.40 - - 0.70 0.40 Max 0.18 0.35 1.00 0.025 0.015 1.15 0.60
The mechanical properties are based on the thickness of the piece, the tensile strength ranges from 420-590 MPa and the yield strength ranges from 240-295 MPa. Material A doesn’t have any requirements for the hardness of the steel.
Material A contains the lowest chromium of all of the materials used for diaphragms. The main problem that occurs when the inner and outer diaphragm are made out of Material A is that the electron beam deflects because of magnetism and does not cover the whole joint. This is difficult to control but during the welding process the beam is monitored and kept as stable as possible. Other defects experienced using this material are cavities and cracks in the joint. A sample of how the beam deflects can be seen in figure 1.2. There the actual joint is marked with a blue straight line and the diameter of the joint is?1270 but the deflected weld, marked Svets 1, has deflected about 3mm inwards. The first repair weld, marked Rep1 is located between the first weld and the joint.
Material B - Inner and Outer Diaphragm Rings
Material B is a creep resistant martensitic stainless steel. The material requirements can be seen in table 1.2
Table 1.2: Chemical composition requirements for Material B
% wt C Si Mn P S Cr Mo Ni V Min 0.18 0.10 0.30 - - 11.00 0.80 0.30 0.25 Max 0.24 0.50 0.80 0.035 0.35 12.50 1.20 0.80 0.35
The mechanical properties of the steel are that yield strength should be between 540 MPa and 640 MPa, the tensile strength between 690 and 830 MPa and the hardness of
Figure 1.2: Example of Beam Deflect.
the material should be from 220 HB to 260 HB. The main defects that occur when using Material B are cracking and the formation of cavities.
Material C Guide Vanes & BAPS Guide Vanes
Material C is a creep resistant martensitic stainless steel. It has specified properties at elevated temperatures for use in exacting applications. The material requirements can be seen in table 1.3
Table 1.3: Chemical composition requirements for Material C
% wt C Si Mn P S Cr Mo Ni V Min 0.18 0.10 0.40 - - 11.00 0.80 0.30 0.25 Max 0.24 0.50 0.90 0.025 0.015 12.50 1.20 0.80 0.35
The mechanical properties are based on the thickness of the piece. The yield strength should be at least 700 MPa for all thicknesses. The tensile strength ranges from 850 MPa to 1050 Mpa and the hardness should be between 265 HB and 315 HB.
Material D - Inner and Outer Band
Material D is a corrosion resisting martensitic steel. The material requirements can be seen in table 1.4.
Table 1.4: Chemical composition requirements for Material D
% wt C Si Mn P S Cr Ni Min 0.08 - - - - 11.50 -Max 0.15 1.00 1.50 0.04 0.030 13.50 0.75
The mechanical properties for that material are that yield strength needs to exceed 250 MPa, tensile strength cannot be higher than 730 MPa and maximum hardness is 220 HB.
1.3.7 Welding Alternatives
The two most common welding alternatives in Siemens are Alternative 1 used when the rings are from Material A and Alternative 2 used when the rings are from Material B. Both of the welding alternatives use the same physical welding parameters, even if they are welding two di↵erent material combinations. Both of the alternatives require pre-heating the diaphragm up to 300 °C before welding but they have slightly di↵erent ordering.
Alternative 1
First the outer ring is stapled to the BAPS or band guide vanes from the top, and then a full butt weld is done on that joint. Next, the inner ring is stapled to the assembly from the top, and welded half way into the material. At last the inner ring is stapled to the assembly from the bottom, and then welded so that the it creates a 10mm overlap with the previous weld.
Alternative 2
First the inner ring is stapled to the BAPS or band guide vanes from the bottom, and then weld is done half way into the material. Next the outer ring is stapled to the assembly from the top, and then welded a full butt weld. Last the inner ring is stapled from the top, and welded so that it creates a 10mm overlap with the previous weld.
Literature Review
When steel compositions, welds or welding parameters are the topics of a research they are generally like this one, specific on exactly what steels are used, how they are welded and even how the parts are shaped. It can therefore be difficult to find literature on exactly the steels that are in your own study, and almost impossible to find the same steels analysed with respect to the same welding method. Below, the general e↵ect the elements have on steels and weld quality will be described briefly as well as all the statistical methods used in the analysis.
2.1
Material
2.1.1 ElementsThe composition of the steel will always be a compromise. To achieve desired weldability and hardness can be expensive, but a little compromise in those areas can decrease the prize [6]. Below, the elements and their e↵ect on weldability and other properties of the steel will be discussed.
Carbon, C
Carbon is necessary for the steel to gain strength, and it is cheap. Increased carbon content increases tensile strength and hardness but ductility and weldability decreases. When relatively low carbon content (0.10% wt) is combined with manganese (1.4%wt) it gives better weldability and ductility than steels with high carbon content (0.22% wt), and without sacrificing the strength. [6]
Silicon, Si
Silicon is present in all steel to some extent, usually between 0.20% wt and 0.45% wt [6]. Low carbon and silicon can be detrimental to the surface quality of welding, and high carbon and silicon combined can aggravate cracking tendencies [7]. Silicon also a↵ects the steel toughness and can reduce the risk of pores and cavities in the weld [7].
Manganese, Mn
Manganese content of steel depends on the carbon content - lower carbon content requires higher manganese content and vice versa [6]. If carbon and manganese are too high, the steel can become brittle [7]. To increase weldability the manganese should be between 0.30% wt and 0.80% wt and the ratio manganese to sulfur should be at least 50 to decrease the risk of hot cracking [? ]. But in most cases manganese has a good e↵ect on the weldability [6].
Phosphorus, P
In modern steels the phosphorus level is usually kept below 0.030 % of weight [6]. It increases strength and hardness but decreases impact to toughness and ductility [7]. Phosphorus can improve machinability in some steels but content over 0.04 % wt can increase the tendency to crack as well as make the weld brittle [7] [8].
Sulfur, S
Sulfur can decrease ductility but it improves machinability. It is usually below 0.05% wt in steels. Increased sulfur content results in decreased weldability. [7]
Chromium, Cr
Increased chromium up to 1.5 % wt can have negative e↵ect on reheat cracking, but after that, the risk of reheat cracking decreases with increased chromium content [8]. Chromium has magnetic properties when heated [9]. It increases resistance to oxidation and it can increase response to heat treatment. Stainless steels usually contain about 11 % wt chromium. [8]
Molybdenum, Mo
Molybdenum improves pitting corrosion resistance in austenitic steels and improves strengths and hardness at high temperatures in low alloy steels [8].
Nickel, Ni
Nickel over 8 %wt combined with high chromium is added to steels to form corrosion and heat resistant austenitic steels. It increases the corrosion resistance in steels and increases strength. Nickel is added in low alloy steels in small amounts to increase toughness. [8]
Vanadium, V
Vanadium is used to achieve finer grain structure in steels [7]. It a↵ects the grain growth during heat treatment and can increase hardenability of steels in contents up to 0.05 % wt [7]. Vanadium combined with molybdenum can increase the risk of stress relief cracking and reheat cracking [6].
2.1.2 Mechanical Properties
The mechanical properties are the Yield Strength (M P a), the Tensile Strength (M P a) and the Hardness (HB). Those three properties all depend on the chemical composition of the steel, and can be controlled with heat treatment and other processing methods.
2.1.3 Weld Composition
Three of the materials (B, C & D) are martensitic and two of them are creep resistant steels with chromium content from 9-13 %. Martensite can be sensitive to cold cracking and therefore need slow cooling after welding [6].
Nakamuras Rule for Reheat Cracking
Material A is a ferrous alloy special steel which can be sensitive to reheat cracking. Nakamuras introduced a formula to calculate whether a metal is sensitive to reheat cracking [6].
G = 10(%C) + %Cr + 3.3(%M o) + 8.1(%V ) 2 (2.1) where G should not exceed 2 to eliminate the risk of reheat cracking [6]. Whether it is reheat cracking or not that occurs when Material A is used for the rings can never be known for sure since it is welded with martensitic materials who need to be heat treated before the temperature of the workpiece after welding goes below 150°C. This sensitivity will though be examined in the results.
Schae✏er Diagram
The Scae✏er diagram is used to determine the weld composition when two or more di↵erent steels are welded together. The Scae✏er diagram can be used for for materials with up to 0.14% carbon content. For the materials in the analysis that only complies to Material A and D. Their location on the Scae✏er diagram will be shown in the results, in chapter 4.
2.2
Statistical Methods
Below, the statistical tools relevant to the database and the analysis are presented and explained.
2.2.1 Descriptives
Descriptives are used to describe the basic features of a data in a quantitative way, and are usually the first step in any statistical analysis. They are vital when choosing what methods can and should be used to analyze the data. [10]
Mean
The mean describes the central tendency of the data. It measures the centroid of the distribution. The formula for mean is
X = 1 n k X n=1 Xn (2.2)
where Xnis a measurement within the sample X and k is the number of measurements
in X. [11]
Standard Deviation
Standard deviation is a measure of variability in a sample or population.Usually, s is used for sample standard deviation and for population standard deviation.
s = v u u t 1 n k X n=1 (Xn X)2 (2.3)
where Xn is a measurement within the sample X, k is the number of measurements in
X and X is the mean of the sample. [11] Median
The median is the middle measurement when the measures are arranged in order of size. For odd number of measurements X1, X2, X3...X2k 1
M edian = Xk (2.4)
for even number of measurements X1, X2, X3...X2k
M edian = Xk Xk+1
2 (2.5)
[11] Skewness
Skewness is a measure of the lack of symmetry in the data. Symmetric data should have skewness close to zero while negative values indicate that the data is skewed to the left and positive values that the data is skewed to the right. Skewed data is often referred to as heavy tailed data. For sample data X1, X2, X3, ...Xk the formula for skewness is
skewness = Pk
n=1(Xn X)3
where X is the mean of the sample and s is the standard deviation. [12] Kurtosis
The kurtosis measures whether the data is peaked or flat. High values for kurtosis indicate that the data has a peak near the mean and declines rather rapidly. Low kurtosis indicates that the data has a rather flat top near the mean. For sample data X1, X2, X3, ...Xk the formula for kurtosis is
kurtosis = Pk
n=1(Xn X)4
(k 1)s4 3 (2.7)
where X is the mean of the sample and s is the standard deviation. Normally distributed data has a kurtosis of 3 so when the 3 is subtracted from the formula it is called excess kurtosis. [12]
2.2.2 Analysis of Variance
ANOVA stands for Analysis of Variance and refers to the statistical analysis of variations between and among two or more groups. Several di↵erent tests exist to test whether the means, medians, variances or ranks di↵er. Those tests can be used for example to test if a sample follows a certain distribution or if the variances are homogeneous. The typical ANOVA tests whether the means are the same between k groups. Di↵erent tests are used for two samples and for k samples, and di↵erent tests are used for normally distributed data and non-normally distributed data. All of those tests have di↵erent assumptions and give di↵erent robustness in di↵erent situations. Violating the tests assumptions can lead decreased statistical power, increased error rates and misinterpreted results. [12] Hypothesis Testing
The first step in hypothesis testing is to state they hypothesis itself [12]. If H0: The null hypothesis, e.g. two population means are equal &
The null hypothesis can only be rejected, never approved. If the calculated p-value from a test statistics is below a significance level ↵, typically 0.05% the null hypothesis can be rejected. If the p-value is higher than ↵ we fail to reject the hypothesis. [12]
Type I & Type II errors
Type I error or the statistical significance level is the probability of rejecting the hypothesis, when it is in fact true. This ↵ is chosen and typical values are 0.1, 0.05 and 0.01. The ability of a test to reject a null hypothesis are dependant on the sample size, so for a large enough sample, almost any di↵erence can be statistically significant. Therefore it is also important to look at the di↵erences in a practical sense. [12] Type II error is the probability of failing to reject a false null hypothesis. This probability is called and the statistical power is defined as (1 ). Statistical power is the probability of correctly rejecting a null hypothesis. The statistical power is usually used when deciding the sample size when designing an experiment. [12]
The usefulness of post- hoc power calculations have been debated between statisticians and John M. Strong & Dennis M. Heisey explained it as: ”Power calculations tell us how well we might be able to characterize nature in the future given a particular state and statistical study design, but they cannot use information in the data to tell us about the likely states of nature” [13]. Analysts working with historical data should therefore rather try to maintain power through choosing the right statistical tests.
2.2.3 Normality
Many statistical techniques assume that the data are at least approximately normal. The two most common ways of testing whether the data are normally distributed are graphical evaluation and analytical methods. [11]
Graphical Evaluation
Graphical evaluation of data consists of ”eyeballing” how the data looks like. Plotting the data in a histogram gives clear image on the center of the data, the spread, the skewness, the presence of outliers and modes. Normally distributed data, plotted in a histogram will take the shape of a bell, so that just by looking at a histogram one can see if the data is approximately normal or not. An example of a normally distributed data can be seen in figure 2.1and an example of skewed data can be seen in figure2.2
Figure 2.1: Normal distribution.
Figure 2.2: Skewed distribution.
In normal probability plots, the plot is scaled so that the normal distribution appears as a straight line. In these plots it is easy to see how the tails of the data are behaving and gain more insight on the nature of the di↵erences. The more the points vary from the normal distribution line, the bigger indication of departures from normality. [12] In figure 2.3 M anganese content in the steel M aterial A has been fitted to a normal
distribution. As the points vary a lot from the straight line, it is clear that the data is not normally distributed.
Figure 2.3: Normal Probability plot.
Analytical methods
Analytical methods are based on statistical tests and analysing the descriptives. It can be seen on the skewness and kurtosis if a data set is going to fail the normality assumption. But for further analyses, there are statistical hypothesis tests that test whether the distribution di↵ers significantly from the normal distribution or not. The two tests that are used in SP SS are the Kolmogorov-Smirnov and Shapiro - Wilk tests. Kolmogorov- Smirno↵
The Kolmogorov - Smirno↵ (K-S) hypothesis test is used to decide if data comes from a population with a specific distribution or not. The K-S test was originally only applicable continuous distribution but has now been extended to discrete distributions as well. The limitations of the K-S test is that it can be more sensitive near the center of the distribution than it is towards the tails. [12]
H1 : The sample does not follow a normal distribution
then the Kolmogorov - Smirnov test statistics is defined as: D = max 1iN ✓ F (Yi) i 1 N , i N F (Yi) ◆ (2.8) where F is the theoretical cumulative distribution of the sample being tested. The critical test value is obtained from a table, for a given ↵. If the test statistics D is greater than the critical value, the hypothesis H0 is rejected. [12]
Shapiro - Wilk
Shapiro- Wilk hypothesis test is used for the same purpose as the Kolmogorov - Smirnov test, to evaluate if a sample comes from a normal distribution. A serious of tests per-formed by Shapiro & Wilk in 1965 have shown that the Shapiro-Wilk test has higher power than the K-S test in a various range of populations for small sample sizes [14]. The limitations of the Shapiro- Wilk test is that it can be sensitive to outliers but oth-erwise has done well in comparison studies with other goodness of fit tests [12] [14] .
If H0: The sample follows a normal distribution &
H1 : The sample does not follow a normal distribution
then the Shapiro - Wilk test statistics is defined as:
W = ⇣PN k=1aiXi ⌘2 PN k=1(Xi X)2 (2.9) where Xi is the sample under consideration, where the values are ordered (X1 is the
smallest value), X is the mean of that sample and N the number of values. ai are
constants generated from the order statistics of a normally distributed sample. [12]
2.2.4 Homogeneity of Variances
Many statistical hypothesis tests also assume that the variances in the groups that are being compared are equal. Violations of the homogeneity assumptions can lead to increased Type I error rates. [15]
Levene Test
The Levene test is used in SP SS to test the homogeneity of variance assumption. It is an alternative to the Barltett test and it is more robust when the data departures from normality[12]. It is though based on the assumption that the distribution is symmetric [15].
If H0: The samples variances are equal for all groups &
H1 : The samples variances are not equal for at least one pair of groups
then the Levene’s test statistics is defined as: W = (N k) (k 1) Pk i=1Ni(Zi Z)2 Pk i=1 PN j=1(Zij Zi)2 (2.10) where N is the sample size and Ni the sample size of the ith subgroup. k is the number
of groups, Zi are the subgroup means and Z is the overall mean of the Zij.
Zij can be defined in three di↵erent ways, either based on the mean of the sample, the
median of the sample or of the trimmed mean of the sample. The null hypothesis is rejected if the test statistics are higher than the critical value F↵,k 1,N k where ↵ is the
significance level chosen [12]. Non-parametric Levene test
The typical Levene test that is described above loses its power when the distribution is asymmetric. David W. Nordstokke et al. introduced a new non-parametric Levene test in 2011 [15]. That test is based on the ranked scores of the data sample and the groups ranked mean.
Anova(|Rij Xij|) (2.11)
where Rij is the ranked score and Xij is the mean of the ranks for each group. Even
though the non-parametric Levenes test has greater statistical power than the traditional Levenes in asymmetric distributions, it looses its power quickly with highly kurtotic data. For highly kurtotic data, David W. Nordstokke et al. recommend using Data Mining methods like bootstrapping, permutations or randomization to determine if the variances are homogeneous or not. [15].
Some statisticians use a rule of thumb to decide if variances are homogeneous: The ratio between the biggest variance and the smallest variance cannot exceed 1.5 [16]. This rule
will be used as an reference when the kurtosis is too high to evaluate the homogeneity with the non-parametric Levene test.
2.2.5 Non-Parametric Comparison Between k Groups
Non-Parametric comparison tests do not assume normality of the groups, but it does not necessarily mean that they do not assume anything about the distribution. The two most common methods for k groups used in SP SS are explained below.
Kruskal Wallis
Kruskal-Wallis test whether all groups being tested come from the same population and it is based on the rank of the combined data [12]. Non-parametric tests are more powerful than parametric tests when the distributions are asymmetric but the Kruskal-Wallis test is rather sensitive to the violation of the homogeneity of variances assumption, specially when combined with unbalanced groups [17].
If H0: The k groups come from the same population distribution &
H1 : At least two of the groups do not come from the same population distribution
then the Kruskal-Wallis test statistics is defined as:
H = 12 N (N + 1) k X i=1 R2 i Ni 3(N + 1) (2.12) where N is the total number of observations, k is the number of groups, Ri is the sum
of ranks within group i and Ni is the number of observations within group i. [12]
Some sources recommend to use a correction of the test statistics if there are many ties in the data. The corrected test statistics C are slightly higher than the regular H value providing a more powerful test. The corrected test statistics is defined as:
C = 1 Ps
i=1(t3i ti)
N3 N (2.13)
where s is the number of sets of ties and ti is the number of tied scores in the ith set of
Mood’s Median Test
Mood’s Median Test is based on the median of the data. It is more robust than the Kruskal - Wallis test to outliers but not useful where there are a lot of ties in the data. This is because each observation is categorized with respect of whether it is below or above the median. The median test requires a minimum cell frequency i.e a minimum number of values below or above the median in each cell. If the minimum cell frequency is not met the median test has limited power. [18]
Post - Hoc Analysis
The MannWhitney test is a rank based hypothesis test for two groups. The Kruskal -Wallis test described above only tells if there are di↵erences between at least two the k groups being tested. To be able to know exactly which groups di↵er all possible pairs of groups need to be compared. If n1 and n2 are the number of observations in the two
groups and
if H0: The two groups have the same central tendency &
H1 : The two groups do not have the same central tendency
then the Mann-Whitney U statistics is defined as:
U1= n1n2+ 0.5(n1)(n1+ 1) T1 (2.14)
U2= n1n2+ 0.5(n2)(n2+ 1) T2 (2.15)
where T1 is the sum of the ranks in group 1 and T2 is the sum of the ranks in group 2.
The test statistics is then:
z = [U E(U )]/ (2.16) where U is the smaller value of U1 and U2 and
E(U ) = 0.5(n1)(n2) (2.17)
= [n1n2(n1+ n2+ 1)]/12 (2.18)
[12]
Bonferroni Adjustment When performing the post- hoc analysis the original ↵ value should be adjusted to avoid inflated Type I error rates [19]. The Bonferroni adjusted ↵B is defined as:
↵B= ↵
k (2.19)
where ↵ is the original value and k is the number of groups [19]. This adjustment is considered quite conservative and Holm (1979) introduced a new approach to the classical Bonferroni adjustment, the sequentially rejective Bonferroni:
↵B= ↵ k, ↵ k 1, ..., ↵ 1 (2.20) this approach is not as conservative though not loosing any probability of rejecting false hypothesis. [20]
2.2.6 Transformations
If the data is not normally distributed or does not have homogeneity of variances between groups it can sometimes be fixed with transformations. Parametric methods are usually more robust than the non-parametric ones but both can benefit from transformed data. Transformations are usually only applied to the dependant variable. They can lower the skewness and kurtosis of the data, but of course they do not change the number of ties. [21]
Di↵erent transformations are recommended for di↵erent scenarios, for a positive skew it is recommended to use the root of the data or logarithmic transformations, for neg-ative skew one should use square or cubic transformations, negneg-ative kurtosis can often be fixed with the inverse transformation but statisticians seem to struggle to find an e↵ective way of transforming positively kurtotic data [22]. Some common transforma-tions used on percentage and proportion data are the Box-Cox transformation, Logistic transformation, Arcsine transformation and Freeman Tukey transformation [23]
Box-Cox Transformation The Box-Cox transformation is a set of power transfor-mations first introduced by Box & Cox in 1964. For data > 0 applies:
y( ) = 8 < : y 1, if 6= 0 log(y), if = 0 (2.21) where y( ) is the transformed data and y is the original data. is a constant, commonly between 2 and 2. [24]
Transformation can make the interpretation of the data more complex, specially for < 0 because it reverses the order of the data [21]. A simple M AT LAB code to evaluate how di↵erent values e↵ect the skewness and kurtosis of a distribution can be seen in Appendix C.
Logit Transformation Logit transformation is usually used when the data is a pro-portion
y⇤= ln( y
1 y) (2.22) where y⇤ is the transformed data and y is the original data. [23]
Arcsine Transformation Arcsine transformation is also generally used for propor-tional data.
y⇤= 2⇤ Arcsine(py) (2.23) where y⇤ is the transformed data and y is the original data. [11]
Freeman Tukey Transformation Freeman Tukey transformation is used when the data is very small.
y⇤ =py +py + 1 (2.24) where y⇤ is the transformed data and y is the original data. [22]
Other Transformations Other common transformations are simple transformations like square (y⇤ = y2), cubic (y⇤ = y3), other power transformations (y⇤ = yi), square root transformation (y⇤= py) and inverse transformation (y⇤ = 1/y). [22]
Methods
3.1
Database
The database contains information about each and every diaphragm that has been pro-duced in Siemens since the fall 2011. They all have a specific name and are assigned to a specific steam turbine. Some of the diaphragms in the database are service projects, they will not be included in the statistical analysis, mainly because it is difficult to find the material composition of the steel that was used. The database also has informa-tion about the diaphragms size, the date it was welded, the date it was approved, what the quality of the weld was and if and how many repairs the diaphragm needed. The diaphragms are given a quality factor, based upon visual inspection of the ultrasound test. The quality factor ranges from 1 to 5, where 1 is excellent quality and 5 is unac-ceptable quality. The quality factor will be described in further details below in chapter
3.1.1. The database on the other hand didn’t contain any information about the steel composition. The steel composition and mechanical properties were taken from material certificates received from Siemens’ suppliers. Siemens has a requirement for traceability of all materials for the inner rings, outer rings, inner strip and outer strip. There is no requirement of traceability of materials for the guide vanes, which caused several problems. It seemed to be completely dependent on the person who did the purchase if there was a possibility to trace down the correct certificate or not. The guide vanes can be produced in two di↵erent ways. One way is to get them from another Siemens factory, Siemens Brno, located in the Czech Republic. The other way is to outsource the guide vanes. Then Siemens SIT buys the material, and then sends it to an outsourced party who manufactures the guide vanes and sends them back. The database was di-vided into five di↵erent parts, where individual materials used in the diaphragms were
kept separate. The data is kept in a Excel file but all calculations are done in SP SS, a statistical analysis software made by IBM [25].
3.1.1 The Quality Factor
The Quality Factor is a number from 1 to 5 where 1 is excellent weld quality and 5 is not acceptable quality. All diaphragms with a quality factor three or higher must undergo a repair weld and a diaphragm can need more than one repair weld. The Quality Factor is an internal scale, and it is based on visual inspection of the results from the ultrasonic testing of the weld. The quality scale is described below.
• Quality Factor 1: No indications of defects larger than 5mm outside the defect zone.
• Quality Factor 2: Single indications of defects up to 5mm outside the defect zone. • Quality Factor 3: Indications of defects larger than 5mm outside the defect zone. • Quality Factor 4: Many indications of defects larger than 5mm outside the defect
zone.
• Quality Factor 5: Cracked guide vanes.
The defect zone is a small zone located in the middle of the cross-section where defects are not critical and are allowed. The defect zone is marked with A in picture 3.1 and B0 is the size of the guide vanes.
Figure 3.1: Defect Zone - Diaphragm.
Since the collection of repair data started, the fall 2011, only two operators have done this visual inspection of the weld quality. Some of the diaphragms have even been judged by both of them. Their reproducibility has not been tested but since they are only two skilled operators it can be assumed that the quality of the weld has been judged quite
similarly from diaphragm to diaphragm.
Since the percentage of repair welds has been from 10 20% the last few years, there is a great unbalance in the group sizes Repair vs Not Repair. Many statistical tests are not robust in unbalanced situations and therefore it was decided to compare only Quality Factor 1 vs Quality Factor 2 vs Quality Factor 3,4 & 5. So that all diaphragms that need repair, regardless of ”how bad” they were, are put in one group in the analysis. This decision was made in agreement with a statistician and the supervisors of the thesis.
3.1.2 Valitity & Reliability
The validity of the report is based on the assumption that the information in the database are correct. The information about the diaphragms themselves are taken from draw-ings and building descriptions from Siemens’ internal database. The information about the steel composition and mechanical properties are, as previously mentioned, taken from material certificates from suppliers. Siemens does not do any chemical analysis of whether the material composition is correct or not, but several hardness tests have been conducted on materials and they have shown that the hardness of the material lies a little bit higher than is stated in the certificate. Therefore there is no way of knowing for sure if the certificates are 100% correct. If the same statistical methods, who are all described in chapter 2.2, were used on the same database there is no reason to believe that the results would alter.
3.2
Analysis
3.2.1 AssumptionsThe number of values as well as the repair rate may di↵er between the four di↵erent materials. This is because of troubles that occurred when collecting the data. The traceability of some materials wasn’t as good as the others, so basically, the analysis is done on every value that was possible to collect. For example, in one diaphragm you could find the material composition of every component, but the Quality Factor of the inner weld was missing. This would mean that the inner ring and inner band of that diaphragm are not in the analysis. Another example is that the material composition for the guide vanes could not be found. So for that diaphragm only the bands and rings are in the analysis.
In the analysis there are certain assumptions made. The hardness [HB] of the di↵erent materials is presented in a single value and this value is used for calculations. This was
done because di↵erent suppliers present the hardness of the material in di↵erent ways. There are four di↵erent ways the hardness is presented in the certificates:
• The hardness is presented in one value.
• The hardness is presented in two values that are measured at two di↵erent positions on the material.
• The hardness is presented in three values that are measured at three di↵erent positions on the material.
• The hardness is presented in two values that are the maximum and minimum hardness values of many objects of the same material. This only complies for the guide vanes.
As was mentioned in the section about homogeneity 2.2.4the non - parametric Levene test is not robust when the kurtosis is high. But what is too high kurtosis? John R. Slate & Ana Rojas-Lebouef suggest that when using SP SS one should divide the kurtosis with the standard error of the kurtosis. If the value is within ±3 the kurtosis isn’t too high. [26]
This rule applies when checking for normality but will be used as a rough benchmark when deciding if the data needs transformation or not. The rule of thumb about the ratio of the biggest and smallest variance not exceeding 1.5 will also be used in the statistical analysis when the non-parametric Levene test cannot be used.
3.2.2 Step-By-Step Description of the Analysis
Before analysing whether there is any di↵erence between the di↵erent quality groups it is important to make sure if it actually can be analysed with statistics. The materials will be analysed separately and the di↵erent elements within each material as well as the mechanical properties will all be analysed on their own.The statistical analysis is described step-by-step below.
First step is to calculate the descriptives and check for normality. The normality was checked with the Shapiro-Wilk test. In all cases the distribution was non-normally distributed, so the scenario of normal data will not be discussed. Next step is to check if the kurtosis is too high for the non-parametric Levene test. If the kurtosis is ok the test is used, otherwise the ratio of the variances is checked. If the variances are homogeneous the analysis can be continued with Kruskal -Wallis test between all 3 groups and post-hoc analysis using Mann-Whitney test with Bonferroni adjusted ↵.
kurtosis needs to be found. Maybe there are some obvious extreme outliers. In the case of outliers they will be evaluated individually for every case. There is rarely any good reason to remove outliers, because they do represent a certain variation in the data that is important in this analysis. They will though be removed to see how the data looks like without them, but any results will be interpreted with caution. Maybe there is no obvious reason for the high kurtosis or the heterogeneity of variances. Then the data can sometimes be transformed. There are a number of transformation that can be tried, and it is mostly a ”trial and error” study. There are some recommendations on which transformations to use in certain situations, but unfortunately very few (if any) are useful when the kurtosis is positively high. The problem with the transformations is that the number of ties will never be reduced even though their e↵ect (kurtosis) can be reduced. That means that the non-parametric Levene test cannot be used, even on the transformed data. So to be able to test the homogeneity with statistical tests the transformation needs to make the data normally distributed. That is achievable for skewed distributions but kurtotic distributions are a bigger problem. So only the ratio rule of thumb for the variances will be used on all transformed data. If the transformation makes the variances homogeneous the analysis can be continued with the Kruskal -Wallis test and a post -hoc analysis if needed. If no transformation is successful there is unfortunately nothing more that can be done.
Results
4.1
Material A
4.1.1 Material Composition
Material A is a ferrous alloy special steel. The material specification can be seen in table
1.1. The statistical descriptives for the material can be seen in table4.1
Table 4.1: Statistical Descriptives for Material A
Descriptive %C %Mn %Si %P %S %Cr %Mo Mean 0.1533 0.8004 0.2399 0.01123 0.005484 0.9648 0.4343 Median 0.16 0.83 0.24 0.012 0.0060 0.96 0.43 Standard Deviation 0.0092 0.0625 0.0243 0.0020 0.0025 0.0811 0.0156 Minimum 0.13 0.64 0.20 0.009 0.002 0.79 0.41 Maximum 0.16 0.85 0.35 0.025 0.015 1.09 0.46 Skewness -0.983 -1.411 0.746 3.942 0.745 -0.218 0.497 Kurtosis -0.446 0.846 3.090 25.257 1.376 0.009 -1.081 All values in the chemical composition are within the limits of the material requirements. The material certificates for inner rings and outer rings are combined, but the inner rings are compared with the Quality Factor on the inner weld and the outer rings are compared to the Quality Factor of the outer weld. The total number of rings is N = 384 and the number in each group is N1 = 291, N2 = 59 and N3 = 34, with the total percentage of
repairs 8.85%.
4.1.2 Mechanical Properties
The mechanical properties of Material A are presented in table 4.2. 30
Table 4.2: Mechanical Properties of Material a
Descriptives Yield Strength [MPa] Tensile Strength [MPa] Mean 338.21 559.93 Median 335 570 Standard Deviation 26.85 23.50 Minimum 314 505 Maximum 443 588 Skewness 2.70 -1.32 Kurtosis 6.468 0.230
The material requirements are met in the tensile strength, but the minimum of the yield strength even exceeds the limit which is 240 290M P a.
4.1.3 Schae✏er Diagram
The Schae✏er Diagram is usually used to see how the weld composition will be. It is only applicable to Material A and D, but it is already known that the other materials are martensitic, so knowing where material A and D are positioned can still be helpful. The nickel equivalent is presented in table 4.3and the chromium equivalent in table4.4
Table 4.3: Nickel Equivalent - Material A
Nickel Equivalent Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 5.033 4.894 4.895 Median 5.210 4.925 4.893 Standard Deviation 0.286 0.321 0.314 Minimum 4.27 4.27 4.27 Maximum 5.23 5.23 5.23 Skewness -1.140 -0.48 -0.386 Kurtosis -0.242 -1.047 -1.154
Table 4.4: Chromium Equivalent - Material A
Chromium Equivalent Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 1.774 1.708 1.716 Median 1.770 1.695 1.740 Standard Deviation 0.120 0.123 0.119 Minimum 1.51 1.51 1.51 Maximum 1.96 1.96 1.96 Skewness 0.131 0.110 -0.126 Kurtosis -0.494 0.048 0.167
In figure 4.1material A is represented on the diagram as a red rectangle (material D is the blue rectangle). The diagram shows that the material is positioned in the martensitic zone. Since the material is within the martensitic zone at all times no further calculations
will be done on the chrome and nickel equivalents. The weld composition when material A is in the rings is therefore somewhere in the martensitic and martensitic-ferrite zone.
Figure 4.1: Position of Material A - Schae✏er Diagram
4.1.4 Hypothesis Testing
Here the results of the hypothesis testing will be presented. The general hypothesis for all elements and mechanical properties is:
If H0: The groups are the same (in terms of medians, means, distribution ect. based on
the data) &
H1 : The groups are not the same
groups meaning the di↵erent quality factors, 1, 2 and 3. The assumptions of the hypoth-esis tests will be checked with the statistical tests presented in the section Statistical Tools2.2. The hypothesis is rejected if the p value is lower than the significance level ↵ = 0.05. The standard error of the mean, skewness and kurtosis is presented in brackets in the tables.
Reheat Cracking The factor should be below 2 to limit the chances of reheat crack-ing. As can be seen in table4.5 the factor is actually lower for Quality Factor 2 and 3 than it is for Quality Factor 1.
Table 4.5: Reheat Cracking Factor - Material A
Reheat Cracking Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 1.957 (.01) 1.845 (.03) 1.860 (.03) Median 1.946 1.786 1.932 Standard Deviation 0.1787 0.192 0.179 Minimum 1.576 1.576 1.576 Maximum 2.208 2.208 2.208 Skewness -0.122 (.143) 0.293 (.311) 0.012 (.403) Kurtosis -0.873 (.29) -0.768 (.613) -0.908 (.708)
All Quality Factor groups di↵er significantly from normality (Shapiro-Wilk test p < 0.05). The kurtosis isn’t too high and the variances are homogeneous (non-parametric Levene test p < 0.05). The groups do not come from the same population (Kruskal-Wallis p < 0.05). Mann- Whitney test post-hoc analysis shows that group 1 di↵ers significantly from both group 2 (p = 0.00) and 3 (p =0.01). Groups 2 and 3 are not significantly di↵erent (p = 0.656). The mean of group 1 is significantly higher than than the other two groups.
Carbon The carbon content in material A is in table4.6, since the range is only 0, 03% for all three groups.
Table 4.6: Carbon Content - Material A
%C Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.1544 (.00) 0.1498 (.00) 0.1503 (.00) Median 0.16 0.15 0.15 Standard Deviation 0.0088 0.0101 0.0097 Minimum 0.13 0.13 0.13 Maximum 0.16 0.16 0.16 Skewness -1.190 (.143) -0.487 (.311) -0.486 (.403) Kurtosis -0.059 (.285) -0.998 (613) -0.963 (.788)
All three groups di↵er significantly from normality (Shapiro-Wilk p < 0.05). The groups have homogeneous variances (Non-parametric Levene test p < 0.978). The distribution of the transformed data is not the same across all groups (Kruskal-Wallis p = 0.00). Mann-Whitney for post-hoc analysis shows that group 1 di↵ers significantly from group 2 (p = 0.00) and from group 3 (p = 0.006). Groups 2 and 3 do not di↵er (p = 0.866).
Manganese The mean of the groups suggest that manganese content is lower in the rings with bad quality than that it is for rings with good quality. The manganese content is in table4.7.
Table 4.7: Manganese Content - Material A
%Mn Qualitys Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.8040 (.00) 0.7986 (.00) 0.7724 (.00) Median 0.83 0.82 0.805 Standard Deviation 0.0609 0.0597 0.0743 Minimum 0.64 0.64 0.64 Maximum 0.85 0.85 0.85 Skewness -1.562 (.143) -1.285 (.311) -0.690 (.403) Kurtosis 1.341 (.285) 0.867 (.613) -1.039 (.788)
All three groups di↵er significantly from normality (Shapiro-Wilk p < 0.05). The first group has too high kurtosis (1.341/0.285 > 3) and the variances break the rule of thumb as well (0.07432/0.05972 > 1.5). No transformation made the data normally distributed
or the variances homogeneous so no further calculations can be done.
By looking at the means we can see that they are decreasing with worse quality, but there is no way of knowing whether that is happening randomly or if lower manganese contents cause worse weld quality.
Silicon It is interesting that both group 1 and group 3 have a wider range than group 2 and also a lot higher kurtosis. By looking at the histogram of the silicon content in Appendix D, it can be seen that the rings containing 0.35% silicon are an obvious outlier. Those rings are only five, four rings in group 1 and one in group 3. The next highest silicon content is 0.27%.
Table 4.8: Silicon Content - Material A
%Si Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.2418 (.00) 0.2339 (.00) 0.2344 (.00) Median 0.24 0.24 0.24 Standard Deviation 0.02543 0.02213 0.02798 Minimum 0.20 0.20 0.20 Maximum 0.35 0.27 0.35 Skewness 0.700 (.143) -0.278 (.311) 2.037 (.403) Kurtosis 3.023 (.285) -0.936 (.613) 7.985 (.788)
All three groups in table4.8 di↵er significantly from normality (Shapiro-Wilk p < 0.05) and because of the high kurtosis in group 1 and 3 no further calculations can be done. No transformations could lower the kurtosis enough. The data without outliers can be seen in table 4.9
Table 4.9: Silicon Content - Material A- Without Outliers
%Si Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.2402 (.00) 0.2339 (.00) 0.2309 (.00) Median 0.24 0.24 0.24 Standard Deviation 0.02213 0.02213 0.01942 Minimum 0.20 0.20 0.20 Maximum 0.37 0.27 0.27 Skewness -0.364 (.144) -0.278 (.311) -0.136 (.409) Kurtosis -0.623 (.287) -0.936 (.613) -0.554 (.798)
This data still failed the normality test (Shapiro - Will p < 0.05) and the variances are not homogeneous (non-parametric Levene test p < 0.05). The variances could not be made homogeneous with any transformation.
Phosphorus Both group 1 and group 3 have a wider range than group 2 and also an extremely high kurtosis. By looking at the histograms of the groups, AppendixE, it can be seen that the rings containing 0.025% phosphorus are an obvious outlier. The rings are the same ones that were outliers in the silicon content. The next highest phosphorus content is 0.013%.
Table 4.10: Phosphorus content - Material A
%P Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.0113 (.00) 0.0106 (.00) 0.0114 (.00) Median 0.012 0.010 0.0115 Standard Deviation 0.0020 0.0013 0.0027 Minimum 0.009 0.009 0.009 Maximum 0.025 0.013 0.025 Skewness 4.048 (.143) 0.179 (.311) 3.81(.403) Kurtosis 25.807 (.285) -1.340 (.613) 18.993 (.788)
All three groups from table 4.10 di↵er significantly from normality (Shapiro-Wilk p < 0.05) and no transformation can reduce this kurtosis enough. The data without the outliers is presented in table4.11.
Table 4.11: Phosphorus content - Material A - Without Outliers
%P Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.0112 (.00) 0.0106 (.00) 0.0105 (.00) Median 0.012 0.010 0.011 Standard Deviation 0.0013 0.0013 0.0013 Minimum 0.009 0.009 0.009 Maximum 0.013 0.013 0.013 Skewness -0.199 (.144) 0.179 (.311) -0.194 (.409) Kurtosis -1.364 (.285) -1.340 (.613) -1.364 (.798)
The data still di↵ers significantly from normality (Shapiro-Wilk p < 0.05) and the variances are not homogeneous (non-parametric Levene test p < 0.05). There is not any di↵erence in the standard deviation until the fifth decimal so this result is somewhat surprising. Without knowing for sure the power of the Kruskal Wallis and the Mood’s median test they both deliver the same result, the hypothesis cannot be rejected (p > 0.05).
Sulphur The range of the sulphur content is proportionally high, the highest value is 7.5 times higher than the lowest value. The standard deviation is even higher than the lowest value in groups 1 and 3.
Table 4.12: Sulphur content - Material A
%S Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.0054 (.00) 0.0054 (.00) 0.0059 (.00) Median 0.006 0.006 0.006 Standard Deviation 0.0026 0.0019 0.0023 Minimum 0.002 0.002 0.002 Maximum 0.015 0.009 0.015 Skewness 0.705 (.143) 0.312 (.311) 1.699 (.403) Kurtosis 1.001 (.285) -0.218 (.613) 6.144 (.788)
All three groups from table 4.12 di↵er significantly from normality (Shapiro-Wilk p < 0.05). Groups 1 and 3 have to high kurtosis. Again it is the same five rings that are outliers and causing the high standard deviation and kurtosis. The data without the outliers is presented in table4.13.
Table 4.13: Sulphur content - Material A- Without Outliers
%S Quality Factor 1 Quality Factor 2 Quality Factor 3 Mean 0.0053 (.00) 0.0054 (.00) 0.0057 (.00) Median 0.006 0.006 0.006 Standard Deviation 0.0023 0.0019 0.0017 Minimum 0.002 0.002 0.002 Maximum 0.009 0.009 0.009 Skewness 0.160 (.144) 0.312 (.311) 0.049 (.409) Kurtosis -0.989 (.287) -0.218 (.613) 0.711 (.798)
The three groups from table 4.13 still di↵er significantly from normality (Shapiro-Wilk p < 0.05). Group 1 has too high kurtosis for the non - parametric Levene test so the data needs to be transformed to be able to continue the analysis. Unfortunately no transformation worked.
Chromium All three groups have the same range but the means di↵er slightly. The kurtosis is not preventing any analysis and there are no obvious outliers.
