Laboratory of Fatigue and Strength
MASTER’S THESIS
STATISTICAL VARIATION OF WELD PROFILES AND THEIR EXPECTED INFLUENCE ON FATIGUE STRENGTH
The subject of the master’s thesis has been confirmed by the Department Council of the Department of Mechanical Engineering on October 5th, 2005.
The thesis has been examined by Prof. Gary B. Marquis and Dr. Timo Nykänen
Lappeenranta, May 23rd, 2006
Arjun Seshadri
Teknologiapuistonkatu 4 B 12 53850 Lappeenranta
+358 50 9239099 DMWT 0273470
ABSTRACT
Lappeenranta University of Technology Department of Mechanical Engineering
Arjun Seshadri
Statistical variation of weld profiles and their expected influence on fatigue strength
Master’s thesis 2006
47 Pages, 27 Figures, 2 Tables and 7 Appendices
Supervisors: Prof. Gary B. Marquis and Dr. Timo Nykänen
Keywords: Cruciform joints; Non-load carrying welds; Fatigue strength; Statistical analysis; Extreme value theory, Fracture mechanics
The general objective of this study was to conduct a statistical analysis on the variation of the weld profiles and their influence on the fatigue strength of the joint. Weld quality with respect to its fatigue strength is of importance which is the main concept behind this thesis. The intention of this study was to establish the influence of weld geometric parameters on the weld quality and fatigue strength. The effect of local geometrical variations of non-load carrying cruciform fillet welded joint under tensile loading was studied in this thesis work.
Linear Elastic Fracture Mechanics was used to calculate fatigue strength of the cruciform fillet welded joints in as-welded condition and under cyclic tensile loading, for a range of weld geometries. With extreme value statistical analysis and LEFM, an attempt was made to relate the variation of the cruciform weld profiles such as weld angle and weld toe radius to respective FAT classes.
ACKNOWLEDGEMENT
The work reported in this thesis was carried out in the Department of Mechanical Engineering at Lappeenranta University of Technology, Finland.
It gives me immense pleasure to express my deepest gratitude to those who have helped me all along my master’s thesis.
I am grateful to my supervisor and mentor, Prof. Gary B. Marquis, who has not only been a source of inspiration all through my studies in Lappeenranta University of Technology, but also has been a good advisor. I sincerely thank him for his ever-patient guidance and counsel, which was helpful during my thesis work.
I also would like to extend my appreciation to Dr. Timo Nykänen and Dr. Timo Björk for their expertise and advices which assisted me during my thesis work.
My further appreciation is towards the aid given by the staffs and assistants of Fatigue and Strength, Material Science and Welding Laboratories of Lappeenranta University of Technology.
I owe a special gratitude to my close and dear ones, far and near to me. My parents and my elder brother, who have been the backbone of my life, have encouraged and supported throughout my studies here at LUT.
TABLE OF CONTENTS
ABSTRACT... i
ACKNOWLEDGEMENT ... ii
TABLE OF CONTENTS... iii
List of Figures ... v
List of Tables... v
List of Abbreviations and Symbols... vi
1 INTRODUCTION ... 1
1.1 Objectives and approach ... 4
1.2 Thesis Outline ... 4
2 TECHNICAL BACKGROUND... 5
2.1 Geometrical weld parameters... 5
2.2 Overview of weld imperfections... 6
2.2.1 Embedded weld discontinuity ... 7
2.2.2 Geometrical imperfections ... 7
2.2.3 Surface weld discontinuity... 7
2.3 Stress concentration ... 9
2.3.1 Stress concentration factor - Kt... 10
2.3.2 Effect of weld geometric profile on stress concentration... 10
2.3.3 Improvement techniques ... 11
2.4 Extreme Value Theory ... 13
2.4.1 Order statistics... 13
2.4.2 Distribution of smallest values... 14
2.4.3 Distribution of largest values ... 15
3 DATA COMPILATION ... 16
3.1 Cruciform weld geometry ... 16
3.2 Weld profile evaluation methods ... 17
3.3 Labeling the specimens ... 18
3.4 Data acquisition... 19
3.5 Measurement of geometric parameters ... 20
3.5.1 Evaluation of weld toe radius... 22
3.5.2 Evaluation of weld toe angle... 22
3.5.3 Evaluation of other parameters ... 23
3.6 Non destructive testing... 23
4 STATISTICAL ANALYSIS OF DATA ... 25
4.1 Extreme value probability plot... 25
4.2 Determination of minimum weld toe radius ... 26
4.3 Determination of minimum weld toe angle ... 27
4.4 Grouping the data... 27
5 FRACTURE MECHANICS APPROACH ... 28
5.1 Introduction ... 28
5.2 Linear-elastic fracture mechanics (LEFM) ... 29
5.3 Stress intensity factor calculation... 31
5.4 Stress concentration magnification factor, Mk... 31
5.5 Calculation of fatigue life... 32
5.6 Fatigue strength ... 33
5.7 Analysis... 34
6 RESULTS AND DISCUSSION ... 35
6.1 Geometric variations ... 35
6.2 Grouping of weld specimens... 36
6.3 Influence of weld profile variation on stress concentration ... 38
6.4 Influence of weld profile variation on fatigue strength... 39
6.5 Comparison with Structured Light Measurement ... 40
7 CONCLUSION ... 43
REFERENCES... 44 APPENDICES
List of Figures
Figure 1.1: Toe and root side cracks of a fillet weld [4] ... 1
Figure 1.2: Fatigue strengths of plain, notched and plate with fillet welded attachment. [6] 2 Figure 1.3: Fatigue Design S-N curves for welded joints in steel [7]... 3
Figure 2.1: Cross-sectional view of a groove and fillet weld [9]... 5
Figure 2.2: Common weld imperfections [14]... 8
Figure 2.3: Influence of weld toe angle on SCF at the weld toe [2] ... 11
Figure 2.4: A stainless steel welded joint after hammer peened... 12
Figure 2.5: S-N curves of steel fillet welds treated by improvement techniques [18]... 12
Figure 3.1: A model of the cruciform weld showing the specimen pieces order... 17
Figure 3.2: Order of the welds in both sets ... 18
Figure 3.3: Diagrammatic view of the surfaces selected ... 19
Figure 3.4: Sequence of welds on selected surfaces as in the micrographs (Set 1) ... 20
Figure 3.5: Sequence of welds on selected surfaces as in the micrographs (Set 2) ... 20
Figure 3.6: An example of a measured weld specimen ... 21
Figure 3.7: Measurement of weld toe radius [24] ... 22
Figure 3.8: Measurement of weld toe angle... 23
Figure 3.9: Schematic representation of non destructive measurement... 24
Figure 5.1: Basic modes of crack surface displacement [30]... 29
Figure 5.2: Basic joint geometry and a typical mesh used in finite element analysis [8].... 31
Figure 6.1: Extreme value distribution for weld toe angle (a side) ... 35
Figure 6.2: Extreme value distribution for weld toe radius (a side) ... 36
Figure 6.3: Extreme value distribution of weld toe angle for different groups (a side) ... 37
Figure 6.4: Extreme value distribution of weld toe radius for different groups (a side) ... 38
Figure 6.5: Weld profile variation with respect to stress concentration factor (a side) ... 39
Figure 6.6: Weld profile variation with respect to fatigue strength (a side) ... 40
Figure 6.7: Comparison of measurements for weld toe angle (a side) ... 41
Figure 6.8: Comparison of measurements for weld toe radius (a side) ... 42
List of Tables Table 3.1: Specimen sets with their steel grades... 18
Table 5.1: Parameters for a/T =1.0 [8]... 34
List of Abbreviations and Symbols
FAT Fatigue class HAZ Heat affected zone
IIW International Institute of Welding LEFM Linear-elastic fracture mechanics NDT Non destructive testing
SCF Stress concentration factor SLM Structured light measurement
a Throat thickness
x Crack depth
C Fatigue capacity
K Stress intensity factor Kt Stress concentration factor
Mk Stress concentration magnification factor
N Fatigue life
T Thickness of the plate
α Weld toe angle
β Flank angle (180° - α)
∆σno Nominal stress range
ρ Weld toe radius
σno Nominal stress
1 INTRODUCTION
Normally all heavy machinery, ships, bridges, buildings and railroads, consists of extensive frameworks and intricate angles in the structure. They may be joined by many kilometers of welded joints. The welded joints are an integral part of these load-carrying structures and at the same time also the weakest points. The strength or the integrity of a structure thus depends on these welded joints. A study which would relate the weld profile to the strength of the structure will consequently help in predicting the life of the structure.
The structures subjected to variable or repeated cyclic loads may fail in service, where as they may withstand static loading. This type of damage, which consists of the formation of crack or cracks under the action of varying loads, is known as fatigue. [2] Fatigue damage generally occurs at stress-concentrated regions where the localized stress exceeds the yield stress of the material. [3] Sharp changes in direction, which occur generally at the toes of butt welds and at the toes or roots of fillet welds, causes local stress concentration. These points will therefore react to be in a highly stressed state and will cause cracks to grow at a faster rate. The toe and root side cracks of a fillet weld are shown in Figure 1.1. [4]
Figure 1.1: Toe and root side cracks of a fillet weld [4]
It is usually found that design stresses in repeatedly loaded structures are limited by the fatigue strength of the welded details; hence the presence of a weld in a member can
drastically reduce its fatigue strength. [5] This comparison of un-notched, notched and welded specimen can be seen in Figure 1.2.
Figure 1.2: Fatigue strengths of plain, notched and plate with fillet welded attachment. [6]
Common welded details have been assigned fatigue categories or classes and corresponding S-N curves according to fatigue recommendations by International Institute of Welding (IIW). These curves are derived from large amounts of experimental data and have been verified with analytical studies. They include effects of weld imperfections, local stress concentration due to the weld geometry, stress direction, residual stresses, etc. Each fatigue strength curve (fatigue class or FAT class) is identified by the characteristic fatigue strength of the detail at two million cycles. [7] The fatigue life (N) of a detail mainly affected by the stress range (∆σ) is usually expressed as, N = C / ∆σm; where, m is a constant, which for most welded details is equal to 3 and C is the fatigue capacity.
The wide variation in the fatigue strengths of welded joints, illustrated in Figure 1.3 recommended by IIW, arises as a result of variations in the severity of stress concentrations for different weld types and loading directions. [5] Though fillet welds are considered as poorer fatigue strength detail, they are widely used and preferred as they are economical, simple to prepare from the standpoint of edge preparation and fit-up. [1]
Figure 1.3: Fatigue Design S-N curves for welded joints in steel [7]
Considering a fillet weld as shown in Figure 1.1, the weld root constitutes severe stress concentration since the lack of weld penetration is equivalent to a crack. The weld toe can be even more severe as the abrupt change in section at the weld toe introduces a stress concentration. In steels, the fillet weld has much lower fatigue strength because fatigue cracks propagate from pre-existing crack-like defects and the majority of the fatigue life of the weld is spent propagating a crack. In contrast, a significant part of the life of a plate with a hole in it is spent initiating the crack. [5]
The existence of crack-like defects is considered to eliminate the crack initiation stage of fatigue life. Clearly the absence of a significant crack initiation period in the fatigue life of welded joints is disadvantageous and considerable improvement in fatigue life would result if measures could be taken to reintroduce a crack initiation period by removing the crack- like defects. Therefore, emphasis of fatigue assessment must be focused on the crack growth stage of fatigue life within the framework of Linear Elastic Fracture Mechanics (LEFM). [5] [8]
The design of any structure is very important to be good as it influences the life of the structure. The fatigue behavior of welded joints with consideration of geometrical factors that produce locally high stresses is thus essential to be determined.
1.1 Objectives and approach
The general objective of this study was to conduct a statistical analysis on the variation of the weld profiles and their influence on the fatigue strength of the joint and therefore to link weld quality to its fatigue strength. The effect of local geometrical variations of non-load carrying cruciform fillet welded joint under tensile loading was studied. The cruciform fillet welded joints in the as-welded condition and under cyclic tensile loading, the crack initiation period was considered non-existent and Linear Elastic Fracture Mechanics was used to calculate fatigue strength for a range of weld geometries. With extreme value statistical analysis and LEFM, an attempt was made to relate the variation of the cruciform weld profiles such as weld angle and weld toe radius to respective FAT classes. A second objective was to compare destructive and non-destructive weld geometry measurement methods.
1.2 Thesis Outline
The purpose of this paper was to statistically analyze the variation of non-load carrying cruciform fillet weld profile and its impact on the fatigue strength of the joint. In the beginning, a brief introduction to fatigue of welded structures and need for assessment of fatigue behavior is discussed. A brief background of the weld defects, cruciform joint and extreme value statistics are covered in the next section. It is followed by a description of the measurements and generally the work involved in collection of the data.
The next task to compute statistical distributions describing the variability in non-load carrying cruciform fillet welded joint details is also explained. The method involved in linking the obtained statistical analysis to the fatigue strength of the welded joint, i.e., LEFM is discussed later. The results of the analysis and then discussed. Here, the fatigue strength or the FAT class of the given structure is predicted. Finally, the results thus studied are discussed and concluded in the final part of the paper.
2 TECHNICAL BACKGROUND
There are many factors that affect the fatigue strength and life of a welded structure. Weld imperfections or flaws, weld profile, loading and the application of the structure together play a role in determining the fatigue life of the structure. The basic step is to understand the parts of a simple weld.
2.1 Geometrical weld parameters
The main parts of a groove or butt weld and fillet weld are shown in Figure 2.1. These parts are briefly explained below.
Figure 2.1: Cross-sectional view of a groove and fillet weld [9]
The toe is the junction between the face of the weld and the parent metal. The root of a weld includes the points at which the back of the weld intersects the base metal surfaces.
When we look at a triangular cross section of a fillet weld, as shown in view B of Figure 2.1, the leg is the portion of the weld from the toe to the root. The throat thickness ‘a’ is the minimum distance from the root to a point on the face of the weld. Theoretically, the face forms a straight line between the toes. [9]
The heat affected zone (HAZ) is a part of the parent metal which has not been melted with the filler metal, but which undergoes fast heating and cooling during the passage of the welding arc. In this zone, the parent metal is subject to a hardening treatment and can consequently become brittle. [10]
Fatigue strength is mainly controlled by weld imperfections and geometrical weld parameters. The main parameters considered in weld designing for fatigue are basically the weld angle ‘α’, weld toe radius ‘ρ’ and throat thickness ‘a’.
2.2 Overview of weld imperfections
Many structures, when subjected to repeated cyclic loading will weaken and eventually initiates cracking. If repeated loading continues, the cracks will grow through the member thickness and increase in length. The development of these cracks through this process is termed fatigue crack initiation and growth. Fatigue damage in large structures normally accumulates most rapidly at joints or discontinuities, where stresses are raised above those in the surrounding structure by local effects. [11]
Understanding weld imperfections will help in identifying them and, more important, prevent them from occurring. A preventive tool within the quality system is more efficient than sorting bad welds from good welds. [12]
Welded structures or joints might be considered to have many kinds of defects and imperfections. A discontinuity could be the result of a defect, but not necessarily a defect.
A defect, on the other hand, is a discontinuity that by nature or accumulated effect. Total crack length is a defect. The three main types of imperfections could be embedded weld discontinuity, geometrical imperfections and surface weld discontinuity. [11]
2.2.1 Embedded weld discontinuity
Embedded weld discontinuity are mostly non-planar imperfection. They could be in the form of cavities, solid inclusions or porosity. Porosity in welds is a void formed by gas which is entrapped during solidification process, and the size of these cavities may vary widely. [2] Various studies have been conducted to determine the severity or the effect of porosity on the welded joint or structure. The results of these studies as shown by Gurney, indicates clearly that the fatigue strength decreases rapidly as the percentage of porosity or cavities increases. Slag or metallic inclusions also have the same effect, and also the most common form of defect to be encountered. Considerable attention has been put in this aspect to reduce the amount of slag inclusions during welding and cleaning. [2]
2.2.2 Geometrical imperfections
Geometrical imperfections include axial misalignment, angular misalignment angular distortion and imperfect weld profile. Misalignments do not change the fatigue strength of the weld but increases the geometric stress on the weld. The stress concentration regions at the weld toe are enhanced, therefore decreasing the fatigue strength of the structure. These types of imperfections can be measured relatively easily. [11] [13]
2.2.3 Surface weld discontinuity
Surface weld discontinuity mainly represent cracks, lack of fusion or penetration, undercut and overlap. These are mainly categorized as planar imperfections.
Lack of fusion between the weld metal and parent base metal or adjoining weld beads can occur at any location within the weld joint and be present in fillet welds or groove welds.
Incomplete fusion may result when the temperature of the base material is not elevated to its melting point during the welding process. [12] This is shown in Figure 2.2.
Figure 2.2: Common weld imperfections [14]
Lack of penetration is a discontinuity in which the weld metal does not extend through the joint thickness. In other words, it is the failure of the filler metal to fill the root of the weld completely. Some common causes of incomplete joint penetration are a bad groove weld design or a fit-up that is unsuitable for the welding conditions. [12]
Undercut is defined as a groove melted into the base metal adjacent the weld toe, or weld root, and left unfilled by weld metal. The first is the melting away of the base material at the side wall of a groove weld at the edge of a bead, which produces a sharp recess in the area where the next bead is to be deposited. The second condition is reduction in thickness of the base material at the toe of the weld (Figure 2.2). This condition can occur on a fillet weld or a butt joint. Excessive undercut can seriously affect the performance of a weld, particularly if it is subjected to fatigue loading in service. [12]
Cold-lap or overlap as shown in Figure 2.2 is a protrusion of weld metal beyond the weld toe, or weld root. This condition occurs in fillet welds and butt joints and produces notches at the toe of the weld that are undesirable because of their resultant stress concentration under load. [12]
Insufficient throat thickness usually occurs in fillet weld and butt joint profiles due to excessive concavity, which considerably reduces weld strength. On the other hand, excessive convexity can also produce a notch effect in the welded area and, consequently, concentration of stress under load. [12]
Cracks in welded joint are probably the most dreaded of all the weld discontinuities. The crack sensitivity of a base material may be associated with its chemistry and its susceptibility to the formation of elements that reduce its ductility. The welding operation can produce stresses in and around the weld, introducing extreme localized heating, expansion, and contraction. Cracking often is caused by stress concentration near discontinuities in welds and base metal and near the notches of the welded joint. [12]
2.3 Stress concentration
Fatigue is a complex problem primarily related to structural geometry, with secondary links to material properties. The influence of notches and other discontinuities on the fatigue strength of a component can be at least partially explained by the effect of such discontinuities on the stress distribution. [2]
When a smooth uniform flat plate is subjected to axial tension, the stress distribution on the cross-section will be uniform. But if you consider the same loading with a hole drilled at the centre of the plate, we observe non-uniformity and a stress peak at the edge of the hole.
Such stress peaks produced by any abrupt changes in shape or discontinuity are known as stress concentrations. [2] In welded structures, we see these stress concentrations originating at weld toes and weld roots.
2.3.1 Stress concentration factor - Kt
If the discontinuities in the component have well-defined geometries, it is usually possible to determine a stress concentration factor (SCF), Kt, for these geometries. Then the local maximum stress can be accounted using the well known relation between the local peak stress to the average stress as shown in equation (1). [2] [15]
σmax = Kt · σno (1)
However in the case of a sharp crack in which the radius of the crack tip approaches zero, the stress concentration being severe, the use of fracture mechanics becomes necessary to analyze structural performance. [15] Weld toe radius and angle are a few parameters defining SCF at the weld toe which is our main concern in this thesis work.
2.3.2 Effect of weld geometric profile on stress concentration
Welds are associated with having pre-existing discontinuities that act as initiation sites for fatigue. These weld discontinuities include slag intrusions, undercut, and lack of penetration (among others). With these fatigue crack initiation sites already present in the structure, crack growth can begin almost immediately. [16]
Extensive research [2] has been made to determine the effect of various weld geometry parameters on fatigue life of welded joints. Analytical models have been developed to predict fatigue strength of welded joints influenced by the variation of weld geometry parameters. Researchers have concluded that the SCF is strongly influenced by weld profile geometry factors such as weld toe radius and flank angle. [17]
Gurney [2] has showed the variation of SCF for different weld angles of cruciform joints, for both load-carrying welds and non-load carrying welds (Figure 2.3). We can observe that when the flank angle increases and the toe become sharper, it in turn increases the SCF at the weld toe. This finally results in decrease of fatigue strength and life of the joint.
Figure 2.3: Influence of weld toe angle on SCF at the weld toe [2]
2.3.3 Improvement techniques
Fatigue is often noted to start at the weld toe due to the presence of the slag intrusions as well as the geometry of the weld toe (angle, radius and undercut), both of which act as stress concentrators. Grinding, hammer peening, or dressing of the weld toe can reduce the incidence of discontinuities. [16] [5]
Figure 2.4 shows a stainless steel lap joint which is hammered to create a smooth radius so as to eliminate the stress concentrators and increase the crack initiation life.
Figure 2.4: A stainless steel welded joint after hammer peened
Higher fatigue lives can be obtained from performing post-weld improvement techniques.
These techniques help to remove weld toe imperfections and reduce the stress concentration by improving the weld profile. [18][19] A comparison of S-N curves obtained for transverse non-load carrying cruciform fillet welded joint treated by different improvement techniques and for as-welded is illustrated in Figure 2.5.
Figure 2.5: S-N curves of steel fillet welds treated by improvement techniques [18]
2.4 Extreme Value Theory
The branch of statistics dealing with the extreme deviations from the median of probability distributions is called Extreme value theory. [20] Applications of extreme value theory include predicting the probability distribution of extreme floods, day to day market risks, fracture or fatigue of materials, aeronautics, corrosion of metals, etc.
Extreme value distributions are the confined or limiting distributions for the minimum or the maximum of a very large collection of random observations from the same arbitrary distribution. Gumbel has shown that for any well-behaved initial distribution (i.e., F(x) is continuous and has an inverse), only a few models are needed, depending on whether you are interested in the maximum or the minimum, and also if the observations are bounded above or below. [20]
Extreme value distributions can be better understood with a paradigm. For example, if a system consists of n identical components in series, and the system fails when the first of these components fails, then system failure times are the minimum of n random component failure times. [21] In general, its applicability can be justified whenever the phenomenon causing failure depends on the smallest or the largest value of variable, the underlying distribution of which is of the exponential type. [22]
2.4.1 Order statistics
Let X1, X2… Xn be a random sample from a probability density function, and if these n observations are arranged in ascending order so that X(1) ≤ X(2) ≤ … ≤ X(n), where X(1) is the smallest observation and X(n) the largest. Clearly X(1) can be any one of the n Xi’s. Then X(1) is called the first order statistic, whereas X(n) is called the nth order statistic.
Consider a chain consisting of n small links. It is obvious that the strength of chain cannot be greater than the strength of the weakest link. Therefore, the chain breaks when its weakest link breaks. Suppose that the life- length distribution of each link, fX(x;λ), is an exponential distribution. Then the life distribution of the chainFX1
( )
x will be given by the distribution of the smallest order statistic. [22] Thus,FX
( )
x =1−[
1−FX( )
x]
n =1−e−λnx1 (2) It can also be shown that the life-length distribution of the strongest link, X(n), is given by,
FX
( )
x[
FX( )
x]
n(
e x)
nn
λ
− −
=
= 1 (3)
2.4.2 Distribution of smallest values
Using the exact distribution of the first order statistic shown in equation (2) and applying to a case of an exponential distribution, the limiting distribution of the first order statistic X(1)
is obtained. It is thus stated by Gumbel that there are three possible types of distributions for the smallest order statistics. [22] The cumulative distribution functions are:
Type I ( ) 1 exp exp ; 0
) 1
( ⇒−∞< <∞ >
−
−
−
= α
α
γ x
x x
FX (4)
Type II ( ) 1 exp ; , 0
) 1
( ⇒−∞< ≤ >
−
−
−
−
=
−
β α α γ
γ β x x
x
FX (5)
Type III ( ) 1 exp ; , 0
) 1
( ⇒ ≤ <∞ >
−
−
−
= γ α β
α γ β x x x
FX (6)
2.4.3 Distribution of largest values
Using the exact distribution of the largest order statistic shown in equation (3) and applying to a case of an exponential distribution, the limiting distribution of the largest or nth order statistic X(n) is obtained. Three possible types of distributions for the largest order statistics are stated by Gumbel. [22] The cumulative distribution functions are:
Type I ( ) exp exp ; 0
)
( ⇒−∞< <∞ >
−
−
−
= α
α
γ x
x x
FX n (7)
Type II ( ) exp ; , 0
)
( ⇒ ≥ >
−
−
=
−
β α α γ
γ β x x
x
FX n (8)
Type III ( ) exp ; , 0
)
( ⇒ ≤ >
−
−
−
= γ α β
α γ β x x x
FX n (9)
3 DATA COMPILATION
As mentioned, the fatigue strength of a welded structure is mainly controlled by the imperfections in weld and also the geometrical weld parameters. The measurement of the various weld parameters such as the weld angle ‘α’, weld toe radius ‘ρ’ and throat thickness
‘a’ are thus essential while designing for fatigue. [13]
Therefore, by knowing the dimensions of the weld profile, it is possible to estimate both remaining life of the component and extent of degradation using the fracture mechanics concepts. The variation of these parameters can be studied then be related to the fatigue strength of the joint. The welded connection considered for this work was cruciform welded joint.
3.1 Cruciform weld geometry
The cruciform joint specimens were fabricated from mostly S355 and S650 steel plates of 6 mm thickness and connected by metal active gas welding (MAG) process. Each specimen was cut to four smaller pieces beside the fatigue specimen. The order in which they were cut and the welding direction is shown in the model of cruciform joint shown in Figure 3.1.
Two points must be noted here;
• The specimen pieces on the left of the fatigue specimen were labeled L1 and L2 in the direction of the welding.
• The specimen pieces on the right of the fatigue specimen were labeled R1 and R2 against the direction of the welding.
Figure 3.1: A model of the cruciform weld showing the specimen pieces order
3.2 Weld profile evaluation methods
Many cruciform welded joints were produced in the Welding Laboratory of Lappeenranta University of Technology for the purpose of fatigue testing and as well as a part of this thesis work. Measurements were done manually using micrographs. The specimens were numbered and then longitudinally cut to smaller pieces through weldment. The surfaces of these specimen pieces were then polished and etched to measure the required dimensions.
The stages involved in measurement and data acquisition are thus briefly explained in this section. Later the final results were compared with the data obtained by Non destructive testing (NDT) using computer-based image processing software. This method is also briefly described later in this section.
3.3 Labeling the specimens
The specimens were divided into two sets, according to chronological sequence of welded sides. The sequences of labeling of welded sides, i.e., a, b, c and d, of both sets are shown clearly in Figure 3.2. The specimens were labeled as ‘X01, X02…’ and so on. Table 3.1 shows each specimen with their steel grades.
Figure 3.2: Order of the welds in both sets
Table 3.1: Specimen sets with their steel grades
First Set Second Set
Specimens Steel Grade Specimens Steel Grade
X01…X30 S355 X31…X34 S355
X39 S355 X35…X37 S650
X40 S355 X41…X47 S650
X59 S650 X51…X53, X55, X56 S650
X66…X70 S650 X60…X65 S650
X71…X77 S650
Note: Specimen numbers X38, X48, X49, X50, X54, X57and X58 were not used
Here again, two points must be noted;
• The specimens were labeled as X01, X02… and so on.
• The specimen pieces were labeled as, for example, X01L1 and X01L2 for the pieces on left side of fatigue specimen and X01R1 and X01R2 for the pieces on right side of the fatigue specimen.
3.4 Data acquisition
The weld parameters can be measured from the cut specimen pieces only after the surfaces were ground, polished and etched. Selection of surfaces for measurement is important as the variation of weld profiles should be considered. Hence, the nearest and the farthest surfaces to the fatigue specimen were selected so as to obtain reasonable values. The surfaces selected are shown in Figure 3.3. These surfaces were ground, polished and then etched so that the extent of the heat affected zone (HAZ), the weld material and the base metal could be seen.
Figure 3.3: Diagrammatic view of the surfaces selected
The sequence of welds on the surfaces of the specimen pieces of both the sets as in the micrographs are shown in Figure 3.4 and 3.5. The micrographs of all the specimens are shown with measurements in Appendix II.
Figure 3.4: Sequence of welds on selected surfaces as in the micrographs (Set 1)
Figure 3.5: Sequence of welds on selected surfaces as in the micrographs (Set 2)
3.5 Measurement of geometric parameters
Measurement of the weld profile was the next step before making a statistical analysis. The important weld parameters such as the weld angle ‘α’, weld toe radius ‘ρ’, cold laps and undercuts were measured on every weld side a, b, c and d. The throat thickness ‘a’ and weld penetration were also measured only for the ‘a’ weld side. The measurements were done from micrographs manually using the program AutoCAD 2002. [23] One of the measured weld specimen used in this thesis is shown in Figure 3.6.
Figure 3.6: An example of a measured weld specimen
21
3.5.1 Evaluation of weld toe radius
The weld toe radius ‘ρ’ has influence of the stress concentration and therefore it is important to measure it. It is not an easy task to fit a circle at the weld toe. In some of the cases, it was hard to decide the radius to be measured wherein a circle with a local radius and a circle with a much bigger radius could be fitted. These extreme cases illustrate the difficulties with measuring the weld toe radius.
The methods used for measuring the weld toe radius were;
• Method of tangent at a distance d from the intersection of weld bead and base metal (Figure 3.7 (a)). [24]
• Choosing the smallest circle that best fits the weld toe (Figure 3.7 (b)). [24]
Figure 3.7: Measurement of weld toe radius [24]
3.5.2 Evaluation of weld toe angle
The measured weld toe angle ‘α’ in all the weld specimens on every weld side showed less variation, unlike weld toe radius measurement. The definition used for the weld toe angle corresponds to the angle between the base metal and the tangent line drawn on the weld bead is 1 mm above the base metal. This is shown in Figure 3.8.
Figure 3.8: Measurement of weld toe angle
3.5.3 Evaluation of other parameters
Cold lap: The length of protrusion of weld metal beyond the weld toe, if any, was measured perpendicular to the base metal.
Undercut: The depth of the groove melted into the base metal adjacent to the weld toe was measured parallel to the base metal.
Throat thickness: If the weld is convex or concave, the shortest distance between the line connecting the weld toes and the intersection of the flange to the base metal was measured to be the throat thickness of the weld. [1] If the weld bead is uneven, then a tangent which is nearer to the intersection was chosen for measurement. For calculation purposes, this measurement is taken approximately as 5 mm.
3.6 Non destructive testing
Any form of testing or inspection that can verify the structural integrity of a structure or component without affecting its ability to perform in service is termed as Non destructive testing (NDT). Non destructive tests require sophisticated equipments and interpretation of results by skilled, well trained personnel.
Visual tests by human eye might be slower and prone to errors; therefore they are being replaced by automated visual testing using optical instruments. These methods are usually referred to as machine vision, where it acquires data and analyses images to reach conclusion automatically. A typical machine vision system consists of a light source, a video camera, digitizer, a computer and an image display. Usually, the test object is illuminated and the image is captured using a video camera for processing by computer.
[25] The method used in determining the weld profile is shown in Figure 3.9.
Figure 3.9: Schematic representation of non destructive measurement
The computer first enhances the contrast of the image and later, the image is segmented for feature extraction and then it records a huge amount of data. The extracted 3-dimentional weld profile from the data is then used for further analysis such as measurement of weld toe profile, throat thickness, weld leg length, etc. The non destructive measurement used for the analysis will be hereby known as Structured Light Measurement (SLM). [13]
This method gives more accurate final values than manually measuring with micrographs.
The various geometrical weld parameters were measured using SLM, but only in the fatigue specimen and not on the specimen pieces. An attempt is made to compare the fatigue strength of the cruciform welded joint obtained from measuring with the help of micrographs and from SLM.
4 STATISTICAL ANALYSIS OF DATA
The collected data of the geometric weld parameters as explained above for both sets were then tabulated and the detail list is shown in Appendix I. The cruciform welded joint under cyclic tensile loading was considered to have non-load carrying welds. The data was used for a statistical analysis of the local geometrical variations. Later LEFM was used to calculate fatigue strength or FAT classes for the range of weld geometries. The task involved in analyzing the data is briefly described in this section.
Extreme value probability plot was first used to analyze the values of the geometric weld parameters of each weld side. These parameters were the measured weld toe radius ‘ρ’ and weld toe angle ‘α’. Since the minimum values of these parameters are considered to increase the stresses and affect the fatigue strength, the concept of extreme value of smallest values was therefore used. Later a plot of weld toe radius versus weld toe angle was used to make probable groups of classes of welds.
Work carried out by Murakami [26] to study the effects of small defects and nonmetallic inclusions and guidelines by TWI [27] show extensive usage of extreme value theory which helped in understanding the concept.
4.1 Extreme value probability plot
As the number of values were not too large, the cumulative function, F(x), used in the extreme value approach was determined by,
F(x) = (j-0.3) / (n+0.4) (10)
where, j is the number of weld specimens arranged in ascending order
n is the total number of weld specimens
The extreme value distribution of Type I for the smallest values, as explained in Section 2.4.2 in this report, was used for the analysis. The equation is given by,
0
; exp
exp 1 )
)(
1
( ⇒−∞< <∞ >
−
−
−
= α
α
γ x
x x
FX (11)
The reduced variates, yi, were then calculated from the above equations. Taking natural logarithms twice for equation (11), we are able to derive,
+
− −
−
= 0.4
3 . ln 0
ln n
yi j (12)
Extreme value probability plot was thus obtained by plotting equation (12) against weld toe radius and weld toe angle.
4.2 Determination of minimum weld toe radius
The method described by Gumbel gives us an estimation of minimum weld toe radius. The minimum weld toe radii obtained from the data of all weld sides of the specimen pieces were ordered, and then plotted on an extreme value probability plot. The right hand axis indicates the extreme value probability in percentage and the left hand axis will represents the extreme value probability obtained from equation (12). A line can be fitted through the data points and so that it is helpful to estimate the probability of minimum weld toe radius.
In other words, this line would indicate the minimum weld toe radius at any chosen probability level.
4.3 Determination of minimum weld toe angle
Similar to the minimum weld toe radius, the minimum weld toe angles were also plotted.
Here again, the minimum weld toe angles obtained from the data of all weld sides of the specimen pieces were ordered, and then plotted on an extreme value probability plot. The right hand axis indicates the extreme value probability in percentage and the left hand axis will represents the extreme value probability obtained from equation (12). The fitted line would indicate the minimum weld toe radius at any chosen probability level.
4.4 Grouping the data
Fitting a line in the extreme value probability plots may however be difficult in this case as the specimens had been welded under various different parameters. Hence, grouping them as different weld qualities such as A, B, C and D; and then an appropriate value from each group can be further estimated from the probability plot. This grouping was done in this thesis work firstly by plotting minimum weld toe radius versus minimum weld toe angle.
Using fracture mechanics, as explained in detail in following Section, Mk – factors or stress concentration magnification factors were calculated and helped in grouping the data points.
5 FRACTURE MECHANICS APPROACH
5.1 Introduction
Fracture mechanics may be defined as the application of the techniques and analyses of applied mechanics to the problem of the extension of an existing crack or crack-like defect.
Fracture mechanics may be used for fatigue analyses as supplement to S-N data. The purpose of such analysis is to document, by means of calculations, that fatigue cracks, which might occur during service life, will not exceed the crack size corresponding to unstable fracture. [2] [28]
Since fracture always consists of the progressive growth of a crack, it is only the material which is instantaneously adjacent to the crack tip which is actually breaking at any particular moment. Thus fracture mechanics is concerned mainly with the situation existing at the crack tip, and it is assumed that the discrete volume of material there will break when some critical condition is reached. The size of any plastic zone at the crack tip will tend to be small compared with the crack length, and it is possible to calculate the conditions around the crack by linear elastic stress analysis. [2]
According to fracture mechanics, defects present in materials lead to failure by growing to a critical, self propagating size. The fracture mechanics concepts allow one to calculate the critical sizes of defects as a function of their depth, length, active stress system and stress intensity and such properties of the material as its elastic modules, yield strength and fracture toughness. Therefore, by knowing the dimensions of defects present in a component, it is possible to estimate both remaining life of the component and extent of degradation using the fracture mechanics concepts. [25]
5.2 Linear-elastic fracture mechanics (LEFM)
The important region from the point of view of crack propagation is the crack tip. In the analysis of stresses near the tip of a sharp crack, where a stress singularity is presumed to exist, the concept of using elastic stress concentration factors breaks down. LEFM manages to overcome this problem by analyzing the stress field surrounding the crack tip, rather than the infinite stress in this region. [29]
A cracked body may be loaded in any one or a combination of the three displacement modes shown in Figure 5.1. Mode I is called the opening mode and consists of the crack faces simply moving apart due to tension. Mode II is called the sliding mode where the crack faces slide relative to one another in a direction normal to the leading edge of the crack due to shear and Mode III is called the tearing mode which involves relative sliding of the crack faces in a direction parallel to the leading edge due to shear. Mode I is the predominant stress situation in most practical cases. [30] [29]
Figure 5.1: Basic modes of crack surface displacement [30]
If the load applied to a member containing a crack is too high, the crack may suddenly grow and cause the member to fail by fracturing in a brittle manner, i.e., with little plastic deformation. From the theory of fracture mechanics, a quantity called stress intensity factor, K, can be defined that characterizes the severity of the crack situation as affected by crack size, stress, and geometry. Here, the material is assumed to behave in a linear-elastic manner, according to Hooke’s Law, so that the approach being used is called linear-elastic fracture mechanics (LEFM). [30]
The stress intensity factor is used to describe the elastic stress field in a cracked structure.
For case of mode I loading, it is typically expressed in the form, [29]
x Y
KI = σno π⋅ (13)
In the above equation, σno represents nominal stress, x is the crack depth and Y accounts for the effects of crack shape and geometry. In welded joints, fatigue failures initiate at the weld toe on the plate surface. The crack then propagates for the major part of the life as semi-elliptical cracks. In more detail, equation (13) can be written as, [29] [2]
M x M
K = Mk s tσno π⋅
φ0 (14)
The factor Mk allows for the fact that the crack is situated at a position of stress concentration factor Kt. The factor Ms is a correction to allow for the fact that the mouth of the crack is situated at a free surface. The factor Mt is a correction for plate thickness, to allow for the fact that there is a free surface ahead of the crack and φ0 is a correction term given by the complete elliptic integral of the second kind and depends on the crack front shape. [2] [31]
In the analysis of fatigue crack growth, where stresses are cyclic in nature, it becomes necessary to define a stress intensity factor range which is a difference between maximum and minimum stress intensity factors in the cycle. [29] It is given by,
min
max K
K
K = −
∆ (15)
Equation (13) then takes the form,
x Y
K = ⋅∆ no ⋅
∆ σ π (16)
5.3 Stress intensity factor calculation
Numerous 2-D non-load carrying fillet welded cruciform joints were modeled and finite element analyses (FEA) were performed by Nykänen et al. [8] in order to study the effect of local geometrical variables parametrically. The investigated finite element model is shown in Figure 5.2.
Figure 5.2: Basic joint geometry and a typical mesh used in finite element analysis [8]
When appropriate, weld toe radius, weld flank angle and weld size were altered and tensile cyclic loading was applied at the end of the main plate. The opening mode and sliding mode stress intensity factors KI and KII were calculated. The equivalent opening mode stress intensity factor, which is used in crack growth predictions, was also calculated. [8]
5.4 Stress concentration magnification factor, Mk
The stress concentration magnification factor, Mk, is defined as the ratio of the stress intensity factor of a cracked plate with a stress concentration to the stress intensity factor with the same cracked plate without the stress concentration. [8] [31] From equation (14), the range of the stress intensity factor for an elliptical crack at the toe of a fillet welded joint, ∆K, can be written as,
Y x K Mk u σno
φ ∆
=
∆
0
(17)
where, ∆σno is the nominal tensile stress range in the main plate.
Yu = Ms Mt
The Mk - factors were based on continuous edge cracks, hence, the crack depth: width ratio is zero, i.e., x/2c = 0 and φ0 = 1. The correction term Yu for a double-edge crack in a plate under tensile loading applied is given as, [8]
95 . 2 0 0 2 , 42 . 2 3
12 . 2 2 36 . 0 98 . 1
3 2
<
<
+
−
+
= T
x T
x T
x T
Yu x (18)
5.5 Calculation of fatigue life
In order to predict fatigue crack propagation, the Paris-Erdogan relationship is commonly accepted and used in practice for a wide range of mode I cracks. This relationship is also recommended by the International Institute of Welding (IIW) [7] for calculating the fatigue crack propagation rate of welded joints is given as, [8]
Km
dN C
dx = ∆ (19)
where dx/dN is the crack growth rate per cycle (mm/cycle)
C and m are material constants, and
∆K is the range of the stress intensity factor (Nmm-3/2)
If the crack length is normalized by the plate thickness, 2x/T, equation (17) and equation (19) can be combined and integrated to give the expected number of cycles to failure. [8]
∫
∫
∆
=
∆
=
−
−
T x
T x
m
u k m
T x
T x
f f
T d x T T
x Y T
C M T
d x K T N C
2
2 2
2 1 1
2 2 2
2 1
2 2
1 σ
By separating the constants from the integration
1 ) 2 /
( −
= ∆ m m
T C N I
σ , where
∫
=
T − x
T x
m
u k
f
T d x T
Y x M I
2
21
2
2 (20)
The value of crack growth integral, I, in the equation (20) depends on the initial and final crack lengths, xi and xf. [8]
5.6 Fatigue strength
The fatigue strength of a welded joint is normally characterized by its fatigue class, FAT, which identifies the range of stress corresponding to 2×106 cycles to failure with a 95%
survival probability. [7] Using the values, m = 3, Cmean = 1.7 ×10-13 andCchar = 3×10-13, the cyclic life corresponding to any stress range can be evaluated. The theoretical FAT can then be determined by adjusting the result according to the S-N curve, equation (20), so as to give the stress range corresponding to the fatigue life of two million cycles. On the basis of equation (20) the corresponding mean fatigue strength is:
FAT C FAT
C
mean char
mean =3 =1.208
∆σ (21)
where, Cmean is the mean fatigue crack growth rate coefficient and
Cchar is the characteristic value corresponding to 95% survival probability.
5.7 Analysis
The analysis was carried out for different a/T ratios, weld flank angles β and ρ/T ratios. For each a/T-ratio, the corresponding Mk-factors were curve fitted as a function of 2x/T, ρ/T and β using nonlinear regression analysis. For very small cracks, it was assumed that Mk ≈ Kt. The Mk-factors (equation (22)) were therefore set equal to Kt when 2x/T = 0. [8]
( )
( )
3,
,
4 3, ,
1 1, 2, , ,
,
2 3 4, ,
π 180
π 180
1 2
, where and , , 1...4
2
i
i j
i i j
r
c
i i i j i j
k s i r i j c
i i j
s Tx r r T a b
M s r i j
x d
s s r
T T
ρ β
ρ β
+ + +
= = = =
+
+ +
(22)
For calculation purposes, a/T ratio was assumed to be 1 and the parameters ai,j, bi,j, ci,j, and di,j used for calculation of Mk values for this thesis work are shown in Table 5.1. These parameters holds good for flank angle ranging between β = 15o – 60o and ρ/T = 0 – 1.0.
These Mk values were calculated for every weld side of each specimen and then Kt curves were used to group the specimens into good and bad welds. [8]
Table 5.1: Parameters for a/T =1.0 [8]
j 1 2 3 4 1 2 3 4
i ai,j bi,j
1 0.7895 0.0115 0.4315 0.0640 0.9701 0.0151 0.2322 -0.00886 2 4.3057 0.3420 0.4481 1.1479 7.2975 1.4614 10.5044 0.8728 3 0.7453 2.0673 -1.9094 0.0540 -0.5478 0.2716 2.0890 0.4897 4 0.7627 0.7800 0.7646 0.0667 0.3925 0.6089 -0.2288 -0.00409
ci,j di,j
1 0.2242 -3.7736 1.1454 1.1104 1.3543 0.9466 1.0889 1.2440 2 -0.4701 2.8158 1.6045 0.2199 0.2448 0.2231 0.7981 0.7760 3 0.6263 -2.2588 -0.5329 5.2627 2.0248 3.9828 1.7797 0.0963 4 2.0902 3.8035 0.9644 1.2096 0.1057 1.0331 1.9074 0.0351
6 RESULTS AND DISCUSSION
6.1 Geometric variations
The measured weld toe angle on each weld side were ordered and then plotted on extreme value probability plot. Figure 6.1 shows the extreme value distribution for weld toe angle of the weld side a. The extreme value distributions of minimum weld toe angle for other weld sides b, c and d are shown in Appendix III. An estimate of the probability for minimum weld toe angle can be obtained if a line is fitted through the data points.
0.995 0.99
0.95 0.9 0.8
0.5
0.1 -1.6
-0.6 0.4 1.4 2.4 3.4 4.4 5.4
80 90 100 110 120 130 140 150
Minim um w eld toe angle 'α' (deg)
-ln(-ln(F)) F
All w elds (70) - a Side
Figure 6.1: Extreme value distribution for weld toe angle (a side)
Since the cruciform joints were welded with many different welding parameters, the distribution fails to fall in a straight line. In other words, the distribution helps us to understand that the specimens have different weld quality and that they have to be grouped into different classes or groups. Hence making an estimation of minimum weld toe angle for different groups is more sensible.
Similar to the minimum weld toe angle, the minimum weld toe radii were also plotted on extreme value probability plot. Here again, the minimum weld toe radius obtained from the data of all weld sides of the specimen pieces were ordered. Figure 6.2 shows the extreme value distribution for weld toe radius of the weld side a. The extreme value distributions of minimum weld toe radius for other weld sides b, c and d are shown in Appendix III.
0.995 0.99
0.95 0.9 0.8
0.5
0.1 -1.6
-0.6 0.4 1.4 2.4 3.4 4.4 5.4
0 0.5 1 1.5 2 2.5
Minim um w eld toe radius 'ρ' (m m )
-ln(-ln(F)) F
All w elds (70) - a Side
Figure 6.2: Extreme value distribution for weld toe radius (a side)
From the above distribution of weld toe radius we can observe that a line is difficult to be fitted through all the points. Unlike weld toe angles, the measured weld toe radii have lesser accuracy and hence many data points fall on the same radius values. These distributions are also grouped into different weld quality and then estimation for minimum weld toe radius is made for required probability level.
6.2 Grouping of weld specimens
As observed in the above extreme value probability plots fitting a line was difficult as the specimens had been welded under various different parameters. Hence grouping them as
