consist of steel plates and/or steel angles, which are connected with each other by .... 4.4 LOADING TRUCK (LT). 119. 4.5 TRUCK DIRECTION AND POSITION.
Zagazig University Faculty of Engineering Structural Engineering Dept.
BEHAVIOR OF STEEL CONNECTIONS UNDER FATIGUE LOADING A THESIS Submitted to the Faculty of Engineering Zagazig University FOR The Degree of M.Sc. in Structural Engineering BY Alaa-Eldin Abd-El-kader Mohamed (B.Sc. Civil Engineering, Zagazig University 2005) Supervised By Prof. Dr.
Prof. Dr.
Ossama M. El-Hussieny
Hossam Eldin M. Sallam
Professor of Steel Structures and Head of Structural Engineering Department. Zagazig University
Professor of Materials Engineering Zagazig University
Assoc. Prof. Dr.
Ehab Boghdadi Matar Ass. Professor of Steel Structures
Zagazig University
2009
ACKNOWLEDGEMENTS
First and foremost, praise and thanks be to Almighty ALLAH for His limitless help and guidance and peace be upon His last messenger Mohammed.
This research work is supervised throughout by Prof. Dr. Ossama El-Hussieny , Prof. Dr. Hossam Eldin Mohamad Sallam and Ass. Prof. Dr. Ehab Boghdadi Matar to them I am very grateful for generous supervision, guidance and criticism. I am also greatly indebted to my family especially to my father and mother for their continuous support and non-stop encouragement.
Finally, I wish also to thank, my friends, H. El-Emam, M. Zaghlal and all those who lend me their hands during executing this research.
i
Abstract Fatigue failure of steel connections is a well-known failure mechanism, especially for structures which sustain heavy cyclic loads like steel bridges. Great numbers of these bridges were constructed at the beginning and the middle of the first half of the last century, where the most steel fabrication fashions were the riveted connections and the riveted built-up members. These bridge elements consist of steel plates and/or steel angles, which are connected with each other by hot driven rivets, which are subjected to shear stress, so that they can be simplified to single or double lapped joints. This thesis is consisted of two parts, in the first part, single and double lapped joints were studied numerically. Load transfer, stress concentration, stress intensity, crack path and crack tip plasticity in the lapped joints under axial load were analyzed. Finite element method was adopted to simulate the static, and fatigue behaviors of these joints. A parametric study for different bolt/rivet diameters and numbers is developed, to show the behavior of these connections. It was observed that for lapped joint the crack path is approximately perpendicular to the loading direction as found experimentally in the literature. Small deviation in the path was detected, which may be due to the bolt contact pressure, which has a component perpendicular to the loading direction. In the case of lapped joints, the crack emanated from bolted hole closed to the applied load has a high stress intensity factor (SIF). Therefore, the failure of the multiple bolted/riveted joint structures may be occurred at the first hole. This situation cannot be treated by the arrangement of more bolts/rivets in the plate, since the critical condition of the first hole will not be changed by increasing the number of bolts/rivets.
ii
It was found that, the extents of both monotonic and cyclic plasticity accommodated at the tip of a crack artificially advancing from the bolted hole showed transition behavior from the bolted hole-affected short crack regime to the relatively long crack regime. This transition behavior of monotonic and cyclic crack tip plastic zones can explain the phenomenon of initiated but not propagating crack which occasionally found in the bolted/riveted steel connections. The effects of clamping force and friction coefficient were presented and studied on the single and double lapped joints for cracked and un-cracked connections. Different parameters such as, clamping force (0, 10, 20, 100, 200 kN) and friction coefficient (0.0, 0.3, 0.5) were studied. It was observed that, high clamping force causes a decrease in stress concentration for un-cracked connectionS. For cracked lapped joint, a SIF range (∆KI) decreases i.e. increase in fatigue life. The second part of this thesis is a partial fatigue evaluation for an existing steel roadway bridge (El-Ministerly Bridge) which is still in service. El-Ministerly Bridge is a riveted steel bridge located in El-Sharkia governorate in Egypt. Field strain measurements and a numerical modeling were carried out for El-Ministerly Bridge. The experimental results are used to develop and validate the numerical finite element model. Evaluate the bridge is executed using the developed numerical model. Different design codes were presented to understand the fatigue evaluation. Further, the Egyptian code and S-N curves from literature were used to carry out the evaluation of the bridge. It was observed that, all stress ranges for this bridge are less than the ECP limits (Fsr) and the remaining fatigue life is about 22 years, if it is environmentally protected.
iii
LIST OF CONTENTS ACKNOWLEDGEMENTS
i
ABSTRACT
ii
LIST OF CONTENTS
v
LIST OF TABLES
ix
LIST OF FIGURES
x
CHAPTER1: INTRODUCTION
1
1.1 INTRODUCTION
1
1.2 OBJECTIVES
3
CHAPTER 2: LITERATURE REVIEW
4
2.1 INTRODUCTION
4
2.2 STRESS CONCENTRATION
4
2.3 MODES OF CRACK TIP DEFORMATION
6
2.4 LINEAR ELASTIC FRACTURE MECHANICS (LEFM)
8
2.4.1 Elastic Stress Field near the Crack Tip
8
2.4.2 Strain-Energy Release Rate
10
2.5 ELASTIC PLASTIC FRACTURE MECHANICS (EPFM)
10
2.5. 1 Crack Tip Plastic Zone
10
2.5.2 Crack Tip Opening Displacement
15
2.5.3 J Contour Integral
17
2.6 FATIGUE LIFE
19
2.6.1 Crack Initiation
19
2.6.2 Micro-Crack, Stage I Propagation
20
2.6.3 Stage II Propagation
21
2.6.4 Stage III Propagation
22
2.7 CRACK INITIATION CRITERIA
23
2.8 FATIGUE CRACK EMINATED FROM A NOTCH
27
2.9 STRESS AND STRAIN LIFE FATIGUE ANALYSIS
28
2.10 FATIGUE IN BRIDGES
29
2.10.1 Field Strain Measurements
29
2.10.2 Static and Fatigue Strength of Riveted/Bolted Connections
32
-v-
2.10.2.1 Tests of small scale riveted connections.
34
2.10.2.2 Tests of full-scale riveted members
40
2.10.2.3 Finite element modeling of riveted and bolted connections.
45
2.11 COMMON METHOD OF FATIGUE EVALUATION
52
2.11.1 Review of Current Standards
`54
2.11.2 Evaluation of Fatigue Life by Linear Elastic Fracture Mechanics.
62
2.12 REMAINING FATIGUE LIFE
64
2.13 CYCLES COUNTING
65
2.13.1 Level-Crossing Counting
65
2.13.2 Peak Counting
66
2.13.3 Simple Range Counting
66
2.13.4 Rainflow Cycle Counting
66
CHAPTER 3: NUMERICAL MODELING
69
3.1 INTRODUCTION
69
3.2 FINITE ELEMENT MODELING OF LAPPED JOINTS
69
3.2.1 Types of Elements Used in The Finite Element Analysis
69
3.2.1.1 PLANE82 for 2D modeling of the parts of steel lapped joints
70
3.2.1.2 SOLID187 element for edge cracked steel connection parts
70
3.2.1.3 SOLID95 element for through thickness cracked steel connection parts 3.2.1.4 CONTA174 element for 3D contact pairs between steel connection parts 3.2.1.5 TAGRE170 for 3D contact pairs between steel connection parts 3.2.1.6 CONTA172 element for 2D contact pairs between steel connection parts
71
71 72 72
3.2.1.7 TAGRE169 for 2D contact pairs between steel connection parts
73
3.2.1.8 PRETS179 element for the simulation of the clamping force
74
3.2.2 Material Modeling
74
3.2.3 Contact Properties
75
3.2.4 Nonlinear Solution
76
3.2.5 Two Dimensional Modeling (2DM)
77
3.2.5.1 2D Modeling Of Un-bolted Holes (2DUH)
-vi-
77
3.2.5.2 2D modeling of bolted holes with rigid bolts (2DR)
79
3.2.5.3 2D analysis of lapped joint (2DL)
80
3.2.6 Three Dimensional (3D) Modeling of Single (3DS) and Double (3DD) Lapped Joint 3.2.7 Modeling of Crack.
82 85
3.2.7.1 Modeling of the crack tip.
85
3.2.7.2 Modeling of crack propagation
86
3.2.8 Verification of The Finite Element Models For A Bolted Joint 3.2.8.1 Verification of the 2D FEM. 3.2.8.2 Verification of the 3D FEM of steel connections.
3.3 BRIDGE MODELING 3.3.1 Types of Elements Used In the Present Finite Element Analysis
88 88 88 93 93
3.3.1.1 SHELL63 for 3D modeling of the bridge members and RC deck.
93
3.3.1.2 SOLID45 element for steel connection parts of the bridge.
94
3.3.1.3 BEAM188 element for the bridge flanges angles and the bracing elements
94
3.3.2 Material Properties
95
3.3.3 Frame Element Model
95
3.3.4 The Global FE Model
96
3.3.5 The Local FE Model
99
3.3.6 Bridge Model Correlation.
101
CHAPTER 4: Case Study: El-Ministerly Bridge
104
4.1 INTRODUCTION
104
4.2 BRIDGE DESCRIPTION
104
4.2.1 Main Girder
107
4.2.2 Cross Girders
107
4.2.2.1 Cross girder type one (XG 1)
109
4.2.2.2 Cross girder type two (XG 2)
109
4.2.3 Stringers
110
4.2.3.1 Stringer type one (STR1)
110
4.2.3.2 Stringer type two stringer (STR2)
111
-vii-
4.2.4 Connection
111
4.3 STRAIN GAUGES INSTALLATION
113
4.4 LOADING TRUCK (LT)
119
4.5 TRUCK DIRECTION AND POSITION
119
CHAPTER 5: RESULTS AND DISSCUTION
123
5.1 INTRODUCTION
123
5.2 FREE HOLES MODEL
123
5.3 BEHAVIOR OF UN-CRACKED LAPPED JOINT
129
5.3.1 Load Transfer Analysis
129
5.3.1.1 2D rigid bolt model (2DR)
129
5.3.1.2 2D lapped joint (2DL)
135
5.3.1.3 3D single (3DS) and double (3DD) lapped model
139
5.3.1.4 Comparison between the three modeling types (Bolts supported load ratios) 5.3.2 Stress Concentration Kt.
146 148
5.3.2.1 2D rigid bolt model (2DR)
148
5.3.2.2 2D lapped joint (2DL)
152
5.3.2.3 3D single (3DS) and double (3DD) lapped model
155
5.3.2.4 Comparison between the three modeling types (Maximum bolt hole SCF)
163
5.3.3 Effect of Clamping Force
165
5.3.4 Monotonic and Cyclic Plastic Zones
168
5.4 BEHAVIOR OF CRACKED LAPPED JOINT 5.4.1 Crack Path and Crack Advance.
172 172
5.4.1.1 Crack path of crack emanation from lapped joint
172
5.4.1.2 Crack tip plasticity
174
5.4.1.3 Stress intensity factor (SIF)
176
5.4.2 Stress Concentration for Un-Cracked Bolts Holes (SIF)
184
5.4.3 Load Transfer Analysis for Cracked Component
185
5.4.4 Effect of Clamping Force and Friction Coefficient on Cracked Bolted Holes.
5.5 RESULTS OF EXPERIMENTAL TEST, BRIDGE MODELING
-viii-
186 194
AND BRIDGE EVALUATION 5.5.1 Comparison between Numerical and Experimental Results
194
5.5.2 Model correlation
206
5.5.3 Evaluation of Bridge Components
207
5.5.3.1 Fatigue evaluation.
207
5.5.3.2 Remaining Fatigue Life
213
CHAPTER 6: CONCLUSIONS AND FUTURE WORKS
215
6.1 CONCLUSIONS
215
6.2 FUTURE WOKS
217
REFEREN78CES
218
LIST OF TABLES Page
Table 3-1: Parameters of Un-bolted Holes models
78
Table 3-2: Parameters of (2DR)
80
Table 3-3: Parameters of (2DL)
81
Table 3-4: Parameters of (3DD and 3DS)
84
Table 3-5: Comparison between Load transfer in current and previous FEM
92
Table 5-1: Evaluation of stress range according to ECP
212
Table 5-2: Fatigue life of El-Ministerly bridge elements.
214
-ix-
LIST OF FIGURES Page
Fig. 2-1 An elliptical hole in a two-dimensional infinite solid under a remote 5
tension Fig. 2-2 Modes of crack tip deformation
7
Fig. 2-3 stress components ahead of the crack tip
8
Fig. 2-4 The shape of Mode I plastic zone
11
Fig. 2-5 the displacements between crack faces for R=0
17
Fig. 2-6 The coordinate system and the contour path when using the J integral 18
methodology Fig. 2-7 Slip band intrusion and extrusion
21
Fig. 2-8 Typical crack propagation behavior of metals
22
Fig. 2-9 Kink angle for to subsequence crack steps
23
Fig. 2-10 Fatigue tests of riveted member from literature
33
Fig. 2-11 some pictures of the test specimen of Lewin et al.
39
Fig. 2-12 Surface texture after cracking in the fatigue tested specimens
44
Fig. 2-13 Fatigue categories for riveted members and connections [89]
44
Fig. 2-14 Shell element model of imam et. al.
45
Fig. 2-15 Local finite element model by Imam et. al.
46
Fig. 2-16 Part of the finite element model and test specimen by Yang and Lee
47
Fig. 2-17 Part of the finite element model and test setup by Hong et. al.
48
Fig. 2-18 Components of the finite element model of Osama et al.
49
Fig. 2-19 Experimental results of the effect of clamping force.
51
Fig. 2-20 Finite element model of Gunther et al.
51
Fig. 2-21 Finite element model of Ju et. al.
52
Fig. 2-22 AASHTO variable amplitude fatigue rules
59
Fig. 2-23 AREA Constant and variable amplitude fatigue rules
59
Fig. 2-24 ECCS Constant and variable amplitude fatigue rules
60
Fig. 2-25 ECP fatigue categories
61
-x-
Fig. 3-1 PLANE82 element for 2D steel plates
70
Fig. 3-2 SOLID187 element for 3D edge cracked steel plates
70
Fig. 3-3 SOLID95 element for through thickness cracked steel plates
71
Fig. 3-4 CONTA174 element for 3D contact
72
Fig. 3-5 TAGRE170 element for 3D contact
73
Fig. 3-6 CONTA172 element for 2D contact
73
Fig. 3-7 TARGE169 element for 2D contact
74
Fig. 3-8 Bilinear plasticity model
75
Fig. 3-9 Newton-Raphson iterative solution
77
Fig. 3-10 Details of Unbolted Hole model (2DUH)
78
Fig. 3-11 Details of Rigid bolt model (2DR).
79
Fig. 3-12 The details and the components of 2DL model.
81
Fig. 3-13 FE mesh of 2DL components with 3 bolts.
82
Fig. 3-14 The details and the components of 3DS and 3DD models.
83
Fig. 3-15 3D Finite element mesh of 3DS with 3 bolts/rivets
84
Fig. 3-16 2D singular element and division a round the crack tip
85
Fig. 3-17 3D singular element and division a round the crack tip
86
Fig. 3-18: 2D Monotonic an cyclic plastic zone
87
Fig. 3-19 Comparison between finite element results and theoretical results
88
Fig. 3-20 Details of FEM in Ref [104]
90
Fig. 3-21 Deformed shape of current FEM
91
Fig. 3-22 Present and previous finite element modeling results
92
Fig. 3-23 SHELL63 element for 3D steel plates
93
Fig. 3-24 SOLID45 element for 3D steel plates
94
Fig 3-25 Frame element model
95
Fig 3-26 Global Finite element model
97
Fig 3-27 Finite element mesh of the main girder
98
Fig 3-28 Bridge supporting system
99
Fig. 3-29 FEM of Bridge Connection
100
Fig. 4-1 El-Ministerly bridge photos.
105
-xi-
Fig. 4-2 El-Ministerly Bridge structural system and supporting system.
106
Fig. 4-3 Details of main girder
108
Fig. 4-4 Type one Cross Girder
109
Fig. 4-5 Type two cross girder
110
Fig. 4-6 Type one stringer
111
Fig. 4-7 Type two stringer
111
Fig. 4-8 Details of connection
112
`Fig. 3-9 Data Logger
113
Fig. 4-10 Selected position of strain gauges on main girder
114
Fig. 4-11 Position of installed strain gauges on main girder
115
Fig. 4-12 Position of installed strain gauges on XG1
115
Fig. 3-13 Selected position of strain gauges of XG I
116
Fig. 4-14 Selected position of strain gauges of STR1
117
Fig. 3-15 Position of installed strain gauged on STR1
118
Fig. 3.16 Selected positions of strain gauges of Connection
118
Fig. 4-17 Position of installed strain gauges in connection
119
Fig. 3-18 Truck characteristics
120
Fig. 3-19 The bridge testing
121
Fig. 3-20 Truck positions
122
Fig. 5-1 Stress gradient beside the first hole for plate contains 6 holes.
124
Fig. 5-2 Kt for a model contains 6 holes
125
Fig. 5-3 The stress distribution around the surface of the hole.
125
Fig. 5-4 Distribution of Kt at each hole for the case of 7 holes for different holes diameters.
126
Fig. 5-5 Ratio between cyclic and monotonic plastic zone for circular notch
127
Fig. 5-6 Stress strain curve for the point of maximum total strain.
127
Fig. 5-7 Development of cyclic plastic zone.
128
Fig. 5-8 Monotonic plastic zone size for 7 holes models.
129
Fig. 5-9 Load ratio at the first bolt, Elastic analysis
130
Fig. 5-10 Load ratio for each bolt for 2DR, Elastic analysis.
131
-xii-
Fig. 5-11 Maximum bolt load ratio, Elastic-Plastic analysis.
132
Fig. 5-12 Load ratio for each bolt for 2DR, Elastic analysis
133
Fig. 5-13 Deformed shape of 2DR contains two bolts.
134
Fig. 5-14 Load ratio of the first bolt, Elastic analysis.
135
Fig. 5-15 Load ratio for each bolt for 2DL, Elastic analysis.
136
Fig. 5-16 load ratio for each bolt for 2DL, Elastic-Plastic analysis.
137
Fig. 5-17 Deformed shape of 2DL contains two bolts.
138
Fig. 5-18 The differences between single and lapped joints.
139
Fig. 5-19 Bolt load ratio of the first bolt (3DD)
140
Fig. 5-20 Bolt load ratio for each bolt (3DD).
141
Fig. 5-21 Deformed shape of a quarter of a model with two M22 bolts of 3DD.
142
Fig. 5-22 Bolt load ratio of the first bolt (3DS)
143
Fig. 5-23 Bolt load ratio for each bolt (3DS).
144
Fig. 5-24 Deformed shape of a half model with two M22 bolts of 3DS.
145
Fig. 5-25 Comparison between 2DL, 2DR, 3DD, and 3DS results (RT).
147
Fig. 5-26 Effect of ratios between plates thicknesses in lapped joint (7 bolts).
148
Fig. 5-27 the SCF (Kt) in the first bolt hole for all cases (2DR).
149
Fig. 5-28 Relation between the bolt number and the Kt at every bolt hole (2DR).
150
Fig. 5-29 σy stress contours in 2DR with 2 bolts
151
Fig. 5-30 the SCF (Kt) in the first bolt hole for all cases (2DL).
152
Fig. 5-31 Relation between the bolt number and the Kt at every bolt hole (2DL).
153
Fig. 5-32 σy stress contours in 2DL with 2 bolts.
154
Fig. 5-33 Position of distance X in single and double lapped joints.
156
Fig. 5-34 The Kt distribution for inner plate in a 3DD contains two bolts.
157
Fig. 5-35 The Kt distribution for outer plate in a 3DD contains two bolts.
157
Fig. 5-36 The distributions of Kt at the first bolt hole in 3DD middle plate.
158
Fig. 5-37 The distributions of Kt at the first bolt hole in 3DD outer plate.
159
-xiii-
Fig. 5-38 The Kt distribution in a 3DS contains one bolt.
160
Fig. 5-39 The distributions of Kt at the first bolt hole in 3DD outer plate.
161
Fig. 5-40 Deformed shape and position of maximum σy in 3DS.
162
Fig. 5-41 Deformed shape and position of maximum σy in 3DD.
162
Fig. 5-42 Comparison between 2DL, 2DR, 3DD, and 3DS results (SCF)
164
Fig. 5-43 The effect of clamping force and friction coefficient on 3DS.
166
Fig. 5-44 The effect of clamping force and friction coefficient on 3DD.
167
Fig. 5-45 Bearing load as a function of clamping force and friction coefficient.
168
Fig. 5-46 Monotonic plastic zone and tension zone.
169
Fig. 5-47 Cyclic plastic zone at R=0 and compression zone.
169
Fig. 5-48 Development of monotonic plastic zone.
170
Fig. 5-49 Development of cyclic plastic zone.
171
Fig. 5-50 The crack path of crack emanated from bolted/riveted joint.
172
Fig. 5-51 The crack mixity of crack emanated from bolted/riveted joint.
173
Fig. 5-52 Effect of friction coefficient on kink angle for stationary crack.
174
Fig. 5-53 Extents of plasticity accommodated at the tip of a mode I crack artificially advancing from the bolted joint.
175
Fig. 5-54 The sites of the propagated cracks.
177
Fig. 5-55 the values of for 2DR for all cases
179
Fig. 5-56 the values of for 2DL for all cases.
180
Fig. 5-57 The variation of geometry correction factor of short crack emanating from bolted hole.
181
Fig. 5-58 Effect of secondary stresses on KeI in 3DS.
181
Fig. 5-59 Through thickness crack in the connection plate.
182
Fig. 5-60 Comparison between SIF of single and double edge crack models results.
183
Fig. 5-61 Stress concentration for un-cracked bolt holes.
184
Fig. 5-62 Load transfer analysis for cracked component.
185
Fig. 5-63 Effect of clamping force for µ =0.0 for cracked 3DS model.
187
-xiv-
Fig. 5-64 Effect of clamping force for µ=0.3 for cracked 3DS model.
188
Fig. 5-65 Effect of clamping force for µ =0.5 for cracked 3DS model.
189
Fig. 5-66 Effect of clamping force for µ =0.0 for cracked 3DD model.
191
Fig. 5-67 Effect of clamping force for µ =0.3 for cracked 3DD model.
192
Fig. 5-68 Effect of clamping force for µ =0.5 for cracked 3DD model.
193
Fig. 5-69 Bending moment diagram due to dead load.
194
Fig. 5-70 Location from the first axis.
195
Fig. 5-71 Comparison between PCM and NCM.
196
Fig. 5-72 Results of strain gauges at Section 2 on the main girder.
198
Fig. 5-73 Results of strain gauges at Section 3 on the main girder.
199
Fig. 5-74 Results of strain gauges at Section 1 on the stringer.
200
Fig. 5-75 Results of strain gauges at Section 1 on the cross girder.
201
Fig. 5-76 Results of strain gauges on connection
202
Fig. 5-77 Deformed shape of the global finite element model.
203
Fig. 5-78 The development of the deformed shape of the main girder with the truck movement. Fig. 5-79 Deformed shape for the local finite element model and the stress contours of the upper plate.
204 205
Fig. 5-80 Maximum and minimum σx stress contours.
208
Fig. 5-81 Maximum stress range contours on main girder.
209
Fig. 5-82 Maximum stress range contours on cross girders.
210
Fig. 5-83 Maximum stress range contours on stringers.
211
-xv-
Chapter(1)
Introduction
Chapter 1 INTRODUCTION
1.1 INTRODUCTION Fatigue occurs in structures that sustain cyclic moving loads, for example, bridges. Under the application of repeated stresses, microscopic cracks in structural members can propagate into large cracks [1, 2]. The presence of large cracks will reduce the load-carrying capacity of these structural members, connections, and, in extreme circumstances, can cause total failure of these parts [2]. Three major factors affect fatigue crack growth. These factors are: 1) the magnitude of the applied stress range, 2) the number of stress cycles, and 3) the length and shape of the initial flaw [3]. Several steel bridges in Egypt were built during the late nineteenth to early twentieth centuries. At this time, riveted construction was the method used for building up members and for connection of one member to another. Of course, the intention of the designers of these bridges was that they have a long useful life. However, since their construction, changes in traffic legal loads and bridges aging deterioration have continually required the re-evaluation of these bridges. When changes of this nature occur, steel bridges must be re-evaluated in terms of strength, serviceability, and fatigue. Even under constant loading, the fact that many of these structures are now over 100 years old, means that the question of their remaining fatigue life must be addressed. Remaining fatigue life of these bridges is directly influenced by the changing bridge traffic. The age of the bridges and the increasing axle loads are the major factors that govern the fatigue life. The increasing vehicle axle loads
1
Chapter(1)
Introduction
applied to the bridge increase the stress range at a given structural detail. This affects the fatigue life by accelerating any pre-existing fatigue crack growth and increasing the number of structural details where fatigue cracks could initiate [4]. Second, the age of the bridges means the number of load cycles that have been imposed on them since their construction. A fatigue evaluation of a steel bridge connections or members, generally involves gathering information about the three major factors that cause fatigue. Structural details that are thought to be susceptible to fatigue crack growth are evaluated in terms of the stress range, they experience and the number of cycles for which the stress range is applied. Structural details are selected either on the basis of previous experience with a certain type of detail, or by some sort of analytical justification. The stress range at a detail can be calculated analytically using the appropriate influence lines, and/or finite element method, or by direct measurement of strains while the bridge is in service. The number of cycles can be estimated using traffic histories of the particular traffic assembly. Once all this information are gathered, they are compared with the allowable limits; of the fatigue resistance [3, 6]. When a defect is present, the fracture mechanics approach provides the most insightful analysis of fatigue crack propagation. Most of the early work in fracture mechanics was motivated by the unexpected and catastrophic failure of the Liberty shipp in 1940s. Since this time the fracture mechanics enters all fields which deal with cyclic loads, as an accurate method for the fatigue life prediction. Notches as stress concentrators are usual sources of fatigue crack initiation at their roots. A numerous work has been carried out to investigate the early growth behavior of such cracks [6]. A fatigue crack initiated at the root of a notch, experiences transition behavior when its tip is advancing within the notch-affected zone. Early fatigue crack growth (FCG) rate from notches may initially decrease with crack length to achieve a minimum value before it can increase or stop.
2
Chapter(1)
Introduction
1.2 OBJECTIVES The objectives of this research work are: 1- Developing a parametric study for the bolted/riveted joints i.e. changing numbers of bolts/rivets (1-7) , bolt/rivet diameters (10, 16, 22 mm) and crack length to understand the behavior of cracked and un-cracked steel riveted/bolted lapped joints such as, load transfer, stress concentration factor, stress intensity factor, crack tip plasticity. 2- Examine the notch affected zone which was studied in the literature for free notch to bolted holes in the steel lapped joints. 3- Study the crack path of a crack emanated from bolted connection. 4- Developing a parametric study to study the effect of clamping force and friction coefficient on SCF and SIF for the riveted/bolted connections. 5- Developing a validated numerical finite element model for El-Ministerly Bridge through comparing their results with the field strain measurements, to validate it. 6- Evaluating El-Ministerly bridge fatigue life through the field strain measurements and the numerical modeling.
3
Chapter(2)
Literature Review
Chapter 2 LITERATURE REVIEW
2.1 INTRODUCTION In this chapter, the behavior of cracked elements under monotonic and cyclic mixed-mode loadings will be introduced. Different methods of determination of fatigue life are reviewed. Different criteria for the prediction of the direction of crack initiation due to the non-similar crack growth of cracks under mixed mode are reviewed. The different parameters that correlate the direction and rate of fatigue crack growth are described. Field tests, experimental and numerical studies of evaluation of riveted steel bridges are surveyed.
2.2 STRESS CONCENTRATION Most structural members have discontinuities of some type, for example, holes, fillets, notches,….etc. If these discontinuities have well-defined geometries, it is usually possible to determine a stress-concentration factor, Kt for these geometries [7]. Then, the engineer can account for the local elevation of stress using the well-known relation between the local maximum stress and the applied nominal stress σnom [8]: σmax = Kt . σnom
(2-1)
Consider an elliptical hole in a two-dimensional infinite solid under remote uniaxial tension. The major and minor axes for the elliptical hole are denoted as c and b, as shown in Fig. 2-1. The elliptical hole surface can be described by the following equation:
4
Chapter(2)
Literature Review
x2 y2 + =1 c2 b2
(2-2)
Fig. 2-1 An elliptical hole in a two-dimensional infinite solid under remote tension
5
Chapter(2)
Literature Review
The radius of curvature at x = ±c and y = 0 is denoted as ρ, as shown in the figure. The radius of curvature ρ can be related to c and b as
ρ=
b2 c
(2-3)
The two-dimensional infinite solid is subjected to a uniformly normal stress σ∞ (σo) in the y direction at y = ±∞. The linear elasticity solution gives the stress solution [8], at x = ±c and y = 0 as:
σ yy ( x = ±c, y = 0) = σ ∞ (1 +
2c ) b
(2-4)
Based on Eq. (2-3), Eq. (2-4), [8] can be rewritten as:
σ yy ( x = ±c, y = 0) = σ ∞ (1 + 2
c
ρ
)
(2-5)
It can be defined the stress concentration factor Kt for the elliptical hole under the remote uniaxial tension as: Kt =
σ yy ( x = ±c, y = 0) c = (1 + 2 ) σ∞ ρ
(2-6)
2.3 MODES OF CRACK TIP DEFORMATION Structures frequently have sizeable existing cracks that may or may not grow, depending on the load level. When a material has an existing crack, flaw, inclusion or defect of unknown small size. The local stress concentration factor approaches infinity, making it practically useless for predicting stress. The
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singular stress field in the region of the crack tip can be completely defined for three modes of deformation [8-10] as shown in Fig. 2-2. In the opening mode, Mode I, the two fracture surfaces are displaced perpendicular to each other in opposite direction. In the sliding in-plane shear mode, Mode II, the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. In the tearing shear mode, Mode III, the crack surfaces move relative to one another and parallel to the leading edge of the crack.
Fig. 2-2 Modes of crack tip deformation. Traditional applications of fracture mechanics have been concentrated on cracks growing under an opening or mode I mechanism. However, many failures occur from cracks subjected to mixed mode loadings. Examples for a crack loaded under mixed mode loadings can be found in various engineering applications. Many uniaxially loaded materials, structures and components often contain randomly orientated defects and cracks which develop a mixed mode state by virtue of their orientation with respect to the loading axis. 7
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2.4 LINEAR ELASTIC FRACTURE MECHANICS (LEFM) 2.4.1 Elastic Stress Field near the Crack Tip A modern approach to the understanding of the resistance to fracture of materials containing defects is to study the stress-strain and displacement at the tip of the defect when it begins to grow. It was found theoretically [10] that the elastic stress distribution at small distances from the crack tip, see Fig. 2-3, of an infinite two-dimensional (2D) plate subjected to mixed mode loading, i.e. mode I and mode II can be expressed in Cartesian coordinates as written in Eqs. (2-7).
Fig. 2-3 stress components ahead of the crack tip.
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σx =
⎡⎧ θ⎛ θ 3θ ⎞⎫ ⎤ ⎟⎬ − ⎥ ⎢⎨K I cos ⎜1 − sin sin 2 ⎠⎭ ⎥ 2⎝ 2 1 ⎢⎩ ⎥ ⎢ 2πr ⎢⎧ θ⎛ θ 3θ ⎞⎫ ⎥ K sin ⎜ 2 + cos cos ⎟⎬ ⎢⎨ II 2 2 2 ⎝ ⎠⎭ ⎥⎦ ⎩ ⎣
σy =
⎡⎧ θ⎛ θ 3θ ⎞⎫ ⎤ ⎟⎬ + ⎥ ⎢⎨K I cos ⎜1 + sin sin 2⎝ 2 2 ⎠⎭ ⎥ 1 ⎢⎩ ⎥ 2πr ⎢⎧ ⎥ ⎢⎨K sin θ cos θ cos 3θ ⎫⎬ ⎥⎦ ⎢⎣⎩ II 2 2 2 ⎭
τ xy =
⎡⎧ ⎤ θ θ 3θ ⎫ ⎬+ ⎢⎨K I cos sin cos ⎥ 2 2 2 ⎭ 1 ⎢⎩ ⎥ ⎢ 2πr ⎧ θ θ 3θ ⎞⎫⎥ ⎢⎨K cos ⎛⎜1 − sin sin ⎟ ⎬⎥ 2⎝ 2 2 ⎠⎭⎥⎦ ⎢⎣⎩ II
σ z = ν⎛⎜ σ x + σ y ⎞⎟ ⎝ ⎠ σz = 0
(2-7)
for plane strain for plane stress
KI and KII are respectively mode I and mode II stress intensity factors. These equations imply that the distribution of the elastic stress field in the vicinity of the crack tip is invariant in all structural components subjected to this type of deformation and the magnitude of the elastic stress field can be described by KI and KII designated as the stress intensity factors.
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2.4.2 Strain-Energy Release Rate The first analysis of fracture behavior of components that contain sharp discontinuities was developed by Griffith, as mentioned by Barsom and Rolfe [8]. The analysis was based on the assumption that incipient fractures in ideally brittle materials occur when the magnitude of the elastic energy supplied at the crack tip during an incremental increase in crack length is equal to or greater than the magnitude of the elastic energy at the crack tip during an incremental increase in crack length. One of the most important relations in the field of linear fracture mechanics is Eq. (2-8) which suggests that, the strain energy release rate, GI represents the material’s resistance (R) to crack extension and it is known as the crack driving force [11]. GI =
KI E
2
(2-8)
2.5 ELASTIC PLASTIC FRACTURE MECHANICS (EPFM) 2.5. 1 Crack Tip Plastic Zone Plastic zone size developed near the tip of a growing crack is regarded as a measure of the material resistance against the crack driving force that is scaled by K or J-integral. The larger plastic zone size yields the higher toughness via the plastic energy dissipation [12]. Upon ductile fracture of a homogeneous material, the plastic zone size gets larger under in-plane mixed mode loading than under the mode-I loading [13]. Figure 2-4 shows the shape of plastic zone due to mode I loading. When cracked components are subjected to either monotonic or cyclic loading, the material ahead of the crack tip suffers different stress strain response. The singularity of stress field at the crack tip is suppressed by the formation of plastic deformation.
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rp a Crack tip Plastic zone
Crack surface
Fig. 2-4 The shape of Mode I plastic zone.
a) Plastic zone due to mode I loading For a homogeneous material under mode I loading and small scale yielding condition, the size and shape of this plastically deformed zone can be estimated by three common approaches. The modified approach developed by Irwin [14] assumed that the plastic zone is circular and its radius, rp, for the plane stress state, is approximately given by: 1 rp ≈ 2π
⎛ KI ⎜ ⎜σ ⎝ yield
⎞ ⎟ ⎟ ⎠
2
(2-9)
Where σyield is the material yield stress. For condition of plane strain where the triaxial stress field suppresses the plastic deformation, the plane strain plastic zone radius is smaller, i.e.
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rp ≈
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1 6π
⎛ KI ⎜ ⎜σ ⎝ yield
⎞ ⎟ ⎟ ⎠
2
(2-10)
Irwin argued that the presence of plasticity at the crack-tip makes the crack behaves as if it was longer than its physical size, since the displacements are longer and the stiffness is lower than that of an elastic crack of comparable size. Irwin then introduced the concept of notional crack that extended to the center of plastic zone. Thus the extent of the plastic zone is 2rp. Dugdale [15] developed an alternative approach to that proposed by Irwin [14]. It was assumed that the plastic deformation is concentrated in a localized strip in front of the crack. In this case, the crack is surrounded by an entirely elastic stress field, while the notional crack increment in the plastic region was assumed to carry the yield stress. The plastic zone size for the plane stress state was estimated as follows: ⎛ πσ = sec⎜ ⎜ 2σ a ⎝ yield
rp
⎞ ⎟ −1 ⎟ ⎠
(2-11)
While the plastic zone size in the plane strain condition was found to be as follows:1 = a 18π
rp
⎛ KI ⎜ ⎜σ ⎝ yield
⎞ ⎟ ⎟ ⎠
2
(2-12)
The maximum size of the zone parallel to, but not in the plane of the crack was equal to three times the predicted value. The model due to Irwin and Dugdale were based upon an over-simplified yield criterion and an assumed plastic zone shape. The first order approximation of plastic zone shapes was provided by the use of the von Mises yield criterion. Two assumptions were made (i) the material is of an elastic-perfectly-plastic type; (ii) no account is taken of stress redistribution due to plastic deformation. The following expression for the radius
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of the plastic zone can be obtained from [10]:1 rp (θ ) = 4π
⎛ KI ⎜ ⎜σ ⎝ yield
⎞ ⎟ ⎟ ⎠
2
⎛3 2 ⎞ ⎜ sin θ + 1 + cos θ ⎟ ⎝2 ⎠
(2-13)
For plane stress, and 1 rp (θ ) = 4π
⎛ KI ⎜ ⎜σ ⎝ yield
2
⎞ ⎛3 2 ⎟ ⎜ sin θ + (1 − 2ν ) 2 (1 + cos θ ) ⎞⎟ ⎟ ⎝2 ⎠ ⎠
(2-14)
For plane strain, where v is the Poisson's ratio By the use of the numerical method, the mathematical difficulties arising from the relaxation of some of the simplifying assumptions will be overcome. One of the most successful numerical methods is the finite element method. A unique feature of this method is that it permits to follow the plastic zone development with the applied stress. Using this method, the development and spread of plasticity ahead of the crack tip for a range of applied stress ratios was performed by Meguid and Coufopanos [17]. Upon unloading, the elastically stressed regions remote from the crack tip tend to recover their original dimensions, but the continuity of the material in the elastoplastic regions does not allow this to occur. Consequently a residual stressfield will be trapped in the component. The residual compressive stress field leads to the formation of a reversed plastic zone. It was found that the size of this cyclic zone, which experiences alternate tensile and compressive yielding may be estimated by substituting ∆KI and σcy ( = 2σyield ) respectively in place of KI and σyield in Eqs. (2-13) and (2-14) [18]. Therefore, the size of reversed plastic zone is smaller than the monotonic plastic zone. Based on the work of Kang and Liu [19] the plane strain cyclic plastic zone was found to be five times smaller than that measured at the specimen surface, plane stress state. McClung [20] found that, there was no significant difference between the forward plastic zone extents for stationary and fatigue loaded cracks. 13
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On the other hand he found that the width of the reversed plastic zone of a growing fatigue crack is less than that of stationary crack.
b) Plastic zone due to mixed mode loading Crack-tip plasticity for mixed-mode loading will produce a crack-tip opening displacement (CTOD) and a crack-tip sliding displacement (CTSD). The radius of the plastic zone accompanying the crack tip for mixed-mode, mode I and mode II, loading under plane strain conditions is given by Eq. (2-15) [10], as a function of the angle from the crack plane, θ. Eq. (2-15) assumes no redistribution of stress due to crack tip plasticity. Rather, this radius is that of the elastic solution whose von Mises equivalent stress corresponds to yielding.
rp =
1 2 2πσ yield
⎡ 2 ⎧ 2⎛θ ⎞ ⎡ 2⎤ ⎫ 2⎛θ ⎞ ⎢ K I ⎨cos ⎜ ⎟ ⎢3 sin ⎜ ⎟ + (1 − 2ν ) ⎥ ⎬ ⎝2⎠ ⎝2⎠ ⎣ ⎦⎭ ⎩ ⎢ ⎢ ⎧ ⎢+ K 2 ⎨3 + sin 2 ⎛⎜ θ ⎞⎟ ⎡(1 − 2ν )2 − 9 cos 2 ⎛⎜ θ ⎞⎟⎤ II ⎢ ⎥ ⎢ ⎝ 2 ⎠⎦ ⎝2⎠ ⎣ ⎩ ⎢ 2 ⎢+ K I K II sin θ 3 cos θ − (1 − 2ν ) ⎢ ⎣
[
]
⎤ ⎥ ⎥ ⎥ ⎫⎥ ⎬ ⎭⎥ ⎥ ⎥ ⎥ ⎦
(2-15)
The increase in plastic zone due to stress redistribution was accounted for by increasing the plastic zone of Eq. (2-16) by a factor of 1.828 [21]. This constant was chosen such that the CTOD for a straight crack matched the theoretical value (K² / E σyield). This relation was used to determine the monotonic plastic zone (MPZ) associated with Kmax. The cyclic plastic zone (CPZ) is a subset of the MPZ, where reverse plasticity occurs during unloading. The size of the CPZ is proportional to ∆K² and the shape can be calculated using Eq. (2-16), by inserting one-half of the cyclic stress intensity factors in place of the maximum values. Reversed plasticity in the CPZ leads to a reduction in residual CTOD and CTSD [21].
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Soh and Bain [22] investigated the behaviour of fatigue crack propagation of rectangular aluminum alloy plates, each consisting of an inclined semi-elliptical crack, subjected to axial loading both experimentally and theoretically. The inclined angle of the crack with respect to the axis of loading varied between 0° and 90°. The von Mises yield criterion was applied to define the core region, instead of assuming a core region with a constant distance r from the crack tip. They adopted von Mises elastoplastic boundary to modify the maximum tensile stress and the minimum strain energy density fracture criteria. They concluded that the direction of the fatigue crack extension from the inclined crack was not along the original plane of the crack, but roughly perpendicular to the loading axis. 2.5.2 Crack Tip Opening Displacement Wells [23] introduced the concept of a critical crack opening displacement as a fracture criterion for the study of crack initiation in situations where significant plastic deformation proceeds fracture. Under such conditions he argued that the stresses around the crack tip reached the critical value and therefore fracture was controlled by the amount of plastic strain. A measure of the amount of crack-tip plastic strain is the separation of the crack faces or crack opening displacement (COD), especially very close to the crack tip. Using the Irwin [14] models, the crack tip opening displacement, CTOD, δ, is expressed by δ=
4 K2 π E σ yield
For plane stress
(2-16a)
δ=
4 (1 − ν 2 ) K 2 3π E σ yield
For plane strain
(2-16b)
Where, E is the modulus of elasticity. According to Dugdale model as mentioned in [10, 24], the value of δ can be estimated from:
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δ =
Literature Review ⎡ ⎛ πσ ⎞ ⎤ 8 σy a ⎟⎥ ln ⎢ sec ⎜ ⎜2σ ⎟ π E ⎢⎣ yield ⎠ ⎥ ⎝ ⎦
(2-17)
Under small scale yielding (SSY), δ can be approximated as: δ =
K I2 π σ2 a = E σ yield E σ yield
(2-18)
Equations (2-17) to (2-18) introduce the CTOD for a crack under mode I loading. However, for a crack under mixed mode loading, another component representing the relative sliding displacement of the crack tip sliding, CTSD, appears since mode II stress intensity factor, KII, is not zero. Since the asymptotic limiting value for CTOD at the crack tip is zero, an extension of the CTOD criterion was then proposed by Rice [25] for a stationary crack. The CTOD (δ45) was defined to be the opening displacement where 45o lines drawn from the crack tip intercept the crack faces. Hellman and Schwalbe [26, 27] used this idea to develop δ5-∆a resistance curves, where the δ5 parameter is the relative displacement of two points 5 mm above and below the original fatigue crack tip location. Because all rough cracks are subjected to a mixed-mode stress state, both opening and sliding mode crack face displacements are expected. Sliding mode displacements are needed to create misalignment of crack face asperities upon unloading. Elastic crack face displacements for straight cracks with mixed-mode loading (I and II) are shown in Eq. (2-19), where E is the elastic modulus, ν is Poisson's ratio, and r is the distance behind the crack-tip [21,29]. The total relative displacements between the two crack faces are 2U and 2V. Figure 2-5 shows the displacements between crack faces for loading ratio R (σmax/ σmin) equals zero. ⎧U ⎫ 4 (1 − ν 2 ) ⎨ ⎬= E ⎩V ⎭
r ⎧ K II ⎫ ⎨ ⎬ 2π ⎩ K I ⎭
(2-19)
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Node 2 y x
crack tip
Node 2 y x crack tip
2U 2V
Node 1
Crack while unloading
Node 1
Crack while loading
Fig. 2-5 the displacements between crack faces for R=0. In the search for practical fracture parameters that can be used to predict mixed-mode crack growth, it is worth noting that the crack-tip opening displacement (CTOD) has been used successfully in predicting the onset of Mode I crack growth in ductile materials. In addition, CTOD also provides a convenient length scale for describing the deformation in the crack tip region. Naturally, as a practical fracture parameter that can be easily monitored in experiments and finite element calculations, CTOD has received increasing attention [29,30,31]. 2.5.3 J Contour Integral Another way to describe the conditions near the crack tip is the J contour integral, named after Jim Rice. This integral can be used as a stress intensity parameter just like the stress intensity factor, K. The main advantage with the J integral is that it does not require linear material behavior [32]. The value for J is determined in two dimensions through:
J = ∫ ωdy − Ti Γ
∂ui ds ∂x
(2-20)
Where:- Γ is a counter clock wise path at an arbitrary distance from the crack tip shown in Fig. 2-6.
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ω is the strain energy density: ε IJ
ω = ∫ σ IJ dε IJ
(2-21)
0
Ti is the components of the traction vector acting on the path, uij is the displacement vector.
Fig. 2-6 The coordinate system and the contour path when using the J integral methodology. Just like the stress intensity factor, K, the J integral can be used to describe the stress-strain field near the crack tip. The origin for the integral was a way to describe the energy release rate when a crack grows under non linear material behavior. In the special case when used under linear material properties, it can be shown that the J integral equals the elastic energy release rate G. This parameter is normally used when studying linear elastic materials. This leads to a relationship between the two crack intensity factors K1 and J under linear premises:
K I2 J= E
(2-22)
J can also be used to define a material toughness, Jc, a value which defines when unstable crack growth occurs. For two dimension problem Gerben.et.al, as mentioned in Ref. [32], developed a method based on FEM as the quantities in the Eq. (2-20) are easily obtained in any commercial FEM software.
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2.6 FATIGUE LIFE Fatigue in metallic structures is caused by plastic deformation. Local plastic deformation taking place at a single site may be sufficient to cause fatigue failure. Such sites may be grain boundaries, inclusions or other material inhomogeneities. Fatigue failures can therefore occur even when the alternating stress is well below the yield stress particularly with the existence of a stress concentrator. The basic mechanisms of fatigue can be described originally in terms of dislocation movement. If fatigue arises in an undamaged material, the following stages are observed [33]: - Crack Initiation - Micro-crack, stage I propagation. -Macro-crack, stage II and stage III propagation. - Final fracture. 2.6.1 Crack Initiation Three nucleation sites are possible. Fatigue slip bands are the most frequent site of nucleation with its nature of slip concentration within a grain. Nucleation at grain boundaries is typical for high-strain fatigue, especially at higher temperature. Surface inclusions can be sources of nucleation for alloys containing large enough particles. Common to all types of nucleation is local plastic strain concentration at or near the surface. Nucleation in the fatigue slip band is a basic type of nucleation not only because this is the most frequent case, but mainly because the cyclic slip process and formation of fatigue slip band that may also precede nucleation at grain boundaries or at surface inclusions. The inclusion type nucleation can be thus understood as cyclic slip localization. There is strong evidence that grain boundary type nucleation is also conditioned by cyclic slip processes. Larid [34] concluded that it is the nature of the cross-slip deformation within the grains, and the compatibility of slip at the grain boundaries, that are the
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most important factors in defining the crack site. The higher stresses at inclusions due to the different elastic properties of inclusions and matrix are responsible for the preferred crack formation at inclusions. Special forms of fatigue slip bands are extrusions and intrusions, as shown in Fig. 2-7. Extrusions arise when the material is extruded out of the surface with the intrusions represented by the corresponding depressions and they form due to the cyclic irreversibility of plastic deformation [35]. Various mechanisms were proposed to explain the origin of intrusions and extrusions [33].
2.6.2 Micro-Crack, Stage I Propagation The nucleation stage ends in the formation of surface micro-cracks. These micro-cracks lie along the active slip planes in which the shear stress has maximum values. In uni-axial loading, the maximum shear stress lies in planes oriented at 45o to the direction of the applied stress. In the course of further cyclic loading the micro-cracks grow and link together. A large majority of these microcracks stop propagating quite early and only some achieve a length greater than a few tens of microns. With increasing length, the growing cracks leave the original near the 45ο oriented slip planes and tend to propagate perpendicular to the stress axis. This is the transition from stage I to stage II crack propagation. The crack length at which the stage I to stage II transition occurs depends mainly on the material, the geometry, the stress amplitude and the environment. Sometimes the transition does not take place and the whole propagation right through to fracture is stage I type [33].
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EXTRUSIONS
INTRUSIONS
Fig. 2-7 Slip band intrusion and extrusion. 2.6.3 Stage II Propagation In stage II of fatigue crack propagation, only one crack usually propagates. A primary feature of stage II crack propagation is the formation of fatigue striation with one to one correspondence between striations and stress cycles. This means that FCG is a repetitive process [36]. As the fatigue crack growth rate in stage I is generally much lower than that in stage II, the number of loading cycles spent in stage I propagation may be much higher than that spent in stage II propagation. This is the case of unnotched specimens at low stresses. The number of cycles necessary for stage I propagation in sharply notched or pre-cracked bodies becomes negligible and the whole fatigue crack propagation is of the stage II – type [36].
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2.6.4 Stage III Propagation Stage III of FCG is the transition in the final fracture. The number of load cycles in stage III relative to the total lifetime is small, as the crack propagates very rapidly in this stage. This stage also shows an increased extent region of
CRACK GROWTH RATE PER CYCLE,da/dN, LOG SCALE
overloading fracture [36]. Fig. 2-8 shows the three stages of crack propagation.
STAGE I
STAGE II
STAGE III
STRESS INTENSITY RANGE, LOG SCALE
Fig. 2-8 Typical crack propagation behavior of metals.
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2.7 CRACK INITIATION CRITERIA There have been numerous crack initiation and/or propagation theories proposed over the years. However, only a few of them have been proven to be viable and producing results that agree with some experimental observations. The maximum circumferential stress theory, minimum strain energy density theory and maximum strain energy release rate theory are considered to be the most important ones among these theories. Various criteria for crack growth direction under mixed mode loadings have been proposed. Some of these criteria are reviewed in this section. Figure 2-9 shows the kink angle for two subsequence crack steps.
Next step y x
Next crack tip θ
Present crack tip Present step
Fig. 2-9 Kink angle for to subsequence crack steps.
Griffith [37] proposed the first criterion for fracture initiation. It states that fracture occurs when the energy stored in the structure overcomes the surface energy of the material. This criterion is defined for mode I loading only. Hussain et al. [38] proposed the G-criterion based on the Griffith energy principle, and showed that crack initiation occurs at the direction of the maximum value of the energy release rate. However, Hussain et al. [38] based the G-criterion on the
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assumption that the crack under combined loading moved along its own initial plane, and assumed that the combined energy release rate Γ was given by Eq. (223). Γ=
1 ( K I2 + K II2 ) E`
(2-23)
Awaji [39] presented energy release rate under combined mode loading (g), and extended the Griffith energy criterion to mode II fracture. For a given applied load, KI and KII, the mode I and mode II stress intensity factors, at the tip of the kink, k1 and k2, can be evaluated for any kink angle, θ, by using coefficients, KII, derived by quadratures following the analysis of Khrapkov [40]. The coefficients were calculated by Howard [41]. The relevant equations are: ⎡⎛3 ⎤ ⎛θ⎞ 1 ⎛ 3θ ⎞ ⎞ ⎢ ⎜⎜ cos ⎜ ⎟ + cos ⎜ ⎟ ⎟⎟ K I + ⎥ ⎝2⎠ 4 ⎝ 2 ⎠⎠ ⎝4 ⎥ k1 = ⎢ ⎢⎛ 3 ⎥ ⎛θ⎞ 3 ⎛ 3θ ⎞ ⎞ ⎢ ⎜⎜ − sin ⎜ ⎟ − sin ⎜ ⎟ ⎟⎟ K II ⎥ ⎢⎣ ⎝ 4 ⎥⎦ ⎝2⎠ 4 ⎝ 2 ⎠⎠ ⎡ ⎛1 ⎤ ⎛ 3θ ⎞ ⎞ ⎛θ⎞ 1 ⎢ ⎜⎜ sin ⎜ ⎟ + sin ⎜ ⎟ ⎟⎟ K I + ⎥ 4 2 4 2 ⎠ ⎠ ⎝ ⎝ ⎝ ⎠ ⎥ k2 = ⎢ ⎢ ⎛1 ⎥ ⎞ θ 3 3θ ⎛ ⎞ ⎛ ⎞ ⎢ ⎜⎜ cos ⎜ ⎟ + cos ⎜ ⎟ ⎟⎟ K II ⎥ ⎝2⎠ 4 ⎝ 2 ⎠⎠ ⎣⎢ ⎝ 4 ⎦⎥
(2-24)
Since the determination of the angle, θc (θ which satisfy Eq. 2-24) , of the actual crack extension is the goal, any further incipient extension from the tip of the kink having this direction must be co-planar with the plane of the kink. Therefore in the investigation for the kink angle that will maximize the energy release rate one is justified in limiting the search to extensions from the tip of the kink that are in the same direction as the kink. i.e. co-planar extensions with the plane of the kink. For such co-planar extensions the following equation applies. g = g1 + g2
(2-25)
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g1 =k12 / E' and g2 = k22 / E' Where: E`=E for plane stress and E`=E/(1-ν2) for plane strain. E and ν are respectively the modulus of elasticity and Poisson’s ratio. The maximum energy release rate criterion then takes the form: gm = gc
(2-26)
The maximum value of g is gm and gc (= KIc2/ E') is the energy release rate fracture toughness of the material and KIc is the fracture toughness of the material. The kink angle, θc, is determined by the maximization of the ratio g/gc as a function of the angle θ, i.e. of the normalized value of g with respect to gc. As it stands the fracture toughness is looked upon as a material property that is independent of the different stress modes prevailing in the region of the crack tip or their ratios [42]. The maximum circumferential (tangential) stress criterion MTS-criteria proposed by Erdogan and Sih [28] is a commonly recognized hypothesis for crack extension in a brittle material under slowly applied in-plane loads. This criterion has later been derived based on the principle of minimum potential energy [43]. The theory states that the direction of crack extension is in the radial direction from the crack tip and is normal to the maximum tangential (circumferential or hoop) stress (σθ) at the original crack tip. The results of this theory are in remarkable agreement with slit cracks; however, the agreement is not so great for elliptical cracks. Following the developments of the Westergaard stress function, formulation for a crack in an infinite plane close tip stress field for mixed mode cracks is given in the literature as follows [44].
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∂σ θ = 0, ∂θ
∂ 2σ θ
